Advances in Psychology Volume 16 Decision Making Under Uncertainty: Cognitive Decision Research, Social Interaction, Development and Epistemology

DECISON MAKING UNDER UNCERTAINTY ADVANCES IN PSYCHOLOGY 16 Editors G . E. STELMACH P. A. VROON NORTH-HOLLAND AMSTER...

Author: Roland W. Scholz

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DECISON MAKING UNDER UNCERTAINTY

ADVANCES IN PSYCHOLOGY 16 Editors

G . E. STELMACH P. A. VROON

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD .TOKYO

DECISION MAKING UNDER UNCERTAINTY Cognitive Decision Research Social Interaction Development and Epistemology

Edited by

Roland W. SCHOLZ IDM University of Bielefeld FRGermany

1983

AMSTERDAM

NORTH-HOLLAND NEW YORK OXFORD .TOKYO

@

ELSEVIER SCIENCE PUBLISHERS B.V., 1983

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyrightownes

ISBN: 0 444 86738 4 First edition: 1983 Second edition: 1987

Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands Soledistributorsfor the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VanderbiltAvenue NewYork, N.Y. 10017 U.S.A.

PRINTED IN THE NETHERLANDS

V

PREFACE

The e d i t o r o f t h i s volume l i k e s t o ask t h e question "What for whom" when being confronted w i t h any new text; and he f e e l s t h a t t h e reader would be i n t e r e s t e d i n a b r i e f answer t o t h i s question

What is to be found in this votwne? I n t h i s volume one w i l l f i n d t h e r e v i s e d papers o f an i n t e r n a t i o n a l symposium on Decision Making under Uncertainty which was h e l d a t B i e l e f e l d U n i v e r s i t y from November l O t h t o 12th, 1982. The p a r t i c i p a n t s a t t h i s conference, psychol o g i s t s (working mainly i n t h e f i e l d o f c o g n i t i v e and s o c i a l psychology), economists, philosophers, mathematicians and mathematics educators were asked t o present new f i n d i n g s o r t o review t h e s t a t e o f research on f a l l a c i e s , biases and t h e development o f d e c i s i o n behavior. The book i s organized i n f i v e chapters. The I n t r o d u c t i o n attempts t o o u t l i n e t h e conceptual framework and how t h e t h r e e sections Coqnitive Decision Researck. w e .r a c t i o n and Development and Epistemoloqy a r e i n t e r r e l a t e d , i n p a r t i c u l a r how t h e research i n t o s o c i a l i n t e r a c t i o n s might be reconnected w i t h t h e research i n t o i n d i v i d u a l d e c i s i o n making, and how new f i e l d s , such as research on developmental questions (i.e. fran mathematics education and epistemology), can be p r o d u c t i v e l y integrated. As already shown by t h e scope o f d i s c i p l i n e s r e presented and by t h e chapter headings, both t h e s u b j e c t i v e c o g n i t i v e and t h e normative t a s k - a n a l y t i c a l sides o f decision making a r e covered by t h e various chapters. Within t h e f i f t h and f i n a l chapter (short-)Comments a r e c o l l e c t e d which evaluate t h e impact o f t h e c o n t r i b u t i o n s on d e c i s i o n research i t s e l f , and a l s o on c o g n i t i v e psychology, developmental psychology, s o c i a l psychology, economic theory and t h e d i s c i p l i n e o f mathematics education.

vi

PREFACE

who should be interested i n the contents? The contents should be o f i n t e r e s t t o t h e o r e t i c a l o r empirical researchers i n those d i s c i p l i n e s t h a t are concerned w i t h the adequate modelling o f d e c i s i o n making. Apart from t h e above mentioned (sub-)disciplines (i.e. cognitive, developmental and soci a1 psychology, economy, phi1osophy , mathematical s t a t i s t i c s and p r o b a b i l i t y , mathematics education) c e r t a i n f i e l d s o f medicine, law, educational and social sciences should a l s o be mentioned, as fundamental t h e o r e t i c a l issues p e r t a i n i n g t o these f i e l d s are discussed. However, t h e subject m a t t e r ’decision making under uncertainty’, i t ’ s conceptuatization,

modelling, and improuement does n o t make i t possible t o provide e i t h e r a closed l i s t o f f i e l d s o f i n t e r e s t , o r even f o r t h e weighing o f i t ’ s importance t o t h e various f i e l d s o f research. This i s due t o t h e f a c t t h a t decision theorists and decision a m t y s t s as w e l l as decisions and uncertainties are t o be found i n many places. Acknowl edqements Science has i t s s o c i a l dimensions. These become p a r t i c u l a r l y important when i n t e g r a t i n g competing approaches o r when t r y i n g t o e s t a b l i s h linkages between p r e v i o u s l y seperate branches.

I t h i n k , t h a t t h e frame and perspectives taken

by t h i s volume on decision research could not have been possible i f t h e symposium on ‘Decision MakinQ under Uncertainty’ a t B i e l e f e l d U n i v e r s i t y had n o t taken place. This symposium was arranged by t h e I n s t i t u t f u r D i d a k t i k der Mathematik ( I D M ) i n cooperation w i t h t h e Zentrum fur i n t e r d i s z i p l i n a r e Forschung ( Z i F) and was sponsored by t h e Deutsche Forschungsgemeinschaft (DFG). Without t h e existence and t h e technical equipment o f these i n s t i t u t e s and without a grant from the Science Foundation t h i s volume and i t ’ s perspecti v e s could not have been produced. The author wishes t o thank t h e technical and a d m i n i s t r a t i v e s t a f f o f these i n s t i t u t e s , e s p e c i a l l y A. Bentrup, M. Onitshabo, and C. Buschmeyer, who d i d most o f t h e t e c h n i c a l e d i t o r i a l work, f o r t h e i r assistance i n t h e arrangement o f the conference and production o f t h i s book. ROLAND W.

SCHOLZ

B i e l e f e l d , Spring 1983

vii

CONTENTS

Preface

V

DECISION MAKING UNDER UNCERTAINTY: Cognitive Decision Research, Social Interaction, Development and Epistemology Introduction to Decision Making under Uncertainty: Biases, Fallacies, and the Development of Decision Making

3

Roland W . Scholz

COGNITIVE DECISION RESEARCH

19

The Theoretical Status of Judgmental Heuristics s. Wallaten

21

The Base Rate Fallacy Controversy

39

Iphma

Maya Bar-HiZleZ

The Two Camps on Rationality

63

Helmut J u n q e m n

The Role o f Heuristics in Models o f Decision e l f Groner, Mzmh~G m e r & Walter F . Biechof

a7

On the Testability of the Availability Heuristic Klaus Fiedler

109

Problem Structuring Cal cul i and Level s o f Know1 edge Representation in Decision Making

121

Patrick liwnphrqs & Dim Berkeley

Verbal vs. Numerical Processing o f Subjective Probabilities A l f C. Zhner

159

CONTEN73

viii

Subjective Expected Utility: A Model o f Decision Making

1.83

Baruch Fischhoff, Bernard Goitein & Zur Shapka

SOCIAL INTERACTION

209

Conditions of Violating Individual Rationality

211

Werner H . Tack

Biases in Group Decision Making

229

Helmut W. Crott & Johrmnes A . Zuber

The Dependance of Cognition and Behavior in 2x2 Matrice Games

253

Ulrich Schulz & The0 May

On the Prominence Structure of the Decimal System

271

I;h*lf Albers & Oisela Albers

289

DEVELOPMENT AND EPISTEMOLOGY Conceptual and Theoretical Issues in Developmental Research on the Acquisition of the Probability Concept Roland W. Scholz & bhnfred Waller

291

The Perception of Probability Situations in Pupils Aged 12 to 14

313

Georges GZaeser

The Psychology of Inference and the Teaching of Probability and Statistics: Two Sides of thesame Coin?

325

J . Michael Shaugnessy

(Mis-) Interpretation of Stochastic Models Georg Schrage

-

Randomness and Stochastic Independence On the Relationship Between Intuitive Notion and Mathemat i cal Defi nit i on

351

363

Gerd v . Harten & Heinz Steinbring

Deci sion-Theoretic Paradoxes Georg Pfzug

€QMMENTS The Possible and the Profitable Some remarks on the symposium's impact on decision research Baruch Fischhoff

375

385 387

CONTENTS

The Importance of the Fictional for Theories of Knowledge Some remarks on the symposium's impact on cognitive psycho1 ogy

ix

391

A l f C. Z6mer

Why are Developmental Psychologists Biased with Regard to the Biasedness o f the Human Subject? Some remarks on the symposium's impact on developmental psychology

397

Manfred WaZZer

Individual Decision Making and Social Psydhology Some remrks on the symposium's impact on social psychology Katrin Borcherding

Towards a Theory of Limited Rationality Some remarks on the symposium's impact on economic theory

403

409

Reinhard Selten

Towards an Education for Decision Behavior Some remarks on the symposium's impact on mathematical education

413

Hans-Georg Steiner

ADDRESSES OF CONTRIBUTORS

421

ABSTRACTS

425

AUTHOR INDEX

437

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INTRODUCTION

This Page Intentionally Left Blank

DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers 8.V. (North-Holland), 1983

3

I NT R OD UCT 10N TO

DECISION MAKING UNDER UNCERTAINTY: BIASES, FALLACIES, AND THE DEVELOPMENT OF DECISION MAKING ROLAND W. SCHOLZ Universitat Bielefeld IDM F. R. Germany Decision Theory and Research are genuinely i n t e r d i s c i p l i n a r y . Thus, i t i s i m possible t o i d e n t i f y e i t h e r a s i n g l e f i e l d o f study o r an i n d i v i d u a l s c i e n t i f i c d i s c i p l i n e which could serve t o s u f f i c i e n t l y explain t h e genesis o f decision theory. While i t i s o f t e n s t a t e d t h a t t h e r o o t s o f decision theory are t o be found i n mathematics and economy, t h e approaches, i n these d i s c i p l i n e s , have h i s t o r i c a l l y always been c l o s e l y l i n k e d w i t h t h e debate on r a t i o n a l i t y i n philosophy. These d i s c i p l i n e s have c o n t r i b u t e d c e n t r a l concepts of c l a s s i c a l normative decision theory such as p r o b a b i l i t y , u t i l i t y , and h e u r i s t i c s . Decision theory, however, has a l s o borrowed central concepts from t h e p o l i t i c a l and s o c i a l sciences' mode o f thought, such as, f o r instance, t h e concept o f strategy. The i n t e r e s t i n analyzing human decision behavior has considerably increased i n recent years. Statements l i k e t h a t o f MARTIN SHUBIK: "One o f t h e most i m p o r t a n t problems o f our world i s t h e trembling hand" (1982) have, i n my opinion, d r a m a t i c a l l y pointed out t h e importance o f t h e human factor, and e s p e c i a l l y o f t h e consequences t h a t are connected w i t h t h e fuzzyness o f human action. However, t h i s fuzzyness i s not r e s t r i c t e d t o motoric actions, but a t l e a s t according t o one 'camp' o f decision researchers (see JUNGERMANN, t h i s volume) i s a l s o inherent i n c o g n i t i v e operations such as memorizing, o r concept formation etc.. Within such f i e l d s as technology assessment, t h e p o s s i b l i t i e s and l i m i t s o f t h e a b i l i t y o f man t o decide sometimes create

4

INTRODUCTION

sources o f u n c e r t a i n t y t h a t may even threaten our very existence. T r a d i t i o n a l l y decisions under u n c e r t a i n t y are defined by incomplete i n f o r m a t i o n o r knowledge about a s i t u a t i o n , i.e. t h e p o s s i b l e a l t e r n a t i v e s , o r t h e p r o b a b i l i t y o f t h e i r occurrence, o r t h e i r outcomes, a r e not known by t h e subjects. The sources o f u n c e r t a i n t y a r e not homogeneous. Uncertainty may be i n t e r n a l l y ( " I am not sure whether I have l e f t my umbrella a t John's house") o r e x t e r n a l l y a t t r i b u t e d ("It i s not c e r t a i n whether i t w i l l stop r a i n i n g by tomorrow morning"). D i f f e r e n t variants o f these two types o f u n c e r t a i n t y may be distinguished (cf. KAHNEMAN & TVERSKY 1982a) and f u r t h e r types may be defined when one considers t h e process o f decision making (cf. BERKELEY and HUMPHREYS, 1982; HOGARTH, 1975). During t h e l a s t t e n years, t h e scope o f t h e 'theory o f decisions' has been expanded by t h e development o f several psychological approaches i n human decision behaviour research. The t h e o r e t i c a l framework has been s i g n i f i c a n t l y extended by t h e elaboration o f judgemental h e u r i s t i c s by DANIEL KAHNEMAN and AMOS TVERSKY. They took t h e fundamental ideas o f HERBERT SIMON'S treatment o f

h e u r i s t i c s contained i n h i s o u t l i n e o f a theory o f bounded o r l i m i t e d r a t i o n a l i t y . This extension shows i n f a c t t h a t some k i n d o f co-mitiue decision science which attempts t o grasp t h e ( c o g n i t i v e ) psychological foundations o f d e c i s i o n behaviour, i s beginning t o ' w o l v e from t h e h i t h e r t o more q u a n t i t a t i v e l y oriented behauiomi! decision theory. The t h e o r e t i c a l approach o f KAHNEMAN and TVERSKY was based on a series o f experimental studies conducted during t h e seventies. While on t h e one hand these studies were widely, and sometimes e n t h u s i a s t i c a l l y received, on t h e o t h e r hand t h e r e has been pronounced c r i t i c i s m o f t h i s work. I n p a r t i c u l a r , various objections have been voiced against t h e f a l l a c y and b i a s argument which s t a t e s t h a t judgment and decision making show serious systematic errors, and t h e v a l i d i t y o f t h e experimental r e s u l t s supporting these arguments has been c r i t i c i z e d . Although t h e f a l l a c y concept i s widely used i t i s hard t o f i n d a general d e f i n i t i o n o f it, and t h e r e a r e obviously d i f f e r e n t comprehensions o f what i s meant by a fallacy. This i s why I want t o suggest t h e f o l l o w i n g d e f i n i t i o n o f a f a l l a c y before s t a r t i n g t o o u t l i n e t h e contents o f t h e main chapters o f t h i s volume and describing how t h e d i f f e r e n t papers are i n t e r r e l a t e d .

BIASES, FALLACIES, AND DEVELOPMENT OF DECISION MAKING

AfaZZaey

5

i s t h e r e s u l t o f a c o g n i t i v e process which, on t h e basis o f

information represented i n memory, has l e d t o a wrong conclusion, o r t o a wrong decision. A f a l l a c y may c o n s i s t o f both t h e a p p l i c a t i o n o f an inadequate model which has l e d t o a d e v i a t i n g s o l u t i o n instead o f a (possibly) e x i s t i n g d e f i n i t e formal solution, o r o f t h e a p p l i c a t i o n o f rough r u l e s o f inference (such as i n t u i t i t i v e estimates) used as a s u b s t i t u t e , which systematically r e s u l t i n an inadequate, o r i n c o r r e c t solution. To me, t h i s attempt t o g i v e a d e f i n i t i o n would seem t o provide a s a t i s f a c t o r y c l a r i f i c a t i o n of t h e f a l l a c y o r b i a s phenmenon which enables us t o formulate a couple o f questions and c r i t e r i a w i t h regard t o it. The core o f t h i s d e f i n i t i o n l i e s i n t h e ( e x t e r n a l l y evaluated, see SCHOLZ, 1981, p. 14 f . ) incorrectness o f t h e fallacy.')

This leads t o the question: (1) How

a r e t h e c o r r e c t conclusions o r inferences conceptualized? It should be noted, t h a t according t o t h i s d e f i n i t i o n f a l l a c i e s are n o t established by s p e c i f i c c o g n i t i v e processes. A l l degrees o f inferences which a r e inherent t o knowledge a p p l i c a t i o n may be p o s s i b l y judged t o be f a l l a c i o u s , depending on t h e frame o f reference o r model o f comparision (see below). Information represented i n memory i s the s t a r t i n g p o i n t f o r an i n f e r e n t i a l f a l l a c y . The questions then are: (2) What i n f o r m a t i o n does an i n d i v i d u a l have represented i n h i s mind w i t h regard t o a task he/she i s t o solve? (Under l a b o r a t o r y conditions o f experimental study, t h i s question can be formulated as follows: what s t r u c t u r e s are b u i l t up by subjects i n the small world t h e experimenter has introduced? ( c f . BERKELEY & HUMPHREYS, 1982).) (3) Which methods can be reasonably used i n order t o assess task understanding and the i n d i v i d u a l ' s l e v e l o f i n f o r mation and knowledge? With regard t o t h e f i r s t p a r t o f t h i s question, I should l i k e t o note t h a t the method t h a t has dominated t h e l i t e r a t u r e on f a l l a c i e s f o r a long time tends t o assume t h a t the subjects have understood t h e task i n e x a c t l y the same way as the experimenter did. Even more profound than the question p e r t a i n i n g t o task representation i s the problem of an adequate model t o be used f o r comparison, i.e.: For which tasks can we be sure o f having a generally accepted s o l u t i o n o r model. I n a very small number o f cases we are able t o draw on a model t h a t i s u n i v e r s a l l y

'1 I

want t o thank my colleague R. BROMME, B i e l e f e l d , f o r h i s comments on the proposed f a l l a c y d e f i n i t i o n .

6

INTRODUCTION

accepted as i n t h e instance of t h e f o l l o w i n g problem task, which I should l i k e t o quote as an example: Example :

Imagine a c i r c l e t h e s i z e o f an orange. Now take a tape which i s one metre longer than t h i s c i r c l e ’ s circumference, t i e i t s ends together and place i t so as t o form a concentrical c i r c l e around t h e smaller one. What w i l l be t h e distance o f t h i s l a r g e r c i r c l e from t h e smaller, orange-sized one? Now do t h e same, beginning w i t h a c i r c l e having t h e circumference o f t h e Earth. When a c i r c l e w i t h a circumference t h a t i s one meter longer, i s then placed c o n c e n t r i c a l l y around the earth sized c r c l e , what i s then the distance between both circles? The s o l u t i o n o r mode

f o r comparison, i s i n t h i s case, t h e c a l c u l a t i o n o f

c i r c l e circumference. One i s required t o compare two r a d i i o f c i r c l e s whose circumferences d i f f e r by one metre. It i s a well-known f a c t o f geometry t h a t t h e difference between r a d i i f o r a constant d i f f e r e n c e o f circumferences i s invariant w i t h regard t o the s i z e of t h e smaller c i r c l e . The definiteness o f t h e model f o r c i r c l e circumference, i n my opinion, i s due t o t h e f a c t t h a t t h e model f o r c a l c u l a t i n g t h e c i r c l e circumference f o r the object sought, i s established as a mathematical theory, and a generally accepted theory of area calculus. Leaving student e r r o r s i n attempts a t formal s o l u t i o n aside, subjects t a c k l i n g t h i s problem i n t u i t i v e l y w i l l generally assume t h a t t h e distance between t h e two c i r c l e s depends on t h e s i z e o f t h e smaller one. Comparison o f the circumference of a c i r c l e w i t h i t s r a d i u s thus s e a s t o be subject t o a systematical proportion o f e r r o r which involves a c o n t r a d i c t i o n between i n t u i t i o n and i n t u t i t i v e s o l u t i o n on t h e one hand, and formal s o l u t i o n on the other. The c o n t r a d i c t i o n i n t h e given example i s maintained (- a t l e a s t f o r a whtle -) if t h e subject i s confronted w i t h t h e f o r m a l l y c o r r e c t

solution. I n t u i t i o n , t h a t i s i n t h i s example t h e rough r u l e o f inference which

7

BIASES, FALLACIES, AND DEVELOPMENT OF DECISION MAKING

says:

"the bigger t h e c i r c l e s , t h e l e s s e f f e c t a constant d i f f e r e n c e o f c i r -

cumferences has on t h e d i s t a n c e between them" l e t s us down here. The q u e s t i o n now i s whether t h i s m i s l e a d i n g i n t u i t i o n can be c o r r e c t e d . LIETZMANN, who d i s c u s s e d t h i s problem i n h i s 1917 problem c o l l e c t i o n under t h e t i t l e "Wo s t e c k t d e r F e h l e r ? " p o i n t e d o u t t h a t t h e r e a r e , however, h e u r i s t i c s which can l e a d t o a f o r m a l s o l u t i o n which i s c o m p a t i b l e w i t h i n t u i t i o n .

In a

w o r d i n g r e m i n i s c e n t o f S o c r a t i c d i a l o g u e , h e g i v e s suggestions o f t h e f o l l o w i n g k i n d ; "When a h i k e r t r a v e l s around t h e world, how much f a r t h e r does h i s head t r a v e l , t h a n h i s f e e t ? ' ' (LIETZMANN, 1969, p.8,

t r a n s l a t i o n by t h e

a u t h o r ) . He goes on t o conclude t h a t t h e r e w i l l i n s t a n t l y b e t h e f e e l i n g t h a t t h e r e cannot be much o f a d i f f e r e n c e .

O t h e r S o c r a t i c d i a l o g u e s c a n be imagined

such as f i r s t t a k i n g , as g e o m e t r i c a l f i g u r e s , s t r a i g h t l i n e s o f v a r i o u s l e n g t h which a r e s i t u a t e d w i t h i n r e c t a n g l e s w i t h p o i n t s o f equal minimal d i s t a n c e , t h e n c o n t i n u i n g w i t h squares and r e c t a n g l e s and o t h e r convex c o n c e n t r i c polygons, e n d i n g up w i t h c o n s i d e r i n g t h e c i r c l e . Even i f i n s i g h t as t o t h e c o r r e c t s o l u t i o n can be a c h i e v e d i n t h e s e ways, t h e t y p i c a l t e n s i o n between mathematical, f o r m a l - o p e r a t i v e problem s o l v i n g , and i n t u i t i v e - c o n c r e t e problem s o l v i n g w i l l be p r e s e r v e d o r reappear a f t e r a p e r i o d o f t i m e f o r many o f t h e f a l l a c i e s f o r which an " i d e a l m a t h m a t i c a l model" i s a comparative measure.

I s h o u l d l i k e t o d i s t i n g u i s h between f a l l a c i e s o f t h e c i r c l e t a s k k i n d , and other fallacies.

I n t h e German language, t h e r e i s t h e f a m i l i a r d i s t i n c t i o n

between F e h l s c h l u s s e ( f a 1 l a c i e s ) and Trugschlusse (paralogisms). i s f o u n d i n r i d d l e mathematics.

The l a t t e r

Paralogisms a r e something l i k e problem

formulatlons " i n disguise". The f o l l o w i n g , r a t h e r s o p h i s t i c a t e d problem t a s k w i l l s e r v e as an example ( c f . WOLFF 1930, p. 124): "An a e r o p l a n e r e q u i r e s one hour and twenty minutes t o c o v e r t h e d i s t a n c e Berlin-Hamburg.

For t h e r e t u r n f l i g h t from

Hamburg t o B e r l i n , t h e same aeroplane, t r a v e l l i n g a t e x a c t l y t h e same speed on e x a c t l y t h e same path, r e q u i r e s o n l y e i g h t y minutes.. Why?"

8

INTRODUCTION

While a t a s k a n a l y s i s o f t h e decoding process which confuses t h e c o g n i t i v e s t r u c t u r e s , may be i n t e r e s t i n g from a l i n g u i s t i c p o i n t o f view, such s t o r i e s a r e u n s u i t a b l e f o r t h e study o f human e r r o r s and f a l l a c i e s with t h e p e r s p e c t i v e s f o r m u l a t e d by KAHNEMAN & TVERSKY (1982, p.494):

" F i r s t , we expose

some o f o u r i n t e l l e c t u a l l i m i t a t i o n s and suggest ways o f i m p r o v i n g t h e q u a l i t y o f o u r t h i n k i n g . Second, e r r o r s and b i a s e s o f t e n r e v e a l t h e p s y c h o l o g i c a l processes and t h e h e u r i s t i c procedures t h a t govern judgment and i n f e r e n c e . T h i r d , m i s t a k e s and f a l l a c i e s h e l p t h e mapping o f human i n t u i t i o n s by i n d i c a t i n g which p r i n c i p l e s o f s t a t i s t i c s o r l o g i c a r e n o n - i n t u i t i v e o r c o u n t e r -

int u it ive

.''

The "Cognitiue Decision Research" Section I n t h e l a r g e s t c h a p t e r " C o g n i t i v e D e c i s i o n Research",

some c o n t r i b u t i o n s t o

t h i s volume a t t e m p t t o d e f i n e t h e p o s i t i o n o f p s y c h o l o g i c a l research w i t h r e g a r d t o judgemental heuristics, b.iases, and faUaeies. I n t h e l i t e r a t u r e on judgment and d e c i s i o n s , t h e base r a t e f a l l a c y i s most c e r t a i n l y t h e most ext e n s i v e l y d i s c u s s e d phenomenon o f human performance among t h o s e which a r e l a b e l l e d as f a l l a c i e s . The c h a p t e r about "THE BASE RATE FALLACY CONTROVERSY" (BAR HILLEL) summarizes t h e v a r i o u s p s y c h o l o g i c a l e x p l a n a t i o n s g i v e n f o r t h i s f a l l a c y . While t h e s e c o n s i d e r a t i o n s about f a l l a c i e s cannot o f c o u r s e c l a i m t o be complete, t h e y can be considered, i n my o p i n i o n , t o be r e p r e s e n t a t i v e i n t h e way i n which t h e y t r e a t t h e fundamental problems o f t h e p r e s e n t r e s e a r c h i n p s y c h o l o g i c a l d e c i s i o n theory. Among these, t h e t a s k s o f d e v e l o p i n g a b r o a d e r p s y c h o l o g i c a l t h e o r y and o f i n t e g r a t i n g v a r i o u s a l t e r n a t i v e approaches t o e x p l a n a t i o n a r e notable. T h i s problem i s d e a l t w i t h i n t h e c o n t r i b u t i o n "THE THEORETICAL STATUS OF JUDGMENTAL HEURISTICS" (WALLSTEN).

A t p r e s e n t , n o concept seems t o be so f r e q u e n t l y quoted i n d e c i s i o n t h e o r y a s t h e concept o f h e u r i s t i c s .

Heuristics a r e g e n e r a l l y understood t o b e rough c o g n i t i v e r u l e s o f i n f e r e n c e As p r i n c i p l e s o f d e c i s i o n which

which reduce t h e c o m p l e x i t y o f s i t u a t i o n s .

reduce t h e c o m p l e x i t y o f s i t u a t i o n s , h e u r i s t i c s a r e c o n c e i v e d o f as models o f c o g n i t i v e a c t i v i t y which make a l l o w a n c e f o r t h e o p e r a t i v e and c a p a c i t i v e l i m i t a t i o n s o f human c o g n i t i o n s .

The s t a t u s o f h e u r i s t i c s (such as r e p r e -

BIASES, FALLACIES, AND DEVELOPMENT OF DECISION MAKING

9

sentativness h e u r i s t i c s o r a v a i l i b i l i t y h e u r i s t i c s ) as psychological explanatory constructs i s not clear. The d e c r i p t i o n o f h e u r i s t i c s i s o f t e n as WALLSTEN (1980, p. 219) r a t h e r imprecise, and h e u r i s t i c s may f o r instance

-

-

p a r t l y be used a r b i t r a r i l y as post hoc explanations o f emphasizes, phenomena. Whether these shortcomings o f research on h e u r i s t i c s can be overcome, depends, i n my opinion, on two things: F i r s t , i t depends on whether h e u r i s t i c s can be defined i n such a way t h a t an e m p i r i c a l l y s a t i s f a c t o r y t e s t can be made. Second, i t depends on whether t h e various h e u r i s t i c s t h a t are t r e a t e d more o r less separately and s i d e by side, can be i n t e g r a t e d i n t o a model encompassing a l l c o g n i t i v e a c t i v i t y thus solvi n g t h e e l i c i t a t i o n problem o f h e u r i s t i c s (cf. SCHOLZ, 1981, p. 80 ff). Theoretical explanations f o r t h e d e v i a t i o n o f human performance frm normative models are connected w i t h e s s e n t i a l l y d i f f e r e n t assumptions about the nature of human rationality. w i t h i n t h e paper on "THE TWO CAMPS ON RATIONAI I T Y " (JUNGERMANN) a d i s t i n c t i o n i s made between a pessimistic and an o p t i m i s t i c position. The pessimists consider judgmental biases, representional f a u l t s and coping e f f e c t s / d e f e c t s t o be an essential p a r t o f human a c t i v i t y . The o p t i m i s t s argue t h a t human performance i s r a t i o n a l when regarding t h e c o g n i t i v e costs o f a decision (rnetu rutionaZity argument), t h a t t h e behavior i s o f t e n w e l l reasoned when decision behavior i s conceptualized as a continuous process i n time (continuity argwnent), o r i s not judged t o be fallaciouwwhen the manner i n which i n d i v i d u a l s have s t r u c t u r e d a problem

(structure argument) i s taken i n t o account. I n t h i s context, t h e p o s i t i o n s taken by t h e c o n t r i b u t e r s t o t h i s volume are by no means uniform. The concept o f h e u r i s t i c does n o t cane from the theory o f judgments and decisions. "THE ROLE OF HEURISTICS I N MODELS OF DECISION" (GRONER, GRONER & BISCHOF) studies t h e o r i g i n s o f t h i s concepts w i t h i n mathematics and p h i l -

osophy w h i l e analyzing applications t o t h e 'Psychology o f Thinking' and 'Artificial

Intelligence'.

I n doing so, i t c l e a r l y shows which s p e c i f i c a t i o n s

and t h e o r e t i c a l i n t e g r a t i o n s are required i n order t o subject t h i s widely used concept t o a s a t i s f a c t o r y empirical t e s t i n t h e research f i e l d o f judgment and decisions. These methodological problems are a l s o elaborated i n a study on judgments o f subjects' c o n t r i b u t i o n s t o various a c t i v i t i e s which i n q u i r e s i n t o

10

INTRODUCTION

fundamental questions "ON THE TESTABILITY OF THE AVAILABILITY HEURISTIC" (FIEDLER). Whether the a s p i r a t i o n t o develop a c o g n i t i v e decision theory o u t o f a behavioral decision theory w i l l be f u l f i l l e d depends mostly on t h e models o f i n t e r n a l representations, on t h e memory models, and on t h e c o g n i t i v e operations used i n psychological models and theories. The c o n t r i b u t i o n s "PROBLEM STRUCTURING CALCULI AND LEVELS OF KNOWLEDGE REPRESENTATION I N CECISION

MAKING" (HUMPHREYS AND BERKELEY) and "VERBAL VS. NUMERICAL PROCESSING OF SUBJECTIVE PROBABILITIES" (ZIMMER) present approaches which d i s t i n g u i s h between d i f f e r e n t l e v e l s o f a b s t r a c t i o n o r modes o f task representation. S t a r t i n g out from q u a n t i t a t i v e l y d i f f e r e n t 'decision making schemes' brings about, i n my opinion, t h e p o s s i b i l i t y o f more adequately grasping, t h e o r e t i c a l l y , t h e broad interpersonal, b u t also intrapersonal variance o f problem treatment. The chapter on 'Cognitive Decision Research' concludes w i t h an extensive c r i t i c a l review o f "SUBJECTIVE EXPECTED UTILITY (SEU)" (FISCHHOFF, GOITEIN, SHAPIRA). The SEU approach, together w i t h simple l i n e a r models, has d m i n a t e d

modelling i n decision theory f o r more than a decade without ever r e a l l y questioning t h e v a l i d i t y o f t h i s approach as a model o f c o g n i t i v e s t r u c t u r i z a t i o n . This c o n t r i b u t i o n reveals the t h e o r e t i c a l shortcoming o f t h e SEU-model but also takes t h e value o f i t s a p p l i c a t i o n i n t o account.

The "Social Interactionr' Section I n s o c i a l i n t e r a c t i o n s , subjects a l s o have t o make i n d i v i d u a l decisions. Withi n t h e normative decision theory t h a t i s o r i e n t e d towards formal modelling, i t can be observed t h a t since JOHN v. NEUMANN's and OSKAR MORGENSTERN's 1944

book "Theory o f Games and Economic Behaviour", research i n t o I n d i v i d u a l Decis i o n Making has separated i t s e l f from the experimental and t h e o r e t i c a l r e search i n game theory, which i s concerned w i t h " s o c i a l i n t e r a c t i o n " . T h i s i s remarkable i n s o f a r as t h e roots o f modelling, t h a t i s the concept of u t i l i t y , the concept o f p r o b a b i l i t y , and the economic p r i n c i p l e s o f r a t i o n a l i t y such as maximizing p r o f i t and expectation, a r e i d e n t i c a l i n both f i e l d s , and t h a t

BIASES, FALLACIES, AND DEVELOPMENT OF DECISION MAKING

11

fundamental d e v i a t i o n s o f behavior from these p r i n c i p l e s have been found n o t o n l y i n research i n t o behavioral d e c i s i o n theory, b u t a l s o i n research i n t o s o c i a l i n t e r a c t i o n . V i o l a t i n g so-called i n d i v i d u a l r a t i o n a l i t y i s one o f these deviations.

I t occurs when a subject co-operates w i t h others, o b t a i n i n g l e s s

by t h i s co-operation than he would o b t a i n by n o t co-operating a t a l l . The s e c t i o n "CONDITIONS OF VIOLATING RATIONALITY" (TACK) s p e c i f i e s c o n d i t i o n s under which i n d i v i d u a l s v i o l a t e t h i s p r i n c i p l e o f r a t i o n a l i t y , p e r m i t t i n g i n s i g h t s i n t o t h e s o c i a l s t r u c t u r e o f behavior and i t s determinedness, which i s p a r t i c u l a r l y d e a l t w i t h i n t h e s e c t i o n "THE DEPENDANCE OF COGNITION AND I O R I N 2 x 2 MATRICE GAMES"(SCHULZ & MAY). Understanding and modelling s o c i a l i n t e r a c t i o n and group decisions w i l l r a i s e , even more p o i n t e d l y than i n t h e study o f SEU v a l i d i t y t h e question whether a comprehensive model should be studied, o r r a t h e r i t s components. The s e c t i o n on IIBIASES I N GROUP DECISION MAKING" (CROTT & ZUBER) t r e a t s t h e phenomenon o f t h e Group Choice S h i f t s f r e q u e n t l y r e f e r r e d t o as r i s k y s h i f t s , r e p o r t i n g t h r e e experimental studies. By means o f these experiments, t h e authors show how f a r t h a t which i s p o s s i b l y s u p e r f i c i a l l y i n t e r p r e t e d as r i s k y s h i f t o r group d e c i s i o n b i a s depends on t h e t h e o r e t i c a l l y assumed (and p o s s i b l y bargained f o r w i t h i n groups) group d e c i s i o n r u l e s , on assumptions about i n d i v i d u a l systems o f preferences, and on t h e d i s t r i b u t i o n o f preferences w i t h i n t h e group. E a r l y work on c o g n i t i v e psychology ( c f . MILLER 1956) has already pointed o u t t h e l i m i t a t i o n s o f human i n f o r m a t i o n processing. F o l l o w i n g t h e p r i n c i p l e s o f bounded r a t i o n a l i t y ( c f . SIMON 1955) i n bargaining t h e o r y f o r instance, t h e a s p i r a t i o n l e v e l approach (see e.g.

SAUERMANN & SELTEN 1962, TIETZ 1983,

SCHOLZ 1980, ALBERS & ALBERS 1983) i s based on t h e assumption t h a t subject's b a r g a i n i n g behavior i s governed by an a s p i r a t i o n scale which i s given by t h e s e t s o f bargaining a l t e r n a t i v e s .

I n t h e chapter "ON THE PROMINENCE STRUCTURE

OF THE DECIMAL SYSTEM" (ALBERS & ALBERS) a model f o r t h e c o g n i t i v e schemata o p e r a t i n g on number systems i s introduced which shows an e s s e n t i a l resemblance t o t h i s approach. The model i s i l l u s t r a t e d and supported by e m p i r i c a l data

from branded and unbranded department s t o r e a r t i c l e s . An e v a l u a t i o n o f t h i s model i s given by applying i t t o f o u r resp. f i v e person bargaining games.

12

INTRODUCTION

The rrDeveZoprnentand Epistemology" Section I n my opinion, one o f t h e essential d e f i c i t s o f d e c i s i o n research, i s t h a t i t mostly conceives o f human i n t u i t i o n s , cognitions, and t h e a b i l i t y t o decide as i f they were s t a t i c constructs. This i s n o t only t r u e i n regard t o t h e con-

c e p t u a l i z i n g o f a subject's

decision behavior i n a given s i t u a t i o n (see above,

and HUMPHREYS & BERKELEY, t h i s volume) but a l s o an invariance o f f a l l a c y i s assumed i n general human a b i l i t y i t s e l f . TVERSKY's statement, reported by NISBETT & ROSS (1980, p. 14), " t h a t t h e r e i s no i n f e r e n t i a l f a i l u r e t h a t can

be demonstrated w i t h untrained undergraduates t h a t cannot a l s o ( a t l e a s t w i t h a l i t t l e i n g e n u i t y ) be demonstrated i n somewhat more s u b t l e form i n t h e h i g h l y t r a i n e d s c i e n t i s t " i s a very pronounced pessimistic statement formulating t h e developmental invariance o f f a l l a c i e s i n decision making. Undoubtedly, many o f t h e f a l l a c i e s presently l i n k e d w i t h t h e concept o f p r o b a b i l i t y , f o r instance, t h e TUROFF problem (cf. TUROFF, 1972, KAHNEMAN & TVERSKY, 1979, SCHOLZ, 1981 pp. 8 8 ) , are arranged so as t o c r e a t e tensions even i n persons f a m i l i a r w i t h t h e formal solutions, thus b r i n g i n g about serious f a l l a c i e s . But: both t h e ontogenetic and t h e phylogenetic invariance o f f a l l a c i e s , however, would seem t o be a fundamental misconception o f t h e nature o f c o g n i t i v e a c t i v i t y , even i f t h e l a t t e r i s l i n k e d t o d i s c o n t i n u i t i e s . The child's

naive f a l l a c y , termed a r t i f i c i a l i s m , which says f o r instance t h a t t h e

rays o f seeing go out from t h e eye i s regarded by some developmental psychologists t o be a universal phenomenon. To assume however, t h a t t h e misconception associated w i t h a r t i f i c i a l i s m cannot be overcome, and t h a t renowned p h y s i c i s t s , f o r instance, f a l l back t o t h e naive conception o f t h e c h i l d i n c e r t a i n s i t u a t i o n s , i s a very far-reaching and unsubstantiated proposition. Another example which could serve t o demonstrate such a d i s c o n t i n u i t y i s t h e conception o f t h e world as a disc, o r a globe. Here again, as i n the case o f t h e 400 year-old concept o f p r o b a b i l i t y , d i s t i n c t d i s c o n t i n u i t i e s can be found. These examples show, on t h e one hand, t h e developmental dependency o f c o g n i t i v e a c t i v i t y . On t h e other hand, they i l l u s t r a t e how f a r an educational and s o c i a l - h i s t o r i c a l shaping o f c o n s t i t u t i v e features o f knowledge and t h i n k i n g schemes influences c o g n i t i v e activity.

13

BIASES, FALLACIES, AND DEVELOPMENTOF DECISION MAKING

W i t h o u t any doubt t h e concept of probability i s c r u c i a l f o r an a n a l y s i s and a c o n c e p t u a l i z a t i o n o f uncertainty. One s h o u l d however t a k e n o t e t h a t f r o m a mathematical p o i n t o f view t h e r e a r e s e v e r a l concepts and c a l c u l i o f p r o b a b i l i t y , and t h a t one s h o u l d d i s t i n g u i s h mathematical f r o m p s y c h o l o g i c a l o b j e c t s . However, t h e r e a r e o f c o u r s e r e l a t i o n s h i p s between t h e i n d i v i d u a l s ' i n t u i t i v e n a i v e knowledge and t h e body o f f o r m a l s c i e n t i f i c knowledge.

This

i n t e r r e l a t i o n s h i p i s n o t o n l y a problem o f c o g n i t i v e psychology b u t a l s o an e p i s t e m o l o g i c a l and e d u c a t i o n a l one. I n d i v i d u a l knowledge, f o r i n s t a n c e about t h e p r o b a b i l i t y concept, depends on t h e i n t e r f e r e n c e o f school and t h e c u l t u r a l background. As LEVY-BRUHL (quoted a c c o r d i n g t o PIAGET B INHELDER, 1975, p. X I V ) p o i n t e d o u t , t h e absence o f t h e i d e a o f chance i s an e s s e n t i a l c h a r a c t e r i s t i c o f t h e p r i m i t i v e mind. Today, t h e r e i s a d i s p u t e among s c i e n t i s t s as t o which scope o f s i t u a t i o n s and which t y p e o f u n c e r t a i n t y may reasonably be h a n d l e d by t h e p r o b a b i l i t y concept, F o r which s i t u a t i o n s a mathematical model based on p r o b a b i l i t y may be m e a n i n g f u l l y c o n s t r u c t e d i s a m a t t e r o f d i s p u t e among mathematicians and s t a t i s t i c i a n s ( c f . UINGES & ROST, 1982). Furthermore, p s y c h o l o g i s t s have found o u t t h a t t h e modes o f c o p i n g w i t h u n c e r t a i n t y i n e v e r y day d e c i s i o n making a r e c u l t u r a l l y dependent ( c f . PHILLIPS & WRIGHT 1Y77, WRIGHT et. a l . 1978). The f i r s t paper p r e s e n t e d i n t h e c h a p t e r on "Development and Epistemology", "CONCEPTUAL AND THEORETICAL ISSUES I N DEVELOPMENTAL RESEARCH ON THE ACQUISITION OF THE PROBABILITY CONCEPT" (SCHOLZ B WALLER), o u t 1 i n e s t h e main v a r i a n t s o f t h e p h i l o s o p h i c a l i n t e r p r e t a t i o n s o f t h e p r o b a b i l i t y concept and a n a l y s e s how t h e s e concepts a r e t o be d e a l t w i t h i n p s y c h o l o g i c a l r e s e a r c h i n t o t h e development o f t h e p r o b a b i l i t y concept. It i s suggested t h a t when b o t h t h e o b j e c t i v i s t i c concept, and t h e i n t u i t i v e component o f p r o b a b i l i t y l e a d i n g t o s u b j e c t i v i s t i c p r o b a b i l i t y a r e j o i n t l y t a k e n i n t o account c o n t r a d i c t i o n s between d i f f e r e n t t h e o r e t i c a l approaches i n psychology w i t h r e g a r d t o t h e a c q u i s i t i o n o f t h e p r o b a b i l i t y concept may b e resolved. I n o r d e r t o understand t h e c a l c u l u s o f p r o b a b i l i t i e s used i n s i m p l e u r n exp e r i m e n t s i t i s necessary t o m a s t e r more t h a n f r a c t i o n a l c a l c u l u s . Also, t h e a b i l i t y t o s o l v e s i m p l e p r o b a b i l i t y t a s k s does n o t emerge spontaneously between t h e ages o f 12 and 14. These a r e two o f t h e r e s u l t s r e p o r t e d i n t h e

INTRODUCTION

14

c o n t r i b u t i o n "THE PERCEPTION

OF

PROBABILITY SITUATIONS I N PUPILS AGED 12 TO

14" (GLAESER) which r e f e r s t o t h e doctoral thesis o f t h e Mexican researcher

_ I

ALAR CO N.

The paper on "THE PSYCHOLOGY OF INFERENCE AND THE TEACHING OF PROBABILITY AND STATISTICS: TWO SIDES OF THE SAME COIN?" (SHAUGHNESSY) discusses conclusions

which may be drawn from t h e t h e o r e t i c a l and empirical research i n psychology and mathematics t h a t could lead t o an improvement o f d e c i s i o n behavior i n t h e process o f teaching p r o b a b i l i t y and s t a t i s t i c s i n t h e classroom. Suggestions a r e made f o r both Psychology and Mathematics Education. Based on some empirical studies already concluded w i t h college courses, a couple o f concrete recommendations can be made such as: 1) "Teach i n t h e f o l l o w i n g order from guesses, t o experiments, t o simulation, t o theory." 2 ) "Emphasize t h e confounding r o l e t h a t b e l i e f systems can play i n judgment and decision making." 3) "Sensitize students t o misuses o f s t a t i s t i c s . ' ' 4) "Give students challenge problems which may r e q u i r e them t o gather o r simulate t h e i r own data, and then make a decision." 5) "Expose students t o s t a t i s t i c a l paradoxes." 6 ) 'I... expose both teachers and students o f p r o b a b i l i t y and s t a t i s t i c s t o the r e s u l t s i n t h e l i t e r a t u r e o f judgment and decision making" (cf. SHAUGHNESSY, t h i s vol ume). The current courses i n s t a t i s t i c s and p r o b a b i l i t y do n o t f o l l o w these ideas and obviously a r e not s u i t a b l e t o overcome the shortcomings t h a t prevent an adequate modelling o f random events. This i s one o f t h e propositions presented i n t h e paper ,"(MIS-) lNTERPRETATION OF STOCHASTIC MODELS" (SCHRAGE). FurtherI

more, i t r e p o r t s serious misapplications o f p r o b a b i l i t y calculus i n a murder t r i a l , medical diagnoses and even i n a mathematical p r o o f on Fermat's conj e c t u r e (an unsolved problem o f number theory) t h a t has been conducted and published by professional mathematicians, and t h a t has a l s o been received awards from an e d i t o r i a l committee ( i n c l u d i n g u n i v e r s i t y teachers). The linkage o f preconcepts and i n t u i t i o n s o f p r o b a b i l i t y on the one hand and m a t h h a t i c a l c a l c u l i and techniques on t h e other hand i s d e c i s i v e f o r an i n s t r u c t i o n o f p r o b a b i l i t y i n schools t h a t would lead t o more successful d e c i s i o n making. The c o n t r i b u t i o n "RANDOMNESS AND STOCHASTIC INDEPENDENCE ON THE RELATIONSHIP BETWEEN I N T U I T I V E NOTION AND MATHEMATICAL DEFINITION"

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15

BIASES, FALLACIES, AND DEVELOPMENT OF DECISION MAKING

(v.HARTEN & STEINBRING) p o i n t s out f i r s t l y , t h a t t h e d i s t i n c t i o n between t h e t h e o r e t i c a l concept o f p r o b a b i l i t y and i t ' s a p p l i c a t i o n i s c r u c i a l f o r an adequate modelling and i n t e r p r e t a t i o n o f stochastic events, and secondly, t h a t t h e r e i s no randomness i n an absolute sense. What i s judged t o be s t o c h a s t i c a l l y independent depends on t h e body o f techniques being used, and on t h e d i f f e r e n t mathematical and i n t u i t i v e approaches t o independence and randomness. The t h e o r e t i c a l and conceptual frame o f reference o f decision t h e o r e t i c modelling and w i t h i t t h e i n t u i t i v e foundations are i n a s t a t e o f continuous change. That which i s perceived t o be paradoxical a t one stage o f s c i e n t i f i c development o r by one school o f thought may be completely c l e a r and undisputed from another p o i n t o f view. Within t h e paper "DECISION THEORETIC

-

PARADOXES" (PFLUG) one can f i n d a v a r i e t y o f examples p a r t l y taken from t h e h i s t o r y o f s t a t i s t i c s which have established o r are s t i l l e s t a b l i s h i n g severe deviations from widely accepted and followed normative c r i t e r i a o f s t a t i s t i c s . The impact of t h i s volume on various disciplines

The t o p i c s discussed i n t h i s volume a r e o f i n t e r e s t f o r several d i s c i p l i n e s . The impact of t h e c o n t r i b u t i o n s presented i n t h i s volume on DECISION RESEARCH (ZIMMER)

, DEVELOPMENTAL PSYCHOLOGY

(WALLER) ,

SOCIAL PSYCHOLOGY (BORCHERDING), ECONOMIC THEORY (SELTEN) , and MATHEMATICS EDUCATION (STEINER) i s o u t l i n e d by researchers from these d i s c i p l i n e s who were

present a t t h e Symposium. O f course, o t h e r d i s c i p l i n e s , e.g. medicine, s o c i a l sciences, mathematics, t h a t were n o t represented, might a l s o be a f f e c t e d o r challenged by t h e r e s u l t s and propositions documented i n t h i s volume. Naturall y , t h e representatives o f t h e d i f f e r e n t d i s c i p l i n e s emphasize d i f f e r e n t aspects. Although from a decision t h e o r e t i c a l perspective, methodoZogica2 pro-

blems and questions o f research s t r a t e g i e s (e.9. t o p down vs. bottom up) seem t o be most s i g n i f i c a n t , conceptual issues about the nature o f human knowledge ( f o r instance "THE IMPORTANCE OF THE FICTIONAL FOR THEORIES OF KNOWLEDGE", see ZIMMER's comment, o r t h e concepts o f l i m i t e d r a t i o n a l i t y i n economic theory, see SELTEN's comment) a r e regarded as c r e a t i n g important research problems i n o t h e r f i e l d s . The comments made by t h e mathematics educational view show the innovative power o f a growing d i s c i p l i n e . The methodology o f mathematization i s shown t o be i n e x t r i c a b l y connected w i t h the s o c i a l dimensions o f l e a r n i n g and i n s t r u c t i o n . Mathematics i n general (and not o n l y s t a t i s t i c s and p r o b a b i l -

16

INTRODUCTION

i t y theory) i s l o o s i n g i t s unique f e a t u r e o f always being e i t h e r r i g h t or

wrong when p u t i n t o a s o c i a l context (e.g.

the classroom). Furthermore, as

several o f t h e papers p o i n t out, i f t h e dynamic views were t o be emphasized, we may n o t o n l y expect decision research t o have an impact on mathematics r e search b u t a l s o the o t h e r way round.

Looking ahead

When considering the o u t l i n e o f contents the question might a r i s e as t o whether i t i s a t a l l possible t o b r i n g such d i f f e r e n t subjects together. The e d i t o r o f t h i s volume i s r a t h e r o p t i m i s t i c i n t h i s respect. The arguments supporting t h i s optimism are a t l e a s t threefold. F i r s t , t h e r e i s t h e necessit y f o r a m u l t i t u d e o f approaches due t o t h e cognitive, s u b j e c t i v e and t o t h e

normative t i e s o f modelling and understanding decisions. These t i e s imply a methodological and d i s c i p l i n a r y multitude. Secondly, broadening t h e frame need n o t necessarily lead t o an increase o f i n f o n a t i o n , complexity and confusion. A theory b u i l d i n g o r i e n t a t i o n increases t h e p o s s i b i l i t y o f r e s t r u c t u r a l i z i n g and r e i n t e g r a t i n g . T h i r d l y , some experiences w i t h regard t o t h e i n t e r a c t i o n w i t h i n t h i s ’broad frame’ were already gained during t h e Symposium about ’Decision Making under Uncertainty’.

The e n t e r p r i s e o f c o n f r o n t i n g

research(ers) from economics w i t h developmental psychology, o r those from mathematics (education) w i t h those from group d e c i s i o n making i n s o c i a l psychology f o r instance may seem t o be adventurous but, as we found out a t B i e l e f e l d , i t worked w e l l , and seems t o be very promising (e.g. t h e second argument) f o r f u t u r e research i n decision making.

w i t h regard t o

References Albers, W. & Albers, 6.: 1983. Prominence and a s p i r a t i o n adjustment i n l o c a t i o n games. In: Tietz, R. (Ed.) A s p i r a t i o n l e v e l s i n Bargaining and Economic Decision Making. Heidelberg: Springer, 243-259 Berkeley, D. & Humphreys, P.C.: 1982. S t r u c t u r i n g Decision Problems and t h e Bias H e u r i s t i c . I n : Acta Psychologica 50, 201-252 Dinges, H. & Rost. H.:

1982. P r i n z i p i e n der Stochastik. S t u t t g a r t : Teubner

Hogarth, R.M.: Cognitive Processes and t h e Assesment o f Subjective P r o b a b i l i t y D i s t r i b u t i o n s . I n : Journal o f t h e American S t a t i s t i c a l Association, June 1975, Vol. 70, No. 350, 271-289

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Kahneman, D. & Tversky, A.: 1982a. V a r i a n t s o f u n c e r t a i n t y . I n : Kahneman, D./ S l o v i c , P. & Tversky, A. (Eds.), Judgement under u n c e r t a i n t y ; H e u r i s t i c s and biases. Cambridge: Cambridge U n i v e r s i t y Press, 509-520 Kahneman, D. & Tversky, A.: 1982b. On t h e study o f s t a t i s t i c a l i n t u i t i o n s . Kahneman, D./Slovic, P. & Tversky, A. (Eds.), Judgement under u n c e r t a i n ty; H e u r i s t i c s and biases. Cambridge: Cambridge U n i v e r s i t y Press, 493-508 Lietzmann, W.:

1969 ( 1 s t ed. 1917). Wo s t e c k t d e r Fehler. L e i p z i g :

M i l l e r , G.A.: 1956. The magical number seven, p l u s o r minus two: Some l i m i t s on our c a p a c i t y f o r processing information. I n : Psychological Review 63, 81-97 N i s b e t t , R.E. & Ross, L.: 1980. Human inference: S t r a t e g i e s and shortcomings o f s o c i a l judgment. New York: Prentice-Hall, Englewood C l i f f s 1977. C u l t u r a l d i f f e r e n c e s i n viewing P h i l l i p s , L.D. & Wright, G.N.: u n c e r t a i n t y and assessing p r o b a b i l i t i e s . I n : Jungermann, H. & deZeeuw, G. (Eds.), D e c i s i o n making an change i n human a f f a i r s . Amsterdam: 0. Reidel, 507-519 Piaget, J. & Inhelder, B.: 1975. La genhse de l ’ i d 6 e de hasard chez l’enfant. P a r i s 1956: Presses U n i v e r s i t a i r e s de France. T r a n s l a t e d as: The O r i g i n o f t h e Idee o f chance i n Children. New York 1975: Norton Sauermann. H. & Selten. R.: 1962: AnsDruchsanDassunastheorie der Unternehmung. I n : i e i t s c h r i f t f u r ’ d i e gesamte Staatswissenschaft 118, 577-597 1980. Dyadische Verhandlungen; Eine t h e o r e t i s c h e und Scholz, R.W.: e x p e r i m e n t e l l e Untersuchung von Vorhersagemodellen. Meisenheim am Glan: Hain Scholz, H.W.: 1981. Stochastische Problemaufgaben: Analysen aus d i d a k t i s c h e r und psycho1 o g i scher Perspekti ve. B i e l e f e l d: U n i v e r s i t a t B i e l e f e l d Shubik, M.: 1982. Personal communication d u r i n g a l e c t u r e h o l d a t B i e l e f e l d University Simon, H.A.:

1955. Models o f Man. New York: Wiley

T i e t z , R. (Ed.): 1983. A s p i r a t i o n l e v e l s i n Bargaining and Economic Decision Making. B e r l i n , Heidelberg, New York, Tokyo: Springer Turoff, M.: 1972. An a l t e r n a t i v e approach t o cross-impact analysis. Technological F o r e c a s t i n g and S o c i a l Change, 309-339

In:

Tversky, A. & Kahneman, 0.: 1979. Causal Schemas i n Judgments Under Uncertainty. I n : Fishbein, M. (Ed.), Progress i n S o c i a l Psychology. H i l l s d a l e : L.E.A., 49-72

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INTRODUCTION

Wallsten, Th.S.: 1980. Processes and models t o describe choice and inference behavior. I n : Wallsten, Th.S. (Ed.): Cognitive processes i n choice and decision behavior. H i l l s d a l e : L.E.A. Wolff, Th.: 1981. Vom lachenden Denken. E i n Buch von Wundern und Problemen. B e r l i n : Scherl

I. & Wright, G.N./Phillips, L.O./Whalley, P.C./Choo, G.T.G./Ng., K.O./Tan, Wisudha, A.: 1978. C u l t u r a l d i f f e r e n c e s i n p r o b a b i l i s t i c thinking. I n : Journal o f Cross-Cul t u r a l Psychology 9, 285-299

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

21

THE THEORETICAL STAlUS OF JUDGMENTAL HEUR ISTICS1

THOMAS S.WALLSTEN University of North Carolina L. L. Thurstone Psychometric Laboratory USA

The recent marked s h i f t i n empirical research on judgment from t e s t i n g normative models t o i n v e s t i g a t i n g people's biases and h e u r i s t i c s has been extremely valuable i n many respects. However, i t a l s o has c e r t a i n shortcomings, as would be expected w i t h any major change i n research focus. There are an almost excessive concern w i t h the t h r e e t h r e e i n t e r r e l a t e d problems: (i) h e u r i s t i c s o f representativeness, a v a i l a b i l i t y , and anchoring and adjustment; ( i i ) a tendency toward t r e a t i n g c l a s s i f i c a t i o n o f h e u r i s t i c s as explanation; and ( i i i ) a l a c k o f rigorous theory development. The l a s t i s t h e most serious problem, o f course, and t h e r e are recent, encouraging trends suggesting t h a t i t may be temporary. The plan o f t h i s paper i s t o f i r s t discuss the problems j u s t o u t l i n e d . Then, having concluded t h a t t h e r e i s a serious need f o r theory development, I w i l l provide a very selected review o f t h e l i t e r a t u r e t o summarize some of the f i n d i n g s t h a t any complete theory must e x p l a i n and t o i n d i c a t e l i m i t a t i o n s t h a t I perceive w i t h some o f the c u r r e n t experimental work.

I am g r a t e f u l t o Austen Clark, Samuel Fillenbaum, and Barbara Forsyth f o r c r i t i c a l l y commenting on an e a r l i e r d r a f t o f t h i s paper, and t o A n i t a Bergmann, Gerda Wallsten, and Herman Wallsten f o r t r a n s l a t i n g chapters by Scholz (1981).

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The problems

The three h e u r i s t i c s o f representativeness, a v a i l a b i l i t y , and anchoring and adjustment were, o f course, f i r s t i d e n t i f i e d by Tversky and Kahneman (1971, 1973, 1974; Kahneman & Tversky, 1972a, 1973), and by now the l i t e r a t u r e on these h e u r i s t i c s i s enormous. T h e i r existence i s probably t h e major c l a i m t h a t people outside o f the f i e l d now know about judgment under uncertainty. However, t h e r e are many other i n t e r e s t i n g features o f judgment which, w h i l e they might n o t be termed h e u r i s t i c s , c e r t a i n l y provide i n s i g h t i n t o t h e judgment process. For example, t h e r e are commonly documented primacy and r e cency e f f e c t s i n judgment. I n t h e former case, e a r l y information e i t h e r causes subsequent information t o be s u b j e c t i v e l y d i s t o r t e d (e.g. Anderson Barrios, 1961; Asch, 1946; Wallsten, 1981), o r causes t h e seeking o n l y o f information which supports o r confirms already e x i s t i n g opinion (Mynatt, Doherty, & Tweney, 1977). I n the l a t t e r case, subsequent information i n a sequence exerts an undue influence on judgment (e.g. Anderson & Hubert, 1963). Another well-documented bias i s t h a t o f value-induced d i s t o r t i o n s ( I r w i n , 1953; I r w i n & Snodgrass, 1966; Delaney & Wallsten, 1977). I n t h i s case, p o s i t i v e and negative values o f events, o r outcomes, i n f l u e n c e t h e i r judged p r o b a b i l i t i e s , and cause information t o be d i s t o r t e d f o r o r against the events.

I am n o t proposing t h a t s i x h e u r i s t i c s and biases ought t o be i n v e s t i g a t e d instead o f three. Rather, the p o i n t i s t h a t research which focusses on any number o f h e u r i s t i c s , simply asking the conditions under which each appears, stops s h o r t o f some o f the most i n t e r e s t i n g and useful questions. C l a s s i f i c a t i o n i s c l e a r l y very important, b u t t h e r e i s t h e danger i n such research, frequently succumbed to, t h a t the d e s c r i p t i o n o f t h e phenomenon i s taken as i t s explanation, and underlying t h e o r e t i c a l p r i n c i p l e s are not sought. A search f o r underlying p r i n c i p l e s r a t h e r than a f i x a t i o n on disparate phenomena w i l l lead t o conclusions o f greater g e n e r a l i t y and usefulness. Related t o the tendency towards explanation by c l a s s i f i c a t i o n i s t h e considerable debate concerning whether o r n o t c e r t a i n judgments are r a t i o n a l (see e.g. Cohen, 1981, and the subsequent commentaries). Cohen (1979) has

JUDGMENTAL HEURISTICS

23

pointed o u t t h a t such evaluations assume independent d e f i n i t i o n s and consensus as t o what i s r a t i o n a l . The question o f what c o n s t i t u t e s r a t i o n a l i t y i s c l e a r l y i n t e r e s t i n g and important, b u t my p o i n t here again i s t h a t the arguments on evaluation focus research away from underlying processes. When the processes by which people encode and judge p r o b a b i l i s t i c i n f o r m a t i o n are s u f f i c i e n t l y w e l l understood, then i t w i l l be p o s s i b l e t o p r e d i c t t h e conditions under which i n d i v i d u a l s use s p e c i f i c h e u r i s t i c s o r are subject t o s p e c i f i c biases. Fortunately, theories along these l i n e s are developing, although t h a t work i s s t i l l i n i t s e a r l y stages and i s q u i t e incomplete. See, f o r example, Groner's chapter i n t h i s volume, Tversky and Kahneman's (1980) o r Einhorn and Hogarth's (1982a, b ) theories on judging c a u s a l i t y , o r my work (Wallsten 1976, 1977; Wallsten & Barton 1982) on sequential process-

i n g o f d i f f e r e n t i a l l y s a l i e n t dimensions. I n t e r e s t i n g l y , both Einhorn and Hogarth's and my t h e o r i e s r e l y s t r o n g l y on an anchoring and adjustment mechanism. To be clear, one does n o t d e s i r e a separate theory f o r each heurist i c o r bias, b u t r a t h e r a s i n g l e t h e o r e t i c a l framework t o p r e d i c t the range o f judgmental e f f e c t s t h a t are observed.

Literature review

I t u r n now t o a selected review o f the l i t e r a t u r e t o summarize some o f the e f f e c t s f o r which any complete theory must account, and t o i n d i c a t e problems w i t h some experiments t h a t become apparent when considering t h e o r e t i c a l explanations. The research s p e c i f i c a l l y t e s t i n g a v a i l a b i l i t y o r anchoring and adjustment i s r e l a t i v e l y sparse ( b u t see F i e d l e r ' s chapter on a v a i l a b i l i t y i n t h i s volume). A possible reason f o r the r e l a t i v e l y l i t t l e empirical work and good agreement among r e s u l t s i s t h a t the conditions f o r e l i c i t i n g these two h e u r i s t i c s appear t o be straightforward and t i e d r a t h e r d i r e c t l y t o well-defined aspects o f the task. A t l e a s t s u p e r f i c i a l l y , i t i s c l e a r when the r e q u i s i t e conditions are present, and when they are not, the h e u r i s t i c s are n o t expected. Indeed, as already indicated, the anchoring and adjustment process i s s u f f i c i e n t l y w e l l s p e c i f i e d t h a t i t can serve as a basis f o r theory development. Matters are d i f f e r e n t w i t h the representativeness h e u r i s t i c . Here t h e r e has

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been a l o t o f research w i t h a wide spectrum o f r e s u l t s . There are a t l e a s t three reasons f o r t h i s s t a t e o f a f f a i r s . F i r s t , representativeness has n o t been, and perhaps cannot be, defined a p r i o r i i n terms o f s p e c i f i c features o r r e l a t i o n s . Second, p r e d i c t i o n s from the representativeness h e u r i s t i c a r e generally i n terms of features o f the stimulus t h a t w i l l n o t i n f l u e n c e judgment. Thus, the predicted experimental outcome i s f r e q u e n t l y t h a t o f t h e n u l l r e s u l t . Third, many o f the experiments have been conducted i n t h e cont e x t o f word problems. The phrasing and choice o f words i s c r u c i a l and can have a substantial impact on t h e way the information i s encoded.

Base Rates An example o f the p r e d i c t i o n o f a n u l l r e s u l t concerns the use o f base r a t e information. According t o Kahneman and Tversky (1973), a person's judgment o f the p r o b a b i l i t y t h a t event A belongs t o class B, r a t h e r than t o some other class, depends o n l y on how representative A i s o f

B, and n o t on t h e

base r a t e o f B i n the population. A chapter by B a r - H i l l e l i n t h i s volume r e views the l i t e r a t u r e on use o f base rates, and i t i s t h e r e f o r e n o t necessary t o provide another d e t a i l e d review here. However, the perspective i n t h i s chapter i s d i f f e r e n t from hers, so some a d d i t i o n a l discussion i s warranted. F i r s t , r e c a l l the basic phenomenon as demonstrated by Kahneman and Tversky (1973). I n one experiment subjects read b r i e f p e r s o n a l i t y sketches, and f o r each one r a t e d the l i k e l i h o o d t h a t the described person was a graduate student i n each o f nine f i e l d s . These r a t i n g s c o r r e l a t e d r e l a t i v e l y p o o r l y w i t h independently judged base rates, i.e., judged proportions o f graduate students i n each o f t h e f i e l d s (mean ~ = . 1 7 ) , b u t c o r r e l a t e d h i g h l y w i t h independently r a t e d s i m i l a r i t i e s o f each p e r s o n a l i t y sketch t o the stereotype graduate student i n each f i e l d (mean ~ = . 9 1 ) . I n another experiment subjects r a t e d persona l i t y sketches according t o whether the described i n d i v i d u a l s were more 1i k e l y engineers o r lawyers, knowing t h a t each sketch came from a population c o n s i s t i n g o f 70 % engineers and 30 % lawyers ( o r the converse). Ratings v a r i e d w i t h the p e r s o n a l i t y sketches, b u t were independent o f the population base rates. Although base r a t e s are c l e a r l y important i n a l l Bayesian c a l c u l a t i o n s , t h e r e are s i t u a t i o n s o r modes o f t h i n k i n g w i t h i n which t h e i r use i s questionable,

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25

as Cohen (1979), Shafer (1981), and a few others have pointed out. An example provided by Cohen (1979) i s one i n which 1,000 people are watching a rodeo; i t i s known t h a t 400 o f them have paid and assumed t h a t the remainder have entered through a h o l e i n the fence. Under those conditions, i t would generally be agreed t h a t an a r b i t r a r i l y selected person should n o t be prosecuted f o r f a i l u r e t o pay simply because the p r o b a b i l i t y t h a t he f a i l e d t o do so i s .6. To expand the example, given a f i x e d amount o f evidence against the person, he would n o t be considered t o be more l i k e l y g u i l t y when 700 people entered through the fence than when 600 people did. Nevertheless, under many conditions base r a t e s do provide useful information, regardless o f the mode o f t h i n k i n g , and i t i s o f i n t e r e s t t o i n q u i r e i n t o t h e i r r o l e i n judgment, p a r t i c u l a r l y when the representativeness h e u r i s t i c suggests t h a t they w i l l be ignored. I n f a c t , t h e r e are various conditions under which base r a t e s are attended to, and thus f a r a t l e a s t t h r e e f a c t o r s have been i d e n t i f i e d t h a t a f f e c t t h e i r use. These are ( i ) e x p e r t i s e o f the judge, ( i i ) relevance o r s p e c i f i c i t y o f the information, and ( i i i ) salience o f the information.

Expertise.

There i s evidence t h a t experts working i n t h e i r f i e l d do n o t

demonstrate many o f t h e same l i m i t a t i o n s shown by subjects i n experimental s i t u a t i o n s . Thus, physicians appear t o be s e n s i t i v e to- base r a t e s when judgi n g t h e p r o b a b i l i t i e s o f diseases. I n a recent experiment (Wallsten 1981), experienced physicians and f o u r t h year medical students r a t e d l i k e l i h o o d s f o r 50 diseases based on i n i t i a l information, then f o l l o w i n g h i s t o r i c a l and physical data, and then again f o l l o w i n g l a b o r a t o r y data, f o r each o f a l a r g e number o f hypothetical patients. The medical students were q u i t e i n s e n s i t i v e t o base rates; they i n i t i a l l y gave a very l a r g e number o f .5 l i k e l i h o o d r a t i n g s , which then propogated throughout t h e examination. The physicians, on t h e other hand, used considerably fewer .5 estimates. Instead, they provided estimates t h a t were much c l o s e r t o zero, which then tended t o decrease f u r t h e r as the examination continued, as was appropriate f o r t h e i n f o r m a t i o n they were receiving. Christensen-Szalanski and Bushyhead (1981) a l s o found evidence i n ongoing c l i n i c a l s i t u a t i o n s t h a t physicians were s e n s i t i v e t o base rates.

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S p e c i f i c i t y of infomution.

B a r - H i l l e l (1980) suggested t h a t base r a t e s

tend t o be ignored when subjects do n o t perceive them t o be r e l e v a n t t o t h e judgment task a t hand. Without f u l l y d e f i n i n g the concept o f relevance, she suggested, c o r r e c t l y I think, t h a t the smaller the subset o f i n d i v i d u a l s o r events t o which the information r e f e r s , the more r e l e v a n t i t i s deemed t o the judgment a t hand. Thus, f o r example, t o l e a r n t h a t a given i n d i v i d u a l enters through a hole i n the fence 60 % o f the time t h a t he goes t o an event i s more r e l e v a n t than t o l e a r n t h a t 60 % o f the people a t the event have snuck i n . Using a series of problems based on Kahneman and Tversky's (1972b) t a x i cab problem, B a r - H i l l e l (1980) manipulated the subset o f t h e population t o which the base r a t e o r diagnostic information referred. She e i t h e r lowered the r e levance o f the diagnostic information by making i t apply l e s s s p e c i f i c a l l y t o the event i n question, o r increased the relevance o f t h e base r a t e by making i t apply more s p e c i f i c a l l y t o the event. I n these cases, s u b j e c t s ' judgments represented an i n t e g r a t i o n o f base r a t e s w i t h diagnostic informat i o n . I n an e a r l i e r experiment using Tversky and Kahneman's (1973) lawyer/ engineer paradigm, C a r r o l l and S i e g l e r (1977) demonstrated t h a t subjects do use base rates, b u t do so t o a greater degree when the base r a t e r e f e r s t o a small sample, each member o f which i s t o be judged, than when i t r e f e r s t o a l a r g e population, o n l y a small f r a c t i o n o f which i s t o be judged. This can be thought o f as another way t o increase the s p e c i f i c i t y , o r relevance o f base rates.

SuZience of base rates.

Nisbett, Borgida, Crandall , and Reed (1976) have

suggested t h a t base r a t e information tends t o be ignored because i t i s " r e mote, p a l l i d , and abstract", i n c o n t r a s t t o d i a g n o s t i c information, which i s " v i v i d , s a l i e n t , and concrete". This hypothesis was supported i n an experiment by Borgida and N i s b e t t (1977) i n which prospective psychology majors r a t e d psychology courses based e i t h e r on catalog i n f o r m a t i o n only, o r on t h a t information supplemented by mean student course evaluations o r by faceto-face comments from o l d e r students. The " p a l l i d " course evaluations had l i t t l e impact on ratings, w h i l e the " v i v i d " comments had s u b s t a n t i a l e f f e c t . Indeed, i t may be t h a t a major reason why base r a t e s have been i n e f f e c t i v e

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i n experiments i s because they are s t a t e d once, and then allowed t o fade i n t o the background w h i l e diagnostic o r t a r g e t i n f o r m a t i o n varies over t r i a l s . Consistent w i t h t h i s suggestion, Fischhoff, Slovic, and Lichtens t e i n (1979) demonstrated t h a t subjects are very responsive t o base r a t e s when the base r a t e s vary from t r i a l t o t r i a l . Base rates may a l s o be more s a l i e n t i f they are a c t u a l l y experienced over time, r a t h e r than simply s t a t e d i n summary form. This was demonstrated i n f o u r experiments by Manis, Dovalina, Avis, and Cardoze (1980), i n which subj e c t s experienced base rates by seeing a sequence o f photographs o f some i n d i v i d u a l s i n the population w h i l e being t o l d the r e l e v a n t behavior o r b e l i e f o f each one. These base r a t e s then a f f e c t e d subjects’ judgments o f t h e behavi o r o r b e l i e f s o f o t h e r members o f t h e population. Manis e t a l . a l s o attempted t o manipulate concreteness o f the base r a t e s through the format i n which they were presented, b u t found t h i s manipulation t o add nothing beyond the e f f e c t o f experience i t s e l f . Christensen-Szalanski and Beach (1982) a l s o gave subjects experience w i t h base rates, b u t then, i n a switch from o t h e r experiments, manipulated whether diagnostic information was simultaneously experienced w i t h t h e base r a t e s o r was subsequently presented i n summary s t a t i s t i c a l form. I n t e r e s t i n g l y , o n l y the subjects simultaneously experiencing both types o f information i n t e g r a t e d them when l a t e r judging the p o s t e r i o r p r o b a b i l i t y o f an event. Judgments by the o t h e r subjects r e f l e c t e d only t h e summary d i a g n o s t i c data and n o t t h e experienced base rates. Also, 10 % o f t h i s l a t t e r group f a i l e d t o respond, i n d i c a t i n g they d i d n o t understand o r could n o t answer the question, whereas none o f the former group f a i l e d t o respond. Christensen-Szalanski and Beach (1982) suggest t h a t base rates are more r e a d i l y used when they are experienced i n r e l a t i o n t o diagnostic data. I f t h i s conclusion i s supported i n f u t u r e r e search, then simple notions o f salience w i l l n o t e x p l a i n under u t i l i z a t i o n o f base rates. I n another study attempting t o make base r a t e s more concrete, C a r r o l l and S i e g l e r (1977) t e s t e d the hypothesis t h a t subjects were more l i k e l y t o use base r a t e s if they could be t r a n s l a t e d i n t o concrete numbers o f people. One group o f subjects judged 20 p e r s o n a l i t y sketches from a population w’ith a

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70:30 s p l i t , which could e a s i l y be t r a n s l a t e d i n t o 14 lawyers and 6 engineers, o r the opposite; w h i l e another group had a 75:25 base r a t e which could n o t be t r a n s l a t e d i n t o an exact number o f lawyers o r engineers. I n two experiments, one w i t h neutral and one w i t h i n f o r m a t i v e sketches, base r a t e had a s i g n i f i c a n t e f f e c t . T r a n s l a t a b i l i t y o f base r a t e s had an i n t e r a c t i v e e f f e c t w i t h t h e n e u t r a l sketches only. I n t h a t case, base r a t e was more i n f l u e n t i a l i n t h e t r a n s l a t a b l e 70 % c o n d i t i o n than i n t h e non-translatable 75 % condition. To summarize the research on use o f base rates, t h e r e i s no doubt t h a t i n many problems base r a t e s are e i t h e r ignored o r i n s u f f i c i e n t l y weighted. However, the g e n e r a l i z a t i o n t h a t subjects judge according t o representativeness and therefore ignore base r a t e s i s c l e a r l y overstated. A t l e a s t t h r e e f a c t o r s affect the use o f base rates: subject expertise, i n f o r m a t i o n s p e c i f i c i t y , and information concreteness o r salience. Furthermore, t h e r e i s the addit i o n a l r e s u l t by Christensen-Szalanski and Beach (1982) t h a t 10 % o f the subjects d i d n o t understand the problem when p a r t s o f i t were presented i n summary form. Problem comprehension has n o t g e n e r a l l y been assessed, b u t i t may be an important f a c t o r i n determining which i n f o r m a t i o n i s u t i l i z e d and which i s ignored.

Sampling distributions The representativeness p r e d i c t i o n regarding judgments o f sampling d i s t r i b u t i o n s i s also a negative one. On the assumption t h a t when people use sample s t a t i s t i c s t o estimate population parameters, they judge the s i m i l a r i t y o f the s t a t i s t i c t o the parameter, i t i s predicted t h a t judgments w i l l be i n s e n s i t i v e t o sample size. Kahneman and Tversky (1972a) provide many examples o f t h i s h e u r i s t i c a t work, two o f which w i l l be summarized here. I n one experiment, s u b j e c t i v e sampling d i s t r i b u t i o n s were estimated by asking l a r g e numbers o f subjects questions o f the f o l l o w i n g s o r t : On what percentage o f days w i l l the number o f boys among 1000 babies born be as f o l l o w s : up t o 50 boys, 50-150 boys, ..., 850-950 boys, more than 950 boys? Problems d i f f e r e d i n terns o f the s i z e o f the sample, being e i t h e r 10, 100, o r 1000 babies i n the above instance, and a l s o i n terms o f the p a r t i c u l a r v a r i a b l e over which t h e d i s t r i b u t i o n was t o be estimated. I n a l l cases t h e s u b j e c t i v e sampling

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distribution was independent of sample s i z e , a "universal sampling distribution", in Kahneman and Tversky's words. In another experiment utilizing three different problems, some subjects were required to judge whether a large sample or a small sample was more likely t o yield an extreme outcome, or whether the probability was about equal f o r the two samples. Other subjects made the judgment with regard t o a non-extreme outcome. For each outcome, the majority of the subjects judged i t t o be equally likely in the two samples, thereby ignoring the impact of sample size. Studies looking into the generality of these r e s u l t s have indicated t h a t sample size i s n o t always ignored, and have identified a t l e a s t three factors that a f f e c t judgments about samples: ( i ) individual differences; ( i i ) sample t o population r a t i o ; and ( i i i ) the wording of the problems.

IndividuaZ differences. In commenting on the Kahneman and Tverhky (1972a) results, Evans and Dusoir (1977) suggest t h a t "the construction of t h e i r problems seems unnecessarily complex and . . . t h e i r t e s t s are not designed t o look a t individuals, thus f a i l i n g t o provide a powerful t e s t of the hypothesis t h a t the role of sample size i s understood by some b u t not a l l subj e c t s " ( p . 130). In one experiment Evans and Dusoir (1977) used binomial samples that varied in sample proportion as well as sample size. Subjects were shown a l l possible pairs of the binomial samples, and i n each case were required t o s t a t e which sample gave better evidence of bias in the underlying population. .In this context, individual subject analyses showed t h a t a clear minority of the subjects (11 of 48) were responsive t o sample s i z e considerations, while the remainder were generally insensitive t o sample size. Recently, Krantz, Nisbett, and t h e i r associates have also demonstrated strong individual differences in the nature of s t a t i s t i c a l reasoning. In one study (Jepson, Krantz, & Nisbett 1982) subjects were given a s e r i e s of word problems t o solve, where the correct solution required recognition of e i t h e r the importance of sample s i z e o r the principle o f regression t o the mean. For each problem, subjects provided brief descriptions of how they arrived a t t h e i r solution. Each description was then scored as s t a t i s t i c a l i f in some

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fashion i t mentioned the c o r r e c t p r i n c i p l e , o r as d e t e r m i n i s t i c o r intermed i a t e otherwise, according t o c e r t a i n c r i t e r i a . For our purposes t h e importa n t r e s u l t was a s i g n i f i c a n t c o r r e l a t i o n i n d i c a t i n g t h a t subjects w i t h higher SAT'S tended t o g i v e more s t a t i s t i c a l responses. I n another study (Krantz, Fang, & N i s b e t t 1982) t h e tendency t o provide s t a t i s t i c a l descriptions i n creased dramatically w i t h degree o f s t a t i s t i c a l t r a i n i n g . Furthermore, t h e use o f s t a t i s t i c a l reasoning was greater i f t h e wording o f the problem specif i c a l l y i n d i c a t e d a random f o r c e a t work. Sample t o PopuZation ratio.

Bar-Hi1 l e l (1979) suggested t h a t subjects f r e -

quently appear t o be s e n s i t i v e t o sample s i z e as an a r t i f a c t o f responding t o perceived sample t o population r a t i o , r a t h e r than because they are responding t o sample size, per se. To t e s t t h i s notion, she t o l d subjects about nine hypothetical v o t i n g surveys which d i f f e r e d i n terms o f t h e number o f people i n the survey sample and the number o f people i n the v o t i n g popul a t i o n . Subjects had t o judge the accuracy o f each survey. As predicted, they were most responsive t o the sample t o population r a t i o , b u t i n t e r e s t i n g l y , w i t h i n r a t i o s they were responsive t o sample size. Normatively, o f course, when sampling from a f i n i t e population t h e sample t o population r a t i o i s an important consideration. However, given t h e population sizes used i n these problems, sample s i z e alone should have been t h e primary, n o t t h e secondary,

determinant o f confidence.

ProbZem wording.

Returning t o Evans and Dusoir (1977), they thought t h a t i n

Kahneman and Tversky's (1972a) o r i g i n a l experiment subjects may have appeared i n d i f f e r e n t t o sample size, because the problems were worded i n a confusing manner. Thus, f o r example, i n the o r i g i n a l experiment, one problem described a l a r g e and a small h o s p i t a l , and then stated: "For a p e r i o d o f one y e a r each h o s p i t a l recorded the days on which more than 60 % o f t h e babies born were boys. Which h o s p i t a l do you t h i n k recorded more such days?" Evans and Dusoir thought t h a t paragraph might be complicated w i t h regard t o two concepts: "most days included i n one year", and "more than 60 % o f t h e babies". Thus, they gave the o r i g i n a l problem t o one group o f subjects b u t then changed t h e problem f o r another group o f subjects t o read: "Which h o s p i t a l do you t h i n k

i s more l i k e l y t o f i n d on one such day t h a t a l l t h e babies born were boys?" Two other groups o f subjects received problems t h a t were s i m p l i f i e d w i t h r e -

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gard t o only one o r the other o f the two concepts. O f t h e 20 subjects receivi n g each problem, 11 provided the c o r r e c t answer f o r the problem t h a t e x a c t l y duplicated Kahneman and Tversky's, w h i l e 48 o f the remaining GO subjects gave t h e c o r r e c t answer t o s i m p l i f i e d problems. I n a d i f f e r e n t context, BarH i l l e l (1979) found a s i m i l a r r e s u l t . Olson (1976) a l s o questioned the g e n e r a l i t y o f t h e Kahneman and Tversky res u l t s . He designed t h r e e experiments on the assumption t h a t subjects were responding t o concrete features determined by the problem wording, r a t h e r than t o a b s t r a c t representations o f the task. Thus, f o r example, regarding t h e universal sampling d i s t r i b u t i o n , Olson thought t h a t perhaps subjects were responding simply t o the o r d i n a l p o s i t i o n s o f the categories r a t h e r than t o the degree t h a t a sample c h a r a c t e r i s t i c represented a population c h a r a c t e r i s t i c . To t e s t t h i s hypothesis, he obtained s u b j e c t i v e sampling d i s t r i b u t i o n s by having subjects estimate the p r o p o r t i o n o f a sample t h a t would be expected i n each o f 11 categories. Both sample s i z e and category s i z e v a r i e d over subjects, b u t the percentage o f t h e sample p u t i n each category was t h e same i n a l l cases, thereby supporting Olson's hypothesis. Two a d d i t i o n a l experiments i n q u i r e d i n t o the features t h a t govern whether a sample i s judged representative o f a population. Kahneman and Tversky (1972a) suggested t h a t a s a l i e n t feature o f a binomial process i s the i d e n t i t y o f the m a j o r i t y outcome. Thus, a binomial sample t h a t preserves the m a j o r i t y m i n o r i t y r e l a t i o n i n t h e population i s judged t o be more probably than one t h a t v i o l a t e 9 t h i s r e l a t i o n , regardless o f o b j e c t i v e considerations. As an a l t e r n a t i v e hypothesis, Olson thought t h a t perhaps people who are n o t knowledgeable about p r o b a b i l i s t i c sampling may t r e a t t h e s t a t e d percentages as absolute numbers r a t h e r than as proportions. The sample i s then judged accordi n g t o whether those physical numbers could a r i s e from t h e given population. For example, Olson p r e d i c t s t h a t subjects would judge t h a t a sample w i t h

45 % X's i s more l i k e l y t o come from a population w i t h 55 % X ' s than from one w i t h 35 % X's. Kahneman and Tversky would make the opposite p r e d i c t i o n . Olson (1976) obtained support f o r h i s p r e d i c t i o n s u t i l i z i n g numerous word problems. Olson's conclusion was n o t t h a t t h e representativeness was wrong, b u t r a t h e r

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t h a t i t was incomplete w i t h o u t s p e c i f y i n g the dimensions along which representativeness i s judged. He distinguished between what he c a l l e d " s u p e r f i c i a i task c h a r a c t e r i s t i c s and underlying problem c h a r a c t e r i s t i c s " , t h e f o r mer depending on the wording o f the question, the form o f t h e response, etc., and the l a t t e r being t h a t about which a judgment i s sought. He concluded t h a t h i s and Kahneman and Tverskyls subjects were responding t o the s u p e r f i c i a l task c h a r a c t e r i s t i c s r a t h e r than t o the underlying problem c h a r a c t e r i s t i c s . This d i s t i n c t i o n can be employed i n summarizing the research on sampling d i s t r i b u t i o n s . F i r s t o f a l l , t h e impact o f sample s i z e on judgment i s f r e q u e n t l y small, but generally n o t zero, and furthermore, f o r some people i t i s q u i t e large. The primary d i f f i c u l t y w i t h research i n t h i s area i s the l a c k o f spec i f i c i t y concerning the determinants o f representativeness combined w i t h t h e s e n s i t i v i t y o f people's responses t o the wording o f t h e problems.

Discussion The purpose o f t h i s very selected review was t o i n d i c a t e some o f the basic e f f e c t s f o r which a theory must account t h a t have emerged i n t h e study o f h e u r i s t i c s , and some o f the l i m i t a t i o n s i n many o f t h e c u r r e n t experimental designs. The l i m i t a t i o n s , o f course, derive from i n s u f f i c i e n t a t t e n t i o n t o the e f f e c t s , so the two aspects can be discussed simultaneously. The manner i n which people use uncertain i n f o r m a t i o n t o a r r i v e t o a judgment seems t o depend, a t l e a s t i n p a r t , on ( i ) how s p e c i f i c a l l y the information r e f e r s t o t h e event i n question, (ii) f a c t o r s t h a t a f f e c t t h e salience o f t h e i n f o r mation, ( i i i ) i n d i v i d u a l differences, i n c l u d i n g expertise, t r a i n i n g , and q u a n t i t a t i v e aptitude, and ( i v ) the wording o f the problem. Theories are needed t o e l u c i d a t e these phenomena, and experimental designs are r e q u i r e d t h a t pay more a t t e n t i o n t o them. Recognition o f these f a c t o r s i s growing i n the h e u r i s t i c s l i t e r a t u r e , as evidenced, f o r example, by the recent thorough discussion by Kahneman and Tversky (1982). Also , as i n d i c a t e d e a r l i e r , t h e o r i e s regarding use o r i n t e r p r e t a t i o n o f p r o b a b i l i s t i c i n f o r m a t i o n are being developed. However, i t i s d i f f i c u l t a t t h i s p o i n t t o see any r e a l impact o f these developments on the design and i n t e r p r e t a t i o n o f experiments on judgmental h e u r i s t i c s .

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Furthermore, none o f the theories o f f e r e d thus f a r attempt t o e x p l a i n i n d i vidual differences o r the e f f e c t s o f problem wording. As Scholz (1981) has pointed out, the d i f f i c u l t y i n analyzing responses t o

s p e c i f i c word problems i s t h a t one cannot separate semantic considerations unique t o the p a r t i c u l a r problem from the a p p l i c a t i o n o f global p r o b a b i l i s t i c concepts. There i s considerable evidence i n other problem s o l v i n g areas t h a t the wording of a problem c r u c i a l l y determines how i t i s understood and the nature o f the c o g n i t i v e operations t h a t w i l l be used t o solve it. Simon (1978) has suggested t h a t a person t r a n s l a t e s a p a r t i c u l a r s i t u a t i o n i n t o a "problem representation" o r a "problem space", which i s then operated upon o r about which judgments are made. The f i n a l response depends both upon t h e encoding o f the task and the nature o f t h e judgment process. This was demons t r a t e d by tlayesand Simon (1977) i n t h e i r study o f problem isomorphs, i.e., f o r m a l l y i d e n t i c a l problems t h a t d i f f e r only i n terms o f t h e semantic encodi n g o f the r e l a t i o n s h i p s . I n t h a t experiment, t h e nature o f the semantic encoding profoundly a f f e c t e d subjects' s o l u t i o n s .

I n another context, Wason and h i s colleagues (Wason & Johnson-Laird 1970; Wason & Shapiro 1971) have demonstrated t h a t people have d i f f i c u l t y s o l v i n g a simple deductive l o g i c problem when the problem i s s t a t e d a b s t r a c t l y , b u t have no d i f f i c u l t y a t a l l when i t i s reformulated i n terms o f s i t u a t i o n s they are used t o t h i n k i n g about. Very recently, Cox and Griggs (1982) have shown t h a t performance i n the task depends on the degree of s i m i l a r i t y o f the problem content t o subjects' experiences, and they propose a reasoning by analogy explanation. There i s a l s o research w i t h i n t h e judgment area showing t h a t t h e nature o f the wording influences the response, much o f i t summarized and discussed very thoroughly by Fischhoff, S l o v i c and L i c h t e n s t e i n (1980). However, the i m p l i c a t i o n s o f t h i s work have n o t y e t been f e l t i n t h e research on heuristics. Turning t o t h e problem o f i n d i v i d u a l differences, some a t t e n t i o n has been given i n the problem s o l v i n g l i t e r a t u r e t o the nature o f t h e d i f f e r e n c e s between experts and novices, For example, Chi, F e l t o v i c h and Glaser (1981) sug-

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COGNlTl VE DECISION RESEARCH

gested t h a t when faced w i t h a physics problem experts a b s t r a c t t h e basic p r i n c i p l e s from the p a r t i c u l a r problem while novices represent the problem i n terms o f i t s l i t e r a l features. Not s u r p r i s i n g l y , the experts b r i n g a good deal o f t a c i t knowledge t o the task t h a t allows them t o make inferences and d e r i v a t i o n s from the s i t u a t i o n described by the problem statement. I n a medic a l judgment s i t u a t i o n , Johnson, Hassebrock, Duran and M o l l e r (1982) demons t r a t e d considerable differences between experienced physicians and l e s s experienced ones i n how they e s t a b l i s h diagnoses based on a v a i l a b l e information. The lesson behind a l l t h i s i s t h a t i t i s n o t s u f f i c i e n t simply t o ask whether a p a r t i c u l a r h e u r i s t i c i s used o r a p a r t i c u l a r judgment i s r a t i o n a l . The o r i g i n a l work demonstrating h e u r i s t i c s was extremely important i n refocussing research away from o n l y t e s t i n g normative models. However, we have now reached the p o i n t where i t i s necessary t o develop theories o f problem representation and o f judgment t h a t concern both the semantic and t h e s t a t i s t i c a l aspects o f the s i t u a t i o n s . These theories must i n d i c a t e how the problem s t r u c t u r e determines,which dimensions o f the information w i l l be encoded, and how these features w i l l be used. The research on h e u r i s t i c s should r e l y l e s s on i n d i v i d u a l word problems, and more on the systematic manipulation o f features i n a manner determined by the theory under consideration. References

Anderson, N.H. & Barrios, A.A.: 1961. Primacy e f f e c t s i n p e r s o n a l i t y impression formation. I n : Journal o f Abnormal and Social Psychology, 63, 346-350 Anderson, N.H. & Hubert, S . : 1963. E f f e c t s o f concomitant verbal r e c a l l on order e f f e c t s i n p e r s o n a l i t y impression formation. I n : Journal o f Verbal Learning and Verbal Behavior, 2, 379-391 Asch, S.E.: 1946. Forming impressions o f pers'onality. I n : Journal o f Abnormal and Social Psychology, 41, 258-290 B a r - H i l l e l , M.: 1979. The r o l e o f sample s i z e i n sample evaluation. I n : Qrgirnizational Behavior and Human Performance, 24, 245-257 B a r - H i l l e l , M.: 1980. The base-rate f a l l a c y i n p r o b a b i l i t y judgments. I n : Acta Psychologica, 44, 211-233 Borgida, E. & N i s b e t t , R.: 1977. The d i f f e r e n t i a l impact o f a b s t r a c t vs. concrete information. I n : Journal o f Applidd Social Psychology, 7, 258-271

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C a r r o l l , J.S. & S i e g l e r , R.S. : 1977. S t r a t e g i e s f o r t h e use o f base-rate i n formation. I n : Organizational Behavior and Human Performance, 19, 392-402 Chi , M.T.H./Feltovich, P.J. & Glaser, R. : 1981. C a t e g o r i z a t i o n and repres e n t a t i o n o f physics problems by e x p e r t s and novices. In: C o g n i t i v e Science, 5, 121-152 Christensen-Szalanski, J.J.J. & Beach, L.R.: 1982. Experience and t h e baser a t e f a l l a c y . In: Organizational Behavior and Human Performance, 29, 270-278 Christensen-Szalanski, J.J.J. & Bushyhead, J.B.: 1981. P h y s i c i a n s ' use o f p r o b a b i l i s t i c i n f o r m a t i o n i n a r e a l c l i n i c a l s e t t i n g . In: Journal o f Experimental Psychology: Human Perception and Performance, 7, 928-935 Cohen, L.J.: 1979. On t h e psychology o f p r e d i c t i o n : Whose i s t h e f a l l a c y ? I n : Cognition, 7, 385-407 Cohen, L.J.: 1981. Can human i r r a t i o n a l i t y be e x p e r i m e n t a l l y demonstrated? In: The Behavioral and B r a i n Sciences, 4, 317-331 Cox, J.R. & Griggs, R.A.: 1982. The e f f e c t s o f experience on performance i n Wason's s e l e c t i o n task. I n : Memory and Cognition, 10, 496-502 Delaney, H.D. & Wallsten, T.S. : 1977. P r o b a b i l i s t i c i n f o r m a t i o n processing: E f f e c t s o f a biased p a y o f f m a t r i x on choices and b i d s . In: Organiza t i o n a l Behavior and Human Performance, 20, 203-237 Einhorn, H.J. & Hogarth, R.M.: 1982a. A theory o f d i a g n o s t i c i n f e r e n c e : I. Imagination and t h e psychophysics o f evidence. Chicago, I L , Center f o r Decision Research, U n i v e r s i t y o f Chicago Einhorn, H.J. & Hogarth, R.M.: 1982b. A theory o f d i a g n o s t i c i n f e r e n c e : 11. Judging c a u s a l i t y . Chicago, I L , Center f o r Decision Research, U n i v e r s i t y o f Chicago Evans, J.St.B.T. & Dusoir, A.E.: 1977. P r o p o r t i o n a l i t y and sample s i z e as f a c t o r s i n i n t u i t i v e s t a t i s t i c a l judgment. I n : Acta Psychologica, 41, 129-137 F i s c h h o f f , B./Slovic, P. & L i c h t e n s t e i n , S. : 1979. S u b j e c t i v e s e n s i t i v i t y a n a l y s i s . I n : Organizational Behavior and Human Performance, 23, 339-359 F i s c h h o f f , B./Slovic, P. & L i c h t e n s t e i n , S . : 1980. Knowing what you want: Measuring l a b i l e values. In: Wallsten, T.S. (ed.), C o g n i t i v e Processes i n Choice and Decision Behavior. H i l l s d a l e , NJ, Erlbaum, 115-141 Hayes, J.R. & Simon, H.A.: 1977. Psychological d i f f e r e n c e s among problem isomorphs. I n : Castellan, N.J., Jr.,/Pisoni, D.B. & Potts, G.R. (eds.), C o g n i t i v e Theory, Vol. 2, H i l l s d a l e , NJ, Erlbaum, 21-41

Irwin, F.W.: 1953. S t a t e d expectations as a f u n c t i o n o f p r o b a b i l i t y and d e s i r a b i l i t y o f outcomes. I n : Journal o f P e r s o n a l i t y , 21, 329-335

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Irwin, F. & Snodgrass, J . : 1966. Effects of independent and dependent outcome values upon bets. In: Journal of Experimental Psychology, 71, 282-285 Jepson, C./Krantz, D.H. & Nisbett, R . E . : 1982. Inductive reasoning: Competence o r s k i l l ? (Bell Labs Technical Memorandum). Murray H i l l , NJ, Bell Labs Johnson, P.E./Hassebrock, F./Duran, A.S. & Moller, J.H.: 1982. Multimethod study of c l i n i c a l judgment. In: Organizational Behavior and Human Performance, 30, 201-230 Kahneman, D. & Tversky, A . : 1972a. Subjective p r o b a b i l i t y : A judgment of representativeness. In: Cognitive Psychology, 3, 430-454 Kahneman, D. & Tversky, A.: 1972b. On prediction and judgment. (OR1 Research Monograph, 12 ( 4 ) ) Eugene, OR, Oregon Research I n s t i t u t e Kahneman, D. & Tversky, A.: 1973. On t h e psychology of prediction. In: Psychological Review, 80, 237-251 Kahneman, D. & Tversky, A.: 1982. On the study of s t a t i s t i c a l i n t u i t i o n s . In: Cognition, 11, 123-141 Krantz, D.H./Fong, G.T. & Nisbett, R.E. : 1982. Formal t r a i n i n g improves the application of s t a t i s t i c a l h e u r i s t i c s t o every day problems (Bell Labs Technical Memorandum). Murray H i l l , NJ, Bell Labs Manis, M./Dovalina, I./Avis, N.E. & Cardoze, S.: 1980. Base r a t e s can a f f e c t individual predictions. In: Journal of Personality and Social Psychology, 38, 231-248 Mynatt, C.R./Doherty, M.E. & Tweney, R.D.: 1977. Confirmation bias i n a simul a t e d research environment: An experimental study of s c i e n t i f i c inference. In: Q u a r t e r l y Journal of Experimental Psychology, 29, 85-95 Nisbett, R.E./Borgida, E./Crandall, R. & Reed, H . : 1976. Popular induction: Information i s not necessarily informative. In: C a r r o l l , J.S. & Payne, J.W. ( e d s . ) , Cognition and Social Behavior, H i l l s d a l e , NJ, Erlbaum, 113-134 Olson, C.L. : 1976. Some apparent v i o l a t i o n s of t h e representativeness heurist i c i n human judgment. In: Journal of Experimental Psychology: Human Perception and Performance, 2, 599-608 Scholz, R.W. : 1981. Stochastische Problemaufgaben: Analysen aus didaktischer und psychologischer Perspektive. IDM Materialien und Studien Band 23. Bielefel d , Uni versi t a t Biel e f e l d

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Shafer, G.: 1981. Constructive probability. In: Synthese, 48, 1-60 Simon, H.A. : 1978. Information-processing theory of human problem solving. In: Estes, W.K. ( e d . ) , Handbook of Learning and Cognitive Processes Vol. 5: Human Information Processing. H i l l s d a l e , NJ, Erlbaum, 271-296

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Tversky, A. & Kahneman, D. 1971. The b e l i e f i n t h e law o f small numbers. In: P s y c hologic a l Bul e t i n , 76, 105-110 Tversky, A. & Kahneman, D. quency and p r o b a b i l i t

.

1973. A v a i l a b i l i t y : A h e u r i s t i c f o r j u d g i n g f r e I n : C o g n i t i v e Psychology, 5, 207-232

Tversky, A. & Kahneman, D. : 1974. Judgment under u n c e r t a i n t y : H e u r i s t i c s and bias es . I n : Science, 185, 1124-1131 Tversky, A. & Kahneman, 0.: 1980. Causal schemas i n judgments under uncert a i n t y . In: F is h b e i n , M. (ed.), Progress i n S o c i a l Psychology, Vol. 1. H i l l s d a l e , NJ, Erlbaum, 49-72 Wallsten, T.S.: 1976. U s i n g c o n j o i n t measurement models t o i n v e s t i g a t e a t h e o r y about p r o b a b i l i s t i c i n f o r m a t i o n p r o cessing. I n : Journal o f Mathem a t i c a l Psychology, 14, 144-185 Wallsten, T.S.: 1977. P r o c e s s i n g i n f o r m a t i o n f o r d e c i s i o n s . In: C a s t e l l a n , N.J./Pisoni, D.B. & P o t t s , G. (eds.), C o g n i t i v e Theory, Vol. 11. H i l l s dale, NJ, Erlbaum, 87-116 Wallsten, T.S. : 1981. P h y s i c i a n and medical s t u d e n t b i a s i n e v a l u a t i n g d i a g n o s t i c i n f o r m a t i o n . I n : Medical D e c i s i o n Making, 1, 145-164 W allst en, T.S. & Barton, C. : 1982. P r o c e s s i n g p r o b a b i l i s t i c m u l t i d i m e n s i o n a l i n f o r m a t i o n f o r d e c i s i o n s . I n : J o u r n a l o f Experimental Psychology: L earn ing Memory and C o g n i t i o n , 8, 361-384 Wason, P.C. & Johnson-Laird, P.N.: 1970. A c o n f l i c t between s e l e c t i n g and e v a l u a t i n g i n f o r m a t i o n i n an i n f e r e n t i a l t a s k . I n : B r i t i s h J o u r n a l o f Psychology, 61, 509-515 Wason, P.C. & Shapiro, D.: 1971. N a t u r a l and c o n t r i v e d experience i n a reas oning problem. I n : Q u a r t e r l y J o u r n a l o f Experimental Psychology, 23, 63-71

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor)

0 Elsevier Science Publishers B.V. (North-Holland), 1983

39

THE BASE RATE FALLACY CONTROVERSY

MAYA BAR-HILLEL Hebrew University Department of Psychology Israel

I t i s n o t easy t o say something new and e n l i g h t e n i n g a b o u t t h e b a s e - r a t e

f a l l a c y c o n t r o v e r s y , s i n c e o n l y r e c e n t l y t h e r e have appeared v a r i o u s summaries and r e v i e w s o f i t (e.g., B o r g i d a & Brekke 1981; K a s s i n 1979; Tversky & Kahneman 1982a). Moreover, I doubt t h a t t h e r e r e a l l y i s much o f a c o n t r o versy, as such, s u r r o u n d i n g t h e i s s u e . I n what f o l l o w s , I w i l l a t t e m p t t o s k e t c h a conceptual and e x p e r i m e n t a l h i s t o r y o f t h e base r a t e i s s u e , h o p i n g t o show t h a t t h e emergent p i c t u r e i s r e a l l y remarkably coherent.

The base r a t e f a l l a c y and a t t r i b u t i o n s of causality

To begin, I s h a l l g i v e t h r e e p u z z l e s t h a t i l l u s t r a t e t h e base r a t e f a l l a c y , b u t i n a c o n t e x t o f causal a t t r i b u t i o n r a t h e r t h a n p r o b a b i l i t y e s t i m a t e s , t h e s e two o f t e n b e i n g i n t i m a t e l y l i n k e d t o each o t h e r . The t a s k i s t o e x p l a i n each o f t h e f o l l o w i n g f a c t s .

1.

A c h i l d r e n ' s r i d d l e asks: Why i s more grass consumed by w h i t e sheep t h a n

2.

A Rhode I s l a n d newspaper p r o c l a i m e d t h a t i n 1957, 10 % o f a l l p e d e s t r i a n s

3.

crossed on a r d d l i g h t ( H u f f 1959). I n 1973, t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y accepted 44 % o f a l l

by b l a c k sheep? k i l l e d had c r o s s e d an i n t e r s e c t i o n on a green l i g h t , w h i l e o n l y 6 % had

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COGNITIVE DECISION RESEARCH

male applicants, versus o n l y 35 % o f a l l female applicants. Yet t h e females were o v e r a l l the more q u a l i f i e d applicants, and there was.no sex d i s c r i m i n a t i o n p r a c t i c e d (Freedman, Pisani i3 Purves 1978). Most people are n c l i n e d t o look f o r causal explanations o f the above f a c t s , which leads t o a s t a r t l i n g conclusion i n the second example, and bewilderment i n the t h i r d . I n f a c t , however, a l l three can be explained by reference t o base rates ( h i n t t h e r e may j u s t be more white than black sheep).

The base rate fallacy i n the evaluation o f diagnostic t e a t r e s u l t s The importance o f considering base r a t e s before making causal a t t r i b u t i o n s i s one which i s i n s t i l l e d i n us a l l i n t h e course o f our t r a i n i n g i n experimental methodology. But base r a t e s p l a y an important r o l e i n other i n f e r e n t i a l formats, too, e s p e c i a l l y Bayesian ones. Yet there i s evidence aplenty t h a t i n those contexts they a r e l a r g e l y ignored i n f a v o r o f the diagnostic information a t hand. This i s the phenomenon known as t h e base r a t e f a l l a c y . As l a t e as i n 1955, Meehl and Rosen found i t necessary t o c a u t i o n against commiting the base r a t e f a l l a c y i n the context o f e v a l u a t i n g a diagnostict e s t r e s u l t . Even a f a i r l y accurate diagnostic device, they pointed out, could lead t o conclusions t h a t are l e s s o f t e n r i g h t than wrong (and c e r t a i n l y l e s s o f t e n r i g h t than judging s o l e l y by base r a t e s ) , unless i t s r e s u l t s are properly i n t e g r a t e d w i t h o t h e r f a c t o r s , most notably base r a t e considerations."&

according t o Dershowitz (1971), Eddy (1982), F o l t z and Kelsey

(1978) and Lykken (1975), t o c i t e b u t a few, t h i s f a l l a c y i s s t i l l prevalent i n such important areas o f judgment and decision making as preventive detent i o n , mammography, Pap smears, and l i e detection, respectively.

The f i r s t laboratory studies For a psychologist, a natural question t o ask i s "why?". Why are base r a t e s so easy t o disregard i n such s e t t i n g s ? Kahneman and Tversky, founders o f t h e h e u r i s t i c s approach t o judgment under uncertainty, were the f i r s t t o study t h e base r a t e f a l l a c y i n a c o n t r o l l e d l a b o r a t o r y s e t t i n g , and t o attempt a c o g n i t i v e account o f it. I n t e r e s t i n g l y , however, the l i n k wisth the above

BASE RA TE FALLACY

41

types o f problems was o n l y made l a t e r , s i n c e Kahneman and T versky's i n t e r e s t i n t h e base r a t e f a l l a c y o r i g i n a t e d f r o m a p r e d i c t i o n generated by t h e i r r e prese n t a t iv e nes s h y p o t h e s i s (Kahneman and Tversky 1972a), r a t h e r t h a n f r o m an a t t e m p t t o account f o r i t s above-mentioned m a n i f e s t a t i o n s . Accor d ing t o t h i s h y p o t h e s i s , t h e p r o b a b i l i t y o f u n c e r t a i n outcomes i s o f t e n judge d by t h e e x t e n t t o which t h e y r e p r e s e n t t h e i r source o r g e n e r a t i n g process. The f e a t u r e s by which t h i s s i m i l a r i t y o r r e p r e s e n t a t i v e n e s s i s assessed are n o t necessarily those t h a t f i g u r e i n t h e normative d e r i v a t i o n o f t h e reque s t e d p r o b a b i l i t y . From t h i s h y p o t h e s i s , Kahneman and Tversky d e r i v e d t h e f o l l o w i n g p r e d i c t i o n : I f an e v e n t i s t o be j u dged v i s a v i s s e v e r a l a l t e r n a t i v e p o s s i b l e sources, o r s e v e r a l a l t e r n a t i v e p o s s i b l e outcomes, t hese w i l l be ranked by t h e s i m i l a r i t y between them and t h e event ( o r v i c e v e r s a ) , and t h e r a n k i n g w i l l n o t be a f f e c t e d ( s u f f i c i e n t l y , o r a t a l l ) by how l i k e l y each source o r outcome i s initially. T his p r e d i c t i o n was borne o u t by a s e r i e s o f s t u d i e s , one o f t h e b e s t known o f which i s t h e Tom W. s t u d y (Kahneman & Tversky 1973). One Tom W . , known o n l y by a b r i e f and a d m i t t e d l y u n r e l i a b l e p e r s o n a l i t y sketch, was judged t o be more l i k e l y t o be i n Computer Science t h a n i n S o c i a l Sciences by s u b j e c t s who were w e l l aware how much more common t h e l a t t e r k i n d o f s t u d e n t i s , merel y because t h e s k et ch c l e v e r l y evoke'd a c o m p u t e r-buf f s t e r e o t y p e more t h a n a

s o c i a l - w o r k e r s t e re o t y p e . A1 though t h e n o r m a t i v e a n a l y s i s o f t h e Tom W. t a s k proceeds v i a Bayes' Theorem, as i t does i n t h e c o n t e x t s discussed by Meehl and Rosen and t h e o t h e r s , n o l i n k was e s t a b l i s h e d between them i n t h e 1973 paper, and t h e ex pre s s i o n "base r a t e f a l l a c y " was n o t used. A c t u a l l y , though, by t h a t t i m e Kahneman and Tv e rsky a l r e a d y had t h e f i r s t r e s u l t s o f t h e l a t e r n o t o r i o u s Cab Problem (Kahneman & Tversky 1972b). T h i s problem a l s o shows how d i a g n o s t i c e v i d e n c e (an eyewit ness' t e s t i m o n y r a t h e r than a p s y c h o l o g i s t ' s p e r s o n a l i t y s k e t c h ) dominates base r a t e c o n s i d e r a t i o n s . I t was n o t i n c l u d e d i n t h e 1973 paper, though, s i n c e i t r e a l l y was n o t a case o f judgment by r e p r e s e n t a t i v e n e s s . Although t h e 1973 r e s u l t s have s t o o d up remarkably w e l l t o t h e t e s t o f time, r e p l i c a t i o n , and even r e f u t a t i o n e f f o r t s , two p o i n t s made by Kahneman and

42

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Tversky had t o be m o d i f i e d l a t e r . One was t h e i m p l i c a t i o n t h a t base r a t e s were i g n o r e d because t h e y do n o t a f f e c t r e p r e s e n t a t i v e n e s s . The second was based on r e s u l t s o b t a i n e d i n t h e i r Engineer/Lawyer s t u d y . I n t h i s s t u d y , subj e c t s guessed t h e p r o f e s s i o n o f f i v e p e o p l e on t h e b a s i s o f b r i e f p e r s o n a l i t y sketches. One s k e t c h was c o m p l e t e l y n o n d i a g n o s t i c w i t h r e s p e c t t o t h e s e two professions.

S u b j e c t s gave a 50 %

-

50 % chance t h a t i t s owner, Dick, was an

Engineer, r a t h e r t h a n t h e p r o b a b i l i t y i n d i c a t e d by t h e c a t e g o r i e s ' base r a t e . From t h i s , Kahneman and Tversky concluded t h a t even when r e p r e s e n t a t i v e n e s s f a i l s t o be a g u i d e t o p r e d i c t i o n (e.g.,

even when t h e j u d g e d e v e n t i s

e q u a l l y r e p r e s e n t a t i v e o f a l l c o n t e n d i n g outcomes) , i t dominates base r a t e considerations

.

T h a t t h e base r a t e f a l l a c y i s n o t n e c e s s a r i l y caused by t h e r e p r e s e n t a t i v e ness h e u r i s t i c i s shown by t h e r e s u l t s o f t h e Cab Problem, as w e l l as i n t h e c l a s s i c a l d i a g n o s t i c - t e s t - r e s u l t - o n - a c t u a r i a1 -background cases, where t h e necessary p r o b a b i l i s t i c components a r e given, and need n o t be assessed by r e p r e s e n t a t i v e n e s s . The second p o i n t was c h a l l e n g e d by Ginosar and Trope (1980). I n p a r t , t h e i r s t u d y was a d i r e c t r e p l i c a t i o n o f Kahneman and T v e r s k y ' s EngineerlLawyer study, b u t Ginosar and T r o p e ' s s u b j e c t s each j u d g e d o n l y one o f t h e f i v e sketches. I n c o n t r a s t t o Kahneman and T v e r s k y ' s r e s u l t s , Ginosar and Trope f o u n d t h a t t h e n o n d i a g n o s t i c s k e t c h was assessed by t h e base r a t e . On t h e b a s i s o f t h e i r r e s u l t s , t h e y concluded t h a t , when t h e i n d i v i d u a t i n g i n f o r m a t i o n i s u s e l e s s as a g u i d e t o p r e d i c t i o n , " a t t e n t i o n i s s h i f t e d

...

t o t h e base r a t e f r e q u e n c i e s " ( p . 240). I t seems t h a t Tversky and Kahneman's s u b j e c t s j u d g e d D i c k e q u a l l y l i k e l y t o be an Engineer o r a Lawyer under b o t h base r a t e c o n d i t i o n s o n l y because t h e y were g i v e n a m u l t i p l e p r e d i c t i o n t a s k ; t h e u s e f u l n e s s o f r e p r e s e n t a t i v e n e s s t o t h e e v a l u a t i o n o f most sketches c r e a t e d some s o r t o f " s e t " . Tversky and Kahneman have accepted b o t h t h e s e m o d i f i c a t i o n s . They concede t h a t " t h e n e g l e c t o f base r a t e s appears t o be a more g e n e r a l phenomenon, which occurs even when p r o b a b i l i t y assessments a r e n o t mediated by s i m i l a r i t y " ( T v e r s k y & Kahneman 1980, p. 79), and t h a t

"Most t h e a v a i l a b l e d a t a

...

s u p p o r t a more moderate h y p o t h e s i s t h a t i n t u i t i v e p r e d i c t i o n s and p r o b a b i l i t y judgments a r e h i g h l y s e n s i t i v e t o r e p r e s e n t a t i v e n e s s , a l t h o u g h t h e y a r e n o t c o m p l e t e l y dominated by i t " ( T v e r s k y & Kahneman 1982b, p. 88).

BASE RA TE FALLACY

43

So the f i r s t round i n the base r a t e f a l l a c y controversy ends w i t h the con-

c l u s i o n t h a t w h i l e representativeness considerations are o f t e n overweighted and base r a t e considerations are o f t e n underweighted, the issue o f when, why and t o what extent base rates are neglected demands a l o g i c a l l y separate account (Bar-Hi1 l e l 1981). The r e s u l t s s t r o n g l y underscored, however, t h a t people a t no time a c t u a l l y p r e d i c t against a strong representativeness i n d i cation.

Two experimental paradigms

I s h a l l c a l l the study o f problems i n which t h e base r a t e f a l l a c y i s divorced from representativeness "the textbook-probl em paradigm" , whi 1e the paradigm established by Kahneman and Tversky w i l l be c a l l e d " t h e social-judgment paradigm". Let us now f o l l o w the development o f these two paradigms separately, n o t i n g para11e l s as they emerge. Ignoring base rates versus inverting conditional probabilities

Before attempting a theory o f base r a t e usage, i t i s worth n o t i n g t h a t o f t e n judgments may have the appearance o f manifesting the base r a t e f a l l a c y and y e t r e s u l t from a c o g n i t i v e dynamic t h a t has nothing t o do w i t h the i g n o r i n g o f base rates. According t o Bayes' Theorem, which i s the normative formalizat i o n o f most o f the problems developed by both paradigms, the p o s t e r i o r p r o b a b i l i t y o f an event, p(E/D), depends both on data impact, p(D/E):p(D), and on the event's base-rate, p(E). On occasion, however, p(E/D) can be estimated d i r e c t l y , r a t h e r than v i a Bayes' Theorem. I n the Cab Problem, f o r example, subjects were asked f o r t h e p r o b a b i l i t y t h a t the e r r a n t cab was green f o l l o w i n g the witness' testimony t o t h a t e f f e c t , p(G/"G"). This condit i o n a l p r o b a b i l i t y i s e a s i l y confused w i t h i t s inverse, p("G"/G) (perhaps since both are n a t u r a l formalizations o f "the p r o b a b i l i t y o f an accurate Green cab i d e n t i f i c a t i o n " ) . Hence, i t i s q u i t e p o s s i b l e t h a t some o f t h e Cab Problem subjects were n o t i g n o r i n g base rates, b u t merely confusing two cond i t i o n a l p r o b a b i l i t i e s . This i s an e r r o r , and i t leads t o t h e same f i n a l response, b u t i s n o t the e r r o r o f i g n o r i n g base rates. A p a r a l l e l p o s s i b i l i t y emerges i n the Tom W. paradigm. Recently, I gave subj e c t s the Tom W. task, b u t requested them t o rank t h e p r o b a b i l i t i e s t h a t

44

COGNITIVE DECISION RESEARCH

students in different f i e l d s of study would be describable like Tom W., p ( D / F ) , rather t h a n requesting, as Kahneman and Tversky did, a ranking of the probabilities t h a t someone describable like Tom W. i s enrolled in the different f i e l d s of study, p(F/D) (Bar-Hillel 1980). Ten years a f t e r the original study, my subjects' rankings correlated .98 with Kahneman and Tversky's! The point I want t o make i s t h a t i t i s d i f f i c u l t t o say whether Kahneman and Tversky's subjects were judging p(F/D) (by representativeness) directly - i n which case base rates are really irrelevant, o r were judging p(D/F) (also by representativeness), and then f a i l i n g t o integrate i t with base r a t e considerations. The confusion between a conditional probability and i t s inverse i s not limited t o the psychological laboratory. I n a s t a r t l i n g paper, Eddy (1982) documents that the medical l i t e r a t u r e dealing with mammograms i s f u l l of j u s t such confusions (e.g. , between p (disease/positive t e s t - r e s u l t ) , called predictive accuracy, and p (positive test-result/disease), called retrospective accuracy).In other words, some of the judgmental errors in marnogram i n t e r pretation may be the result of the confusion of two different conditional probabilities, rather t h a n the ignoring of base rates. The potential f o r such confusion has somehow been ignored in the published discussions of the base rate fallacy (Bar-Hillel 1980). Perhaps universal explanations were sought, and i t i s so obvious t h a t t h i s i s not a universally occuring confusion. No one, presumably, would f a i l t o distinguish between the probability of being male i f one i s a head of s t a t e (which i s close t o one), and the probability of being a head of s t a t e i f one i s male (which i s close t o zero). Not even a l l empirical results where probability judgments show insensitivity t o base rates can be accounted f o r by such a confusion. In the Suicide Problem (Bar-Hillel 1980), f o r example, subjects were informed that among the causes of death in young adults, p ( s u i c i d e h a r r i e d ) = 5 %, while p (suicide/single) = 15 %.They estimated p (married/suicide) t o be 25 %.This response ignores the base-rate of married people in the relevant population, b u t a t the same time, i t d i f f e r s numerically from the value of the inverse conditional probability. Similarly, "McCauley and S t i t t (1978) gave subjects a l i s t of nine t r a i t s (e.g., " e f f i c i e n t " ) and asked them t o estimate both p (German/Trait) and p (Trait/German) They found a product

...

45

BASE RATE FALLACY

moment c o r r e l a t i o n o f .83 between t h e two

... A t

t h e same time, mean

p(Trait/German) was between 1.7 and 3.1 t i m e s h i q h e r t h a n mean p(German/ T r a i t ) " ( B a r - H i l l e l 1981, p. 7 ) . The causaZity factor So perhaps r a t h e r t h a n a s k i n g why o r when base r a t e s w i l l be i g n o r e d

presupposes t h a t t h e y a r e

-

-

which

i t i s b e t t e r t o ask when t h e y w i l l b e i n c o r p o r a -

t e d . The f i r s t f a c t o r t h a t was i d e n t i f i e d as enhancing t h e impact o f base r a t e s was c a u s a l i t y . W i t h i n t h e s o c i a l - j u d g m e n t paradigm, t h e e f f e c t o f causal base r a t e s was shown by A j z e n (1977). A student, Gary W., was d e s c r i b e d by a b r i e f deSCriFt i o n . S u b j e c t s were asked t o e s t i m a t e G a r y ' s p r o b a b i l i t y o f h a v i n g passed some t e s t . Some s u b j e c t s were t o l d t h a t Gary had been randomly sampled f r o m a group o f s t u d e n t s amongst whom i t j u s t so happened

t h a t 75 % had f a i l e d

t h e t e s t and 2 5 % had passed. These s u b j e c t s i g n o r e d t h e base r a t e o f passi n g , even though i t i s n o r m a t i v e l y r e l e v a n t . O t h e r s u b j e c t s were t o l d t h a t t h i s t e s t was, o v e r a l l , f a i l e d by 75 % o f s t u d e n t s and passed by 25 % o f s t u d e n t s i n G a r y ' s c l a s s . These s u b j e c t s were r e s p o n s i v e t o t h e base r a t e o f t e s t f a i l u r e s , presumably because h e r e t h e b a s e - r a t e c o u l d be c a u s a l l y t i e d t o t h e t a r g e t e v e n t r e a d i l y , as i n t h e i n f e r e n c e t h a t t h e t e s t was a d i f f i c u l t one.

A para1 l e l phenomenon occurs i n t h e textbook-problem paradigm. T y p i c a l l y , t h e Cab Problem s t a t e s t h a t 85 % o f t h e cabs i n some town a r e Blue. T h i s i n f o r m a t i o n i s i g n o r e d . B u t s u b j e c t s who a r e t o l d t h a t b o t h cab companies a r e o f equal s i z e , and y e t t h e B l u e ones account f o r 85 % o f t h e a c c i d e n t s i n t h a t town, no l o n g e r i g n o r e t h e base r a t e i n f o r m a t i o n ( T v e r s k y & Kahneman 1980). The c a u s a l i t y h y p o t h e s i s can be strengthened, f r o m m e r e l y p r e d i c t i n g t h a t causal base r a t e s a f f e c t judgments more t h a n noncausal ones , t o p r e d i c t i n g t h a t i n problems where b o t h a r e presented, causal base r a t e s w i l l dominate noncausal ones, even i f t h e l a t t e r a r e o b j e c t i v e l y d i a g n o s t i c . T h i s , indeed, i s j u s t what happened i n t h e S u i c i d e Problem d e s c r i b e d e a r l i e r . Note t h a t t h e weak v e r s i o n o f t h e h y p o t h e s i s p r e d i c t s t h e r e l a t i v e magnitude o f t h e

46

COGNITIVE DECISION RESEARCH

base r a t e e f f e c t between problems, whereas the strong version makes a pred i c t i o n w i t h i n a problem. There cannot be a t e s t o f t h i s i n the s o c i a l judgment paradigm, since there, by d e f i n i t i o n , one of the items o f informat i o n i s n o t a base r a t e .

The s p e c i f i c i t y factor The second hypothesis was i n i t i a l l y formulated as a within-problem one. It states t h a t more s p e c i f i c information w i l l tend t o dominate a l e s s s p e c i f i c one. Note t h a t i n the social-judgment paradigm, the i n d i v i d u a t i n g i n f o r m a t i o n n o t only lends i t s e l f t o judgment by representativeness, b u t i n a d d i t i o n per-

-

so s p e c i f i c t a i n s t o the t a r g e t case more s p e c i f i c a l l y than t h e base r a t e a l l y , indeed, as t o be i n d i v i d u a t i n g . Perhaps t h i s i s why the output o f the representativeness judgment i s n o t then i n t e g r a t e d w i t h t h e base r a t e : being more s p e c i f i c , i t j u s t dominates i t . There are independent reasons f o r b e l i e v i n g such a hypothesis. I n many cases, i t i s normative t o replace a general base r a t e by a more s p e c i f i c one. For

example, f o r the purpose o f p r e d i c t i n g t h e l i f e expectancy o f a newborn German baby, the l i f e expectancy i n Germany replaces the worldwide l i f e expectancy. Another example i s given i n the Modified Cab problem ( B a r - H i l l e l 1980). I n t h i s problem, subjects were given, i n a d d i t i o n t o the c i t y bhse r a t e s and i n place o f t h e witness, the base r a t e s o f Blue and Green cabs i n the v i c i n i t y o f the accident. The p a t t e r n o f responses given t o t h i s problem was much l i k e t h a t given t o the standard problem, t h a t i s , the median subject ignored the c i t y base r a t e i n f a v o r o f the more s p e c i f i c information - o n l y i n t h e Modif i e d Cab problem, t h i s i s the appropriate t h i n g t o do. I t seems l i k e l y t h a t people, e s p e c i a l l y those untutored i n s t a t i s t i c s and

p r o b a b i l i t y , would f i n d i t hard t o d i s t i n g u i s h between Bayesian problems, where the diagnostic information base r a t e

-

-

even i f i t i s more s p e c i f i c than t h e

does n o t replace it, and between problems such as the ModifCed

Cab problem. This might be the o r i g i n o f the s p e c i f i c i t y f a c t o r

'1.

This f a c t o r can a l s o account f o r the r e s u l t s o f problems such as the Dreamers Problem and the Urn and Beads Problem ( B a r - H i l l e l 1980), which can n o t be accounted f o r by any o f the other explanations (representativeness, c a u s a l i t y , confusion o f c o n d i t i o n a l p r o b a b i l i t i e s , etc.).

BASE RATE FALLACY

47

CausaZity and s p e c i f i c i t y considered j o i n t l y Combining t h e c a u s a l i t y and t h e s p e c i f i c i t y f a c t o r y i e l d s a t e s t a b l e i m p l i c a t i o n : I f two base r a t e s a r e p r e s e n t e d i n a (Bayesian) problem and no one o f them i s e i t h e r more s p e c i f i c o r more causal t h a n t h e o t h e r , t h e n n e i t h e r w i l l dominate, and t h e two w i l l be i n t e g r a t e d . Evidence s u p p o r t i n g t h i s p r e d i c t i o n i s t o be f o u n d i n t h e I n t e r c o m Problem ( B a r - H i l l e l 1980). T h i s i s a cab problem i n which t h e e r r a n t cab i s i d e n t i f i e d as one t h a t has an i n t e r com i n i t , and t h e two cab companies a r e known t o have a d i f f e r e n t p r o p o r t i o n o f cabs equipped w i t h i n t e r c o m s . T h i s d i f f e r e n t i a l base r a t e can n o more be c a u s a l l y t i e d t o a c c i d e n t proneness t h a n company s i z e , n o r does i t p e r t a i n t o a more s p e c i f i c s u b s e t o f t h e o v e r a l l p o p u l a t i o n . And, l o ! , i t does n o t dominate t h e o v e r a l l c i t y base r a t e .

The vividness factor Meanwhile, a n o t h e r f a c t o r had been proposed t o a c c o u n t f o r t h e base r a t e f a l l a c y , which i s o n l y a p p l i c a b l e t o t h e s o c i a l - j u d g m e n t paradigm. T h i s i s t h e f a c t o r o f v i v i d n e s s o r s a l i e n c y . N i s b e t t and B o r g i d a (1975) argued t h a t a base r a t e i s " a b s t r a c t and p a l l i d " , w h i l e i n d i v i d u a t i n g i n f o r m a t i o n i s " c o n c r e t e and v i v i d " and t h e r e f o r e more memorable and u t i l i z a b l e . L i k e t h e p r e v i o u s two, t h i s f a c t o r can a l s o g e n e r a t e two p r e d i c t i o n s : i. By making base r a t e s more v i v i d t h e i r e f f e c t can be increased. ii. Base r a t e s t h a t a r e as v i v i d as t h e i n d i v i d u a t i n g i n f o r m a t i o n w i l l n o t be dominated by it.

A c l a s s i c a l example o f t h e e f f e c t o f v i v i d n e s s i s t h e Volvo e f f e c t ( N i s b e t t , Borgida, C r a n d a l l 81 Reed 1976, p. 129): A person who has decided t o buy a Volvo on t h e b a s i s o f t h e s t a t i s t i c s g a t h e r e d by Consumers Reports, changes h i s mind upon overhearing, i n a c o c k t a i l p a r t y , t h e w o e f u l , b u t l i v e l y , des c r i p t i o n o f t h e demise o f one Volvo purchased by t h e t e l l e r ' s b r o t h e r - i n law. In a

p e n e t r a t i n g l o o k a t t h e v i v i d n e s s f a c t o r , T a y l o r and Thompson (1982)

concluded t h a t , a l t h o u g h "case h i s t o r y i n f o r m a t i o n does habe a p e r s u a s i v e i m p a c t on judgments when i t i s c o n t r a s t e d w i t h b a s e - r a t e o r a b s t r a c t i n f o r m a tion

... t h e e x t e n t

t o which t h i s f i n d i n g speaks t o t h e v i v i d n e s s e f f e c t i s

48

questionable

COGNITIVE DECISION RESEARCH

... (p.

162).

One s t u d y p r o v i d e d d i r e c t evidence s u p p o r t i n g T a y l o r and Thompson's c o n c l u s i o n . Manis, Dovalina, A v i s and Cardoze (1980) t e s t e d whether i n problems where i t has been e s t a b l i s h e d t h a t p e o p l e c o n s i d e r base r a t e s t o be i r r e l e vant, t h e i r c o n s i d e r a t i o n can be a l t e r e d b y making t h e base r a t e s more v i v i d . They used t h e s o c i a l - j u d g m e n t paradigm, p r e s e n t i n g base r a t e s i n t h e s t a n d a r d numerical summary f o r m t o some s u b j e c t s , b u t v i a a s t e p - b y - s t e p l a b e l l e d p r e s e n t a t i l o n o f t h e p o p u l a t i o n t o o t h e r s . The l a t t e r base r a t e s were n o t more i n f l u e n t i a l t h a n t h e former, i n c o n t r a s t t o t h e v i v i d n e s s p r e d i c t i o n . C h r i s t e n s e n - S z a l a n s k i and Beach (1982) gave a n o t h e r t w i s t t o t h e v i v i d n e s s hypothesis. They c o n j e c t u r e d t h a t t o i n c r e a s e s u b j e c t s ' s e n s i t i v i t y t o base r a t e s , n o t o n l y t h e base r a t e s themselves must be v i v i d l y experienced, b u t a l s o t h e i r r e l a t i o n s h i p t o t h e d i a g n o s t i c i n f o r m a t i o n . To t h a t end, t h e i r s u b j e c t s were i n t r o d u c e d , s l i d e by s l i d e , t o a p o p u l a t i o n o f diseased o r h e a l t h y people, who had o b t a i n e d p o s i t i v e o r n e g a t i v e r e s u l t s on a d i a g n o s t i c t e s t f o r d e t e c t i n g t h a t disease. S u b j e c t s were u l t i m a t e l y asked t o assess p ( d i s e a s e / p o s i t i v e t e s t - r e s u l t ) . Assessments w e r e q u i t e reasonable, w h i c h t h e i n v e s t i g a t o r s t o o k t o be evidence f o r t h e i r c o n j e c t u r e . B u t as Beyth-Marom and Arkes (1983) c o r r e c t l y p o i n t e d o u t , i t i s e m i n e n t l y p o s s i b l e t h a t s u b j e c t s were s i m p l y d i r e c t l y a s s e s s i n g t h e r e q u i r e d c o n d i t i o n a l p r o b a b i l i t y f r o m t h e d a t a base t h e y had been exposed t o . Beach h i m s e l f ( P e t e r s o n & Beach 1968) had demonstrated t h a t p e o p l e a r e q u i t e p r o f i c i e n t a t e x t r a c t i n g summary s t a t i s t i c s f r o m data. I f so, t h e y had no need f o r Bayesian r e a s o n i n g i n C h r i s t e n s e n S z a l a n s k i ' s and Beach's t a s k , hence t h e observed improvement i n t h e i r p e r formance has n o b e a r i n g on t h e c o n j e c t u r e t h a t d i r e c t e x p e r i e n c e a i d s i n overcoming t h e base r a t e f a l l a c y .

Multiple judgments tasks Up t o now, a l l t h e f a c t o r s we have e x p l o r e d p e r t a i n t o a s i n g l e judgment. When s u b j e c t s a r e c a l l e d upon t o make a s e r i e s o f judgments, o t h e r f a c t o r s e n t e r t h e p i c t u r e . The most i m p o r t a n t i s t h a t s u b j e c t s may want t h e d i s t r i b u t i o n o f t h e i r m u l t i p l e judgments t o r e f l e c t t h e p o p u l a t i o n base r a t e . T h i s i s a v a r i a n t o f r e p r e s e n t a t i v e n e s s , b u t one w h i c h focusses on base r a t e s

49

EASE RATE FALLACY

r a t h e r than i g n o r i n g them. Another p o s s i b i l i t y i s t h a t subjects e n t e r a response set, namely t h a t they adopt a c e r t a i n f i x e d convenient s t r a t e g y throughout the judged s t i m u l i , though they would have adopted d i f f e r e n t s t r a t e g i e s i f they were making a s i n g l e judgment. This p o s s i b i l i t y was e a r l i e r r a i s e d when e x p l a i n i n g Kahneman and Tversky's "Dick" r e s u l t s . L a s t l y , when considering m u l t i p l e judgments tasks, i t i s important t o remember t h a t t h e i r r e s u l t s are pooled across both subjects and s t i m u l i , a f a c t t h a t might i n f l u e n c e the i n t e r p r e t a t i o n . With these p o s s i b i l i t i e s i n mind, l e t us review the m u l t i p l e judgments studies. Such studies have only been done w i t h i n the social-judgment paradigm.

The probabiZity matching paradigm The f i r s t f a c t o r mentioned above, t h a t subjects making m u l t i p l e judgments may f e e l c a l l e d upon t o d i s t r i b u t e t h e i r judgments i n a manner t h a t represents the population base rate, i s known as " p r o b a b i l i t y matching". I n probability-matching experiments, subjects s e q u e n t i a l l y p r e d i c t the i d e n t i t y o f each item i n a binomial series. T y p i c a l l y , they l e a r n the base r a t e o f the i t e m categories from t h e i r t r i a l - b y - t r i a l feedback, and eventually come t o match i t i n t h e i r p r e d i c t i o n s (see, e.g.,

Estes 1976). I n the p r o b a b i l i t y -

matching experiments, however, there was nothing t o d i s t i n g u i s h one t r i a l from the n e x t - except the p a t t e r n o f preceeding t r i a l s . I n contrast, judgment studies t h a t c a l l f o r sequential p r e d i c t i o n s g i v e an i t e m o f i n d i v i d u a t i n g information on each t r i a l . Thus, the probability-matching paradigm and the social-judgment paradigm e l i c i t d i f f e r e n t response tendencies

-

matching

base r a t e s and judging by representativeness, r e s p e c t i v e l y .

Representing stereotypes versus representing base rates

-a

conjecture

These two tendencies, however, need n o t be incompatible. I f the body o f judged s t i m u l i contains a mixture o f representative and non representative s t i m u l i , a subject can c l a s s i f y the f i r s t according t o representativeness, and simultaneously probability-match by c l a s s i f y i n g the n e u t r a l s t i m u l i i n a manner t h a t w i l l give an o v e r a l l correspondence between p r e d i c t i o n proport i o n s and population proportions. And what i f t h e mixture does n o t a l l o w

50

COGNlTl VE DECISION RESEARCH

b o t h tendencies t o be s a t i s f i e d a t once? T h i s has n o t been t e s t e d y e t , b u t B a r - H i l l e l and F i s c h h o f f (1981) v e n t u r e d t h e h y p o t h e s i s t h a t s u b j e c t s w i l l n o t v i o l a t e r e p r e s e n t a t i v e n e s s u n l e s s t h a t l e a d s t o a sample t h a t i s impossi b l e , r a t h e r t h a n m e r e l y improbable, g i v e n t h e p o p u l a t i o n .

Indirect evidence f o r the conjecture What evidence do we have t o s u p p o r t t h e c o n j e c t u r e t h a t s u b j e c t s s t r i v e s i m u l t a n e o u s l y t o j u d g e by r e p r e s e n t a t i v e n e s s , and t o p r o b a b i l i t y match? A s t u d y by C a r r o l l and S i e g l e r (1977) r e q u i r e d s u b j e c t s t o c l a s s i f y 20 s h o r t p e r s o n a l i t y d e s c r i p t i o n s as b e l o n g i n g t o e i t h e r an e n g i n e e r o r a l a w y e r . The way t h e d e s c r i p t i o n s were c o n s t r u c t e d makes i t u n l i k e l y t h a t any would s t r o n g l y resemble e i t h e r s t e r e o t y p e . When s u b j e c t s were t o l d t h a t , among t h e 20 d e s c r i p t i o n s ,

70 % were members o f one p r o f e s s i o n (engineers, f o r one

group o f s u b j e c t s , and lawyers, f o r a n o t h e r ) and 30 % belonged t o t h e o t h e r , an average o f 62 % o f t h e c l a s s i f i c a t i o n s were i n t o t h e m a j o r i t y p r o f e s s i o n . When t o l d t h a t t h e 20 were sampled a t random f r o m 400 d e s c r i p t i o n s w h i c h were d i v i d e d between t h e two p r o f e s s i o n s i n a 70 %/30 % r a t i o , o n l y 56 % o f t h e c l a s s i f i c a t i o n s , on average, were i n t o t h e m a j o r i t y p r o f e s s i o n . L a s t l y , when t e n d e s c r i p t i o n s were designed t o resemble, r e s p e c t i v e l y , f i v e l a w y e r s and f i v e engineers, t h e y were a l l c l a s s i f i e d i n t o t h e c a t e g o r y t h e y were i n t e n d e d t o resemble by an average 65 % o f t h e s u b j e c t s ( 7 0 % when t h e c a t e g o r y was a m a j o r i t y one, b u t a l s o 60 % when i t was a m i n o r i t y one). Since t h e e x t e n t t o which d e s c r i p t i o n s resembled e i t h e r c a t e g o r y s t e r e o t y p e was n o t i n d e p e n d e n t l y a s c e r t a i n e d i n C a r r o l l and S i e g l e r l s study, i t i s conceivable t h a t t h e i r data r e s u l t e d from having d i f f e r e n t subjects disagree which p r o f e s s i o n a g i v e n d e s c r i p t i o n r e p r e s e n t s , i f i n d e e d any, b u t h a v i n g no s u b j e c t c l a s s i f y d e s c r i p t i o n s c o n t r a r y t o r e p r e s e n t a t i v e n e s s . (Note t h a t when r e s u l t s a r e p o o l e d o v e r b o t h s u b j e c t s and s t i m u l i , what m i g h t seem "weakly r e p r e s e n t a t i v e " s t i m u l i may s i m p l y be s t i m u l i t h a t a r e j u d g e d as r e p r e s e n t a t i v e by f e w e r s u b j e c t s t h a n " s t r o n g l y r e p r e s e n t a t i v e " s t i m u l i , B u t

I have d e l i b e r a t e l y r e f r a i n e d f r o m d i s t i n g u i s h i n g between l e v e l s o f represent a t i v e n e s s , and try, as f a r as p o s s i b l e , t o e x p l a i n a l l r e s u l t s by t h e d i chotomy r e p r e s e n t a t i v e h o t r e p r e s e n t a t i v e ) .

BASE RATE FALLACY

51

A l t h o u g h C a r r o l l and S i e g l e r found a c o n s i s t e n t base r a t e e f f e c t i n t h e i r s t u d i e s , t h e e f f e c t f a l l s s h o r t o f p r o b a b i l i t y matching. T h i s c o u l d b e due t o s e v e r a l reasons. a. The s u b j e c t s were d e a l i n g w i t h a s m a l l number o f p r e d i c t i o n s r e l a t i v e t o t h a t c u s t o m a r i l y rendered i n p r o b a b i l i t y matching experiments. P r o b a b i l i t y matching emerges g r a d u a l l y r a t h e r t h a n i m m e d i a t e l y i n t h o s e s t u d i e s . b. The sample j u d g e d i n C a r r o l l and S i e g l e r ' s s t u d y was a s m a l l p h y s i c a l subset f r o m some f i x e d p o p u l a t i o n , r a t h e r t h a n b e i n g t h e s t e p -

I n p r o b a b i l i t y matching experiments, s u b j e c t s a i m f o r p r e d i c t i o n s t h a t w i l l be " j u s t 1 ike" t h e a c t u a l sequence. P o s s i b l y i n c l a s s i f y i n g samples, e s p e c i a l l y such s m a l l ones, t h e i r b y - s t e p r e a l i z a t i o n o f a random generating-process.

o n l y concern i s n o t t o produce a sample d i s t r i b u t i o n t h a t i s t o o i n c o m p a t i b l e w i t h t h e p o p u l a t i o n , as we know t h a t s u b j e c t s b e l i e v e random samples s h o u l d r e p r e s e n t sampling e r r o r as w e l l as p o p u l a t i o n c h a r a c t e r i s t i c s (Kahneman & Tversky 1972a). Manis e t a l . (1980) f o l l o w e d t h e p r o b a b i l i t y - m a t c h i n g paradigm more c l o s e l y , by f o r c i n g s u b j e c t s t o l e a r n t h e base r a t e s t r i a l by t r i a l , i n a l o n g s e r i e s (50) o f p r e d i c t i o n s . Indeed, t h e y f o u n d a l a r g e r base r a t e e f f e c t . I n t h e l a s t b l o c k o f t e n t r i a l s , 70 % o f t h e i r s u b j e c t s ' p r e d i c t i o n s were i n t o t h e c a t e g o r y e n j o y i n g t h e 80 % m a j o r i t y . Manis e t a l . c o n t r a s t e d t h e i r r e s u l t w i t h what t h e y t o o k t o have been Kahneman and T v e r s k y ' s r e p r e s e n t a t i v e n e s s - b a s e d p r e d i c t i o n . I n f a c t , however, B a r - H i l l e l and F i s c h h o f f (1981) showed t h a t Manis e t a l . I s r e s u l t s a r e perf e c t l y compatible w i t h previous r e s u l t s , according t o t h e conjecture p u t f o r t h above, namely, t h a t e v e r y s u b j e c t c l a s s i f i e d t h o s e s t i m u l i t h a t were c l e a r l y r e p r e s e n t a t i v e o f some c a t e g o r y i n t o t h a t c a t e g o r y , i r r e s p e c t i v e o f t h e base r a t e , whereas n o n r e p r e s e n t a t i v e s t i m u l i were c l a s s i f i e d l a r g e l y i n t o t h e m a j o r i t y category, r e s u l t i n g i n a p r o p o r t i o n o f m a j o r i t y category p r e d i c t i o n s t h a t a l l b u t match t h e c a t e g o r y p r o p o r t i o n . Indeed, w o r k i n g f r o m e s t i m a t e s g i v e n by Manis e t a l . c o n c e r n i n g t h e p r o p o r t i o n o f n e u t r a l versus r e p r e s e n t a t i v e s t i m u l i i n t h e i r study, B a r - H i l l e l and F i s c h h o f f used t h e c o n j e c t u r e t o p r e d i c t t h e p r o p o r t i o n s o f v a r i o u s c l a s s i f i c a t i o n s t o be ex

-

p e c t e d under t h e v a r i o u s base r a t e c o n d i t i o n s , and came remarkably c l o s e t o t h e p r o p o r t i o n s a c t u a l l y observed (see F i g u r e 1).

COGNITIVE DECISION RESEARCH

52

'0

I

60

.

100

so

t

60 40

.

a0 20 . 10.

Figure 1: Predictions ( s o l i d l i n e = actual, broken l i n e = reconstructed) concerning category membership o f s t i m u l i characterized by i n d i v i d u a t i n g information as a f u n c t i o n o f the observed base r a t e and the consensual c l a s s i f i c a t i o n o f t h e s t i m u l i by representativeness. (From "When do base r a t e s a f f e c t p r e d i c t i o n s ? " by M. BarH i l l e l and B. Fischhoff, Journal o f P e r s o n a l i t y and Social Psychology 4 1 (1981) 671-680, Figure 1).

Earlier

I mentioned Manis e t a l . ' s study as one which f a i l e d t o produce a

greater base-rate e f f e c t f o r more v i v i d base rates. Indeed,. we now see t h a t t h e i r e n t i r e e f f e c t can be a t t r i b u t e d t o p r o b a b i l i t y matching tendencies, w h i l e vividness contributes nothing.

Direct evidence for the conjecture We have seen t h a t the data o f previous multiple-judgments studies are compati b l e w i t h our conjecture. The t i m e had come f o r t e s t i n g i t d i r e c t l y . To t h a t end, subjects were given t e n s h o r t p e r s o n a l i t y d e s c r i p t i o n s ( F i s c h h o f f & B a r - H i l l e l 1981). They assessed how business-executive l i k e versus u n i v e r s i t y professor l i k e the descriptions r e a l l y were. Over 75 % o f the subjects agreed on f i v e o f the descriptions ( f o u r o f them executives, one professor), b u t there was considerable d i f f e r e n c e o f opinion concerning the f i v e others. S u r p r i s i n g l y , however, even those descriptions t h a t were judaed executivel i k e by some subjects and p r o f e s s o r - l i k e by others, were judged so w i t h considerable confidence, and were r a r e l y judged as n e u t r a l ( o n l y one descrip-

53

BASE RATE FALLACY

t i o n was judged as n e u t r a l by over 10 % o f the subjects). Even though a t e s t o f our conjecture requires a comparison o f how representative versus n e u t r a l s t i m u l i are judged, and our s t i m u l i turned o u t t o be n o t n e u t r a l enough, the data nevertheless supported the conjecture, as follows. I n the course o f making the p r e d i c t i o n categories i n c r e a s i n g l y s i m i l a r (from executive/professor through engineer/physicist t o d e n t i s t / o r t h o d o n t i s t ) , t h e percentage o f n e u t r a l judgments rose from 8.4 % through 24.0 % t o 31.5 %, respectively. Correspondingly, the base r a t e e f f e c t increased from 4.5 % (which i s much l i k e the magnitude o f e f f e c t reported by Kahneman and Tversky, 1973, f o r t h e i r engineer-lawyer categories) through 8.7 % t o 12.0 %, respecti v e l y . I f subjects were p r o b a b i l i t y matching on the n e u t r a l s t i m u l i , and judging by representativeness elsewhere, t h e s i z e o f the e f f e c t would have gone from 3.4 % through 9.6 % t o 12.6 %, respectively. These f i g u r e s are much l i k e the ones a c t u a l l y observed. We are p r e s e n t l y engaged i n an attempt t o p u t the conjecture t o a stronger t e s t .

The base r a t e f a l l a c y and procedural variables I n t h e i r recent review o f the base r a t e issue, Tversky and Kahneman (1982a) d i s t i n g u i s h between e v i d e n t i a l v a r i a b l e s

-

those t h a t " r e f e r t o the nature

o f the source and the i n t e r p r e t a t i o n o f the evidence" (p. 154), such as t h e r e l a t i v e relevance o f the given base rates, and procedural variables those

-

t h a t " r e f e r t o the p r o p e r t i e s o f the design, the task, and the d i s p l a y " (p. 154), w h i l e l e a v i n g the actual information given unchanged. H i t h e r t o , we have dwelt on e v i d e n t i a l variables. Many d i f f e r e n t procedural v a r i a b l e s have also been studied. Overall, i t i s f a i r t o generalize t h a t they have had very disappointing e f f e c t s , t h a t i s , none a t a l l , o r such t h a t were spurious and f l e e t i n g , n e i t h e r r e s u l t i n g from nor r e s u l t i n g i n greater understanding o f the normative r o l e o f base rates. Let us observe one example i n d e t a i l .

A study by Fischhoff, S l o v i c and L i c h t e n s t e i n ( 1979) attempted t o increase the saliency o f the base r a t e i n the Cab Problem by e x h o r t i n g subjects t o perform a s o r t o f " s u b j e c t i v e s e n s i t i v i t y analysis", c a l l e d SSA. Subjects were asked t o consider how d i f f e r e n t base r a t e values might a f f e c t t h e i r summary judgments. Subjects e x h i b i t i n g t h e base r a t e f a l l a c y i n the extreme

54

COGNl TI VE D EClSlON RESEARCH

would have r e p l i e d c o n s i s t e n t l y w i t h t h e w i t n e s s ' accuracy, 80 %.I n s t e a d , mean responses were m o n o t o n i c a l l y r e l a t e d t o t h e s h i f t i n g base r a t e s , and though t h e y were, a p p r o p r i a t e l y , l o w e r t h a n 80 %, t h e y were n o t n e a r l y l o w enough. Moreover, t h e base r a t e f a l l a c y was r e i n s t a t e d as soon as t h e s e same s u b j e c t s were g i v e n a f o r m a l l y e q u i v a l e n t , b u t d i f f e r e n t l y worded, problem, such as Lyon and S l o v i c ' s (1976) L i g h t B u l h problem. To check whether t h i s poor showing was n e v e r t h e l e s s an improvement, F i s c h h o f f and B a r - H i l l e l (1982) had s u b j e c t s do an SSA on t h e M o d i f i e d Cab problem,

i n which i g n o r i n g t h e c i t y base r a t e i s t h e r i g h t t h i n g t o do, and on t h e Dreamer problem, i n w h i c h t h e r i g h t t h i n g t o do i s t o use t h e Dreamers base r a t e o n l y . S u b j e c t s r e p e a t e d t h e p a t t e r n t h a t had been p r e v i o u s l y observed. They gave mean responses t h a t were l o w e r t h a n 80 %, v a r i e d m o n o t o n i c a l l y w i t h t h e base r a t e - and were t o t a l l y u n r e l a t e d t o t h e n o r m a t i v e s t r a t e g y . "It was as i f t h e v a r i a b l e m a n i p u l a t e d i n a SSA o n l y succeeded i n s a y i n g t o

s u b j e c t s 'use me', r a t h e r t h a n 'comprehend me''' ( F i s c h h o f f & B a r - H i l l e l 1982, p. 16). Two o t h e r "Focussing Techniques", s i m i l a r t o SSA, were t r i e d o u t w i t h s i m i l a r l y dismal r e s u l t s . O t h e r v a r i a b l e s t h a t have y i e l d e d , a t b e s t , mixed and u n r e l i a b l e e f f e c t s , i f any, a r e : o r d e r o f i n f o r m a t i o n p r e s e n t a t i o n ; t r a n s l a t a b i l i t y o f t h e base r a t e i n t h e sample; s e q u e n t i a l r a t h e r t h a n summary p r e s e n t a t i o n o f t h e base r a t e s ;

base r a t e v a l u e s t h a t a r e c o m p a t i b l e w i t h knowledge o r p r i v a t e t h e o r y versus t h o s e t h a t a r e n ' t ; and more.

The normative controversy I t i s d o u b t f u l whether a l l ' o f t h e above r e a l l y d e s c r i b e s a c o n t r o v e r s y . Rather, i t i s an expanding o f knowledge, a deepening o f understanding, an e x p l o r a t i o n

o f boundaries. Here and t h e r e , o f course, t h e r e have been, and s t i l l e x i s t , genuine disagreements, b u t t y p i c a l l y o f t h e k i n d t h a t can be s e t t l e d by r e f e r e n c e t o e m p i r i c a l f i n d i n g s . There does e x i s t , however, a more g e n e r a l controversy, concerning the l e g i t i m a c y o f the e n t i r e h e u r i s t i c s research program. One o f t h e most vocal c r i t i c s o f t h i s approach i s t h e O x f o r d p h i l o sopher L.J. Cohen. The base r a t e f a l l a c y i s b u t one component i n t h i s c o n t r o versy, b u t i t w i l l s e r v e n i c e l y t o demonstrate i t s essence.

BASE RATE FALLACY

55

Cohen c l a s s i f i e s the biases and f a l l a c i e s found i n judgment and reasoning, whether p r o b a b i l i s t i c o r l o g i c a l , i n t o f o u r categories, o n l y the f i r s t two of which acknowledge the existence o f any e r r o r . The base r a t e f a l l a c y enters under "Applications o f inappropriate normative theory", a category t h a t absolves the subjects o f e r r o r , and places i t instead on the experimenters shoulders (as does an e x p l i c i t t i t l e o f Cohen's: "On the psychology o f pred i c t i o n : Whose i s the f a l l a c y ? " , 1979). S p e c i f i c a l l y , Cohen goes through t h e Bayesian reasoning behind the Cab Problem, and then c r i t i c i z e s i t t h u s l y : "The r a t i o 17/29 i s the value o f the c o n d i t i o n a l p r o b a b i l i t y t h a t a cabc o l o r i d e n t i f i c a t i o n by the witness i s i n c o r r e c t , on the c o n d i t i o n t h a t i t ought n o t t o r e l y on i s an i d e n t i f i c a t i o n as green. Jurors, however,

...

...

[ b u t r a t h e r on1 the p r o b a b i l i t y t h a t a statement t o t h e that probability e f f e c t t h a t t h e cab a c t u a l l y i n v o l v e d i n the accident was green, i s f a l s e , on the c o n d i t i o n t h a t the statement i s made by the witness. I f the j u r o r s know t h a t o n l y 20 % o f the w i t n e s s ' s statements about cab c o l o r s a r e f a l s e , The [base r a t e ] i s s t r i c t l y i r they r i g h t l y estimate [p(G/"G")I as 1/5 relevant.

... A

...

probability that

...

i s based on causal properties, such as

...

chance d i s t r i b u t i o n s , t h a t have no causal e f f i c a c y t o t h e i n d i v i d u a l events" (Cohen 1981, p. 328, 329). the physiology o f v i s i o n , cannot be a l t e r e d by

Cohen goes on t o suggest t h a t " t h e l i t e r a t u r e under c r i t i c i s m i s propagating an analysis t h a t could increase the number o f deaths from a r a r e disease" (p. 329). For suppose you are s u f f e r i n g from one o f two diseases, A and B, the base r a t e s o f which are 85 % versus 15 %. Suppose f u r t h e r t h a t t h e r e e x i s t s a diagnostic t e s t t h a t i d e n t i f i e s each disease c o r r e c t l y 80 % o f the t i m e b u t e r r s 20 % o f the time. The t e s t reports you t o be s u f f e r i n g from B, whose base r a t e i s o n l y 15 %."Should you", asks Cohen menacingly, ' I . . . o p t f o r the treatment appropriate t o A, on t h e supposition [reached b y ' c a l c u l a t i n g as the experimenters d i d l t h a t t h e p r o b a b i l i t y o f your s u f f e r i n g from A i s 19/23? O r should you o p t f o r t h e treatment appropriate t o 6, on the suppos i t i o n [reached by c a l c u l a t i n g as t h e subjects d i d l t h a t the p r o b a b i l i t y o f one f o r you

...

B

i s 4/5? I t i s t h e former t h a t would be the i r r a t i o n a l n o t t h e l a t t e r " (p. 329).

your s u f f e r i n g from

Cohen's argument does n o t end here, b u t the fundamentals must by now be c l e a r . The controversy a r i s e s because o f a basic renouncing o f Bayesian reasoning,

56

COGNITIVE DECISION RESEARCH

and an upholding o f those f a c t o r s which we regarded as mere d e s c r i p t i v e explanations o f the base r a t e f a l l a c y as normatively sound. Indeed, Cohen o f f e r s an a l t e r n a t i v e , "Baconian", p r o b a b i l i t y calculus t o replace the fami l i a r , "Pascalian", one as a normative theory o f s u b j e c t i v e b e l i e f , w i t h the a d d i t i o n a l bonus t h a t i t i s more i n l i n e w i t h prevalent human i n t u i t i o n s , and i n p a r t i c u l a r , condones the response patterns t h a t we have termed " t h e base rate fallacy". Cohen attaches a g r e a t deal o f s i g n i f i c a n c e t o the manner i n which various p r o b a b i l i s t i c arguments are regarded i n court. The Law o f Evidence, as he sees it, represents the d i s t i l l e d and r e f l e c t i v e wisdom o f many people over many generations, and cannot be shrugged o f f by psychologists as being mere e r r o r s o f reasoning. Therefore, a p r o b a b i l i s t i c procedure upheld by the law must, by d e f i n i t i o n , be normatively c o r r e c t . And t h e law o f evidence does indeed seem t o "discriminate" against base rates. Sometimes they are r u l e d inadmissible; other times they are given l e s s weight than comparably, o r even less, diagnostic evidence from eyewitness testimony; and they are s e l dom allowed t o e x c l u s i v e l y s e t t l e a case, though they may seem p r o b a t i v e enough t o do so. This does n o t necessarily grant Cohen's point, however. A c a r e f u l examination o f the r a t i o n a l e f o r t h e c o u r t ' s a t t i t u d e s t o base r a t e information would take us too f a r a f i e l d , although I hasten t o say t h a t I do n o t share Cohen's confidence i n the inherent wisdom o f a l l laws. Nevertheless, a couple o f examples are i n order. "Imagine a rodeo i n t o which 400 people are known t o have been admitted through an automatic t u r n s t i l e a f t e r paying t h e proper sum. Then 1000 people are counted on t h e seats, and a hole i s discovered i n the fence" (Cohen 1979, p. 315). I t would be h i g h l y unjust, says Cohen, i f indeed possible, f o r a j u r y t o c o n v i c t a randomly selected person i n t h e rodeo audience on the basis o f t h i s evidence alone, though i t seems t h a t the balance o f the p r o b a b i l i t i e s favors t h e sueing management. Rather than deal w i t h t h i s hypothetical example, consider the f o l l o w i n g example, based on a r e a l case (Smith vs. Rapid Transit, Inc. 1945). P l a i n t i f f was negligentl y run over by a blue bus, and could prove t h a t the defendant operates four f i f t h s o f a l l the blue buses i n town. P l a i n t i f f could n o t t e s t i f y t o t h e actual i d e n t i t y o f the bus. "That datum suggests t h a t , i n the absence

...

BASE RATE FALLACY

o f o t h e r information,

.. . t h e r e

57

i s a " f o u r - f i f t h s c e r t a i n t y " t h a t t h e de-

f e n d a n t ' s bus h i t t h i s P l a i n t i f f " ( T r i b e 1971, p. 1346). The absence o f o t h e r i n f o r m a t i o n i s r a r e , and, e s p e c i a l l y i n a c o u r t o f law, t h e v e r y absence o f any o t h e r evidence "may p r o p e r l y be t r e a t e d as d i s p o s i t i v e i n some situations" (ibid.,

f o o t n o t e 54). B u t even i f t h e absence o f any o t h e r e v i -

dence, on e i t h e r l i t i g a n t ' s p a r t , were t o b e s a t i s f a c t o r i l y e x p l a i n e d , and i n a manner t h a t l e a v e s t h e p r o b a b i l i t y o f 4/5 unchanged, t h e r u l i n g i s " e n t i r e l y s e n s i b l e i f understood

some " i n d i v i d u a l i z e d "

... as

i n s i s t i n g on t h e p r e s e n t a t i o n o f

p r o o f o f i d e n t i t y b e f o r e c o m p e l l i n g a p a r t y t o pay

damages" r a t h e r t h a n i n s i s t i n g on " t h e m i n i m i z a t i o n o f t h e t o t a l number o f j u d i c i a l e r r o r s i n s i t u a t i o n s of t h i s k i n d " ( i b i d . , p. 1341, f o o t n o t e 37). And c l e a r l y , t h e l a w o f evidence has g o a l s t h a t e x t e n d way beyond t h i s l a s t .

The normative roZe of base r a t e s Tversky and Kahneman (1982a) concluded t h e i r d i s c u s s i o n o f t h e e v i d e n t i a l impact o f base r a t e s t h u s : "The m a j o r c o n c l u s i o n o f [ t h e base r a t e ] r e s e a r c h i s t h a t t h e use o r n e g l e c t o f consensus i n f o r m a t i o n i n i n d i v i d u a l p r e d i c t i o n depends c r i t i c a l l y on t h e i n t e r p r e t a t i o n o f t h a t i n f o r m a t i o n " (p. 160). My own c o n c l u s i o n i s i n complete agreement w i t h t h e i r s . Moreover, I would l i k e t o c l o s e w i t h an example t h a t shows, w i t h much elegance, how t h e n o r m a t i v e r o l e o f consensus i n f o r m a t i o n ( i . e . ,

base r a t e s ) a l s o depends c r i t i c a l l y on

i t s i n t e r p r e t a t i o n . T h i s example as w e l l as much o f my knowledge and views on t h e base r a t e i s s u e , i s due t o D a n i e l Kahneman and Amos T v e r s k j .

A young d o c t o r i s m o o n l i g h t i n g by w o r k i n g on h o l i d a y weekends i n t h e emergency ward o f a l a r g e urban h o s p i t a l . A p a t i e n t comes i n , c o m p l a i n i n g o f v a r i o u s symptoms. These symptoms l o o k t o t h e d o c t o r l i k e t h o s e s i g n a l l i n g t h e o n s e t o f h e a r t t r o u b l e , b u t t h e d o c t o r i s a w a r e t h a t amerestomach d i s o r d e r m a y sometimes e x h i b i t t h e same symptoms. Experience has t a u g h t t h e d o c t o r t h a t t h e r e l a t i v e base r a t e o f stomach d i s o r d e r s t o h e a r t t r o u b l e among p a t i e n t s i s c o n s i d e r a b l y h i g h e r d u r i n g o r d i n a r y weekdays t h a n d u r i n g h o l i d a y weekends. How should t h e d o c t o r e v a l u a t e t h e chances t h a t t h e p a t i e n t i s s u f f e r i n g f r o m h e a r t t r o u b l e r a t h e r t h a n a stomach d i s o r d e r ? I n t u i t i v e l y , i t m i g h t seem t h a t i d e n t i c a l symptoms s h o u l d be i d e n t i c a l l y eva-

58

COGNI TI VE DECISION RESEARCH

luated, w i t h no need t o f i r s t check the calender t o see what the date is. B u t the doctor versed in the l i t e r a t u r e on judgment under uncertainty might consider i t fallacious t o ignore the difference in base r a t e s , and conclude that the exact same symptoms should be judged more likely t o result from heart trouble on a weekend than on a weekday. Is t h i s indeed necessary? The surprising answer i s t h a t both procedures may be j u s t i f i a b l e , depending on the causal structure that leads t o the different base rates on weekends and wekkdays. According t o one possible account, hospital emergency wards get more than the regular share of false alarms ( i . e , , mild complaints such as stomach disorders parading as heart trouble) on weekends, because of the relative nonaccessabil i t y of everyday medical " f i l t e r s " such as neighborhood c l i n i c s , family doctors, etc. A different account i s t h a t eating habits on holiday weekends are different enough from everyday habits t o genuinely increase the overall occurence of stomach disorders. If one believes the former account, one may as well use the weekday s t a t i s t i c s , i f a patient with the symptoms a t hand would have been sent t o a hospital on a weekday anyway (making i t irrelevant that others would be diagnosed as suffering from stomach disorders and never sent on t o a hospital). I f , on the other hand, one believes the l a t t e r account, then the population of weekend diseases i s truely distributed differently than the weekday population, and doctors would be wise t o take t h a t into consideration (as they would take, say, the p a t i e n t ' s sex into consideration, i f the two sexes have different rates of heart trouble). The indetemninaoy o f normativeness?

I t would seem, on the basis of the case of the moonlighting doctor, that any use of base rates could be j u s t i f i e d , provided one can come up w i t h the proper scenario - a somewhat baffling conclusion. Happily, however, we are not faced with complete normative indeterminacy. "Proper scenarios", i f by "proper" we mean "plausible", too, may n o t be easy to come by (see, e.g. , Fischhoff, Slovic & Lichtenstein 1978). Indeed, among people with a certain shared cultural and intellectual background, a consensus typically e x i s t s concerning w h a t i s o r i s n ' t a proper scenario, therefore what i s o r i s n ' t an acceptable solution t o some problem. The example does show t h a t we may need t o relax o u r concept of a "normative solution". Sometimes a problem may have

BASE RATE FALLACY

59

no unique normative s o l u t i o n , sometimes what consensually appears t o be t h e b e s t s o l u t i o n may d i f f e r from t h e formal , "textbook", s o l u t i o n , e t c . Nevert h e l e s s , some s o l u t i o n s a r e unacceptable by any s e n s i b l e standard (see d i s cussion i n Kahneman & Tversky 1982). Some o f t h e attempts t o v i n d i c a t e t h e base r a t e f a l l a c y have, u n f o r t u n a t e l y , f a l l e n i n t h i s category. The i g n o r i n g o f base r a t e c o n s i d e r a t i o n s i s a common phenomenon, and o f t e n u n j u s t i f i a b l e . A t t h e same time, i n t u i t i o n i s o f t e n s e n s i t i v e t o c o n s i d e r a t i o n s t h a t a r e n o t captured by s i m p l i s t i c formal models. The a s t u t e judge under u n c e r t a i n t y would do w e l l t o r e l y on n e i t h e r o f them t o t h e t o t a l e x c l u s i o n o f t h e o t h e r . References

Ajzen, I.: 1977. I n t u i t i v e t h e o r i e s o f events and t h e e f f e c t s o f base r a t e i n f o r m a t i o n on p r e d i c t i o n . I n : Journal o f P e r s o n a l i t y and S o c i a l Psychology 35, 303-314 B a r - H i l l e l , M.: 1980. The base-rate f a l l a c y i n p r o b a b i l i t y judgments. I n : Acta Psychologica 44, 211-233 Bar-Hi1 l e l , M. : 1981. Representativeness Reconsidered. Decision Research Report 81-1 B a r - H i l l e l , M. & F i s c h h o f f , B.: 1981. When do base r a t e s a f f e c t p r e d i c t i o n s ? I n : Journal o f P e r s o n a l i t y and S o c i a l Psychology 41, 671-680 Beyth-Marom, R. & Arkes, H.R.: 1983. Being accurate b u t n o t n e c e s s a r i l y Bayesian: Comment on Christensen-Szalanski and Beach. I n : O r g a n i z a t i o n a l Behavior and Human Performance, i n press Borgida, E. & Brekke, N.: 1980. The base r a t e f a l l a c y i n a t t r i b u t i o n and p r e d i c t i o n . In: J.H. Harvey/W.J. I c k e s & R.F. K i d d (Eds.), New D i r e c t i o n s i n A t t r i b u t i o n Research, Vol. 3. H i l l s d a l e , N.J.: Erlbaum C a r r o l l , J.S. & S i e g l e r , R.S.: 1977. S t r a t e g i e s f o r t h e use o f base r a t e i n f o r m a t i o n . I n : Organizational Behavior and Human Performance 19, 392 402

-

Cohen, L.J.: 1979. On t h e psychology o f p r e d i c t i o n : Whose i s t h e f a l l a c y ? I n : Cognition 7, 385-407 Cohen, L.J. : 1981. Can human i r r a t i o n a l i t y be e x p e r i m e n t a l l y demonstrated? I n : The Behavioral and B r a i n Sciences 4, 317-331 Christensen-Szalanski, J . J . J . & Beach, L.R.: 1982. Experience and t h e base r a t e effect. I n : Organizational Behavior and Human Performance 29, 270-278 Dershowitz, A. : 1971. Imprisonment by j u d i c i a l hunch. I n : American Bar Assoc i a t i o n Journal 57, 560-564

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Eddy, D.M.: 1982. P r o b a b i l i s t i c reasoning i n c l i n i c a l medicine: Problems and o p p o r t u n i t i e s . In: D. Kahneman/P. S l o v i c & A. Tversky, Judgment under Unc e r t a i n t y : H e u r i s t i c s and Biases. New York: Cambridge U n i v e r s i t y Press Estes, W.K.: 1976. The c o g n i t i v e s i d e o f p r o b a b i l i t y l e a r n i n g . I n : Psycholog i c a l Review 83, 37-64 Fischhoff, B. & B a r - H i l l e l , M.: 1981. D i a g n o s t i c i t y and t h e base r a t e e f f e c t . I n : Decision Research Report 81-9 F i s c h h o f f , B. & B a r - H i l l e l , M.: 1982. Focussing techniques as a i d s t o i n f e r ence. Unpublished manuscript Fischhoff, B./Slovic, P. & L i c h t e n s t e i n , S.: 1978. F a u l t t r e e s : S e n s i t i v i t y o f estimated f a i l u r e p r o b a b i l i t i e s t o problem r e p r e s e n t a t i o n . I n : Journal o f Experimental Psychology: Human Perception and Performance 8, 330-344 F i s c h h o f f , B./Slovic, P. & L i c h t e n s t e i n , S.: 1979. S u b j e c t i v e s e n s i t i v i t y analysis. I n : Organizational Behavior and Human Performance 23, 339-359 F o l t z , A.M. & Kelsey, J.L.: 1978. The annual Pap t e s t : A dubious p o l i c y success. In: Milbank Maemorial Q u a r t e r l y / H e a l t h and S o c i e t y 56(4), 426-462 Freedman, D./Pisani,

R. & Purves, R.:

1978. S t a t i s t i c s . New York:

W.W. Norton

Ginosar, Z. & Trope, Y.: 1980. The e f f e c t s o f base r a t e s and i n d i v i d u a t i n g i n f o r m a t i o n on judgments about another person. I n: Journal o f Experiment a l S o c i a l Psychology 16, 228-242 H u f f , D.:

1959. How t o take a chance. Harmondsworth: P e l i c a n Books

Kahneman, D. & Tvecsky, A.: 1972a. S u b j e c t i v e p r o b a b i l i t y : A judgment o f r e presentativeness. I n : C o g n i t i v e Psychology 3, 430-454 Kahneman, D. & Tversky, A.: search Monograph 12(4)

1972b. On p r e d i c t i o n and judgment. I n : OR1 Re-

Kahneman, D. & Tversky, A.: 1973. On t h e psychology o f p r e d i c t i o n . I n : Psycholo g i c a l Review 80, 237-251 Kahneman, D. & Tversky, A.: C o g n i t i o n 11, 123-141

1982. On t h e study o f s t a t i s t i c a l i n t u i t i o n s . I n :

1979. Consensus i n f o r m a t i o n , p r e d i c t i o n , and causal a t t r i b u t i o n : Kassin, S.M.: A review o f t h e l i t e r a t u r e and issues. I n : Journal o f P e r s o n a l i t y and S o c i a l Psychology 37, 1966-1981 Lykken, D.T.: 1975. The r i g h t way t o use a l i e d e t e c t o r . I n : Psychology Today 8, 56-60 Lyon, D. & S l o v i c , P.: 1976. Dominance o f accuracy i n f o r m a t i o n and n e g l e c t o f base r a t e s i n p r o b a b i l i t y e s t i m a t i o n s . I n : Acta Psychologica 40, 287-298 Manis, M./Dovalina,

I./Avis,

N.E.

& Cardoze, S.:

1980. Base r a t e s can e f f e c t

BASE RATE FALLACY

i n d i v i d u a l predictions. I n : Journal o f Personality and Social Psychol-

O W 38, 231-248 Meehl, P.E. & Rosen, A.: 1955. Antecedent p r o b a b i l i t y and the e f f i c a c y o f psychometric signs, patterns o r c u t t i n g scores. I n : Psychological B u l l e t i n 52, 194-216 Nisbett, R.E. & Borgida, E.: 1975. A t t r i b u t i o n and the psychology o f pred i c t i o n . I n : Journal o f P e r s o n a l i t y and Social Psychology 32, 932-943 Nisbett, R.E./Borgida, E./Crandall, R. & Reed, H. : 1976. Popular i n d u c t i o n : Information i s n o t always informative. I n : J.S. C a r r o l l & J.W. Payne (Eds.), Cognition and Social Behavior Tversky, A. & Kahneman, 0.: 1980. Causal schemas i n judgment under uncert a i n t y . I n : M. Fishbein (Ed.), Progress i n Social Psychology. H i l l s d a l e , N.J.: Erlbaum Tversky, A. & Kahneman, 0.: 1982a. E v i d e n t i a l impact o f base rates. I n : D. Kahneman/P. S l o v i c & A. Tversky, Judgment under Uncertainty: Heurist i c s and Biases. New York: Cambridge U n i v e r s i t y Press Tversky, A. & Kahneman, D.: 1982b. Judgments o f and by representativeness. I n : D. Kahneman/P. S l o v i c & A. Tversky, Judgment under Uncertainty: Heur i s t i c s and Biases. New York: Cambridge U n i v e r s i t y Press

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor)

0 Elsevier Science Publishers B.V. (North-Holland), 1983

THE TWO CAMPS ON RATIONALITY

HELMUT JUNGERMANN Technische Universitat Berlin lnstitut fur Psychologie F. R. Germany

The s t a t u s of the r a t i o n a l i t y concept I f one postulates t h a t people are g e n e r a l l y r a t i o n a l , one meets

today u s u a l l y o b j e c t i o n or, a t best, scepticism; biases, e r r o r s and f a u l t s are described and i l l u s t r a t e d t o prove t h a t the p o s t u l a t e has l i t t l e empirical j u s t i f i c a t i o n . I f one declares a person i r r a t i o n a l , however, one equally w i l l meet u s u a l l y p r o t e s t from t h i s person; he o r she w i l l e x p l a i n t h a t there were good reasons f o r the judgment o r decisions questioned. The c o n t r a d i c t i o n might be due t o the words "generally" and " u s u a l l y " i n the above statements: People a r e n o t always r a t i o n a l and people w i l l n o t always c l a i m t o be r a t i o n a l . However, the present debate about the q u a l i t y o f human judgment and decis i o n i n d i c a t e s t h a t the controversy i s n o t o n l y about generalizations and exceptions (e.g., Einhorn and Hogarth, 1981; N i s b e t t and Ross, 1980; Cohen, 1979; Kahneman and Tversky, 1982; Edwards, 1983; Fischh o f f , 1983; Berkeley and Humphreys, 1982; P h i l l i p s , 1983). The p a r t i cipants i n t h i s debate tend t o avoid the term " r a t i o n a l i t y " because i t i s overloaded w i t h many connotations, b u t j u s t f o r t h a t reason I f i n d i t useful as an umbrella under which most c o n t r i b u t i o n s f i t f a i r l y w e l l ; they a l l t r e a t one o r another d e f i n i t i o n o r meaning of rationality.

63

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COGNITIVE D EClSlON RESEARCH

R a t i o n a l i t y i s n o t a genuine term o f s c i e n t i f i c psychology b u t r a t h e r a concept o f p h i l o s o p h y and economics. The most common, and i n t h i s c o n t e x t most r e l e v a n t , d e f i n i t i o n says t h a t an a c t i o n i s r a t i o n a l i f i t i s i n l i n e w i t h t h e values and b e l i e f s o f t h e i n d i v i d u a l concerned; o r more p r e c i s e l y , i f i t i s " l o g i c a l " o r " c o n s i s t e n t " as s t a t e d i n a s e t o f axioms. T h i s d e f i n i t i o n s p e c i f i e s r a t i o n a l b e h a v i o r n o r m a t i v e l y . E m p i r i c a l r e s e a r c h can s t u d y whether a c t u a l human b e h a v i o r i s r a t i o n a l i n t h e sense t h a t i t obeys t h e norm.

To measure human b e h a v i o r w i t h a n o r m a t i v e y a r d s t i c k i s n o t t h e r u l e b u t r a t h e r t h e e x c e p t i o n i n c o g n i t i v e psychology. I n t h e psychology o f perception, f o r i n s t a n c e , t h e p h y s i c a l w o r l d i s t a k e n as a s t a n d a r d when p e r c e p t u a l i l l u s i o n s a r e i n v e s t i g a t e d . N o r m a l l y , however, t h e p h y s i c a l s t i m u l i a r e used t o provoke responses; one t r i e s t o u n d e r s t a n d and e x p l a i n t h e s e responses and t h e y a r e n o t e v a l u a t e d as ' d e v i a n t f r o m ' o r ' c o n s i s t e n t w i t h ' something. The psychology o f language worked f o r some t i m e on t h e assumption t h a t human language c o u l d be e v a l u a t e d w i t h t h e l i n g u i s t i c y a r d s t i c k o f an i d e a l speaker o r h e a r e r . B u t t h i s assumption t u r n e d o u t t o be n o t v e r y h e l p f u l i n u n d e r s t a n d i n g a c t u a l human language, and r e s e a r c h began t o f o c u s on p s y c h o l o g i c a l models l i k e semantic networks. F i n a l l y , t h e psychology o f thinking used, and t o some e x t e n t s t i l l uses, f o r m a l l o g i c as a y a r d s t i c k f o r t h e s t u d y o f deductive reasoning. But n o t o n l y i s t h i s merely a small area w i t h i n t h e psychology o f t h i n k i n g , t h e approach i t s e l f has i n c r e a s i n g l y been d i s p u t e d as mistaken. S i m i l a r l y

t o t h e development i n psycho-

1 i n g u i s t i c s , t h e f o c u s has been d i r e c t e d i n c r e a s i n g l y on t h e c o n t e n t r a t h e r than on t h e f o r m a l c h a r a c t e r i s t i c s o f t h i n k i n g . I n a l l t h e s e areas o f r e s e a r c h , t h e use o f a p h y s i c a l o r l o g i c a l y a r d s t i c k p l a y s o n l y a m i n o r r o l e t o d a y o r has been abandoned a l t o g e t h e r .

I n t h e psychology o f judgment and decision, on t h e o t h e r hand, n o r m a t i v e models have been t h e most i m p o r t a n t r e s e a r c h t o o l s s i n c e i t s b e g i n n i n g s i n t h e l a t e f i f t i e s . Research has been p r i m a r i l y been concerned w i t h s t u d y i n g , e x p l a i n i n g and i n t e r p r e t i n g d i s c r e p a n c i e s between p r e d i c t i o n s d e r i v e d f r o m n o r m a t i v e models and a c t u a l judgments

THE TWO CAMPS ON RATIONALITY

and decisions. The most prominent models o f judgment are probably Bayes' theorem and the m u l t i a t t r i b u t e u t i l i t y models, and the normat i v e model o f decision i s the SEU model. Most research i s s t i l l o r i e n t e d towards these models and the debate about the q u a l i t y o f human judgment and decision i s s t i l l centered around the idea o f r a t i o n a l i t y embodied i n these model s

.

I n a somewhat exagerated manner, I w i l l d i s t i n g u i s h two camps i n t h i s debate, one t h a t p o i n t s t o the d e f i c i e n c y and one t h a t argues f o r the e f f i c i e n c y o f human judgment and decision, The pessimists, as I w i l l c a l l the members o f the f i r s t camp, c l a i m t h a t judgment and d e c i s i o n making under u n c e r t a i n t y o f t e n show systematic and serious e r r o r s , due t o i n - b u i l t c h a r a c t e r i s t i c s o f the human c o g n i t i v e system. V i o l a t i o n s o f r a t i o n a l i t y , p a r t i c u l a r l y o f the SEU model, a r e i n t e r p r e t e d as t r u e d e f i c i t s o f the decision maker. The optimists o f the other camp c l a i m t h a t judgment and decision a r e h i g h l y e f f i c i e n t and f u n c t i o n a l even i n complex s i t u a t i o n s . Observed v i o l a t i o n s o f r a t i o n a l i t y axioms are i n t e r p r e t e d as u n j u s t i f i e d evaluations based on inappropriate t h e o r e t i c a l assumptions o r empirical approaches on the p a r t o f the researcher. I n the f o l l o w i n g , I w i l l describe the approach t o r a t i o n a l i t y taken i n both camps, each o f which has various f a c t i o n s , and

discuss some p o i n t s t h a t make the debate so complicated and, sometimes, obscure. The d e s c r i p t i o n cannot be as d i f f e r e n t i a t e d as the approaches are, o f course, b u t I hope t h a t i t i s e s s e n t i a l l y c o r r e c t ; the i n t e r ested reader i s r e f e r r e d t o the o r i g i n a l sources.

The pessimists: Biases are i n peopZe Since Simon (1955) proposed t h e concept o f "bounded r a t i o n a l i t y " , t h e strongest attacks against human r a t i o n a l i t y came from Tversky and Kahneman (e.g., 1974), S l o v i c (1972), Janis and Mann (1977), and N i s b e t t and Ross (1980). The general tendency o f t h i s research was t h a t human judgment and decision making a b i l i t y and capacity i s indeed l i m i t e d , leading t o v i o l a t i o n s o f r a t i o n a l i t y p r i n c i p l e s . There a r e t h r e e v a r i a n t s i n t h i s camp i n e x p l a i n i n g these v i o l a t i o n s : As r e s u l t s

65

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COGNITIVE DECISION RESEARCH

of judgmental biases, of representationaZ faults, and of coping defects. Judgmental biases: The most extensive and influential work of i n t e r e s t

in t h i s context has been the research of Tversky & Kahneman on probabil i s t i c thinking. While in the s i x t i e s , Peterson & Beach (1967) i n t h e i r review came t o the conclusion that people form and revise t h e i r beliefs according t o the normative principles of s t a t i s t i c s . Tversky & Kahneman (e.g. , 1974) offered a different conception: Judgments of probabilities are often severely biased because people i n many situations rely on heuristics which, although generally e f f i c i e n t , can sometimes lead t o systematic errors. For example, people often judge the probability of an event according to the representativeness of the event for the underlying population or for the generating process. However, " t h i s approach t o the judgment of probability leads t o serious errors, because similarity or representativeness i s n o t influenced by several factors that should a f f e c t judgments of probability" (Tversky & Kahneman, 1974, p.1124). Such factors a r e , f o r instance, base r a t e s , sample s i z e and the r e l i a b i l i t y of information. Another heuristic people use i s the saliency o r availabiZity of information, i . e . , how easily instances of the event whose probability i s t o be assessed can be retrieved from memory. Since ease of retrieval i s also influenced by other factors t h a n the actual frequency of the event (e.g., recent occurence) the use of t h i s heuristic can likewise r e s u l t in systematic judgmental errors. Other heuristics described in t h i s research are "anchoring and adjustment" and "simulation" ( f o r a collection of the most important papers on biases and heuristics see the recent book by Kahneman, Slovic and Tversky, 1982). Judgmental biases may produce inconsistent decisions. Such inconsistencies would then not r e f l e c t violations of assumptions of the model (e.g., SEU model) b u t rather incorrect i n p u t . However, the d i s cussion has mostly been restricted t o the quality of the judgment i t s e l f . The judgment i s compared to a standard l i k e r e l a t i v e frequency o r the inference based on a s t a t i s t i c a l model f o r which the researcher has defined the relevant data ( e . g . , what the base rates a r e ) . In both cases, deficiency means deviation of the judgment from some

THE TWO CAMPS ON RATIONALITY

" o b j e c t i v e " q u a n t i t y , and the " o b j e c t i v i t y " i s r o o t e d i n t h e r e a l world, o r r a t h e r t h e r e s e a r c h e r ' s view o f t h e r e a l world, The i m p l i c i t n o t i o n o f r a t i o n a l i t y , then, i s n o t consistency b u t r e a l i s m ; i t i s l e s s forma2 r a t i o n a l i t y t h a t i s questioned than substantive

r a t i o n a l i t y : People use c o g n i t i v e i n f e r e n c e and r e t r i e v a l s t r a t e g i e s which o f t e n l e a d t o ( i n t h e r e s e a r c h e r ' s view) s u b s t a n t i v e l y i n c o r r e c t judgments about t h e world.

ImpZieitZy, t h i s meaning o f r a t i o n a l i t y pervades t h e d i s c u s s i o n on biases and h e u r i s t i c s i n p r o b a b i l i s t i c t h i n k i n g . I t i s i n t e r e s t i n g t o f i n d i t made expZicit w i t h r e s p e c t t o judgments which a r e e q u a l l y i m p o r t a n t f o r d e c i s i o n making b u t much l e s s debated, namely, u t i l i t y judgments (March 1978). Tversky & Kahneman (1981) q u e s t i o n t h e r a t i o n a l i t y o f u t i l i t y judgments t h a t t u r n o u t t o have been u n r e a l i s t i c

a n t i c i p a t i o n s o f s a t i s f a c t i o n a t t h e time when t h e consequences a c t u a l l y i n c u r . But biases and h e u r i s t i c s i n t h e judgment o f u t i l i t y a r e s t i l l unexplored, maybe due t o t h e l a c k o f a y a r d s t i c k l i k e r e l a t i v e frequency i t o f f e r s f o r p r o b a b i l i t y judgments, A d i f f e r e n t approach i s needed here,

maybe t a k i n g t h e work o f A i n s l i e (1976) on preferences as a f u n c t i o n o f time delay o f s a t i s f a c t i o n and t h e economic approach t o ' r a t i o n a l e x p e c t a t i o n s ' as a s t a r t i n g p o i n t . R a t i o n a l i t y i n t h i s sense, i.e., as realism, c e r t a i n l y comes c l o s e t o t h e meaning o f t h i s term i n common sense: We o f t e n c a l l hopes and f e a r s " i r r a t i o n a l " which we c o n s i der as extremely u n r e a l i s t i c . The question, o f course, i s who i s t o d e f i n e r e a l it y

.

h'epresentationai! f a u l t s : A more r e c e n t argument a g a i n s t r a t i o n a l i t y , r a i s e d by t h e same group o f researchers, concerns t h e e f f e c t o f d i f f e r e n t frames f o r t h e d e c i s i o n problem on p e o p l e ' s d e c i s i o n making behavior (Tversky and Kahneman, 1981; cp. a l s o Payne, 1982). A s t r i k i n g example i s t h e experiment i n which a m a j o r i t y o f s u b j e c t s behaved r i s k averse when they had t o decide between two o p t i o n s t h a t were f o r m u l a t e d i n terms o f l i v e s saved, b u t behaved r i s k seeking when t h e o p t i o n s were formulated i n terms o f l i v e s l o s t

-

a l t h o u g h t h e (expected) values

o f t h e o p t i o n s were i n both cases i d e n t i c a l . T h i s v i o l a t i o n o f r a t i o n a l i t y , i.e.,

o f t h e SEV model, i s i n t e r p r e t e d by t h e assuvption t h a t

67

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COGNI TI VE DECISION RESEARCH

people code the possible outcomes as gains and losses r a t h e r than as f i n a l states, and the framing o f the options induces i n one case coding i n terms o f gains and i n the losses, i.e.,

other case coding i n terms o f

i t induces subjects t o focus on d i f f e r e n t p a r t s o f t h e i r

u t i l i t y functions. Since the u t i l i t y functions are d i f f e r e n t below and above the reference p o i n t , apparently i n c o n s i s t e n t preferences r e s u l t . Coding outcomes i n terms o f gains and losses i s , according t o Kahneman and Tversky (1979), o n l y one o f several c o g n i t i v e mechanisms which people use t o e d i t , o r represent, decision problems before the options w i l l be evaluated, and these operations can a l s o lead t o v i o l a t i o n s o f r a t i o n a l i t y . Other e d i t i n g mechanisms are, f o r instance, the segregation o f r i s k l e s s components o f prospects and the c a n c e l l a t i o n o f components which are shared by two prospects. The representational e r r o r s r e s u l t i n g from the a p p l i c a t i o n o f such mechanisms are l i k e n e d t o perceptual i l l u s i o n s (Tversky and Kahneman, 1981). This i m p l i e s the assumption t h a t there e x i s t s one and only one c o r r e c t representation o f the problem, as t h e r e e x i s t s o n l y one v e r i d i c a l representation o f the physical world, I n c o n s i s t e n t preferences are understood as r e s u l t s o f a d e f i c i e n t perception and i n t e r p r e t a t i o n o f the d e c i s i o n problem. I t i s i n t e r e s t i n g t o note t h a t w i t h t h i s argument, the context f o r

discussing r a t i o n a l i t y i s being enlarged. Previously, the domain o f t h e o r e t i c a l and empirical research was jugdment and evaluation only, b u t not the c o g n i t i v e a c t i v i t i e s which forego these steps, e.g., how the problem s t r u c t u r e i s generated. Only r e c e n t l y , i n t e r e s t has s h i f t e d t o these e a r l y phases o f the decision making process (e.g., P i t z , Sachs and Heerboth, 1981; Jungermann, von U l a r d t and Hausmann, 1983), b u t t h i s research has n o t been generally l i n k e d t o the d i s cussion about the d e f i c i e n c y / e f f i c i e n c y o f judgment and decision.

Coping defects: The t h i r d a t t a c k on r a t i o n a l i t y comes from Janis and Mann (1977) who look a t people's d e c i s i o n making behavior f r o m a motiv a t i o n a l perspective. They d i s t i n g u i s h various coping patterns t h a t people use i n handling stress o f d e c i s i o n s i t u a t i o n s , and o n l y one o f

69

THE TWO CAMPS ON RATIONALITY

them corresponds t o t h e r a t i o n a l b e h a v i o r as e x p l i c a t e d i n d e c i s i o n t h e o r y , The o t h e r f o u r p a t t e r n s l e a d t o d e f e c t i v e d e c i s i o n making. F o r example, i f t h e c o p i n g p a t t e r n is " d e f e n s i v e avoidance",

t h e person

escapes t h e d e c i s i o n a l c o n f l i c t by p r o c r a s t i n a t i n g , s h i f t i n g r e s p o n s i b i l i t y t o someone e l s e , o r c o n s t r u c t i n g w i s h f u l r a t i o n a l i z a t i o n s t o b o l s t e r the l e a s t objectionable a l t e r n a t i v e , remaining s e l e c t i v e l y i n a t t e n t i v e t o c o r r e c t i v e i n f o r m a t i o n , A l t h o u g h t h e s e p a t t e r n s , as Mann and J a n i s (1982) concede, can o c c a s i o n a l l y be a d a p t i v e i n s a v i n g t i m e and e f f o r t , t h e y o f t e n l e a d t o d e f e c t i v e d e c i s i o n making. " D e f i c i e n c y " i s h e r e n o t d i r e c t l y d e f i n e d i n r e l a t i o n t o some n o r m a t i v e model, e.g.,

t h e SEU model, b u t more g e n e r a l l y by t h e f a i l u r e o f p e o p l e

" t o a s s i m i l a t e and combine i n f o r m a t i o n r e l a t i n g t o outcome e x p e c t a t i o n s and v a l u e s " ( ~ " 3 4 7 )and p a r t i c u l a r l y " t o make use o f t h e r e s o u r c e s a v a i l a b l e t o them f o r engaging i n e f f e c t i v e search f o r and a p p r a i s a l o f alternatives

-

within the l i m i t s o f t h e i r cognitive capabilities

and w i t h i n l i m i t s imposed by p o w e r f u l s o c i a l c o n s t r a i n t s " ( ~ ~ 3 4 6 I) n. a l a r g e number o f e x p e r i m e n t a l s t u d i e s , J a n i s and Mann (1977) have c o l l e c t e d evidence f o r t h e s e c o p i n g d e f e c t s , and t h e y have t r i e d t o i d e n t i f y t h e c o n d i t i o n s under which p e o p l e e x h i b i t d i f f e r e n t t y p e s o f

nforma-

t i o n processing behavior i n decison s i t u a t i o n s . Thus, J a n i s and Mann a l s o emphasize t h e d e f i c i e n c e s o f human

udgment

and d e c i s i o n , b u t i n t h e i r p e r s p e c t i v e , t h e sources a r e m o t i v a t i o n a l , n o t c o g n i t i v e i n n a t u r e : "We see man n o t as a c o l d f i s h , b u t as a warm-blooded mammal, n o t as a r a t i o n a l c a l c u l a t o r always r e a d y t o work o u t t h e b e s t s o l u t i o n b u t as a r e l u c t a n t d e c i s i o n maker -. b e s e t by c o n f l i c t s , doubts, and w o r r y , s t r u g g l i n g w i t h i n c o n g r o u s l o n g i n g s , a n t i p a t h i e s , and l o y a l i t i e s

*..I'

(1977, p.15). T h i s l a t t e r s t a t e m e n t

i l l u s t r a t e s t h a t , a l t h o u g h t h e f o c u s i s e x p l i c i t l y on d e c i s i o n making under s t r e s s , t h e a u t h o r s t e n d t o g e n e r a l i z e t h e i r t h e o r y t o human d e c i s i o n making i n g e n e r a l A s i m i l a r approach, though f r o m a c o g n i t i v e s t a n d p o i n t , has been t a k e n

by Dorner who examined how p e o p l e o p e r a t e i n h i g h l y complex and dynamic s i t u a t i o n s (ems., Dorner, 1983; Dorner, K r e u z i g , R e i t h e r and Staudel, 1983). I n one o f t h e s t u d i e s , f o r i n s t a n c e , he used a computer

COGNI T I VE DECISION RESEARCH

70

model o f a l i t t l e town and had t h e s u b j e c t s r e i g n t h i s town as mayor f o r t e n s i m u l a t e d y e a r s i n i n t e r a c t i o n w i t h t h e computer. Having t o cope w i t h such i l l - d e f i n e d problems, s u b j e c t s o f t e n a p p a r e n t l y use a number of h e u r i s t i c procedures w h i c h m i g h t be c o n s i d e r e d d e f i c i e n t , Dorner e t a l . do n o t compare t h e observed b e h a v i o r w i t h n o r m a t i v e models o f judgment and d e c i s i o n , however, b u t r a t h e r check whether t h e s u b j e c t s ' b e h a v i o r meets c e r t a i n demands o f t h e s i t u a t i o n , F o r example, r e q u i r e m e n t s r e s u l t i n g f r o m c h a r a c t e r i s t i c s o f t h e a s p i r e d goal a r e o f t e n managed by " r e d e f i n i n g " o r " f o r g e t t i n g " t h e f i n a l g o a l , by t h e m a t i c vagabonding (e.g., s w i n g i n g f r o m one a r e a o f p u r s u i t t o another)

, or

by e n c a p s u l a t i o n (e.g.

, sticking

obstinately t o a

theme i r r e s p e c t i v e o f t h e changing s i t u a t i o n ) . A l t h o u g h t h e d e f i c i e n c i e s d i s p l a i d i n these experiments a r e p r o b a b l y l a r g e l y due t o c o g n i t i v e o v e r l o a d , some o f t h e h e u r i s t i c s m i g h t w e l l be used a l s o i n l e s s complex s i t u a t i o n s . I n summary, t h e members o f t h i s camp view human judgment and d e c i s i o n making as d e f i c i e n t i n s e v e r a l r e s p e c t s : Judgments a r e sometimes s y s t e m a t i c a l l y biased, due t o t h e use o f h e u r i s t i c s ; d e c i s i o n s a r e sometimes i n c o n s i s t e n t , due t o e r r o r s i n t h e r e p r e s e n t a t i o n o f t h e problem; and i n f o r m a t i o n search and c o m b i n a t i o n i s o f t e n d e f e c t i v e , due t o m o t i v a t i o n a l f a c t o r s . M o s t l y , t h i s d e f i c i e n c y i s n o t seen s i m p l y as a consequence o f c o g n i t i v e o v e r l o a d i n h i g h l y complex o r u n f a m i l i a r s i t u a t i o n s , b u t as r o o t e d i n mechanisms w o r k i n g w i t h i n t h e human i n f o r m a t i o n p r o c e s s i n g system i t s e l f . People a r e prone t o v i o l a t e p r i n c i p l e s o f r a t i o n a l i t y . A key assumption o f a t l e a s t t h e c o g n i t i v e v a r i a n t s d e s c r i b e d i s t h a t t h e r e i s a r e l i a b l e and v a l i d y a r d s t i c k f o r t h e e v a l u a t i o n o f judgment and d e c i s i o n , namely, t h e o b j e c t i v e r e a l i t y , and c o n s e q u e n t l y t h a t t h e r e i s some k i n d o f o b j e c t i v e l y v e r i d i c a l mode o f i n f o r m a t i o n p r o c e s s i n g which i s mapped

i n t h e r e s p e c t i v e n o r m a t i v e model

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THE TWO CAMPS ON RATIONALITY

The optimists: Biases are i n research I n t h e l a s t few years, a k i n d o f counter-movement has developed.

The members o f t h i s new camp p a r t l y q u e s t i o n t h e v a l i d i t y o f t h e o t h e r camp's f i n d i n g s , and p a r t l y emphasize t h e i m p l i c i t r a t i o n a l i t y o f human judgment and d e c i s i o n behavior. The biases, they argue, a r e n o t i n human behavior b u t i n t h e a n a l y s i s o f t h i s behavior i n t h e o t h e r camp (Berkeley and Humphreys, 1982)

I n t e r e s t i n g l y enough, a l e a d i n g

r e p r e s e n t a t i v e o f t h i s camp has been a founder o f t h e o t h e r camp w i t h h i s research on conservatism i n human i n f o r m a t i o n processing (Edwards, 1968); o t h e r researchers I i n c l u d e i n t h i s camp a r e Beach and M i t c h e l 1 (1978)

, E i nhorn

and Hogarth (1981)

, Berkeley

and Humphreys

(1982), and P h i l l i p s (1983), Three t h e o r e t i c a l arguments a r e r a i s e d i n p a r t i c u l a r , which I w i l l c a l l t h e meta-rationality argument, t h e

continuity argument and t h e structure argument, The meta-rationality argument: The essence o f t h i s argument i s t h a t d e c i s i o n behavior which v i o l a t e s p r i n c i p l e s o f r a t i o n a l i t y as, f o r instance, t h e p r i n c i p l e o f maximizing s u b j e c t i v e l y expected u t i l i t y , can be described as p e r f e c t l y r a t i o n a l i f t h e c o g n i t i v e costs o f being r a t i o n a l a r e taken i n t o account. To i l l u s t r a t e t h e p o i n t , when somebody wants t o buy a book a t t h e t r a i n s t a t i o n f o r a l o n g r i d e , he w i l l probably n o t check a l l a v a i l a b l e books i n o r d e r t o f i n d t h e b e s t o f a l l b u t he w i l l l o o k a t a few and then buy t h e f i r s t t h a t he f i n d s reasonably a t t r a c t i v e . T h i s person can be described as working on t h e s a t i s f i c i n g p r i n c i p l e , i.e., as n o t behaving f u l l y r a t i o n a l from a SEU model perspective. But t o many people, t h a t i s c o u n t e r - i n t u i t i v e . They would probably c a l l a person n o n - r a t i o n a l i f he wouZd check a l l books i n t h e s t o r e . With f i n i t e t i m e and resources a v a i l a b l e , i t i s n o t r a t i o n a l t o spend i n f i n i t e e f f o r t on t h e e x p l o r a t i o n o f a l l potent i a l consequences o f a l l o p t i o n s . Rather, the d e c i s i o n c o s t s a r e weighted a g a i n s t t h e p o t e n t i a l b e n e f i t s r e s u l t i n g from t h e a p p l i c a t i o n o f a d e c i s i o n s t r a t e g y , and t h i s may l e a d t o v i o l a t i o n s o f SEU model r a t i o n a l i t y which a r e , however, p e r f e c t l y r a t i o n a l .

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COGNITIVE DECISION RESEARCH

T h i s p o i n t has been made by v a r i o u s authors (e.g.,

M i l l e r and S t a r r ,

1967; Einhorn and Hogarth, 1981; J a n i s and Mann, 1977; Hogarth, 1980; Montgomery and Svenson, 1976; cp. Payne, 1982), I t i s most e x p l i c i t l y and c l e a r l y represented i n t h e contingency model o f Beach and M i t c h e l l (1978). I n t h i s model i t i s assumed t h a t people make meta-decisions between t h e v a r i o u s d e c i s i o n s t r a t e g i e s they have i n t h e i r r e p e r t o i r e , S t r a t e g i e s d i f f e r w i t h r e s p e c t t o t h e c o g i t i v e e f f o r t r e q u i r e d and t h e p r o b a b i l i t y w i t h which they l e a d t o an optimal s o l u t i o n maximizing, s a t i s f i c i n g , e l imination-by-aspects

, coin

(e.g.,

SEU

flipping)

S t r a t e g y s e l e c t i o n i s seen as c o n t i n g e n t upon a ( c o s t / b e n e f i t ) compromise between t h e d e c i s i o n maker's d e s i r e t o make t h e b e s t d e c i s i o n and h i s o r h e r n e g a t i v e f e e l i n g s about i n v e s t i n g t i m e and e f f o r t i n t h e d e c i s i o n making process. The s t r a t e g y t h a t i s perceived as y i e l d i n g t h e maximum n e t g a i n i s t h e one selected, The s p e c i f i c c h o i c e depends on c h a r a c t e r i s t i c s o f t h e s i t u a t i o n (e.g., blem, time pressure) and o f t h e s u b j e c t (e.g.,

f a m i l a r i t y o f the proknowledge, a b i l i t y ) .

Various hypotheses d e r i v e d from t h i s model have been e x p e r i m e n t a l l y i n v e s t i g a t e d and have g e n e r a l l y been confirmed (e.g.,

Christensen-

Szalanski, 1978; M c A l l i s t e r , M i t c h e l l and Beach, 1979; ChristensenSzalanski

, 1980)

The r a t i o n a l i t y cost, i.e.,

t h e c o g n i t i v e e f f o r t a s s o c i a t e d w i t h each

s t r a t e g y , i s n o t f o r m a l i z e d i n Beach and M i t c h e l l ' s (1978) model. For t h e comparison o f multidimensional a l t e r n a t i v e s , Johnson (1979) has proposed a process model which a l l o w s t h e computation o f t h e number o f mental operations r e q u i r e d . A s i m i l a r method has been proposed by Shugan (1980). I n b o t h models i t i s assumed, however, t h a t t h e r e searcher knows t h e number o f a t t r i b u t e s . For more simple s t r a t e g i e s , a formal approach m i g h t be more d i f f i c u l t . I n t h e model o f Beach and M i t c h e l l (1978), maximizing o f s u b j e c t i v e l y expected u t i l i t y i s o n l y one o f several a v a i l a b l e s t r a t e g i e s each o f which may be chosen r a t i o n a l l y as t h e b e s t s t r a t e g y under t h e g i v e n c o n d i t i o n s , even c o i n f l i p p i n g , V i o l a t i o n s o f ' c l a s s i c a l ' r a t i o n a l i t y a r e considered as e r r o r s t h a t t h e s u b j e c t a n t i c i p a t e s and, more i m p o r t a n t , t o l e r a t e s . People a r e g l o b a l maximizers w i t h l o c a l i n c o n s i s -

THE TWO CAMPS ON RATIONALITY

t e n c i e s ( E l s t e r , 1979). R e n u n c i a t i o n o f c o g n i t i v e e f f o r t m i g h t n o t o n l y r e s u l t , however, f r o m t h e c a l c u l a t i o n t h a t t h e c o s t s outweigh t h e expected b e n e f i t s b u t a l s o from t h e knowledge t h a t "the very act of deZiberuting can m o d i f y t h e c h a r a c t e r f o r t h e worse, and i n ways

judged even more i m p o r t a n t , t h r o u g h t h e s t u l t i f y i n g e f f e c t s on spont a n e i t y " ( E l s t e r , 1979, p.40). Thus, t o save s p o n t a n e i t y , which i n i t s e l f has a v a l u e , a person m i g h t r a t i o n a l l y s e l e c t a s t r a t e g y p o s s i b l y l e a d i n g t o v i o l a t i o n s o f SEU r a t i o n a l i t y . The continuity argument: The c o r e o f t h i s argument i s t h e conceptua-

l i z a t i o n o f judgment and d e c i s i o n as moments i n a c o n t i n u o u s process. They sometimes may appear b i a s e d o r d e f i c i e n t i f t r e a t e d o r t e s t e d as d i s c r e t e events, w h i l e t h e y m i g h t i n f a c t be v e r y f u n c t i o n a l when cons i d e r e d as moments i n a c o n t i n u o u s and changing environment (Hogarth, 1981). An experiment by Ronen (1973), d e s c r i b e d i n H o g a r t h (1981), can s e r v e as an i l l u s t r a t i o n : S u b j e c t s had t o choose between two o p t i o n s w i t h i d e n t i c a l ( p o s i t i v e ) p a y o f f s and p r o b a b i l i t i e s o f success. The outcomes were dependent on t h e r e s u l t s o f two s e q u e n t i a l events. I n one o p t i o n , t h e r e was a h i g h e r p r o b a b i l i t y o f success i n t h e f i r s t s t e p and a l o w e r p r o b a b i l i t y i n t h e second s t e p , w h i l e i t was t h e o t h e r way around w i t h t h e a l t e r n a t i v e o p t i o n , S i n c e t h e SEUs o f t h e two o p t i o n s were equal , r a t i o n a l i t y would i m p l y i n d i f f e r e n c e . However, subjects p r e f e r r e d t h e o p t i o n w i t h the higher p r o b a b i l i t y i n the f i r s t step. Hogarth (1981) p r o v i d e s t h e f o l l o w i n g e x p l a n a t i o n : I f p e o p l e a r e used t o a changing environment,

they migth g i v e l e s s

w e i g h t t o t h e p r o b a b i l i t y o f t h e second s t e p s i n c e t h e s i t u a t i o n m i g h t have changed when t h i s phase has been reached; i t m i g h t be a t t r a c t i v e t o s t a y i n t h e game as l o n g as p o s s i b l e , and t h e r e f o r e t o choose t h e o p t i o n w i t h t h e h i g h e r p r o b a b i l i t y o f success i n t h e f i r s t phase. More g e n e r a l l y , many experiments on p r o b a b i l i s t i c i n f o r m a t i o n p r o c e s s i n g have e x c l u d e d t h e p o s s i b i l i t y o f feedback and redundancy which, however, i s c h a r a c t e r i s t i c o f t h e environment and t h u s r e l e v a n t f o r judgment and d e c i s i o n , Models have been t e s t e d under t h e assumption of stabile environment, w h i l e s u b j e c t s m i g h t have o p e r a t e d under

73

74

COGNITIVE DECISION RESEARCH

t h e assumption o f a changing environment. A number o f b i a s e s , demons t r a t e d w i t h t h e d i s c r e t e approach, can be i n t e r p r e t e d as i n d i c a t o r s o f c o g n i t i v e mechanisms and s t r a t e g i e s which a r e a c t u a l l y v e r y funct i o n a l i n a c o n t i n u o u s environment. Hogarth (1981) d i s c u s s e s s e v e r a l assumptions u n d e r l y i n g p r e s c r i p t i v e models which e x p l i c i t l y do n o t t a k e t h i s c o n t i n u i t y i n t o account, e,g.,

t h e existence of a

stabiZe time horizon: An example g i v e n by Tversky and Kahneman (1973)

f o r t h e use o f a v a i l a b i l i t y r e s u l t i n g i n a b i a s e d judgment i s r e i n t e r p r e t e d by Hogarth (p.206) w i t h t h e argument t h a t i n t h e s p e c i f i c e x p e r i m e n t a l s i t u a t i o n under t i m e pressure, t h e a p p l i c a t i o n o f t h i s h e u r i s t i c c o u l d be c o n s i d e r e d as v e r y f u n c t i o n a l : People use t h e cues which a r e most a v a i l a b l e f i r s t , because i n normal l i f e one can e x p e c t t o have a chance t o c o r r e c t such a judgment. I t i s t h e r e f o r e i m p o r t a n t t o s p e c i f y under w h i c h c o n d i t i o n s t h e a p p l i c a t i o n o f a h e u r i s t i c i s f u n c t i o n a l and under which i t i s n o t . Another assumption d i s cussed i s t h e stabiZity of preferences and goaZs w h i c h maybe wrong f o r v a r i o u s reasons (March, 1978) : F i r s t , p r e f e r e n c e s develop o v e r time. Choices i m p l y , however, a n t i c i p a t i o n s o f f u t u r e preferences, i.e.,

p r e f e r e n c e s a t t h e t i m e when t h e consequences i n c u r ( a p o i n t

I discussed e a r l i e r i n S e c t i o n 2 ) . Secondly, p r e f e r e n c e s a r e formed by experiences, and t h e s e e x p e r i e n c e s a r e o f t e n sought a c t i v e l y b y people, sometimes i n o r d e r t o know more about t h e i r p r e f e r e n c e s . T h i r d l y , p r e f e r e n c e s a r e u s u a l l y c h a r a c t e r i z e d by a h i g h degree o f a m b i g u i t y . Such a m b i g u i t y can be f u n c t i o n a l , because t h e person i s open f o r f u r t h e r i n f o r m a t i o n and saves mental energy t o c l a r i f y t h e a m b i g u i t y . Other assumptions d i s c u s s e d a r e t h e stationarity of probabiZistic processes, t h e independence of judgmentai! effects on consequences, and t h e abstraction from the competition character of action. The c o n t i n u i t y argument i s r a i s e d by Hogarth (1981) n o t

o n l y a g a i n s t t h e c o n t e n t i o n o f judgmental b i a s e s b u t a l s o a g a i n s t t h e contention o f representational e r r o r s . The a t t a c k on t h e p e s s i m i s t s ' camp can t a k e two forms. Demonstrations o f d e f i c i e n t judgment and d e c i s i o n can e i t h e r be a t t r i b u t e d t o i n adequate assumptions o f t h e model, l i k e f i x e d t i m e h o r i z o n s ; t h e observed b e h a v i o r was i n f a c t e f f i c i e n t and f u n c t i o n a l , t h e model

75

THE TWO CAMPS ON RATIONALITY

was wrong. O r , one can c l a i m t h a t t h e experimental s i t u a t i o n s used f o r t e s t i n g t h e model had a very low e c o l o g i c a l v a l i d i t y and t h a t t h e r e f o r e biases and e r r o r s were p o s s i b l y a r t i f a c t s

, encouraged

by t h e researcher.

The structure argwnent: The t h i r d argument a g a i n s t t h e o t h e r camp has

been r a i s e d i n p a r t i c u l a r by Berkeley and Humphreys (1982) and P h i l l i p s (1983). The c l a i m i s t h a t demonstrations o f c o g n i t i v e d e f i c i e n c y a r e questionable due t o t h e n e g l e c t o f t h e s u b j e c t s ' i n t e r n a l s t r u c t u r a l r e p r e s e n t a t i o n o f t h e problem. The c h a r a c t e r i z a t i o n o f t h e observed behavior as biased r e s t s on t h e assumption t h a t t h e s u b j e c t s share t h e experimenter's understanding o f t h e problem s t r u c t u r e , A commom understanding o f t h e problem s t r u c t u r e can be assumed t o

be

e s t a b l i s h e d by

t h e use o f process t r a c i n g methods o r through t h e experimental i n s t r u c t i o n s ; o r , i t can be assumed t o e x i s t a p r i o r i because o f t h e o b j e c t i v e n a t u r e o f t h e task, as i l l u s t r a t e d by Tversky and Kahneman (1981) i n t h e i r use o f t h e analogy o f perception,

As an example, Berkeley and Humphreys (1982) discuss t h e problem p r e sented t o s u b j e c t s by Tversky and Kahneman (1981) i n which two o p t i o n s a r e formulated e i t h e r i n terms o f l i v e s l o s t o r i n terms o f l i v e s saved. I n t h e f i r s t case, s u b j e c t s behaved r i s k averse, i n t h e second case they behaved r i s k prone, i n t e r p r e t e d by Tversky and Kahneman as i n c o n s i s t e n t preferences s i n c e " i t i s easy t o see t h a t t h e two problems a r e e f f e c t i v e l y i d e n t i c a l " (p.453) Berkeley and Humphreys question t h i s i n t e r p r e t a t i o n because t h e " u n c e r t a i n t y , concerning human agency i n a f f e c t i n g subsequent s t a t e s o f t h e world, i s l e f t unresolved"(p.222) i n Tversky and Kahneman's f o r m u l a t i o n o f t h e problem. Berkeley and Humphreys o f f e r a d i f f e r e n t s t r u c t u r a l r e p r e s e n t a t i o n t h e s u b j e c t s m i g h t have developed i n o r d e r t o r e s o l v e t h i s u n c e r t a i n t y which would r e s u l t i n preferences p e r f e c t l y c o n s i s t e n t w i t h SEU t h e o r y . A s i m i l a r reasoning i s found i n Hogarth (1981) and P h i l l i p s (1983). The second approach, i n which t h e problem s t r u c t u r e i s imposed under t h e assumption o f a common understanding, does n o t i n v e s t i g a t e experimental data i n t h e i r own r i g h t as products o f t h e s u b j e c t s ' reasoning, b u t compares them w i t h d a t a r e s u l t i n g from t h e a p p l i c a t i o n

76

COGNITIVE DECISION RESEARCH

o f a n o r m a t i v e model t o t h e problem i n q u e s t i o n , " t h e assumption b e i n g t h a t t h e model r e p r e s e n t s t h e b a s i s f o r r a t i o n a l c h o i c e ( o r i n f e r e n c e ) i n t h e p r e s e n t e d problem" ( B e r k e l e y and Humphreys, 1982, p.230). Two i m p l i c a t i o n s o f t h i s assumptions a r e q u e s t i o n e d by t h e a u t h o r s : F i r s t , t h e n a t u r a l i z a t i o n o f s m a l l w o r l d s , w h i c h means t h e exclusion o f t h e subjects' i n d i v i d u a l "goal-closing" o f h i s o r her l a r g e w o r l d i n t o t h e s m a l l w o r l d o f t h e e x p e r i m e n t a l moment and present i n g t h e -observed d e p a r t u r e s f r o m v e r i d i c a l i t y as cognitive r a t h e r t h a n m o t i v a t i o n a l b i a s e s (p.233).

Second, t h e u t i l i z a t i o n o f n o r m a t i v e

models as i d e a l t y p e s , i.e., as s t a n d a r d s o f comparison w i t h t h e i n t u i t i v e model o f t h e s u b j e c t ; t h i s r e q u i r e s , however, t h e i n v e s t i g a t i o n o f whether t h e r e i s a common u n d e r s t a n d i n g o f t h e way i n which t h e problem u n i v e r s e i s t o be g o a l - c l o s e d and t h u s does n o t p e r m i t i m p o s i t i o n o f a problem s t r u c t u r e (p.234). T h i s argument, then, focuses on t h e s t r u c t u r a l r e p r e s e n t a t i o n o f t h e t a s k and i t s s i g n i f i c a n c e f o r t h e i n t e r p r e t a t i o n o f judgments and d e c i s i o n s . The r e p r e s e n t a t i o n depends s t r o n g l y , however, on t h e c o n t e x t w i t h i n which i t i s developed. D i f f e r e n t c o n t e x t s evoke d i f f e r e n t knowledge

-

elements and s t r u c t u r e s o f knowledge

- and t h u s w i l l

often lead

t o d i f f e r e n t b e h a v i o r s . Each b e h a v i o r m i g h t be c o n s i s t e n t w i t h i n t h e g i v e n c o n t e x t a1 though i n c o n s i s t e n t w i t h some b e h a v i o r towards t h e same t a s k i n a d i f f e r e n t c o n t e x t . The d e s c r i p t i o n o f b e h a v i o r as d e f i c i e n t o r non-rational i s n o t j u s t i f i e d i f i t i s n o t i n v a r i a n t over m u l t i p l e c o n t e x t e s ; t o t h e c o n t r a r y , i n v a r i a n t b e h a v i o r would be d e f i cient. Two views a r e s t i l l p o s s i b l e h e r e : One can go as f a r as P h i l l i p s (1983) and argue phenomenologically t h a t t h e r e i s no c r i t e r i o n f o r d e f i n i n g some " o b j e c t i v e " problem r e p r e s e n t a t i o n as i m p l i e d i n T v e r s k y and Kahneman's (1981) analogy between judgmental e r r o r s and p e r c e p t u a l i l l u s i o n s . O r one can assume such a c r i t e r i o n b u t c o n c l u d e t h a t " t a s k r e p r e s e n t a t i o n may be o f more importance i n d e f i n i n g e r r o r s t h a n t h e r u l e s t h e y ( t h e p e o p l e ) use w i t h i n t h a t r e p r e s e n t a t i o n " ( E i n h o r n and Hogarth, 1981, p.60),

e.g.,

i f one c o n s i d e r s t h e case o f a p a r a n o i d

person. I n b o t h views, however, t h e problem o f s t r u c t u r e i s e s s e n t i a l

THE TWO CAMPS ON RATIONALITY

f o r a discussion o f d e f i c i e n c y / e f f i c i e n c y . I n summary, t h e members o f t h i s camp c h a l l e n g e t h e v i e w t h a t human j u d g ment and d e c i s i o n i s c o g n i t i v e l y d e f i c i e n t , They q u e s t i o n t h i s c o n c l u s i o n w i t h d i f f e r e n t arguments: Because an i m p o r t a n t parameter has been neglected, namely, t h e c o s t o f a d e c i s i o n s t r a t e g y ; because judgments and d e c i s i o n s a r e t r e a t e d as d i s c r e t e e v e n t s and n o t as moments i n a c o n t i n u o u s process; and because no a t t e n t i o n has been p a i d t o t h e i n t e r n a l s t r u c t u r a l r e p r e s e n t a t i o n o f t h e problem as t h e person- and context-dependent b a s i s o f judgment and d e c i s i o n . The emphasis i n a l l t h r e e p o s i t i o n s i s l e s s on d e m o n s t r a t i n g e f f i c i e n c y o f human b e h a v i o r t h a n on s t u d y i n g more c a r e f u l l y under what c o n d i t i o n s p e o p l e show w h i c h k i n d o f b e h a v i o r , and n o t " t o o e a s i l y t o a d o p t a c r u d e v i e w o f human r a t i o n a l i t y " ( S l o v i c , 1972).

ReswnOe and Outlook I t seems t h a t t h e h i s t o r y o f r e s e a r c h on human judgment and d e c i s i o n

making i n t h e l a s t t h i r t y y e a r s r e p r e s e n t s a t y p i c a l example of s c i e n t i f i c progress. . A g a i n s t t h e once dominant model o f r a t i o n a l man, t h e camp I have c a l l e d p e s s i m i s t i c formed and q u e s t i o n e d t h i s c o n c e p t i o n by d e m o n s t r a t i n g d e f i c i e n c i e s i n judgment and d e c i s i o n . Now a f t e r t h i s camp has r u l e d d e c i s i o n r e s e a r c h f o r many years, i t comes under a t t a c k by a new group t h a t I have c a l l e d o p t i m i s t i c w h i c h p o i n t s t o weaknesses i n t h e former concepts and suggests even more d i f f e r e n t i a t e d perspect i v e s of judgment and d e c i s i o n . B o t h camps have g a i n e d t h e i r impetus from t h e c r i t i q u e o f t h e r e s p e c t i v e p r e v a i l i n g c o n c e p t i o n . Some o f t h e c r i t i c i s m s , p a r t i c u l a r l y of t h e o p t i m i s t s ' s camp a g a i n s t t h e p e s s i m i s t s ' camp, have a l r e a d y been mentioned i n t h e p r e v i o u s s e c t i o n . However,

I w i l l b r i e f l y sum up these p o i n t s and add those more g e n e r a l arguments which have been exchanged between t h e two camps more r e c e n t l y . The c r i t i q u e o f t h e p e s s i m i s t s ' p o s i t i o n focuses on t h e f o l l o w i n g p o i n t s : F i r s t , t h e experiments a r e o f l o w e c o l o g i c a l v a l i d i t y i n t h e sense t h a t t h e y do n o t r e p r e s e n t t h e m a j o r i t y o f s i t u a t i o n s i n w h i c h

77

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COGNITIVE DECISION RESEARCH

people have t o g i v e judgments and make d e c i s i o n s . Second, t h e e x p e r i m e n t s were designed such t h a t b i a s e s and e r r o r s a r e n o t s u r p r i s i n g s i n c e t h e s i t u a t i o n s and problems were t a k e n o u t o f any r e a l l i f e c o n t e x t . T h i r d , i m p o r t a n t parameters l i k e c o g n i t i v e e f f o r t have n o t been accounted f o r i n t h e models used as y a r d s t i c k s . F o u r t h , whether s u b j e c t s cogni t i v e l y r e p r e s e n t e d t h e problem i n t h e way t h e r e s e a r c h e r assumed i s r a r e l y i n v e s t i g a t e d . A more g e n e r a l , f i f t h argument i s t h a t t h i s r e s e a r c h has focused t o o much on e r r o r s r a t h e r t h a n on t h e c o g n i t i v e processes p e r se; n o t o n l y i s i t h a r d t o d e f i n e what e r r o r s are, t h e y a l s o c o n s t i t u t e a t y p i c a l b e h a v i o r samples. The r e j o i n d e r o f t h e p e s s i m i s t s ' camp i s : F i r s t , m e t a - t h e o r i e s on t h e s e l e c t i o n o f s t r a t e g i e s according t o c o g n i t i v e e f f o r t considerations a r e m e t h o d o l o g i c a l l y more o r l e s s immune t o charges o f " i r r a t i o n a l i t y " ; any b e h a v i o r t h a t v i o l a t e s t h e SEU model can be " r a t i o n a l i z e d " by r e c u r r i n g t o some h i g h e r l e v e l o f r a t i o n a l i t y . Second, d e m o n s t r a t i o n s o f d i f f e r e n c e s between t h e e x p e r i m e n t a l and r e a l w o r l d s i t u a t i o n s do n o t prove t h a t these d i f f e r e n c e s m a t t e r . T h i r d , s p e c u l a t i o n s t h a t subj e c t s m i g h t have r e p r e s e n t e d a problem c o g n i t i v e l y d i f f e r e n t l y t h a n t h e r e s e a r c h e r s do n o t p r o v e t h a t t h e y have done so. F o u r t h , t h e f o c u s on errors i s

j u s t i f i e d because ( a ) " t h e y expose some o f o u r i n t e l l e c t u a l

l i m i t a t i o n s and suggest ways o f i m p r o v i n g t h e q u a l i t y o f o u r t h i n k i n g " , ( b ) t h e y " o f t e n r e v e a l t h e p s y c h o l o g i c a l processes and t h e h e u r i s t i c procedures t h a t govern judgment and i n f e r e n c e " , and ( c ) t h e y " h e l p t h e mapping o f human i n t u i t i o n s by i n d i c a t i n g

which p r i n c i p l e s o f s t a t i s t i c s

o r l o g i c a r e n o n - i n t u i t i v e o r c o u n t e r - i n t u i t i v e " (Kahneman and Tversky, 1982, p.124). C l e a r l y , t h e s e a r e d e f e n s i v e arguments b u t t h e r e f o r e n o t l e s s r e a s o n a b l e t h a n t h e o f f e n s i v e arguments o f t h e a u t h o r s . It i s n o t t o be expected t h a t t h i s camp w i l l c a p i t u l a t e as easy under t h e a t t a c k as d i d t h e " r a t i o n a l man" when t h a t concept was besieged by t o d a y ' s defenders. As H e r a c l i t s a i d , war i s t h e f a t h e r o f a l l and t h e k i n g o f a l l . The t h e o r e t i c a l and e m p i r i c a l work i n b o t h camps as w e l l as t h e d i s p u t e between them has generated many data, i n s i g h t s and i d e a s . The unders t a n d i n g o f judgment and d e c i s i o n making has been dependend and

79

THE TWO CAMPS ON RATIONALITY

d i f f e r e n t i a t e d and has opened new, more comprehensive r e s e a r c h perspect i v e s . I w i l l c h a r a c t e r i z e b r i e f l y some e s s e n t i a l aspect s o f t h e p r e s e n t s t a t e o f t h e debate and o f t h e d i r e c t i o n f u t u r e research m i g h t t ake. F o r a lmos t a l l arguments, t h e t h e o r e t i c a l p o i n t o f r e f e r e n c e i s s t i l l t h e SEU model o r some v a r i a n t o f i t (e.g., p r o s p e c t t h e o r y ) , d e s p i t e some harsh c r i t i q u e (e.g., F i s c h h o f f , G o i t e i n and Shapira, 1982). W hile one s i d e p r o v i d e s evidence t h a t p e o p l e do n o t behave as t h e model p r e d i c t s , t h e o t h e r s i d e demonstrates t h a t w i t h d i f f e r e n t t h e o r e t i c a l assumptions o r ex pe r i m e n t a l _ s e t t i n g s t h e same behavior m i g h t n o t v i o l a t e t h e model a t a l l . P i t z s a i d ( i n 1977) t h a t "models based upon some v a r i a n t o f expected v a l u e t h e o r y have t h e a t t r a c t i o n o f being f a m i l i a r and easy t o work w i t h , and i n e r t i a i s one o f t h e s t r o n g e s t f o r c e s i n n a t u r e " (p.421); a n o t h e r reason i s p r o b a b l y t h a t t h e SEU model i s h ard t o f a l s i f y s i n c e " w i t h s u f f i c i e n t i n g e n u i t y , one can always f i n d something t h a t a p a r t i c u l a r d e c i s i o n maker has maximized i n a p a r t i c u l a r s i t u a t i o n " (Fischhoff e t al.,

1982, p.317).

Pitz

a l s o s a i d t h a t " u n t i l a w e l l developed, s y s t e m a t ic a l t e r n a t i v e t o t h e normat iv e model i s a v a i l a b l e , i t i s l i k e l y t h a t new approaches w i l l have no more l a s t i n g success t h a n d i d G e s t a l t psychology" (1977, p.421). T h i s s t i l l seems t o be a f a i r l y v a l i d d e s c r i p t i o n o f t h e p r e s e n t s i t u a t i o n . However, t h a t no r e a l a l t e r n a t i v e has been o f f e r e d m i g h t a l s o be taken as an i n d i c a t o r o f t h e s t r e n g h t o f t h e model as a c o r e o f t h e o r i e s o f judgment and d e c i s i o n . The wide use o f expectancyv a l u e models a l s o i n o t h e r a r e a s o f psychology, p a r t i c u l a r l y i n m o t i v a t i o n research, i s a f u r t h e r i n d i c a t o r ( c p . Fe a t her, 1982). The i s s u e i s n o t anymore what t h e l i m i t a t i o n s

o f t h e human c o g n i t i v e system are . A l l approaches assume t h a t t h e r e a r e boundaries f o r r a t i o n a l i t y i n s i t u a t i o n s o f c o g n i t i v e o v e r l o a d due t o t h e c a p a c i t y and p r o c e s s i n g l i m i t s o f t h e system. The i s s u e i s how people f o r m judgments

and make d e c i s i o n s , and p a r t i c u l a r l y whether t h e y o p e r a t e r a t i o n a l l y within t h e c o n s t r a i n t s . While one s i d e focused on e r r o r s , biases and f a l l a c i e s , t h e other s i d e considers t h i s a t h e o r e t i c a l l y questionable and a l s o t o o n e g a t i v e approach. I n t h i s view, r e search on judgment and d e c i s i o n making has been d r i v e n t o o much by a concern f o r e r r o r s r e l a -

80

COGNITIVE DECISION RESEARCH

t i v e t o a n o r m a t i v e s t a n d a r d t h e v a l i d i t y o f w h i c h one can d o u b t w i t h good arguments ( E i n h o r n and Hogarth, 1981). The focus on e r r o r s has r e c e n t l y been defended by Kahneman and T v e r s k y (1982) by p o i n t i n g towards v a r i o u s advantages o f t h i s approach ( s e e above). Kahneman and Tversky a l s o argue t h a t " t h e emphasis on t h e s t u d y o f e r r o r s i s c h a r a c t e r i s t i c o f r e s e a r c h on human judgment, b u t i t i s n o t unique t o t h i s domain: we use i l l u s i o n s t o u n d e r s t a n d t h e p r i n c i p l e s o f normal p e r c e p t i o n and we l e a r n a b o u t memory by s t u d y i n g f o r g e t t i n g " (1982a, p.123). I d o n ' t q u i t e agree, f o r two reasons: F i r s t , because, a l t h o u g h i l l u s i o n s a r e a s u b j e c t o f i n t e r e s t i n t h e psychol o g y o f p e r c e p t i o n , t h e emphasis has c e r t a i n l y n o t been on e r r o r s . Second, memory r e s e a r c h i s n o t i n t e r e s t e d i n e r r o r s as such b u t uses them as a dependent v a r i a b l e f o r t e s t i n g models a b o u t s t o r i n g o r r e t r i e v a l o f i n f o r m a t i o n , i.e.

, about

t h e "normal" memory processes

and s t r u c t u r e s ; r e s e a r c h e r s on memory do n o t demonstrate e r r o r s i n t h e i r research, as do many r e s e a r c h e r s on judgment and d e c i s i o n . Howe v e r , t h e r e c e n t d i s c u s s i o n o f t h e i s s u e seems t o have l e d t o some rapprochment o f p o s i t i o n s . Kahneman and Tversky concede t h a t " a l t h o u g h e r r o r s o f judgment a r e b u t a method by which some c o g n i t i v e processes a r e s t u d i e d , t h e method has become a s i g n i f i c a n t p a r t o f t h e message" (1982a, p.124), and t h e y a c c e n t u a t e s t r o n g e r t h a n b e f o r e , e.g.

, that

t h e n o t i o n o f judgmental h e u r i s t i c s s h o u l d " p r o v i d e a common a c c o u n t f o r b o t h c o r r e c t and i n c o r r e c t judgments" (1982b, p.325).

If it i s

agreed, then, t h a t " t h e o r e t i c a l l y i t i s necessary t o understand a22 reasoning data, regardless o f t h e i r conformity t o a normative r u l e system" (Evans, 1982, p.319), t h e s t u d y o f e r r o r s becomes a q u e s t i o n o f r e s e a r c h s t r a t e g y . Kahneman and Tversky, i n t h e i r r e p l y t o Evans, argue t h a t " e r r o r s may sometimes be more i n f o r m a t i v e t h a n c o r r e c t judgments, which can be produced e i t h e r by an i n i t i a l v a l i d i n t u i t i o n

or by a subsequent c o r r e c t i o n " (1982b, p.325). Another i m p o r t a n t p r o g r e s s i s t o be seen i n t h e now g e n e r a l l y accepted c o n c e p t u a l i z a t i o n o f judgment and d e c i s i o n as p a r t s o f a m u l t i - s t a g e c o g n i t i v e process. A c t u a l l y , a c t i v i t i e s t h a t precede t h e "moment o f d e c i s i o n " seem p r e s e n t l y t o be t h e p r e v a i l i n g o b j e c t o f t h e o r e t i c a l

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THE TWO CAMPS ON RATIONALITY

and e m p i r i c a l work - i n p a r t i c u l a r t h e a c q u i s i t i o n and p r o c e s s i n g o f i n f o r m a t i o n , or, i n o t h e r terms, t h e g e n e r a t i n g and s t r u c t u r i n g o f knowledge. Some work on p r e d e c i s i o n a l c o g n i t i v e processes i s a l r e a d y a v a i l a b l e ( c p . E i n h o r n and Hogarth, 1981), b u t new q u e s t i o n s have come up. F o r i n s t a n c e , what a r e t h e i m p l i c a t i o n s o f u s i n g t h e " c o n v e r s a t i o n a l paradigm" ( B e r k e l e y and Humphreys , 1982; Kahneman and Tversky, 1982a)? How can we e l i c i t o r i n f e r t h e c o g n i t i v e r e p r e s e n t a t i o n o f problems? We need t h e o r i e s o f problem r e p r e s e n t a t i o n t h a t e x p l a i n t h e use and t h e r e s u l t s o f p a r t i c u l a r judgmental and d e c i s i o n making strategies.

-

There i s a l s o i n c r e a s i n g i n t e r e s t i n p o s t - d e c i s i o n a l

processes, p a r t i c u l a r l y t h e i m p l e m e n t a t i o n o f d e c i s i o n s ( c p . Gasparski, 1980). Again, r e s e a r c h has focused p r i m a r i l y on " e r r o r s " , i .e.

, on

t h e phenomenon t h a t p e o p l e o f t e n do n o t implement t h e i r m e n t a l d e c i s i o n s due t o m o t i v a t i o n a l and emotional f a c t o r s ( e . g . , Sjoberg, 1980; E l s t e r , 1979). E l s t e r has analyzed t h e v a r i o u s s t r a t e g i e s t h a t p e o p l e use t o overcome t h i s "weakness o f w i l l " ,

i n t e r f e r i n g between d e c i s i o n and

a c t i o n ( u s i n g U l y s s e s as a p r o m i n e n t example who bound h i m s e l f t o t h e mast i n o r d e r n o t t o g i v e i n t o t h e t e m p t a t i o n by t h e s i r e n s . He p u t t h i s problem i n t h e c o n t e x t o f t h e r a t i o n a l i t y d e b a t e : "Man o f t e n i s n o t r a t i o n a l , and r a t h e r e x h i b i t s weakness of w i Z Z . Even when n o t r a t i o n a l , man knows t h a t he i s i r r a t i o n a l and can bind hirnseZf a g a i n s t t h e i r r a t i o n a l i t y . T h i s second-best o r i m p e r f e c t r a t i o n a l i t y t a k e s c a r e b o t h o f reason and passion. What i s l o s t , perhaps, i s t h e sense o f adventure" ( E l s t e r , 1979, p.111).

A f i n a l l e s s o n t o be l e a r n e d f r o m t h e debate m i g h t be t h a t one s c h o u l d a v o i d t h e t e r m r a t i o n a l i t y i n psychology a t a l l . O b v i o u s l y , t h e conc e p t i s used w i t h d i f f e r e n t meanings; i t i s c e r t a i n l y n o t l o n g e r def i n e d e x c l u s i v e l y by f o r m a l coherence and c o n s i s t e n c y . An a l t e r n a t i v e way o f h a n d l i n g t h i s d i f f i c u l t y i s , however, t o d i s t i n g u i s h e x p l i c i t l y v a r i o u s meanings o f t h e term. Besides f o r m a l r a t i o n a l i t y , c o n c e p t s l i k e s u b s t a n t i v e r a t i o n a l i t y and p r o c e d u r a l r a t i o n a l i t y m i g h t be u s e f u l f o r a t h e o r y o f judgment and d e c i s i o n (Simon, 1978). Substantive r a t i o n a l i t y c a p t u r e s an i m p o r t a n t a s p e c t o f t h e term as i t i s used i n common language, namely, how r e a l i s t i c , c o r r e c t , adaequat some judgment o r decision i s w i t h respect t o t h e r e a l world

-

the paranoid w i t h h i s

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COGNITIVE DECISION RESEARCH

p e r f e c t l y c o n s i s t e n t b e l i e f s and v a l u e s p r o v i d e s an example. R a t i o n a l i t y i s n o t o n l y a q u e s t i o n o f whether a c h o i c e i s i n l i n e w i t h a p e r s o n ' s b e l i e f and p r e f e r e n c e s , b u t a l s o a q u e s t i o n o f what s o r t o f p r e f e r e n c e s and be1 i e f s t h e person h o l d s , a p o i n t e l a b o r a t e d a l s o b y E i n h o r n and Hogarth (1981). D i f f e r e n t l y s t a t e d , a d e c i s i o n m i g h t be c o n s i s t e n t (and t h u s f o r m a l l y r a t i o n a l ) , b u t t h e judgments t h a t p r o v i d e t h e i n p u t f o r t h e d e c i s i o n m i g h t be v e r y poor (and t h u s t h e c h o i c e m i g h t be s u b s t a n t i v e l y n o t r a t i o n a l ) . ProceduraZ r a t i o n a l i t y can be l i n k e d t o s u b s t a n t i v e r a t i o n a l i t y e a s i l y i f one asks what i n f o r m a t i o n i s searched f o r and used by a person t o f o r m v a l u e s and b e l i e f s , whether t h e person t r i e d h a r d enough t o a n t i c i p a t e f u t u r e consequences o f c u r r e n t a c t i o n s and f u t u r e p r e f e r e n c e s f o r t h o s e consequences, e t c . What J a n i s and Mann (1977) d e s c r i b e as " v i g i l a n t i n f o r m a t i o n p r o c e s s i n g " m i g h t be c o n s i d e r e d one a s p e c t o f t h i s k i n d o f r a t i o n a l i t y . P a r t i c u l a r l y i n a s o c i a l context,when d e c i s i o n s have t o be j u s t i f i e d , p r o c e d u r a l r a t i o n a l i t y can be much more i m p o r t a n t t h a n s u b s t a n t i v e m a t t e r s about which agreement o f t e n cannot be reached. My c o n c l u s i o n f r o m t h e debate between t h e two camps w i t h r e s p e c t t o t h e

r a t i o n a l i t y problem i s t h r e e f o l d : First, we s h o u l d

be l i b e r a l

( w h i c h does n o t mean vague) i n o u r use o f t h e r a t i o n a l i t y concept. The d i f f e r e n t meanings t h a t I have b r i e f l y d e s c r i b e d ( a n d o t h e r s , cp. March, 1978) a r e a l l u s e f u l t o o l s i n a n a l y z i n g judgment and d e c i s i o n p r o cesses. Second, we may use t h e concept i n i t s p r e s c r i p t i v e sense l e g i t i m a t e l y i n s i t u a t i o n s where p r e s c r i p t i o n i s asked f o r , l i k e i n decision aiding o r i n decision training. C r i t e r i a f o r defining rat i o n a l i t y i n a p r e s c r i p t i v e sense can be t a k e n f r o m o u r knowledge a b o u t t h e r e a l w o r l d , f r o m our i n t e l l e c t u a l system and f r o m t h e s o c i a l consensus

-

however r e l a t i v e t h e s e c r i t e r i a a r e i n a h i s t o r i c a l o r

c u l t u r a l sense. Third, we s h o u l d use

t h e concept i n d e s c r i p t i v e

r e s e a r c h more c a u t i o u s l y because i t t o o e a s i l y e n t i c e s t o l a b e l a l l b e h a v i o r t h a t does n o t meet t h e c r i t e r i a as d e v i a n t o r d e f i c i e n t , as e r r o r s , b i a s e s , o r whatever. I s u b s c r i b e t o Kahneman and T v e r s k y ' s bal a n c e d a d v i c e t h a t we " s h o u l d a v o i d o v e r l y s t r i c t i n t e r p r e t a t i o n s , which t r e a t r e a s o n a b l e answers as e r r o r s , as w e l l as t o o v e r l y c h a r i t a b l e i n t e r p r e t a t i o n s , which a t t e m p t

t o r a t i o n a l i z e every

03

THE TWO CAMPSON RATIONALITY

response" (1982a, p.124). ~ i n a Z Z y , t h e r e c o g n i t i o n o f t h e c o n d i t i o n a l i t y o f normative mode1,s on assumptions about t h e environment as w e l l as t h e c o g n i t i v e r e p r e s e n t a t i o n o f t h a t environment, most c o n v i n c i n g l y elaborat e d by E i n h o r n and Hogarth (1981) and Payne (1982), i s i n my o p i n i o n t h e synthesis emerging from t h e debate between t h e ( p e s s i m i s t i c ) t h e s i s and t h e ( o p t i m i s t i c ) a n t i t h e s i s . I m p l i c i t l y o r e x p l i c i t l y , i t i s agreed t o by members o f both camps.

References

A i n s l i e , G . : 1975. Specious reward. I n : Psychological B u l l e t i n 82, 463-496 Beach, L.R. & M i t c h e l l , T.R.: 1978. A contingency model f o r t h e s e l e c t i o n o f d e c i s i o n s t r a t e g i e s . I n : Academy o f Management Review 3, 439-449 Berkeley, D. & Humphreys, P.: 1982. S t r u c t u r i n g d e c i s i o n problems and t h e ' b i a s h e u r i s t i c ' . I n : Acta Psychologica 50, 201-252 Christensen-Szalanski, 5.5.3.: 1978. Problem-solving s t r a t e g i e s : A s e l e c t i o n mechanism, some i m p l i c a t i o n s , and some data. I n : O r g a n i z a t i o n a l Behav i o r and Human Performance 22, 307-323 Christensen-Szalanski, J.J.J.: 1980. A f u r t h e r examination o f t h e s e l e c t i o n o f problem-solving s t r a t e g i e s : The e f f e c t s o f deadlines and a n a l y t i c a p t i t u d e s . I n : O r g a n i z a t i o n a l Behavior and Human Performance 25, 107122 Cohen, L.J.: 1979. On t h e psychology o f p r e d i c t i o n : Whose i s t h e f a l l a c y ? I n : C o g n i t i o n 7, 385-407 Dorner, D.: 1983. H e u r i s t i c and c o g n i t i o n i n complex systems. I n : R.Groner, M.Groner & W.F.Bischof (Eds.), Methods o f h e u r i s t i c s . H i l l s d a l e , N.J.: Lawrence E r l baum Assoc. Dorner, D., Kreuzig, H.W., R e i t h e r , F. & Staudel, T. (Hrsg.): sen. Bern/Stuttgart/Wien: Hans Huber

1983. Lohhau-

Edwards, W . : 1968. Conservatism i n human i n f o r m a t i o n processing. I n : B . K l e i n muntz (Ed.), Formal r e p r e s e n t a t i o n o f human judgment. New York: W i l e y & Sons Edwards, W.: 1983. Human c o g n i t i v e c a p a b i l i t i e s , representativeness, and ground r u l e s f o r research. I n : P.C.Humphreys, 0.Svenson & A.Vari (Eds.), Analysing and a i d i n g d e c i s i o n processes. Amsterdam: North-Hol l a n d Einhorn, H.J. & Hogarth, R.M.: 1981. Behavioral d e c i s i o n theory: Processes o f judgment and choice. I n : Annual Review o f Psychology 32, 53-88 E l s t e r , J.: 1979. Ulysses and t h e s i r e n s . Cambridge: Cambrige U n i v e r s i t y

COGNITIVE DECISION RESEARCH

04

Press, and P a r i s : E d i t i o n s de l a Maison des Sciences de 1'Homme Evans, J.St.6.T.: 1982. On s t a t i s t i c a l i n t u i t i o n s and i n f e r e n t i a l r u l e s : A d i s c u s s i o n o f Kahneman and Tversky. I n : C o g n i t i o n 12, 319-323 Feather, N.T. (Ed.): 1982. Expectations and a c t i o n s . H i l l s d a l e , N.J.: rence E r l baum Assoc.

Law-

F i s c h h o f f , B.: 1983. Reconstructive c r i t i c i s m . I n : P.C.Humphreys, 0.Svenson & A.Vari (Eds.), Analysing and a i d i n g d e c i s i o n processes. Amsterdam: North-Hol 1and F i s c h h o f f , B., G o i t e i n , B. & Shapira, Z.: 1982. The experienced u t i l i t y o f expected u t i l i t y approaches. I n : N.T.Feather (Ed.), Expectations and a c t i o n s . H i l l s d a l e , N.J.: Lawrence Erlbaum Assoc. Gasparski, W. (Ed.): 1980. D e c i s i o n making and a c t i o n . Report. Warsaw: Pol i s h Academy o f Sciences Hogarth, R.M.: 1981. Beyond d i s c r e t e biases: F u n c t i o n a l and d y s f u n c s t i o n a l aspects o f judgmental h e u r i s t i c s . In: Psychological B u l l e t i n 90, 197217 Janis, I . L . & Mann, L.: 1977. D e c i s i o n making. New York: The Free Press Johnson, E.J.: I n press. Deciding how t o decide: The e f f o r t o f making a dec i s i o n . I n : L.Sjoberg, J.A.Wise & T.Tyszka (Eds.), Human d e c i s i o n making. Lund: Doxa Jungermann, H., von U l a r d t , I.& Hausmann, L.: 1983. The r o l e o f t h e goal f o r generating a c t i o n s . I n : P.C.Humphreys, 0.Svenson & A.Vari (Eds.), Analysing and a i d i n g d e c i s i o n processes. Amsterdam: North-Hol l a n d Kahneman, D., S l o v i c , P. & Tversky, A. (Eds.): 1982. Judgment under uncert a i n t y : H e u r i s t i c s and biases. Cambridge: Cambridge U n i v e r s i t y Press Kahneman, D. & Tversky, A . : 1979. Prospect theory: An a n a l y s i s o f d e c i s i o n under r i s k . I n : Econometrica 47, 263-291 Kahneman, D. & Tversky, A . : 1982a. On t h e s t u d y ' o f s t a t i s t i c a l i n t u i t i o n s . I n : C o g n i t i o n 11, 123-141 Kahneman, D. & Tversky, A . : 326.

1982b. A r e p l y t o Evans. I n : C o g n i t i o n 12, 325-

Mann, L. & Janis, I . L . : 1982. C o n f l i c t t h e o r y o f d e c i s i o n making and t h e expectancy-value approach. I n : N.T.Feather (Ed.), Expectations and a c t i o n s . H i l l s d a l e , N.J.: Lawrence Erlbaum Assoc. March, J.G.: 1978. Bounded r a t i o n a l i t y , ambiguity, and t h e e n g i n e e r i n g o f choice. I n : The B e l l Journal o f Economics 9, 587-608 M c A l l i s t e r , D.W.,

M i t c h e l l , T.R. & Beach, L.R.:

1979. The contingency model

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f o r the s e l e c t i o n o f decision s t r a t e g i e s : An empirical t e s t o f t h e effects of s i g n i f i c a n c e , a c c o u n t a b i l i t y , and r e v e r s i b i l i t y . I n : Organ i z a t i o n a l Behavior and Human Performance 24, 228-244. M i l l e r , D.W. & S t a r r , M.K.: 1967. The s t r u c t u r e o f human decisions. Englewood C l i f f s , N.J.: Prentice-Hall Montgomery, H. & Svenson, 0.: 1976. On decision r u l e s and information processing s t r a t e g i e s f o r choices among m u l t i a t t r i b u t e a l t e r n a t i v e s . I n : Scandinavian Journal o f Psychology 17, 283-291 N i s b e t t , R. & ROSS, L.: 1980. Human inference: S t r a t e g i e s and shortcomings o f social judgment. Englewood C l i f f s , N.J.: Prentice-Hall Payne, J.W.: 1982. Contingent decision behavior. I n : Psychological B u l l e t i n 92, 382-402 Peterson, C.R. & Beach, L.R.: 1967. Man as an i n t u i t i v e s t a t i s t i c i a n . I n : Psychological B u l l e t i n 68, 29-46. P h i l l i p s , L.D.: 1983. A t h e o r e t i c a l perspective on h e u r i s t i c s and biases i n p r o b a b i l i s t i c t h i n k i n g . I n : P.C.Humphreys, 0.Svenson & A.Vari (Eds.), Analysing and a i d i n g decision processes, Amsterdam: North-Hol land P i t z , G.: 1977. Decision making and cognition. In: H.Jungermann & G.de Zeeuw (Eds.), Decision making and change i n human a f f a i r s . Dordrecht/Boston: D. Reidel P i t z , G.F., Sachs, N.J. & Heerboth, J.: 1980. Procedures f o r e l i c i t i n g choices i n the analysis o f i n d i v i d u a l decisions. I n : Organizational Behav i o r and Human Performance 26, 396-408 Ronen, J.: 1973. Effects o f some p r o b a b i l i t y displays on choices. I n : Organ i z a t i o n a l Behavior and Human Performance 9, 1-15 Shugan, S.M.: 1980. The cost o f t h i n k i n g . I n : Journal o f Consumer Research 7, 99-111 Simon, H.A.: 1955. A behavioral model o f r a t i o n a l choice. I n : Q u a r t e r l y Journal o f Economics 69, 99-118 Simon, H.A.: 1978. R a t i o n a l i t y as process and as product o f thought. I n : American Economical Review 68, 1-16 Sjoberg, L.: 1980. V o l i t i o n a l problems i n c a r r y i n g through a d i f f i c u l t dec i s i o n . I n : Acta Psychologica 45, 123-132

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Slovic, P.: 1972. From Shakespeare t o Simon: Speculations and some e v i dence about man's a b i l i t y t o process information. Research B u l l e t i n 12, no.2. Eugene, Ore.: Oregon Research I n s t i t u t e

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Tversky, A . & Kahneman, 0.: 1973. A v a i l a b i l i t y : A h e u r i s t i c f o r judging frequency and p r o b a b i l i t y . I n : Cognitive Psychology 5, 202-232

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Tversky, A. & Kahneman, 0.: 1974. Judgment under u n c e r t a i n t y : H e u r i s t i c s and bias es . I n : Science 185, 1124-1131 Tversky, A . & Kahneman, D.: 1981. The f r a m i n g o f d e c i s i o n s and t h e psychology o f c hoic e. In : Science 211, 453-458.

DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North.Holland), 1983 I

a7

THE ROLE OF HEURISTICS IN MODELS OF DECISION RUDOLF GRONER MARINA GRONER & WALTER F. BISCHOF Universitat Bern Psychologisches lnstitut Switzerland

There i s an i n c r e a s i n g tendency f o r t h e word " h e u r i s t i c " t o become a f a s h i o n a b l e l a b e l f o r a whole l o t o f q u i t e d i f f e r e n t e n t e r p r i s e s . I t s meanin? reaches f r o m " i m p e r f e c t s o l u t i o n method" up t o " c r e a t i v e achievement". The purpose o f t h i s a r t i c l e is f i r s t t o r e c a p i t u l a t e t h e r o o t s o f t h i s concept i n p h i l o s o phy and mathematics and i t s f u r t h e r development i n psychology and A r t i f i c i a l I n t e l l i g e n c e . F i n a l l y , some a p p l i c a t i o n s o f t h e t e r m i n d e c i s i o n r e s e a r c h a r e discussed. 1.

Early History of Heuristics

The e t h y m o l o g i c a l r o o t o f t h e word goes back t o a n c i e n t Greece where i t i s t h e a d j e c t i v e a s s o c i a t e d w i t h t h e v e r b heuriskein ( = t o f i n d ) . The f i r s t h e u r i s t i c methods have been used by E u c l i d and l a t e r d e s c r i b e d i n d e t a i l by Pappus o f A l e x a n d r i a ( f o r a more d e t a i l e d h i s t o r i c a l r e v i e w see Groner, Groner & B i s c h o f , 1983). They d i s t i n g u i s h e d between AnaZysis (assuming somet h i n g t o be t r u e and w o r k i n g

backwards t o something a l r e a d y a d m i t t e d t o be

t r u e ) and Stinthesis ( w o r k i n g forwards f r o m a l r e a d y known m a t e r i a l towards a new r e s u l t ) . The two methods were m a i n l y a p p l i e d t o c o n s t r u c t i o n s and p r o o f s i n t h e f i e l d o f geometry. I n subsequent epochs, t h e h i s t o r y o f h e u r i s t i c s became i n c r e a s i n g l y concerned w i t h t h e development o f a l g o r i t h m s , i . e . s t e p - b y - s t e p procedures which automat i c a l l y produce s o l u t i o n s t o any problem o u t o f a c l a s s o f problems. The t e r m " a l g o r i t h m " can be t r a c e d back t o t h e name o f a mathematician,

88

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Mohammed ben Musa a1 Khovarezmi, who l i v e d around t h e y e a r 830 a t Baghdad. He had w r i t t e n a book on r u l e s o f a r i t h m e t i c developed by t h e I n d i a n s . I n t h e M i d d l e Ages i t was t r a n s l a t e d i n t o L a t i n under t h e name " A l g o r i t m i de numero Indorum". L a t e r , " a l g o r i t m i " was c o n s i d e r e d a L a t i n possessive case and " l i b e r a l g o r i t m i " was used as a t i t l e f o r books on a r i t h m e t i c . F i n a l l y when t h e o r i g i n o f t h e word was f o r g o t t e n c o m p l e t e l y , i t was m i s t a k e n f o r a Greek word and t h e s p e l l i n g was changed i n t o " a l g o r i t h m i " . (Hankel, 1874). The s u c c e s s f u l a l g o r i t h m i c methods i n a r i t h m e t i c and a l g e b r a s t i m u l a t e d R a i mundus L u l l u s , a Spanish p h i l o s o p h e r o f t h e t h i r t e e n t h c e n t u r y , t o search f o r a general a l g o r i t h m t h a t would produce esery t r u t h . I n h i s famous work "Ars magna" he d e s c r i b e d a system where you s t a r t w i t h a few c l a s s e s o f b a s i c n o t i o n s and by combining them i n a s y s t e m a t i c way y o u s h o u l d i n v e n t e v e r y t h i n g . L u l l u s even i n v e n t e d a mechanical d e v i c e t o g e n e r a t e a l l combinations. He w r o t e t h e s i x c l a s s e s , each c o n t a i n i n g n i n e concepts, on s i x c o n c e n t r i c d i s c s . Now by t u r n i n g t h e d i s c s a g a i n s t each o t h e r he c o u l d have g o t e v e r y p o s s i b l e cornb i n a t i o n . L u l l u s d i d n o t know c o m b i n a t o r i c s and p r o b a b l y u n d e r e s t i m a t e d t h e number o f combinations h i s e x h a u s t i v e search a l g o r i t h m would generate, because he a l s o t h o u g h t o f e x p l o r i n g a l l combinations o f a l l p a i r s o f b a s i c concepts which would amount t o 2 ' 1 7 6 ' 7 8 2 ' 3 3 6 cases. A t l a t e r p e r i o d s two eminent p h i l o s o p h e r - m a t h e m a t i c i a n s became i n v o l v e d i n

t h e area o f h e u r i s t i c s f r o m an a l g o r i t h m i c p e r s p e c t i v e : Descartes and L e i b n i z Descartes b e l i e v e d t h a t a l g o r i t h m s would e v e n t u a l l y be f o u n d f o r a l l rnathem a t i c a l problems, perhaps a l s o f o r t h e p h i l o s o p h i c a l ones. He developed anal y t i c geometry t o t r a n s f o r m g e o m e t r i c a l problems i n t o a l g e b r a i c ones which can t h e n be t r e a t e d by a l g o r i t h m i c procedures. Changing t h e r e p r e s e n t a t i o n o f a problem can be a way o f s o l v i n g i t . The r o l e which r e p r e s e n t a t i o n a l s h i f t s p l a y i n t h e process o f problem s o l v i n g

i s an i m p o r t a n t p a r t o f t h e f i e l d o f

h e u r i s t i c s (Amarel, 1983). Descartes (1908) recommended i n h i s 21 h e u r i s t i c r u l e s t o d i r e c t t h e mind ( " r e g u l a e ad d i r e c t i o n e m i n g e n i i " ) t o reduce e v e r y problem, i f p o s s i b l e , t o a l g e b r a i c e q u a t i o n s . I n p a r t i c u l a r , t h e r u l e s say t h a t one should f i r s t r e duce a problem t o i t s s i m p l e s t f o r m o m i t t i n g a l l s u p e r f l u o u s i n f o r m a t i o n .

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HEURISTICS I N MODELS OF DECISION

one should t r y t o q u a n t i f y t h e problem i n g e o m e t r i c a l

erms, drawing f i g u r e s .

One should determine t h e known and unknown magnitudes. Then, one s h o u l d d i s s e c t t h e c o n d i t i o n s t h a t t h e unknowns have t o s a t i s f y

n as many p a r t s as

t h e r e a r e unknowns i n o r d e r t o g e t as many e q u a t i o n s , and f i n a l l y one s h o u l d reduce a l l e q u a t i o n s t o one. S o l v i n g a problem by means o f a l g e b r a i c e q u a t i o n s i s an example o f t h e a p p l i c a t i o n o f t h e method o f a n a l y s i s . D e s c a r t e s ' s r e m a i n i n g r u l e s a r e more o f a p s y c h o l o g i c a l n a t u r e and a r e concerned w i t h t h e o p t i m a l f u n c t i o n i n g o f t h e mind. They recommend t o s t u d y a problem as l o n g as i t i s necessary t o understand i t c l e a r l y , t o use t h e senses, t h e i m a g i n a t i o n and t h e memory, t o s u p p o r t t h e memory u s i n g e x t e r n a l r e p r e s e n t a t i o n s o f t h e problem, and t o p r a c t i s e t h e mind s o l v i n g a l o t o f problems t h a t have a l r e a d y been s o l v e d by o t h e r s . L e i b n i z c r i t i c i z e d Descartes, s a y i n g t h a t he had g i v e n us b e a u t i f u l r u l e s b u t no means t o execute them. The r u l e s were t o o g e n e r a l t o be o f any use. He even p u t h i s c r i t i c i s m i n t o a sharp s a t i r i c a l form, summarizing D e s c a r t e s ' s r u l e s i n t h e f o l l o w i n g statement: "Sume quod debes e t o p e r a r e u t debes, e t habebis quod o p t a s . " (1880, v o l . I V ,

p . 3 2 9 ) . ( " T a k e what y o u have t o t a k e , and

work t h e way you have t o , and you w i l l g e t what y o u a r e l o o k i n g f o r " . ) L e i b n i z h i m s e l f t o o k t h e g r e a t e s t e f f o r t i n s e a r c h i n g f o r a g e n e r a l method which would s o l v e e v e r y problem i n an a l g o r i t h m i c way. I n h i s y o u t h , when he was s t u d y i n g geometry and algebra, t h e i d e a came t o h i s mind t h a t i t s h o u l d be p o s s i b l e t o f i n d a r e l a t i v e l y s m a l l s e t o f p r i m i t i v e concepts f r o m which a l l known concepts and new ones c o u l d be d e r i v e d ( C o u t u r a t , 1961 b ) . L a t e r , i n h i s d i s s e r t a t i o n "De a r t e c o m b i n a t o r i a " , he worked o u t h i s p l a n . A l l n o t i o n s s h o u l d be decomposed i n t o a few, simple, independent and n o n - c o n t r a d i c t o r y elements. He c a l l e d them " a l p h a b e t o f human t h o u g h t s " . These elements s h o u l d be r e p r e s e n t e d by a p p r o p r i a t e symbols. Combining them i n a s y s t e m a t i c way would t h e n l e a d t o a l l known n o t i o n s , and a l s o g e n e r a t e new ones. T h i s g i g a n t i c p l a n i n v o l v e d s e v e r a l aspects on which L e i b n i z worked d u r i n g h i s whole l i f e . There was t h e a n a l y s i s o r r e d u c t i o n t o p r i m i t i v e s o f t h e i m mense knowledge t h a t had been accumulated, even t h e n , by a l l sciences f r o m mathematics t o t h e o l o g y . There was t h e problem o f choosing t h e r i g h t k i n d o f

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symbols t o r e p r e s e n t t h e p r i m i t i v e s and t h e i r i n t e r r e l a t i o n s . F i n a l l y , a method had t o be developed t h a t would " m e c h a n i c a l l y " s o l v e t h e fundamental problem o f t h e a r t o f d i s c o v e r y as L e i b n i z

has seen i t : t h a t i s , t o f i n d a l l

t r u e p r o p o s i t i o n s i n which a c e r t a i n concept i s e i t h e r s u b j e c t o r p r e d i c a t e . 7

That means t h i s method should n o t o n l y i n v e n t p r o p o s i t i o n s b u t a l s o d e c i d e about t h e i r t r u t h . O f course, t h e s e t a s k s were n o t independent o f each o t h e r . I n h i s search f o r p r i m i t i v e n o t i o n s , L e i b n i z was i n f l u e n c e d by t h e work o f L u l l u s and by s e v e r a l o f h i s own contemporaries who attempted t o c o n s t r u c t an u n i v e r s a l language by d e f i n i n g some g e n e r a l c a t e g o r i e s as b a s i c , r e p r e s e n t i n g them by symbols and combining these. One o f them was Athanasius K i r c h e r ( l 6 6 9 ) who had w r i t t e n a new v e r s i o n o f t h e " a r s magna". A l t h o u g h L e i b n i z esteemed t h o s e systems, he c r i t i c i z e d t h a t t h e number o f b a s i c c a t e g o r i e s had been a r b i t r a r i l y chosen and some c a t e g o r i e s were n o t p r i m i t i v e and u n i v e r s a l . He a l s o s t r e s s e d t h e importance o f choosing a k i n d o f r e p r e s e n t a t i o n o f concepts t h a t would f a c i l i t a t e t h i n k i n g about them. I n h i s o p i n i o n , t h e " c h a r a c t e r i s t i c a u n i v e r s a l i s " was a d e c i s i v e p a r t o f t h e a r t o f d i s c o v e r y . The symbols r e p r e s e n t i n g t h e p r i m i t i v e n o t i o n s s h o u l d be, on t h e one hand as n a t u r a l as poss i b l e , and on t h e o t h e r hand s h o r t and simple, such t h a t t h e y c o u l d be e a s i l y m a n i p u l a t e d i n c a l c u l a t i o n . The complex n o t i o n s s h o u l d be r e p r e s e n t e d b y some k i n d o f c o m b i n a t i o n o f t h e symbols r e p r e s e n t i n g t h e p r i m i t i v e concepts. I n t h i s way, i t s h o u l d be p o s s i b l e t o d e r i v e a l l p r o p e r t i e s o f a complex n o t i o n f r o m i t s r e p r e s e n t a t i o n . The o n l y sample L e i b n i z has g i v e n o f h i s " c h a r a c t e r i s t i c a " i s t h e r e p r e s e n t a t i o n o f concepts o f e l e m e n t a r y geometry. (1880, v o l . IV). L e i b n i z has made s e v e r a l a t t e m p t s a t d e v e l o p i n g an u n i v e r s a l system o f r e p r e s e n t a t i o n , and w i t h i t , a method o f d e r i v a t i o n . L e t us have a c l o s e r l o o k a t one d a t i n g f r o m t h e s i x t e e n - s e v e n t i e s .

I n t h i s system L e i b n i z i n t e n d e d t o use

i n t e g e r s as symbols and t h e b a s i c p r i n c i p l e o f a r i t h m e t i c t h a t e v e r y i n t e g e r can be uniqueZy r e p r e s e n t e d by a p r o d u c t o f p r i m e numbers. The p r i m i t i v e conc e p t s a r e r e p r e s e n t e d by p r i m e numbers. The complex concepts which a r e con

-

c e i v e d as combinations o f p r i m i t i v e concepts, as we remember, a r e r e p r e s e n t e d by p r o d u c t s o f t h e c o r r e s p o n d i n g p r i m e numbers. (1880, v o l . V I 1 ) .

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HEURISTICS IN MODELS OF DECISION

L e t us c o n s i d e r a h y p o t h e t i c a l example. Assume t h a t " a n i m a l " andl'being r a t i o n a l " a r e p r i m i t i v e concepts. Then, t h e y a r e r e p r e s e n t e d by prime numbers, say 2 and 3, r e s p e c t i v e l y . The concept "man" d e f i n e d as r a t i o n a l animal i s r e p r e sented by 2 x 3, which i s 6. B e f o r e we go on, we l i k e t o mention, by t h e way, t h a t L e i b n i z who t h o u g h t o f u s i n g h i s system a l s o as a u n i v e r s a l language, gave r u l e s f o r t r a n s f o r m i n g t h e i n t e g e r s i n t o s y l l a b l e s . The consonants b,c,d,f,g,h,l,m,n t h e d i g i t s 1 t o 9, t h e vowels, a,e,i,o,u,

correspond t o

t o t h e f i r s t f i v e powers o f t e n (0,

1,2,3 and 4 ) . An i n t e g e r i s decomposed i n t o a sum o f powers o f t e n and e v e r y t e r m o f t h e sum i s t r a n s f o r m e d i n t o a s y l l a b l e . So, t h e number 6, r e p r e s e n t i n g "man" i n o u r example, c o n s i s t s o f 1 t e r m : 6 x 10' w h i c h t r a n s f o r m s i n t o

HA. The number 65'134, f o r i n s t a n c e , would be r e a d "HUGOBIDEFA". L e i b n i z ment i o n s as an advantage o f t h e s e t r a n s f o r m i n g r u l e s t h a t t h e o r d e r o f t h e s y l l a b l e s can be changed which s h o u l d g i v e many p o s s i b i l i t i e s t o poets, BIDEFAGOHU o r FABIGODEHU, and so on ( L e i b n i z i n C o u t u r a t , 1961 a ) . The mapping o f concepts i n t o numbers, d e s c r i b e d above, i s n o t o n l y a b l e t o g e n e r a t e new concepts i n a s y s t e m a t i c way. I t a l s o a l l o w s t o d e c i d e about t h e t r u t h o f p r o p o s i t i o n s . F o r i n s t a n c e , an u n i v e r s a l a f f i r m a t i v e p r o p o s i t i o n , a l l A a r e B, i s t r u e i f t h e number r e p r e s e n t i n g t h e concept "A" c o n t a i n s t h e number r e p r e s e n t i n g " B " as a f a c t o r . O f c o u r s e t h e r e a r e many problems w i t h t h i s system. F o r i n s t a n c e , i t would be h a r d t o r e p r e s e n t a d i s j u n c t i v e conc e p t . L e i b n i z was aware o f these d i f f i c u l t i e s . C o n s i d e r i n g t h e s e few samples o f t h e i d e a o f L e i b n i z , we can imagine what enormous d i f f i c u l t i e s he was f a c i n g i n t r y i n g t o r e a l i z e h i s p l a n . He was aware o f t h e consequences h i s g e n e r a l l o g i c would have,

i f i t worked. F o r

i n s t a n c e , t h e r e would be no a r g u i n g l e f t among people, he says, because i t c o u l d be decided by c a l c u l a t i o n who i s r i g h t . (1880, v o l . V I 1 ,

e.g.p

25 f,64 f,

200). The method would be a s u b s t i t u t e f o r t h i n k i n g , and c o u l d be executed by

a machine ( C o u t u r a t , 1961 b ) . L e i b n i z was a l s o among t h e f i r s t t o d e s i g n c a l c u l a t i n g machines. The i d e a , t h a t an a l g o r i t h m c o u l d be f o u n d f o r e v e r y c l a s s o f mathematical

92

COGNlTl VE DECISION RESEARCH

problems, was a p p a r e n t l y c o n f i r m e d a g a i n and a g a i n by t h e f u r t h e r development o f mathematics. However, i t was f i n a l l y d i s p r o v e d i n t h i s c e n t u r y f o r t h e f i r s t t i m e by G o d e l ' s paper,"Ueber f o r m a l unentscheidbare Satze d e r P r i n c i p i a Mathematica und verwandter Systeme",

(1931

. Godel

showed, t h a t t h e r e

a r e c e r t a i n mathematical problems t h a t cannot be s o l v e d by any a l g o r i t h m o u t o f an e x a c t l y d e f i n e d c l a s s o f a l g o r i t h m s . A t t h s p o i n t an e x a c t d e f i n i t i o n o f t h e i n t u i t i v e concept o f a l g o r i t h m had become necessary. Up t o then, t h e r e had been no reason t o have a p r e c i s e d e f i n i t i o n . Whenever an a l g o r i t h m was proposed f o r a c e r t a i n c l a s s o f problems, t h e r e was agreement whether o r n o t i t r e a l l y was an a l g o r i t h m . Since 1935 s e v e r a l p r e c i s e d e f i n i t i o n s o f a l g o -

r i t h m have been g i v e n , f o r i n s t a n c e i n terms o f g e n e r a l r e c u r s i v e f u n c t i o n s by Herbrand, Godel und Kleene, i n terms o f Turing-machine c o m p u t a b i l i t y by T u r i n g , o r i n terms o f Markovian a l g o r i t h m s b y Markov. There i s g e n e r a l agreement t h a t t h e s e d e f i n i t i o n s , which have been proven e q u i v a l e n t t o each o t h e r , c a p t u r e t h e i n t u i t i v e n o t i o n o f an a l g o r i t h m . The emergence o f computers has made t h e aspect o f e f f i c i e n c y o f an a l g o r i t h m more i m p o r t a n t . Although t h i s n o t i o n r e f e r s t o a l l computing r e s o u r c e s needed t o execute an a l g o r i t h m , one n o r m a l l y c o n s i d e r s t h e f a s t e s t among d i f f e r e n t a l g o r i t h m s f o r t h e same c l a s s o f problems as t h e most e f f i c i e n t one. The t i m e r e q u i r e d t o execute an a l g o r i t h m f o r a p a r t i c u l a r i n s t a n c e o f a c l a s s o f p r o b lems can be d e s c r i b e d as some f u n c t i o n o f t h e " s i z e " o r " c o m p l e x i t y " o f t h e i n s t a n c e . Such a f u n c t i o n i s c a l l e d t i m e c o m p l e x i t y f u n c t i o n . D i f f e r e n t a l g o r i t h m s have a v a r i e t y o f d i f f e r e n t t i m e c o m p l e x i t y f u n c t i o n s . W i t h r e s p e c t t o e f f i c i e n c y , a s i m p l e d i s t i n c t i o n i s made by computing s c i e n t i s t s between p o l y nomial t i m e a l g o r i t h m s and e x p o n e n t i a l t i m e a l g o r i t h m s . F o r t h e f i r s t group, e x e c u t i o n t i m e i n c r e a s e s as a p o l y n o m i a l f u n c t i o n o f -the s i z e o f t h e problem i n s t a n c e s , and f o r t h e second group, e x e c u t i o n t i m e i n c r e a s e s as an exponent i a l function. The p r a c t i c a l meaning o f t h i s d i s t i n c t i o n i s t h a t problem i n s t a n c e s w i t h l a r g e " s i z e " can u s u a l l y n o t be handled by e x p o n e n t i a l t i m e a l g o r i t h m s . To show t h i s d r a s t i c a l l y , we t a k e an example f r o m Garey & Johnson (1979, p . 7 ) . Assume two t i m e c o m p l e x i t y f u n c t i o n s , a p o l y n o m i a l one: n5, and an exponent i a l one: 3n; n i s a measure f o r t h e s i z e o f t h e problem i n s t a n c e . Assume

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HEURISTICS IN MODELS OF DECISION

t h a t t h e e x e c u t i o n t i m e f o r n = 10 i s 0.1 sec i n t h e p o l y n o m i a l case. Then i t i s 0.059 sec f o r t h e e x p o n e n t i a l case. Now, f o r n = 40, t h e t i m e f o r t h e p o l y nomial f u n c t i o n goes up t o 1.7 minutes, whereas t h e t i m e f o r t h e e x p o n e n t i a l f u n c t i o n goes up t o 3855 c e n t u r i e s . The a u t o m a t i c i n v e n t i o n system o f L u l l u s we have c o n s i d e r e d b e f o r e i s an example o f an e x p o n e n t i a l t i m e a l g o r i t h m . Most o f t h e s e a l g o r i t h m s a r e some v a r i a n t o f e x h a u s t i v e search. Problems f o r which no p o l y n o m i a l t i m e a l g o

-

r i t h m s a r e known a r e r e f e r r e d t o as i n t r a c t a b l e . C l e a r l y , i t i s v e r y u s e f u l t o know whether a c l a s s o f problems i s i n p r i n c i p l e i n t r a c t a b l e , t h a t means no p o l y n o m i a l t i m e a l g o r i t h m e x i t s . I n such a cage, t h e c l a s s o f problems has t o be t a c k l e d by h e u r i s t i c methods. I n t h e l a s t decade, i t has become poss i b l e t o p r o v e f o r some c l a s s e s o f problems t h a t t h e y a r e d e c i d a b l e b u t t h a t t h e r e e x i s t no p o l y n o m i a l t i m e a l g o r i t h m s . (Garey & Johnson, 1979). A f t e r t h i s r a t h e r l e n g t h y e x c u r s i o n i n t o a l g o r i t h m s l e t us go back t o heur i s t i c methods w i t h t h e s p e c i f i c a t i o n t h a t t h e y do n o t guarantee s o l u t i o n s . Here t h e work o f t h e 1 9 t h - c e n t u r y mathematician Bernhard Bolzano s h o u l d be mentioned who d e d i c a t e d about 250 pages o f h i s " W i s s e n s c h a f t s l e h r e " ( i .e. P h i l o s o p h y o f Science) (1930) t o h e u r i s t i c s . The r u l e s which a r e d e s c r i b e d and comment2d i n g r e a t d e t a i l a r e grouped i n t o two c l a s s e s : g e n e r a l r u l e s which can be a p p l i e d t o any problem and s p e c i a l r u l e s which a r e concerned w i t h s p e c i f i c problems such as t h e a n a l y s i s o f concepts o r p r o p o s i t i o n s , o r t h e casual a n a l y s i s o f events. Space f o r b i d s us m e n t i o n i n g more t h a n a few o f t h e g e n e r a l r u l e s . Many o f them a r e p s y c h o l o g i c a l r u l e s such as t h e f o l l o w i n g ones: b e f o r e one s t a r t s l o o k i n g f o r a s o l u t i o n one s h o u l d g e t a v e r y c l e a r p i c t u r e o f t h e problem. One s h o u l d w r i t e down e v e r y t h i n g r e l e v a n t t o t h e problem i n o r d e r t o overcome t h e d i f f i c u l t y t h a t t h e human mind can d i r e c t i t s a t t e n t i o n t o o n l y a few i d e a s a t t h e same t i m e . One s h o u l d a v o i d s t r o n g s t i m u l a t i o n o f t h e senses w h i l e s o l v i n g a problem, and so on. Several r u l e s concern t h e c h o i c e o f a p p r o p r i a t e symbols t o r e p r e s e n t t h e problem. F u r t h e r , we f i n d a g a i n t h e c l a s s i f i c a t i o n o f problems i n t o p r o o f s and d i s c o v e r i e s , and d e t a i l e d d e s c r i p t i o n o f t h e methods o f a n a l y s i s and s y n t h e s i s . Here, Bolzano recommends combining

t h e two methods, t h a t means, one s h o u l d work f r o m b o t h

ends, f r o m t h e d a t a and f r o m t h e h y p o t h e t i c a l s o l u t i o n , a t t h e same t i m e .

94

COGNI TI VE DECISION RESEARCH

This should help t o d i r e c t t h e l i n e o f reasoning. ' I n another system, t h e e x h a u s t i v e aspect o f t h e o l d e r methods i s t a k e n up a g a i n . I t i s t h e m o r p h o l o g i c a l method o f d i s c o v e r y developed by t h e a s t r o p h y s i c i s t F r i t z Zwicky (1957, 1966) i n t h i s c e n t u r y . The f i r s t s t e p o f h i s method i s t o d e f i n e t h e problem as e x a c t l y as p o s s i b l e , t h e second s t e p t o determine a l l f a c t o r s i n f l u e n c i n g t h e s o l u t i o n , t h e t h i r d s t e p t o combine t h e s e f a c t o r s and t o t e s t each. c o m b i n a t i o n f o r i t s u s e f u l n e s s as a s o l u t i o n and t h e f i n a l s t e p i s t o choose t h e o p t i m a l s o l u t i o n . Such an e x h a u s t i v e method has t h e i m p o r t a n t advantage t h a t even t h o s e p o s s i b i l i t i e s a r e e v a l u a t e d which u s u a l l y a r e missed because o f v a r i o u s k i n d s o f p s y c h o l o g i c a l b l o c k s . Z w i c k y ' s i d e a s c o u l d be summarized i n t h e f o l l o w i n g heuristic r u l e : Do not use heuristic r u l e s t o g u i d e you t h r o u g h t h e s o l u t i o n space, b u t go s y s t e m a t i c a l l y o v e r each p o i n t o f t h e whole space and t e s t i t c a r e f u l l y ! O f course t h i s procedure i s o n l y r e a s o n a b l e i f t h e number o f r e l e v a n t f a c t o r s can be k e p t s m a l l . T h e r e f o r e , t h e c h o i c e o f an o p t i m a l problem r e p r e s e n t a t i o n i s t h e c r u c i a l s t e p i n Z w i c k y ' s method as h i s examples show. We want t o conclude o u r general h i s t o r i c r e v i e w on h e u r i s t i c s w i t h j u s t ment i o n i n g t h e e x t e n s i v e and i n f l u e n t i a l work o f Georg P o l y a (1945, 1954, 1962, 1965; f o r a r e v i e w see a l s o Newell, 1983; van d e r Waerden, 1983). P o l y a gave d e t a i l e d d e s c r i p t i o n s o f t h e v a r i o u s h e u r i s t i c methods and condensed them t o a few general p r i n c i p l e s . By i n t r o d u c i n g b r i l l i a n t examples he s u c c e s s f u l l y a p p l i e d them t o t h e t e a c h i n g o f mathematics. The m i x t u r e o f pedagogical and mathematical i n g e n u i t y made h i s book

"How t o S o l v e

I t " a g r e a t p u b l i c success

and c o n t r i b u t e d t o t h e r e v i v a l o f h e u r i s t i c s as a d i s c i p l i n e . 2. Psychology of Thinking and Heuristics It i s reasonable t o assume t h a t a p s y c h o l o g i c a l t h e o r y o f t h i n k i n g can con-

t r i b u t e a l o t t o a t h e o r y o f h e u r i s t i c s . The s t u d y o f human t h i n k i n g r e v e a l s s p e c i f i c shortcomings and l i m i t a t i o n s o f problem s o l v i n g and can t h e r e f o r e s t i m u l a t e a t t e m p t s t o overcome them.

I f we l o o k a t t h e e a r l y h i s t o r y o f t h e psychology o f t h i n k i n g we n o t i c e g r e a t

95

HEURISTICS IN MODELS OF DECISION

d i f f e r e n c e s between d i f f e r e n t s c h o o l s . There even was no agreement on what t h e b a s i c u n i t s o f t h i n k i n g were. A mode o f r e p r e s e n t a t i o n had t o be f o u n u o r d e f i n e d , b e f o r e one c o u l d r e a s o n a b l y s t a r t t a l k i n g about problem s o l v i n g o r c r e a t i v i t y . Me t h e r e f o r e w i l l f i r s t b r i e f l y r e c a p i t u l a t e t h e a t t e m p t s o f l o o k i n g f o r s u i t a b l e representations. In

t h e p r e - e x p e r i m e n t a l t r a d i t i o n up t o t h e end o f t h e 1 9 t h c e n t u r y , p s y c h o l o -

gy was t r e a t e d as a s u b d i s c i p l i n e o f p h i l o s o p h y . Since H r i s t o t l e a s s o c i a t i o n t h e o r y was t h e d o m i n a t i n g t h e o r y o f t h i n k i n g . The elements o f t h i n k i n g were assumed t o be conscious i d e a s o r mental images. The B r i t i s h e m p i r i s t s

empha-

s i z e d t h e sensory o r i g i n o f t h e s e elements. The c o m b i n a t i o n o f t h e s e elements was governed by one p r i n c i p l e , t h e a s s o c i a t i o n . However, a t t h e b e g i n n i n g o f t h i s c e n t u r y t h i s d o c t r i n e was c h a l l e n g e d by s e v e r a l s c h o o l s o f psychology. F i r s t t h e r e was t h e Wurzburg school, a group o f p s y c h o l o g i s t s around Oswald Kulpe a t t h e u n i v e r s i t y o f Wurzburg, B a v a r i a (see B o r i n g , 1950, o r Humphrey, 1951).

They made t h e f i r s t s y s t e m a t i c a t t e m p t t o s t u d y t h o u g h t process e x p e r i -

m e n t a l l y u s i n g i n t r o s p e c t i v e s e l f - r e p o r t s . To t h e i r s u r p r i s e t h e y f o u n d h a r d l y any evidence f o r p e r c e p t - l i k e images o r c l e a r l y conscious i d e a s . Concequently, t h e Wurzburg school emphasized t h a t t h e elements o f t h o u g h t processes had n e i t h e r c l e a r l y conscious n o r i m a g e - l i k e q u a l i t i e s , ar;d Mayer & O r t h (1901) proposed t o c a l l them 'Bewusstseinslagen' ( ' s t a t e s o f c o n s c i o u s n e s s ' ) . These 'Bewusstseinslagen' were n e g a t i v e l y d e f i n e d as b e i n g n e i t h e r an image n o r a p e r c e p t , n o t an a c t o f w i l l and as n o t b e i n g f u r t h e r r e d u c i b l e . Among t h e o t h e r members o f t h a t group (Ach, B u h l e r , Marbe, Messer anu W a t t ) i t was K a r l B u h l e r (1907) who o u t l i n e d i n most d e t a i l t h e e s s e n t i a l aspects o f a t h e o r y o f t h i n k i n g . He used t h e word 'Gedanke' t o denote t h e fundamental u n i t s o f t h i n k i n g andhe conceived u n d e r s t a n d i n g as t h e process o f ' l o g i c a l - p l a c e - a s s i g n m e n t ' o f d i f f e r e n t 'Gedanken'. Although h i s work c o n t a i n s a m u l t i t u d e o f s u b t l e obs e r v a t i o n s , h i s t h e o r y s u f f e r e d f r o m t h e l a c k o f a more r i g o r o u s f o r m a l i s m f o r e x p r e s s i n g h i s i d e a s . However, B u h l e r had a n t i c i p a t e d inany i d e a s o f c o g n i t i v e o r g a n i z a t i o n which became p o p u l a r more t h a n f i f t y y e a r s l a t e r . Taking up t h e work o f t h e Wurzburg group, O t t o S e l z developed

a more e l a b o -

r a t e t h e o r y o f t h i n k i n g (For a b r i e f o u t l i n e o f h i s arguments and h i s t h e o r y

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COGNITIVE DECISION RESEARCH

se@ Selz, 1924). I n a v i g o r u o u s a t t a c k a g a i n s t a s s o c i a t i o n i s m i n i t s l a t e s t form, t h e c c n s t e l l a t i o n t h e o r y o f a s s o c i a t i o n s by Georg E l i a s M u l l e r (1911 ),he demonstrated

t h a t such a t h e o r y c o u l d n o t even account f o r such s i m p l e

t a s k s as f i n d i n g s u b o r d i n a t e o r s u p e r o r d i n a t e concepts. S e l z r e p l a c e d a s s o c i a t i o n s h a v i n g no q u a l i t i e s by v a r i o u s d i f f e r e n t l i n k s r e p r e s e n t i n g d i r e c t e d

r e l a t i o n s . T h i s n o t i o n , as w e l l as o t h e r s l i k e ' o p e r a t i o n ' o r 'schema'are a l r e a d y v e r y c l o s e t o p a r a l l e l concepts i n C o g n i t i v e Science. Selz developed h i s own t h e o r y o f p r o d u c t i v e and r e p r o d u c t i v e t h i n k i n g and a p p l i e d i t l a t e r t o t h e problem o f i m p r o v i n g i n t e l l i g e n c e ( S e l z , 19351, an i s s u e which i s t a k e n up by

de Groot ( 1 9 8 3 ) .

A p a r t f r o m t h e Wurzburg group and O t t o S e l z t h e r e was a t h i r d group w o r k i n g on new i d e a s about t h e s t r u c t u r e and d i r e c t i o n o f t h i n k i n g , t h e G e s t a l t p s y c h o l o g i s t s . I n s t e a d o f a t o m i s t i c concepts, l i k e a s s o c i a t i o n o r ' B e w u s s t s e i n s l a g e n ' , t h e y proposed a l a r g e r b a s i c u n i t o f o r g a n i z a t i o n , t h e G e s t a l t , which i s def i n e d as a whole and cannot be reduced t o a sum o f i t s p a r t s . W i t h i n t h e school o f G e s t a l t p s y c h o l o g i s t s t h e r e were t h r e e main proponents i n v e s t i g a t i n g t h i n k i n g : Wolfgang Kohler, Max Wertheimer and K a r l Duncker. K o h l e r (1921) and Wertheimer (1945) emphasized t h e c e n t r a l r o l e o f i n s i g h t which t h e y c o n c e i v e d as a sudden r e o r g a n i z a t i o n of a problem. B u t i t was p r o b a b l y K a r l Duncker who has t h e s t r o n g e s t i n f l u e n c e on t h e f u r t h e r developments i n t h e psychology o f t h i n k i n g . F i r s t he banished t h e method o f r e c o r d i n g i n t r o s p e c t i o n s used by t h e Wurzburg group where t h e s u b j e c t s gave an a p o s t e r i o r i i n t e r p r e t a t i o n o f t h e i r i n n e r e x p e r i e n c e s . I n s t e a d , Duncker (1935) i n t r o d u c e d t h e ' s o l v i n g a l o u d ' method where s u b j e c t s v e r b a l i z e i m i i i e d i a t e l y whatever o c c u r s t o them d u r i n g t h e process o f t h i n k i n g . Duncker was a l s o among t h e f i r s t t o use a t r e e t o r e p r e s e n t t h e d i f f e r e n t i n t e r m e d i a t e s t e p s o f t h e s o l u t i o n process. T h i s k i n d o f r e p r e s e n t a t i o n became v e r y p o p u l a r l a t e r a l t h o u g h some d e t a i l s were c l e a r l y d i f f e r e n t . I n D u n c k e r ' s problem s o l v i n g t r e e s successive branches r e p r e s e n t s t a t e s o f i n c r e a s i n g s p e c i f i t y , whereas i n l a t e r a p p l i c a t i o n s t h e y u s u a l l y r e f e r t o s u c c e s s i v e app l i c a t i o n s o f o p e r a t o r s . T h i s modern view o f problem r e p r e s e n t a t i o n can be summarized as f o l l o w s : I n a f i r s t s t a g e o f problem s o l v i n g , t h e s u b j e c t has

HEURISTICS I N MODELS OF DECISION

97

t o f i n d o u t what i s g i v e n , what t h e problem i s and which a c t i o n s , t r a n s f o r m a t i o n s o r o p e r a t o r s can be a F p l i e d . These o p e r a t o r s a r e c a p a b l e o f changing t h e s t a t e s o f t h e problem.

A problem s t a t e i s d e f i n e d as t h e i n i t i a l problem

s i t u a t i o n o r any s t a t e t h a t can be achieved t h r o u g h t h e successive a p p l i c a t i o n o f a l l o w e d o p e r a t o r s . The s e t o f a l l p o s s i b l e sequences o f o p e r a t o r s can be r e p r e s e n t e d by a s o l u t i o n t r e e , where t h e branches r e p r e s e n t t h e a p p l i c a t i o n o f an o p e r a t o r and t h e nodes r e p r e s e n t problem s t a t e s . A s o l u t i o n i s d e f i n e d by c e r t a i n p r o p e r t i e s and a l l s t a t e s o f t h e s o l u t i o n t r e e which have t h e s e p r o p e r t i e s a r e s o l u t i o n s t a t e s . F i n d i n g a s o l u t i o n t h e n corresponds t o a search f o r a s o l u t i o n s t a t e i n t h e s o l u t i o n t r e e . B e f o r e c o n t i n u i n g a l o n g t h i s l i n e , we f i r s t r e t u r n t o t h e h e u r i s t i c methods proposed by Duncker. A c c o r d i n g t o Duncker, t h e process o f t h i n k i n g i s e s L e n t i a l l y che r e s o l u t i o n o f a c o n f l i c t between t h e g i v e n d a t a and t h e g o a l . The r e s o l u t i o n u s u a l l y cannot be reached i n a s i n g l e s t e p b u t must be achieved t h r o u g h a s e r i e s o f p r o g r e s s i v e r e o r g a n i z a t i o n s which a r e c o n t r o l l e d by h e u r i s t i c methods: " I m e c h t e n Denkprozess s p i e l e n e i n e entscheidende R o l l e gewisse h e u r i s t i s c h e 'Methoden', d i e das Entstehen

d e r sukzessiven Losungsphasen v e r m i t t e l n " (Duncker, 1935,

p.28). Some o f t h e methods Duncker proposed a r e (Duncker, 1935, p. 13-45) :

-

Using suggestions f r o m t h e g i v e n m a t e r i a l fur t h e s o l u t i o n

-

Learning from mistakes through c o r r e c t i n g v a r i a t i o n

-'Resonance o f s i g n a l e l e m e n t s ' , i . c . u s i n g p a r t i a l a n a l o g i e s by a c t i v a t i n g some resonance i n memory

-

A n a l y s i s o f t h e c o n f l i c t between t h e s t a r t i n g s i t u a t i o n and t h e g o a l Analysis o f t h e material A n a l y s i s o f t h e r e q u i r e m e n t s o f t h e goal A v a i l a b i l i t y ( " D i s p o n i b i l i t a t " ) o f t l ? e aspects o f t h e s i t u a t i o n , i .e. a v o i d i n g f i x a t i o n s on some aspects o f t h e problem by p a y i n g a t t e n t i o n t o o t h e r s

-

S p e c i a l i z a t i o n , i . e . c o n s t r u c t i n g some s p e c i a l cases i n o r d e r t o o b t a i n a

-

Analysis o f p r e r e q u i s i t e s ,

b e t t e r p i c t u r e o f t h e underlying general p r i n c i p l e As m i g h t be n o t i c e d , s e v e r a l o f t h e s e methods appeared l a t e r again i n t h e works o f o t h e r a u t h o r s , as, f o r example, Polya (19451, DeBono (19681, Adams (19741, o r Wickelgren (19741, t o m e n t i o n j u s t a few.

98

COGNITIVE DECISION RESEARCH

According t o b u n c k e r ' s view t h e most e s s e n t i a l p r o p e r t y o f t h e a v a i l a b l e m a t e r i a l i s i t s fzmctional value : how can something be used t o o b t a i n t h e goal. A t about t h e same t i m e , t h e r e were o t h e r approaches emphasizing t h e temporaZ

course o f t h e problem s o l v i n g process by p o s t u l a t i n g a s e r i e s o f s u c c e s s i v e stages. Here problem s o l v i n g i s c o n c e i v e d as t h e process o f g e n e r a t i n g and t e s t i n g hypotheses. The c e n t r a l concept i s t h a t o f a hypothesis d e f i n e d as t h e t e n t a t i v e a n t i c i p a t i o n o f a r e s u l t ( f o r a r e v i e w see Groner & Groner, 1 9 8 2 ) . F o r i n s t a n c e , Dewey (1933, p. 107 .. 115) d i s t i n g u i s h e d t h e f o l l o w i n g f i v e phases o f t h e problem s o l v i n g processes: ( 1 ) suggestion, t i o n , (31 t h e g u i d i n g idea, h y p o t h e s i s , ( 4 ) r e a s o n i n g

(2) intellectualiza-

, (5)

t e s t i n g t h e hypo-

t h e s i s by a c t i o n . I n a s i m i l a r p r o p o s a l Claparede (1933) i n t r o d u c e d t h e f o l l o w i n g c o n s e c P t i v e 'moments': " l a q u e s t i o n , p o i n t du d e p a r t de l ' o p e r a t i o n i n t e l l e c t u e l l e ; l a recherche, ou decouverte de l ' h y p o t h 6 s e ,

...,

l a verification,

ou c o n t r d l e de l ' h y p o t h e s e " (1933, p 6 ) . S t i l l a n o t h e r well-known d i s t i n c t i o n was i n t r o d u c e d by Wallas (1926): ( 1 ) p r e p a r a t i o n , ( 2 ) i n c u b a t i o n , ( 3 ) i l l u m i n a t i o n and (4) v e r i f i c a t i o n . Claparede who a l s o used s o l v i n g - a l o u d p r o t o c o l s c o u l d h a r d l y f i n d any v e r b a l r e p o r t s about t h e d e t a i l s o f t h e proposed procedures. A p p a r e n t l y t h e r e were p a r t s o f t h e processes i n a c c e s s i b l e t o t h e consciousness. T h i s view was supp o r t e d by t h e o b s e r v a t i o n s

C:f

famous mathematicians (Poincare, 1913; Hadamard,

1945; van d e r Waerden, 1983) t h a t t h e d e c i s i v e i d e a f o r a s o l u t i o n v e r y o f t e n appeared l i k e a si:dden s t r o k e o f i l l u m i n a t i o n . J t i s t h e r e f o r e i n t e r e s t i n g t o n o t e t h a t j u s t one m a j o r h e u r i s t i c r u l e emanated f r o m a l l t h e s e approaches: w a i t and see, l e t i n c u b a t i o n work! As a l r e a d y mentioned, b e h a v i o r i s m chology around t h e f o r t i e s

-

- the

dominating d o c t r i n e

iti

American psy-

was n o t s u i t e d f o r a p p l i c a t i o n i n h e u r i s t i c s . I n

t h e f i f t i e s s e v e r a l newly a r i s i n g approaches had t h e i r i n f l u e n c e on t h e de

-

velopment o f psychology. We j u s t m e n t i o n some o f them ( f o r a more d e t a i l e d r e v i e w see Newell & Simon, 1972, p. 8 7 8 f f , Newell, 1980a) : t h e r e were t h e new mathematical t h e o r i e s o f i n f o r m a t i o n , games and d e c i s i o n s , at;d s i g n a l d e t e c t i o n , and, of course, t h e r e were t h e f i r s t d i g i t a l computers. These d e v e l o p -

HEURISTICS IN MODELS OF DECISION

99

ments c f f e r e n d new t o o l s f o r a more c o n c i s e and l e s s s p e c u l a t i v e t r e a t m e n t o f t h e o l d q u e s t i o n s on how t h i n k i n g m i g h t proceed. The new paradigm o f g r e a t p o t e n t i a l g e n e r a l i t y was t h a t o f i n f o r m a t i o n process i n g . I n f o r m a t i o n was conceived as b e i n g i n v a r i a n t o v e r d i f f e r e n t codes and independent o f t h e p a r t i c u l a r medium o r hardware. A l r e a d y a t an e a r l y stage, programs were d e v e l o p e d such as t h e L o g i c T h e o r i s t (Newell, Shaw & Simon, 1959) and t h e General Problem S o l v e r (Newell & Simon, 1961) o r t h e Elementary P e r c e i v e r and Memorizer (Feigenbaum, 1959) which were conceived as computer s i m u l a t i o n s o f human c o g n i t i v e processes. A new d i s c i p l i n e emerged f r o m the j o i n t e f f o r t s o f s p e c i a l i s t s from various f i e l d s , c a l l e d A r t i f i c i a l I n t e l l i

-

gence. 3 . Heuristics in Artificial Intelligence

A r t i f i c i a l I n t e l i g e n c e comprises a l l e f f o r t s a:

making computers s o l v e p r o b -

lems t h a t r e q u i r e i n t e l l i g e n c e such as, f o r example, p l a y i n g chess, p r o v i n g l o g i c a l theorems, making medical diagnoses, d e s i g n i n g and a n a l y z i n g e l e c t r o n i c c i r c u i t s , understanding

language o r images e t c . A r t i f i c i a l I n t e l l i g e n c e can

be seen f r o m two d i f f e r e n t v i e w p o i n t s , f r o m en e n g i n e e r i n g p o i n t o f v i e w and f r o m a ' s c i e n t i s t s ' p o i n t o f view. From an e1:gineering p o i n t o f view, i t s goal i s t o produce machines t h a t a r e i n c r e a s i n g l y p o w e r f u l i n i n c r e a s i n g l y d i v e r s e areas o f a p p l i c a t i o n . From t h e second, more g e n e r a l p o i n t o f view, A r t i f i c i a l I r i t e l l i g e n c e i s a new approach t o d e t e r m i n e t h e p r e r e q u i s i t e s o f i n t e l l i g e n t b e h a v i o r and t c uncover i t s n a t u r e , independent o f whether t h i s b e h a v i o r i s produced by a n d t u r a l , b i o l o g i c a l o r an a r t i f i c i a l system. I n t h e l a s t few y e a r s t h i s second view hds become v e r y p o p u l a r al;d has l e a d t o a new s c i e n t i f i c paradigm c a l l e d ' C o g n i t i v e S c i e n c e ' which p u r s u e s t h i s goal i n an i n t e r d i s c i p l i n a r y e f f o r t between A r t i f i c i a l I n t e l l i g e n c e , c o g n i t i v e psychology, l i n g u i s t i c s and o t h e r d i s c i p l i n e s . I n C o g n i t i v e Science b o t h man and machine a r e u n i f o r m l y seen as systems t h a t c o n s t r u c t s y m b o l i c d e s c r i p t i o n s o f t h e w o r l d and m a n i p u l a t e t h e s e d e s c r i p t i o n s . One consequence o f t h i s view i s t h e separat i o n o f t h e p l i y s i c a l r e a l i z a t i o n O F t h e symbols f r o m t i l e i r meaning and t i i u s i t l e a d s t o t h e view t h a t t h e s t u d y O F t h e computation proc?sses on t h e symbols

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can be d i s t i g u i s h e d f r o m t h e s t u d y o f t h e p h y s i c a l mechanisms t h a t s u p p o r t t k e computation ( F o r a more d e t a i l e d d i s c u s s i o n o f t h e s e i s s u e s see Newell, 1980b:.

From t h i s p o s i t i o n i t i s t h e n n a t u r a l t h a t t h e r e s u l t s f o u n d i n A r t i -

f i c i a l I n t e l l i g e n c e can be c a r r i e d o v e r t o man and can be used t o u n d e r s t a n d h i s own i n t e l l i g e n t b e h a v i o r . The h i s t o r y o f A r t i f i c i a l I n t e l l i g e n c e i s i n t i m a t e l y connected w i t h t h e n o t i o n u f h e u r i s t i c s , o r t o be more p r e c i s e , t h e n o t i o n o f h e u r i s t i c search. Two i m p o r t a n t ideas l i e behind t h i s notion. F i r s t i t i s t h e idea t h a t i n t e l l i g e n t b e h a v i o r czn be c h a r a c t e r i z e d as search, t h e search f o r s o l u t i o n s t o probiems posed t o t h e system. Secondly i t i s t h e i d e a

t h a t t h e search space a s s o c i a t e d

w i t h problems r e q u i r i n g i n t e l l i g e n c e i s t y p i c a l l y so l a r g e t h a t any system (nust use h e u r i s t i c s t o reduce i t t o a manageable s i z e g i v e n t h e system's lim i t e d r e s o u r c e s o f t i m e and space.

'Time' here r e f e r s t o t h e number o f o p e r -

a t i o n s t h e system must perform t o f i n d a s o l u t i o n w h i l e ' s p a c e ' r e f e r s t o t h e number and s i z e of symbolic d e s c r i p t i o n s o f i n t e r m e d i a t e s o l u t i o n s t h e system produces on t h e way t o a s o l u t i o n . L u l l u s ' a u t o m a t i c i n v e n t i o n system has l e a d us t o c o n s i d e r a l g o r i t h m s w i t h e x p o n e n t i a i t i m e c o m p i e x i t y and we have seen t h a t such a l g o r i t h m s exceed any p r a c t i c a l t i m e l i m i t s f o r l a r g e problem i n stances. We c o u l d a v o i d t h i s problem by u s i n g a p a r a l l e l search scheme b u t t h i s would l e a d t o an e x p l o s i o n o f i n t e r m e d i a t e s o l u t i o n s produced by t h e system and t h u s would v i o l a t e t h e c o n s t r a i n t o f l i m i t e d space r e s o u r c e . H e u r i s t i c s a r e used t o reduce t h e search space o f a g i v e n problem. But what a r e h e u r i s t i c s ? The i d e a l answer would be a f o r m a l i z e d n o t i o n o f h e u r i s t i c s and we m i g h t g a i n as much o u t o f such a f o r m a l i z a t i o n as we g a i n e d f r o m t h e f o r m a l i z e d n o t i o n o f an a l g o r i t h m . U n f o r t u n a t e l y t h i n g s l o o k n o t t h a t p r o m i s i n g : The h i s t o r y o f A r t i f i c i a l I n t e l l i g e n c e shows us t h a t h e u r i s t i c s a r e d i f f i c u l t t c d e l i n e a t e i n a c l e a r c u t manner and t h a t t h e convergence o f i d e a s about t h e i r n a t u r e i s v e r y slow. T h i s i s n o t t h e p l a c e t o go i r i t o a d e t a i l e d d i s c u s s i o n about t h e r o l e o f heuristics i n A r t i f i c i a l Intelligence.,

b u t a few v e r y b a s i c remarks seem

a p p r o p r i a t e t o j u d g e t h e c o n t r i b u t i o n s o f A r t i f i c i a l I n t e l l i g e n c e towards a theory o f heuristics.

HEURISTICS IN MODELS OF DECISION

101

E a r l y work i n A r t i f i c i a l I n t e l l i g e n c e has l e d t o a s e t o f t e c h n i q u e s on how t o use general knowledge t c g u i d e t h e process o f g e n e r a t i n g new s t e p s i n t h e search space. These techniques, known as h e u r i s t i c search techniques, grew o u t o f a l a r g e number o f e m p i r i c a l s t u d i e s and a r e d e s c r i b e d , f o r example, i n N i l s s o n (1971). But t h e s e s t u d i e s have a l s o shown t h a t t h e use o f v e r y g e n e r a l knowledge a l o n e i s u s u a l l y n o t s u f f i c i e n t t o c c n s r r a i n t n e search space clown t o a manaGeable s i z e . As a h e u r i s t i c r u l e one can sdy t k a t any p i e c e o f knowledge, as s p e c i f i c i t may be, should and i n p r a c t i c e must be used t o c o n s t r a i n t h e search space. W i t h t h e i n c r e a s e d s i z e o f knowledge bases we a r e f a c e d w i t h a whole bunch o f d i f f i c u l t problems: How must a l l t h i s knowledge be r e p r e s e n t e d i n a system? Hljw s h o u l d knowledge be used, i . e .

how can a p i e c e o f knowledge

l e a d t o a p p r o p r i a t e a c t i o n s by t h e system? How s h o u l d knowledge be o r g a n i z e d , i . e . how do we manage t h a t a system has access t o a l l knowledge r e l e v a n t t o a s i t u a t i o n b u t i s n o t d i v e r t e d t o o much by knowledge i r r e l e v a n t t o t h a t s i t u a t i o n ? How do we manage t h a t any p i e c e o f knowledge i n c l u d i n g knowledge about p a r t i a l s o l u t i o n s i s a v a i l a b l e t o a l l t h o s e components o f t h e system t h a t may u s e f u l l y a p p l y i t ?How can a system a c q u i r e t h e huge mass o f knowledge i t may need, i . e . how can we t r a n s f e r o u r knowledge t o t h e machine o r how can t h e system d i s c o v e r t h a t knowledge by i t s e l f ?

A p a r t i c u l a r l y p r o m i s i n g aspect o f A r t i f i c i a l l n t e l l i g e n c e i s t h e f a c t t h a t i t p r o v i d e s us w i t h t o o l s f o r e m p i r i c a l l y s t u d y i n g o u r i d e a s on a f u t u r e t h e o r y o f h e u r i s t i c s (e.g.

Lenat, 1983).

4. Common Properties of Heuristic Methods

AS t h i s p o i n t we s h o u l d b r i e f l y summarize t h e conimon aspects o f h e u r i s t i c methods as t h e y a r e used today

.

( 1 ) Problems (and t h e i r s o l u t i o n ) a r e repesentedin term of information processing (computer programs, terms o f automata t h e o r y , c o g n i t i v e models).

( 2 ) H e u r i s t i c methods

restrict the search r;pace t o a manageable s i z e .

( 3 ) They a r e sufficient w i t h r e s p e c t t o a c e r t a i n o u t p u t , t h u s g e n e r a t i n g a sequence o f o p e r a t i o n s f r o m t h e problem t o t h e s o l u t i o n . With ill defined o r open-goal problems t h i s p r o p e r t y cannot be assessed i n advance.

102

COGNITIVE DECISION RESEARCH

( 4 ) They a r e expected t o p r o v i d e a b e t t e r s o l u t i o n t h a n a p u r e t r i a l

and

e r r o r procedure (efficiency). T h i s l i s t i s n o t complete, and s t i l l i d e a l i z e d s i n c e , as has been seen i n e a r l i e r c h a p t e r s and w i l l a l s o be seen below, s e v e r a l approaches c a l l e d "heur i s t i c " do n o t s a t i s f y a l l o f t h e s e p r o p e r t i e s . We a r e s t i l l f a r o f f f r o m an adequate d e f i n i t i o n and comprehensive t h e o r y o f h e u r i s t i c s as i t e x i s t s f o r algorithms

One

consequence o f t h i s f a c t i s t h a t i t i s i m p o s s i b l e t o gener-

a l i z e f r o m a p a r t i c u l a r i n s t a n c e o f a h e u r i s t i c t o c l a s s e s o f problem domains, u n l e s s on a p u r e l y s p e c u l a t i v e b a s i s . I t i s a l s o i m p o s s i b l e t o d e r i v e some t h i n g l i k e an a n a l y t i c s o l u t i o n o v e r a whole parameter space, as e.g. i n s t a t i s t i c a l estimation. Therefore t h e e f f i c i e n c y c h a r a c t e r i s t i c s o f a h e u r i s t i c have t o be e x p l o r e d e m p i r i c a l l y e i t h e r by comparing s y s t e m a t i c a l l y t h e p r e d i c t e d performance o f d i f f e r e n t h e u r i s t i c s w i t h each o t h e r (Groner, 1978; Groner & Groner, 1582) o r by s i m u l a t i o n s t u d i e s ( L e n a t , 1983). 5. Heuristics in Judgment trnd Decision

Since t h e r e i s no u n i v e r s a l l y accepted d e f i n i t i o n f o r d e c i s i o n ( n e i t h e r i n current$esearch

n o r i n t h e prese1:t

book) which would c l e a r l y r u l e o u t n e i g h -

bour areas l i k e judgment, i n f e r e n c e , e t c . , we w i l l a l s o i n c l u d e examples f r o m t h e s e areas i n t h i s chapter, b e i n g w e l l aware t h a t o u r p r e s e n t a t i o n i s h i g h l y s e l e c t i v e and n o t even r e p r e s e n t a t i v e f o r a l l f i e l d s o f d e c i s i o n r e s e a r c h .

I t i s commonly accepted t h a t h e u r i s t i c approaches i n d e c i s i o n have been made p o p u l a r by H e r b e r t A. Simon (1955, 1956) w i t h t h e two r e l a t e d concepts o f "bounded r a t i o n a l i t y " and " s a t i s f i c i n g " .

I n t h e conventional a d d i t i v e u t i l i t y

models o f d e c i s i o n r e s e a r c h i t had been assumed t h a t t h e homo oeconomicus would behave r a t i o n a l l y by o p t i m i z i n g h i s expected u t i l i t y . The amount o f 1 ) I n a conference on methods o f h e u r i s t i c s ( U n i v e r s i t y of Bern, September 1981; proceedings p u b l i s h e d i n Groner, Groner & B i s c h o f , 1983) one o f t h e d i s c u s s i o n themes was t h e q u e s t i o n "How c o u l d h e u r i s t i c s be d e f i n e d ? " The comment o f M a r v i n Minsky t o t h i s q u e s t i o n was "Give me a d e f i n i t i o n o f l i f e " . Needless t o say t h a t n e i t h e r o f t h e concepts were a d e q u a t e l y def i n e d a t t h a t conference!

103

HEURISTICS IN MODELS OF DECISION

computation i n v o l v e d i n a r r i v i n g a t an o p t i m a l d e c i s i o n was n o t t a k e n i n t o account, an a l g o r i t h m i c d e c i s i o n scheme was i m p l i c i t l y assumed. Simon argues a g a i n s t t h i s c o n c e p t u a l i z a t i o n :

"The p o i n t o f d e p a r t u r e i s t h e

o b s e r v a t i o n t h a t human t h i n k i n g powers a r e v e r y modest when compared w i t h t h e c o m p l e x i t i e s o f t h e environments i r t which human b e i n g s l i v e . I f c o m p u t a t i o n a l powers were u n l i m i t e d , a person would s i m p l y c o n s u l t h i s o r h e r p r e f e r e n c e s ( u t i l i t y f u n c t i o n s ) and choose t h e course o f a c t i o n t h a t would y i e l d maximum u t i l i t y under t h e g i v e n circumstances. That i s , o f course, j u s t what t h e " r a t i o n a l man" o f c l a s s i c a l economic t h e o r y does. But r e a l human beings, o f bounded r a t i o n a l i t y , cannot f o l l o w t h i s procedure. f a c e d w i t h c o m p l e x i t y and u n c e r t a i n t y , l a c k i n g t h e w i t s t o o p t i m i z e , t h e y must be c o n t e n t t o s a t i s f i c e

-

t o f i n d "good enough" s o l u t i o n s t o t h e i r probiems and "good enough" courses

o f action. The f i r s t s p e c i f i c a t i o n , t h e n f o r a model o f t h i n k i n g processes i s t h a t i t be a model o f bounded r a t i o n a l i t y , t h a t i t i n c o r p o r a t e mechanisms f o r c o p i n g w i t h c o m p l e x i t y even when i t i s u n a b l e t o d i g e s t i t c o m p l e t e l y

... S a t i s f i -

c i n g p r o v i d e s an escape f r o m t h e d i f f i c u l t y t h a t , i n a complex w o r l d , t h e a l t e r n a t i v e s o f a c t i o n a r e n o t g i v e n b u t must be sought o u t . Since t h e search g e n e r a l l y t a k e s p l a c e i n a space t h a t i s e s s e n t i a l l y i n f i n i t e , some s t o p r u l e n:ust be imposed t o t e r m i n a t e problem s o l v i n g a c t i v i t y . The s a t i s f i c i n g c r i t e r i o n p r o v i d e s t h a t s t o p r u l e : search ends when a good-enough a l t e r n a t i v e i s found" (1979, p.3). The work on judgmental b i a s e s by Tversky & Kahneman bears some r e l a t i o n t o t h e n o t i o n o f bounded r a t i o n a l i t y . "When f a c e d w i t h t h e d i f f i c u l t t a s k o f j u d g f n g p r o b a b i l i t y o r frequency, p e o p l e employ a l i m i t e d number o f h e u r i s t i c s which reduce t h e s e judgn.lents t o s i m p l e r ones" ( T v e r s k y & Kahneman

,

1973,

p. 207). By t h e r e p r e s e n t a t i v e n e s s h e u r i s t i c , "people p r e d i c t t h e outcome t h a t appears most r e p r e s e n t a t i v e o f t h e evidence. Consequently, i n t u i t i v e p r e d i c t i o n s a r e i n s e n s i t i v e t o t h e r e l i a b i l i t y o f t h e evidence or t o t h e p r i o r p r o b a b i l i t y o f t h e outcome, i n v i o l a t i o n o f t h e l o g i c o f s t a t i s t i c e l p r e d i c t i o n " (Kahneman

104

COGNITIVE DECISION RESEARCH

& Tversky, 1973, p.237). However, r e p r e s e n t a t i v e n e s s as an e x p l a n a t o r y conc e p t s u f f e r s f r o m a s e r i o u s d i f f i c u l t y . i t i s much e a s i e r t o s t a t e u t o s t e r -

i o r i t h a t some evidence must have been r e p r e s e n t a t i v e f o r a c e r t a i n outcome, t h a n t o make a pmediction f o r which outcome an evidence w i i l be r e p r e s e n t a tive. The a v a i l a b i l i t y h e u r i s t i c s u f f e r s f r o m s i m i l a r d i f f i c u l t i e s : "A person i s s a i d t o employ t h e a v a i l a b i l i t y h e u r i s t i c whenever he e s t i m a t e s f r e q u e n c y o r p o s s i b i l i t y by t h e ease w i t h which i n s t a n c e s o r a s s o c i a t i o n s c o u l d be b r o u g h t t o mind" (p. 208). I n some sense, a v a i l a b i l i t y can be i n t e r p r e t e d as an a t t e n t i o n a l f a c t o r which r e g u l a t e s t h e search process e i t h e r i n memory o r i n a p e r c e p t u a l domain. T h i s l e a d s i m m e d i a t e l y t o t h e same o b j e c t i o n as w i t h t h e r e p r e s e n t a t i v e n e s s h e u r i s t i c , i . e . c i r c u l a r i t y o f argumentation: i f we do nut hzve d i r e c t access t o t h e a t t e n t i o n a l process b u t o n l y i n f e r f r o m t h e s u b j e c t ' s response what he ought t o have a t t e n d e d t o , t h e model w i l l become u n f a l s i f i a b l e . The amount t o which a t t e n t i o n a l processes can be manipul a t e d e x p e r i m e n t a l l y i s l i m i t e d . A c c o r d i n g t o Tversky & Kahneman (1973) t h e e m p i r i c a l evidence f o r t h e a v a i l a b i l i t y h e u r i s t i c i s t h e p o s i t i v e c o r r e l a t i o n between a frequency e s t i m a t i o n o f some event and t h e number o f i n s t a n c e s p r o duced by t h e same i n d i v i d u a l . We m i g h t p o l e m i c a l l y o b j e c t t h a t Tversky & Kahneman f e l l i n t o t h e t r a p o f t h e i r own h e u r i s t i c "causal scheme" by t a k i n g t h e s e c o r r e l a t i o n s as evidence

f o r causation.

W i t h r e s p e c t t o o u r l i s t o f common p r o p e r t i e s o f h e u r i s t i c methods o u t l i n e d i n c h a p t e r 4, we a l s o n o t i c e t h a t " h e u r i s t i c s " i n t h e sense o f Tversky & Kahneman l a c k almost a l l i m p o r t a n t a t t r i b u t e s s i n c e t h e y a r e n o t s t a t e d i n terms o f an unambiguous language such as i n f o r m a t i o n p r o c e s s i n g t h e o r y . T h i s makes them v u l n e r a b l e t o c r i t i c i s m s l i k e t h e ones s t a t e d above. R e c e n t l y t h e r e i s a s t r o n g t r e n d towards models o f d e c i s i o n based on t h e i n f o r m a t i o n p r o c e s s i n g approach (e.g.

Payne, 1976; Hogarth, 1980; Huber, 1982;

f o r r e v i e w s see E i n h o r n & Hogarth, 1981). The process as a whole i s decomposed i n a s e r i e s o f subprocesses, l i k e i n f o r m a t i o n a c q u i s i t i o n , e v a l u a t i o n , a c t i o n , and f e e d b a c k / l e a r n i n g ( E i n h o r n & Hogarth, 1981) which i n t e r a c t w i t h each o t h e r and a r e b u i l t up o f s t i l l s m a l l e r subprocesses. Three d i f f e r e n t

105

HEURISTICS IN MODELS OF DECISION

sources o f d a t a a r e used as e m p i r i c a l evidence f o r t h e p o s t u l a t e d processes: information

pick-up

f r o m d i s p l a y boards (e.g.

Payne, 19761, r e g i s t r a t i o n o f

eye f i x a t i o n s (Russo & Rosen, 1975; Groner, 1978; Groner & Groner, 19831, and solving-aloud

p r o t o c o l s (Einhorn, Kleinmuntz & Kleinmuntz, 1979; Huber,

1982). A lthough t h e v oc abu l a r y o f most approaches seems t o d i s p l a y much communal i t y on a s u p e r f i c i a l l e v e l , a c l o s e r e x a m i n a t i o n g i v e s a more v a r i e g a t e d p i c t u r e . Appar ent ly i t i s d i f f i c u l t t o p u t t h e d i f f e r e n t r e s e a r c h pieces t o g e t h e r , since

-

as we argued above

-

we s t i l l l a c k even t h e o u t l i n e o f a comprehen-

s i v e t h e o r y o f h e u r i s t i c s . I t can o n l y be hoped t h a t a l l t h e present p r o l i f i c model b u i l d i n g and assiduous e x p e r i m e n t a t i o n w i l l l e a d us somewhat c l o s e r t o t h a t goal.

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Godel, K. : 1931. Ueber formal unentscheidbare Satze d e r P r i n c i p i a Mathematica und verwandter Systeme I. I n : Monatshefte fiir Mathematik und Physik 38, 173-198 Groner, R. : 1978. Hypothesen i m Denkprozess. Bern: Huber Groner, R. & Groner, M. : 1982. Towards a general hypothetico-deductive t h e o r y o f c o g n i t i v e a c t i v i t y . I n : R. Groner & P. F r a i s s e (Eds.), C o g n i t i o n and eye movements. Ams terdam: North-Hol 1and Groner, R. & Groner, M.: 1983. A s t o c h a s t i c hypothesis t e s t i n g model f o r m u l t i t e r m s e r i e s problems based on eye f i x a t i o n s . I n : R. Groner/D.A. F i s h e r & R.A. Monty (Eds.), Eye movements and psychological f u n c t i o n s : I n t e r n a t i o n a l views. H i l l s d a l e , N.J.: Lawrence Erlbaum Groner, M./Groner, R. & Bischof, W.F.: 1983. Approaches t o h e u r i s t i c s : A h i s t o r i c a l reivew. I n : R. Groner/M. Groner & W.F. B i s c h o f (Eds.), Methods o f h e u r i s t i c s . H i l l s d a l e , N.J.: Lawrence Erlbaum de Groot, A. : 1983. H e u r i s t i c s , mental programs, and i n t e l l i g e n c e . I n : R. Groner/M. Groner & W.F. B i s c h o f (Eds.), Methods o f h e u r i s t i c s . H i l l s dale, N.J.: Lawrence Erlbaum Hadamard, J.: 1945. The psychology o f i n v e n t i o n i n t h e mathematical f i e l d . Princeton, N.J.: P r i n c e t o n U n i v e r s i t y Press Hankel, H.: 1874. Zur Geschichte d e r Mathematik i n A l t e r t u m und M i t t e l a l t e r . L e i p z i g : Teubner Hogarth, R.M. : 1980. Judgment and choice: The psychology o f d e c i s i o n . Chicester, U.K.: Wiley Huber, 0. : 1982. Entscheiden a l s Problemlosen. Bern: Huber Humphrey, G.: 1951. Thinking. An i n t r o d u c t i o n t o i t s experimental psychology. London: Methuen, and New York: Wiley

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I n : C.J. Gerhardt

Lenat, D.B.: 1983. Toward a theory o f h e u r i s t i c . I n : R. Groner/M. Groner & W.F. Bischof (Eds.), Methods o f h e u r i s t i c s . H i l l s d a l e , N.J.: Lawrence E r l baum L u l l us, R. : 1656. Ars Magna. Frankfurt: Cornelius Sutorius Mayer, A. & Orth, J.: 1901. Zur q u a l i t a t i v e n Untersuchung der Association. In: Z e i t s c h r i f t f i r Psychologie und Physiologie der Sinnesorgane 26, 1-13 M i l l e r , G.E. : 1911. Zur Analyse der Gedachtnistatigkeit und des Vorstellungsverlaufs. Leipzig: Barth Newel 1 , A. : 1980a. Reasoning, problem s o l v i n g and d e c i s i o n processes : t h e probelm space as a fundamental category. I n : R.S. Nickerson (Ed.), A t t e n t i o n and Performance V I I I . H i l l s d a l e , N.J. : Erlbaum Newell , A. : 1980b. Physical symbol systems. In: Cognitive Science 4, 135-184 Newell, A.: 1983. The h e u r i s t i c o f Georg Polya and i t s r e l a t i o n t o A r t i f i c i a l I n t e l l i g e n c e . I n : R. Groner/M. Groner & W.F. Bischof (Eds.), Methods o f h e u r i s t i c s . H i l l s d a l e , N.J.: Lawrence Erlbaum Newell, A./Shaw, J.C. & Simon, H.A.: 1958. Elements o f a theory o f human problem solving. I n : Psychological Review 65, 151-166 Newell, A. & Simon, H.A.: 1961. GPS, a program t h a t simulates human thought. I n : H. B i l l i n g (Ed.), Lernende Automaten. Munchen: Oldenbourg ( r e p r i n t e d A. Feigenbaum & J. Feldman (Eds.), Computers and thought. New York: McGraw-Hi 11 1963) Newell, A. & Simon, H.A.: Prentice-Hall

1972. Human problem solving. Englewood C l i f f s , N.J.:

Nilsson, N.J.: 1971. Problem-solving methods i n A r t i f i c i a l I n t e l l i g e n c e . New York: McGraw-Hill Payne, J.W.: 1976. Task complexity and contingent processing i n decision making: an information search and protocol analysis. I n : Organizational Behavior and Human Performance 16, 366-387 Poincar@, H.: 1913. Mathematical creation. I n : G.B. foundation o f science. New York: Science Press Polya, G.:

1945. How t o solve it. Princeton, N.J.:

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Polya, G.: 1954. Mathematics and p l a u s i b l e reasoning, Vol. I + 11. P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r s i t y Press Polya, G.: 1965

Mathematical d i s c o v e r y . New York: Wiley. Vol. 1: 1962; Vol. 2 :

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Selz, 0.: 1935. Versuch d e r Hebung des I n t e l l i g e n z n i v e a u s . Ein B e i t r a g z u r T h e o r i e d e r I n t e l l i g e n z und i h r e r e r z i e h l i c h e n B e e i n f l u s s u n g . I n : Z e i t s c h r i f t f u r P s y c h o l o g i e 134, 236-301 Simon, H.A.: 1955. A b e h a v i o r a l model o f r a t i o n a l c h o i c e . I n : Q u a r t e r l y J o u r n a l o f Economics 69, 99-118 Simon, H.A.: 1956. R a t i o n a l c h o i c e and t h e s t r u c t u r e o f t h e environment. I n : P s y c h o l o g i c a l Review 63, 129-138 Simon, H.A.: Press

1979. Models o f t h o u g h t . New Haven & London: Yale U n i v e r s i t y

Tversky, A. & Kahneman, D.: 1973. A v a i l a b i l i t y : A h e u r i s t i c f o r j u d g i n g f r e quency and p r o b a b i l i t y . In: C o g n i t i v e Psychology 5, 207-232 van d e r Waerden, B.L.: 1983. I n s p i r a t i o n and t h i n k i n g i n mathematics. I n : R. Groner/M. Groner & W.F. B i s c h o f (Eds.), Methods o f h e u r i s t i c s . H i l l s dale, N.J.: Lawrence Erlbaum Wallas, G.:

1926. The a r t o f thought. New York: H a r c o u r t & Brace

Wertheimer,

M. : 1945. P r o d u c t i v e t h i n k i n g . New York: Harper

Wickelgren, W.A.: 1974. How t o s o l v e problems: Elements o f a t h e o r y of problems and problem s o l v i n g . San F r a n c i s c o : Freeman Zwicky, F.:

1975. M o r p h o l o g i c a l astronomy. B e r l i n : S p r i n g e r

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

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KLAUS FIEDLER Universitat Giessen Fachbereich Psychologie F. R. Germany

I n t h i s a r t i c l e I am concerned w i t h t h e t h e o r e t i c a l i d e a t h a t s o c i a l j u d g ments a r e mediated o r even determined by t h e avaiZabiZity o f r e l e v a n t i n f o r matio n, t h a t i s , by t h e ease w i t h which d i f f e r e n t pieces o f memorized i n f o r matio n come t o t h e j u d g e ' s mind ( Tv e r s k y & Kahneman, 1973). I w i l l m a i n l y focus

on

one e m p i r i c a l phenomenon which has been e x p l a i n e d i n terms o f an

a v a i l a b i l i t y h e u r i s t i c , namely, t h e e g o c e n t r i c b i a s i n r e s p o n s i b i l i t y a t t r i b u t i o n s , ac c ord ing t o Ross & S i c o l y (1979) o r Thompson & K e l l e y (1981). When t h e p a r t n e r s i n a c l o s e p e r s o n a l r e l a t i o n s h i p j u d g e t h e i r own and t h e i r p a r t n e r ' s c o n t r i b u t i o n t o v a r i o u s a c t i v i t i e s (such as p l a n n i n g l e i s u r e a c t i v i t i e s o r br e ak ing promises), t h e y t e n d t o o v e r e s t i m a t e t h e i r own c o n t r i b u t i o n . T h i s e g o c e n t r i c b i a s has been e x p l a i n e d by r e f e r e n c e t o t h e e m p i r i c a l f a c t t h a t

Ss a r e more l i k e l y t o r e c a l l examples o f t h e i r own a c t i v i t i e s t h a n of t h e i r p a r t n e r ' s a c t i v i t i e s . Hence, d i f f e r e n t i a l r e c a l l p r o d u c t s a r e assumed t o cause t h e judgment b i a s . I am p r i m a r i l y concerned w i t h t h i s causal assumption.

As Ross and S i c o l y (p. 334) c o r r e c t l y admit, t h e r e i s "no d e f i n i t i v e evide nc e t h a t t h e b i a s i n a v a i l a b i l i t y Causes t h e b i a s i n r e s p o n s i b i l i t y . " Indeed, t h e main evidence i s m e r e l y c o r r e l a t i o n a l and i t may be noted, parent h e t i c a l l y , t h a t t h i s i s t r u e o f most a p p l i c a t i o n s o f t h e a v a i l a b i l i t y concept. F o r example, i n Thompson and K e l l e y ' s Study 2 , t h e mean c o r r e l a t i o n between e g o c e n t r i c judgments and t h e tendency t o r e c a l l s e l f - r e l a t e d i n s t a n ces was r=.50. B u t such a p i e c e o f c o r r e l a t i o n a l evidence may be due t o a s p u r i o u s r a t h e r t h a n a causal r e l a t i o n . Thus, i f an S's judgment t h a t he/she was more r e s p o n s i b l e f o r b r e a k i n g promises c o i n c i d e s w i t h r e c a l l i n g t h a t S h i m s e l f / h e r s e l f bro k e a promise, t h i s may s i m p l y r e f l e c t t h e f a c t t h a t he/she

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has r e a l l y broken more promises than h i s / h e r partner. To the e x t e n t t h a t r e c a l l and judgments are v e r i d i c a l , they w i l l tend t o be c o r r e l a t e d even though t h e process o f judgment formation may be independent o f any r e t r i e v a l processes. Indeed, the t h e o r e t i c a l weakness o f e x p l a i n i n g judgmental outcomes i n terms o f the a v a i l a b i l i t y h e u r i s t i c o r i g i n a t e s i n the very d i f f i c u l t y o f d i s t i n g u i s h i n g o p e r a t i o n a l l y between r e c a l l measures and judgmental measures. Memory-based judgments can a l s o be regarded as r e c a l l o f r e l e v a n t information, and v i c e versa. Thus, the two operations involved

--

i.e.,

judging t h a t one

has broken more promises and r e c a l l i n g t h a t one has broken more promises are s i m i l a r t o such a degree t h a t any causal explanation o f the former i n terms o f t h e l a t t e r must appear t h e o r e t i c a l l y empty.

--

T r i v i a l l y , any memory-based judgment involves r e c a l l operations and a c t i v a t e s r e l e v a n t memory content. However, the n o n - t r i v i a l version o f the a v a i l a b i l i t y p r i n c i p l e , as I understand it, s t a t e s t h a t judgmental deviations r e f l e c t r e c a l l biases operating a t the time o f judgment formation. This i s e s s e n t i a l l y an assumption about the process o f judgment formation, emphasizing the r o l e of r e t r i e v a l attempts w i t h i n t h a t process. The present research c o n s t i t u t e s an attempt t o overcome the methodologic a l d i f f i c u l t i e s inherent i n t e s t i n g the causal r o l e o f d i f f e r e n t i a l r e c a l l i n judgment formation. Egocentric biases were i n v e s t i g a t e d by having the part i c i p a n t s perform two c o g n i t i v e operations: (a) a judgment operation, t h a t i s having 5s judge t h e percentage of t h e i r own (vs. t h e i r p a r t n e r ' s ) c o n t r i b u t i o n t o 20 d i f f e r e n t a c t i v i t i e s ; and (b) an elementary r e c a l l operation, t h a t i s , having them r e c a l l one example f o r each a c t i v i t y . Judgment o f an a c t i v i t y item, say breaking promises, e i t h e r preceded o r followed the r e c a l l o f an example o f t h a t item. Thus, the temporal p o s i t i o n o f the two tasks was varied. The main dependent measures were the response l a t e n c i e s , i n p a r t i c u l a r , the f a c i l i t a t i o n e f f e c t (i.e., latency reduction) o f p r i o r r e c a l l on subsequent judgments and of p r i o r judgment on subsequent r e c a l l . O f course, f a c i l i t a t i o n can be expected i n both d i r e c t i o n s . As r e c a l l and judgment operations f o r the same a c t i v i t y r e f e r t o the same knowledge base i n memory, t h e r e should be a priming e f f e c t o f the f i r s t operation and a consequent reduction i n l a tency f o r the second operation. However, an a v a i l a b i l ity-based explanation has i m p l i c a t i o n s f o r the reZatioe degree of f a c i l i t a t i o n i n the two d i r e c tions.

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According to the differential recall explanation, the recall operation can be regarded as,a subroutine, as it were, of the judgment task; that is, a minimal recall operation i s included as part of the judgment process. For the arguments that follow, it is sufficient to state this inclusive relation more loosely as in the left part of Figure 1. The graphical representation means that if recall is a subprocess of judgment formation, more elements of an elementary recall task must be included in the judgment task than vice versa. Thus, if the judgment operation is performed prior to recall, there should be most facilitation, for most elements of a recall opzration should be already activated as part of the preceding judgment task. On the other hand, if recall is performed first, there should be less facilitation in later judgments, for a relatively smaller part of the judgment operation is acti-

Figure 1. Graphical representation of two possible relations between recall and judgment operations. ( J and R are the sets of cognitive elements making up judgment and recall, respectively.)

vated by the preceding recall task. Therefore, the idea that recall of relevant information is a mediating subprocess in the process o f judgment formation entails more facilitation in the judgment-first condition. An alternative theory -- that recall is the more inclusive operation -is depicted in theright part of Figure I. Of course, the predictions are opposite in this case. An implicit assumption underlying this kind of reasoning ought to be

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made e x p l i c i t , The assumption s t a t e s t h a t f a c i l i t a t i o n i s a p o s i t i v e monot o n i c f u n c t i o n of the degree t o which the to-be-primed m a t e r i a l i s included i n the primary task (i.e., the degree o f overlap), and t h a t response time i s independent, w i t h i n c e r t a i n l i m i t s , o f other features o f the priming task (i.e., the non-overlapping p a r t ) . Although there are s t i l l some problems, t h i s assumption i s a t l e a s t c o n s i s t e n t w i t h common p r i m i n g manipulations ( c f . Wyer & S r u l l , 1981; Higgins & King, 1981). Now l e t us consider t h e experimental design. Two independent v a r i a b l e s were crossed., The f i r s t f a c t o r r e f e r s t o the comparison o f the j u d g m e n t - f i r s t and the r e c a l l - f i r s t condition, as already mentioned. The other f a c t o r r e f e r s t o the manipulation depicted i n Figure 2. The two tasks o f making judgments and r e c a l l i n g instances were e i t h e r performed blockwise o r i n an aZtemating fashion. I n the blockwise condition, the f i r s t task (e.g., judgment) was f i r s t completed f o r a l l items before t u r n i n g t o the second task (e.g., r e c a l l ) w i t h the same 20 items i n the same order. I n the a l t e r n a t i n g c o n d i t i o n , however, both tasks were completed f o r the same item before proceeding t o the next item o f the series, thus switching between the d i f f e r e n t tasks from t r i a l to trial.

B1ockwi se

J,

J2

J,

J,

,,,

J20

IAAA4

A1 t e r n a t i ng

Figure 2. Schematic representation o f sequences o f t r i a l s i n the blockwise and a l t e r n a t i n g condition.

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Clearly, the f a c i l i t a t i o n e f f e c t s should be more pronounced i n t h e a l t e r n a t i n g c o n d i t i o n where t h e two operations on the same i t e m are performed i n close temporal succession and are n o t separated by much d i s t r a c t i n g a c t i v i t y , as i n tt.e blocked condition. However, i t might be t h a t f a c i l i t a t i o n i n t h e a l t e r n a t i n g c o n d i t i o n i s so pronounced t h a t d i f f e r e n c e s w i t h resnect t o r e c a l l o r j u d g m e n t - f i r s t cannot be sensibly detected. Hence, the blocked c o n d i t i o n , where much l e s s f a c i l i t a t i o n can be expected,might a f f o r d a more sensible t e s t . Before I come t o the empirical r e s u l t s , l e t me b r i e f l y summarize t h e main p r e d i c t i o n s t o be tested: The phenomenon t o be explained must be r e p l i c a b l e , t h a t i s , egocentric biases should occur and there should be a c o r r e l a t e d b i a s t o r e c a l l s e l f r e l a t e d examples. 0 According t o the usual a v a i l a b i l i t y - b a s e d explanation, f a c i l i t a t i o n i n the

judgment-first c o n d i t i o n ought t o be more pronounced. 0 Further, t h e r e should be substantial c o r r e l a t i o n s , w i t h i n judges across the

20 items, between r e c a l l l a t e n c i e s and judgment l a t e n c i e s , i f r e c a l l i s the determining subprocess i n judgment formation. 0 And these c o r r e l a t i o n s should be higher i n the a l t e r n a t i n g condition, due t o the temporal closeness. The material, c o n s i s t i n g o f a subset o f 20 a c t i v i t i e s selected from Thompson and K e l l e y ' s 36 a c t i v i t i e s , appears i n Table 1 along w i t h the mean percentage judgments o f

Ss' own c o n t r i b u t i o n . Obviously, the egocentric

b i a s was r e p l i c a t e d , t h a t i s , the p a r t i c i p a n t s judged t h e i r own c o n t r i b u t i o n t o be more than 50%, and t h i s tendency i s h i g h l y s i g n i f i c a n t (t=5.48; p<.OOI). Moreover, there i s a c o r r e l a t e d tendency t o r e c a l l s e l f - r e l a t e d examples. O r stated another way, judgments of Ss' own c o n t r i b u t i o n increase f o r those items i n which a s e l f - r e l a t e d example i s r e c a l l e d (t=2.86,~<02) and decrease f o r those items i n which a partner-related example i s r e c a l l e d (t=3.98, p<.OOI). Having r e p l i c a t e d the explanandum, we can now t u r n t o the latency data. Response l a t e n c i e s were transformed t o y i e l d normalized d i s t r i b u t i o n s and then transformed i n t o standard z-scores. Then the mean latency per i t e m was calculated separately f o r each experimental c o n d i t i o n . The r e s u l t i n g 20 blocks o f two by two latency values o f each item were subjected t o analyses o f variance. There were two r e l i a b l e e f f e c t s i n the r e c a l l as w e l l as the judgment data (Table 2 ) : Highly s i g n i f i c a n t main e f f e c t s f o r task order and s u b s t a n t i a l

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Table 1 S o c i a l a c t i v i t i e s used as s t i m u l u s m a t e r i a l a long w i t h mean judgment o f Ss' c o n t r i b u t i o n (I t e m s a r e r e t r a n s l a t e d f r o m German i n t o English) Planning j o i n t l e i s u r e a c t i v i t i e s I r r i t a t i n g the other C a r r y i n g t h e conversation when t h e two o f you a r e alone

52.4 % 56.8 41 58.8 %

C r i t i c i z i n g the other Demonstrating a f f e c t i o n f o r p a r t n e r

57.9 %

Complaining about t h e o t h e r Resolving c o n f l i c t s between t h e two o f you

54.5 %

Causing arguments t h a t occur between t h e two o f you Being s e n s i t i v e t o t h e o t h e r ' s needs A c t i n g possessively toward t h e o t h e r

50.9 % 54.7 % 57.2 % 54.9 %

Expressing f e a r s and w o r r i e s Considering p a r t n e r ' s i n t e r e s t s when p l a n n i n g a c t i v i t i e s

51.2 % 55.4 % 50.8 %

M a i n t a i n i n g s e l f - c o n t r o l d u r i n g an argument

46.3 %

Helping p a r t n e r w i t h h i s / h e r problems w i t h people and work Embarrassing p a r t n e r i n p u b l i c

50.6 % 59.6 %

I n i t i a t i n g d i s c u s s i o n s about t h e r e l a t i o n s h i p Being a s s e r t i v e i n your r e l a t i o n s h i p

56.8 % 54.8 % 56.8 %

Making an e f f o r t t o spend t i m e w i t h p a r t n e r Accepting c r i t i c i s m from p a r t n e r Taking o u t on p a r t n e r one's o u t s i d e f r u s t r a t i o n s

46.5 % 51.4 %

i n t e r a c t i o n s . As Table 2 shows, t h e t i m e r e q u i r e d t o judge i s much s h o r t e r i f r e l e v a n t i n f o r m a t i o n has been r e c a l l e d before, t h a t i s , i n t h e r e c a l l f i r s t c o n d i t i o n ( t h e main e f f e c t ) . However, t h e i n t e r a c t i o n i n d i c a t e s t h a t

t h i s i s l a r g e l y due t o t h e a l t e r n a t i n g c o n d i t i o n . Nevertheless, t h e small f a c i l i t a t i o n e f f e c t o f t h e blocked c o n d i t i o n i s s t i l l s i g n i f i c a n t a t t h e 5%1eve1

.

Turning now t o t h e r e c a l l l a t e n c i e s , t h e r e i s a g a i n much f a c i l i t a t i o n i n t h e a l t e r n a t i n g c o n d i t i o n , t h a t i s , reduced l a t e n c i e s f o r r e c a l l i n t h e judgm e n t - f i r s t c o n d i t i o n . However, t h i s t i m e t h e r e i s no f a c i l i t a t i o n whatsoever i n t h e blocked c o n d i t i o n .

TESTABILITY OF THE AVAILABILITY HEURISTIC

115

Table 2 Mean s t a t i s t i c s and ANOVA r e s u l t s f o r judgment l a t e n c i e s and r e c a l l l a t e n c i e s . (Values i n parentheses r e f e r t o untransformed d a t a )

udgment l a t e n c i e s :

B1 ockwise

A1 t e r n a t i n g

Judgment

Recall

first

first

0.20 (5.47)

-0.29 (4.04)

1.01 (7.82)

B1ockwise

0.16 (12.80)

Alternating

Significant effects Main e f f e c t columns: F1 ,,9=56.99; p<.OOI

-0.92

Interaction:

(3.31)

F

0.06 (12.84)

-1.18

0.96

(7.65)

(16.04)

1 ,I9

=14.99;

p<.OO2

Main e f f e c t columns: F1 ,19=68.63;

p<.OOI

I n t e r a c t i o n: F1 ,19=83.47;

p<.OOI

T h i s p a t t e r n o f r e s u l t s i s i n v a r i a n t w i t h r e g a r d t o t h e data t r a n s f o r mations. Table 2 a l s o i n c l u d e s t h e untransformed values which y i e l d essent i a l l y t h e same p i c t u r e : Reduction o f r e c a l l and judgment l a t e n c i e s by more than 50% i n t h e a l t e r n a t i n g c o n d i t i o n ; and i n t h e blocked c o n d i t i o n , no r e c a l l f a c i l i t a t i o n b u t considerable judgment f a c i l i t a t i o n . Hence, a t a s k s i t u a t i o n has been i d e n t i f i e d where t h e r e i s f a c i l i t a t i o n i n o n l y one d i r e c t i o n . Cont r a r y t o an a v a i l a b i l i t y - b a s e d e x p l a n a t i o n , o n l y p r i o r r e c a l l f a c i l i t a t e s l a t e r judgments, suggesting t h a t an elementary r e c a l l t a s k i s t h e more i n c l u s i v e operation. The r e c a l l task i s more time-consuming (as t h e a b s o l u t e l a t e n c i e s r e v e a l ) and i s experienced as r e q u i r i n g more e f f o r t . Even minimal r e c a l l (recallir!g b u t one example) seems t o be a deeper c o g n i t i v e process than j u d g i n g one's c o n t r i b u t i o n on a percent scale. I f t h e discrepancy between r e c a l l f a c i l i t a t i o n and judgment f a c i l i t a t i o n ( i n t h e blocked t a s k s i t u a t i o n ) were o n l y a m a t t e r o f degree, i t would be more

d i f f i c u l t t o i n t e r p r e t because we do n o t know which t i m e u n i t o f r e c a l l l a t e n -

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COGNITIVE DECISION RESEARCH

c y corresponds t o which t i m e u n i t o f judgment l a t e n c y . F o r t u n a t e l y , however, t h e dis c re panc y i s a q u a l i t a t i v e one i n t h a t t h e f a c i l i t a t i o n e f f e c t i s conf i n e d t o o n l y one d i r e c t i o n . F u r t h e r n e g a t i v e evidence a g a i n s t t h e assumption t h a t r e c a l l mediat es o r even determines t h e process o f judgment f o r m a t i o n stems f rom c o r r e l a t i o n a l analyses. The c o r r e l a t i o n s between r e c a l l l a t e n c i e s and judgmental l a t e n c i e s (computed w i t h i n judges a c r o s s t h e 20 i t e m s ) a r e remarkably l o w and a r e n o t h i g h e r i n t h e a l t e r n a t i n g as compared t o t h e b l o c k w i s e c o n d i t i o n (t =0. 21;

-

ralt = .20, yblock= .18). Ifr e c a l l o p e r a t i o n s were t h e most e s s e n t i a l subr o u t i n e s o f t h e judgmental process, we would have expect ed somewhat h i g h e r c o r r e l a t i ons. The reader may f u r t h e r have r e c o g n i z e d t h a t i n b o t h l a t e n c y ANOVAs t h e r e was no main e f f e c t f o r t h e a l t e r n a t i n g v e r sus blocked comparison. A f i n a l glanc e a t T a b l e 2 c l a r i f i e s why t h i s i s t h e case: I n t h e a l t e r n a t i n g c o n d i t i o n , t h e g a i n i n response t i m e f o r t h e t a s k i n t h e second p o s i t i o n i s

c a n c e l l e d o u t by an increase o f response t i m e f o r t h e first t a s k . F o r example, t h e r e c a l l l a t e n c i e s o f about 17 sec f o r t h e r e c a l l - f i r s t c o n d i t i o n a r e w e l l above t h e 12 sec f o r t h e b l o c k e d c o n d i t i o n s which may be regarded as a basel i n e ( t h e same i n h i b i t i o n appears w i t h r e s p e c t t o judgment l a t e n c i e s ) . Obv i o u s l y , s w i t c h i n g between r e c a l l and judgment t a s k s ( i n t h e a l t e r n a t i n g cond i t i o n ) i n t e r f e r e s w i t h t h e c o g n i t i v e processing. T h i s s u b s i d i a r y f i n d i n g i s a l s o h a r d l y c omp a t i b l e w i t h t h e assumption t h a t r e c a l l i s a n a t u r a l l y o c c u r r i n g subprocess embedded i n t h e process o f judgment f ormat ion. L e t us f i n a l l y c o n s i d e r t h e c o n t e n t o f t h e r e c a l l out put s. As a l r e a d y noted, t h e c o r r e l a t i o n between e g o c e n t r i c a t t r i b u t i o n s and a tendency t o r e c a l l s e l f - r e l a t e d examples s h o u l d n o t be i n t e r p r e t e d t o mean t h a t r e c a l l i s c a u s a l l y i n v o l v e d i n t h e process of judgment f o r m a t i o n . P o o l i n g across a l l judges and items, 19% o f t h e r e c a l l responses a r e s e l f - r e l a t e d , 7% a r e p a r t n e r - r e l a t e d , and 74% have no c l e a r person r e f e r e nce. F u r t h e r , 41% o f t h e same r e c a l l responses can be regarded as c o n c r e t e e p i s o d i c events, 52% as ( p r o t o - ) t y p i c a l examples w i t h o u t e p i s o d i c r e f e r e n c e , and 7% a r e n o t codable i n t h i s respec t . Hence, t h e most t y p i c a l r e c a l l o u t p u t i s u n i n f o r m a t i v e w i t h r e g a r d t o which p a r t n e r c o n t r i b u t e d most i n t h e p a s t . I t seems, t h e r e f o r e , t h a t t h e c o g n i t i v e c o n d i t i o n s f o r an a v a i l a b i l i t y - b a s e d mechanism a r e s i m p l y n o t f u l f illed.

TESTABILITY OF THE AVAILABILITY HEURISTIC

117

Concluding Remarks On the whole; t h e evidence reported here i s c l e a r l y i n c o n s i s t e n t w i t h Ross and S i c o l y ' s (1979) and Thompson and K e l l e y ' s (1981) a v a i l a b i l i t y account f o r the phenomenon o f egocentric a t t r i b u t i o n s . The nature o f the f a c i l i t a t i o n e f f e c t s , the within-judge c o r r e l a t i o n s between r e c a l l l a t e n c i e s and judgment l a t e n c i e s , the contents o f the r e c a l l outputs, and the i n h i b i t i o n e f f e c t s i n the a l t e r n a t i n g c o n d i t i o n a l l seem t o i n d i c a t e t h a t the judgmental process i s n o t causally dependent on r e c a l l operations. One might argue t h a t t h e present o p e r a t i o n a l i z a t i o n i s inappropriate: I m p l i c i t r e c a l l operations t h a t a r e i n volved i n the process o f judgment formation cannot be experimentally simulated by such an e x p l i c i t r e c a l l task. I am p r i n c i p a l l y i n c l i n e d t o agree w i t h t h i s argument. However, i t should be noted t h a t the p o s i t i v e evidence which seems t o support the a v a i l a b i l i t y conception r e s t s on the same questionable opera t i o n a l i z a t i o n s . O f course, a research method t h a t i s acceptable f o r conf i r m a t i o n must a l s o be acceptable f o r disconfirmation. There i s no reason t o generalize the present f i n d i n g s t o a l l kinds o f s o c i a l judgments and t o completely dismiss the a v a i l a b i l i t y theory. We can imagine many s i t u a t i o n s i n which judges have t o r e l y on whatever happens t o be a v a i l a b l e i n memory, e s p e c i a l l y when required t o make r e t r o s p e c t i v e judgments cases,

OR

past issues t h a t they have never thought o f before. I n many o t h e r however, the s o c i a l judge can already a n t i c i p a t e the requirement t o

judge a t the time when information i s perceived and encoded i n memory. I t i s most l i k e l y i n these cases t h a t judgments are already formed a t the e a r l y encoding stage and incorporated as an i n t e g r a l p a r t o f the memory representat i o n ( c f . L i n g l e & Ostrom, 1979). I n t u i t i v e l y , I would suggest t h a t most judgments made i n t h i s experiment about Ss' own and t h e i r p a r t n e r ' s c o n t r i butions belong t o t h i s class o f pre-formed judgments. I n c i d e n t a l l y , i t should be noted how w i l l i n g l y and prematurely the a v a i l a b i l i t y hypothesis has been accepted and used t o e x p l a i n other judgmental phenomena. I t seems curious t o me t h a t the idea has never been r e a l l y challenged. As I understandthe nature o f s c i e n t i f i c growth, strong and f r u i t f u l theories always g i v e r i s e t o t h e o r e t i c a l and empirical controversies. But there i s v i r t u a l l y no controversy around t h e a v a i l a b i l i t y theory ( o r some version o f i t ) . Consider, as another example, the a v a i l a b i l i t y - b a s e d account o f i l l u sory c o r r e l a t i o n s (Chapman & Chapman, 1967, 19691, t h a t i s , the i l l u s i o n t h a t

COGNITIVE DECISION RESEARCH

118

a c o r r e l a t i o n c o u l d be observed i n a s e r i e s o f events where a c t u a l l y none was present. Since Tversky & Kahneman's (1973) suggestion t h a t " t h e v a r i o u s sources o f i l l u s o r y c o r r e l a t i o n s can a l l be e x p l a i n e d by t h e o p e r a t i o n o f a s i n g l e

--

t h e assessment o f a v a i l a b i l i t y " (p. 2241, repeated attempts have been made t o e x p l a i n i l l u s o r y c o r r e l a t i o n s v i a s e l e c t i v e r e c a l l but, t o my knowledge, n o t a s i n g l e p u b l i s h e d experiment has t r i e d t o demonstrate t h a t

mechanism

t h e o r i g i n o f t h e i l l u s i o n may be a l r e a d y l o c a t e d i n t h e p e r c e p t i o n o r encod i n g of t h e observations. I n my own ( s t i l l unpublished) research on t h e judgment of c o v a r i a t i o n , I have r e p e a t e d l y found t h a t judgmental d e v i a t i o n s can be ( p a r t l y ) p r e d i c t e d from i n d i v i d u a l judges' s u b j e c t i v e i n t e r p r e t a t i o n s o f t h e s t i m u l u s events (i.e., from encoding). Moreover, u s i n g a f r e e r e c a l l t e s t i n f i v e experiments as a measure o f which s t i m u l u s observations a r e spontaneously a v a i l a b l e i n memory -- I was never a b l e t o support a d i f f e r e n t i a l r e c a l l mechanism. Hence, my own evidence does n o t f i t v e r y w e l l w i t h t h e

--

p i c t u r e t h a t emerges from t h e l i t e r a t u r e . Here I have recognized again t h a t -- a biased search i n f a v o r o f an a v a i l a b i l i t y ex-

a lack o f controversy

--

planation c h a r a c t e r i z e s t h e research on i l l u s o r y c o r r e l a t i o n s . My i n t e n t i o n s here, and my conclusions, can be summarized as f o l l o w s : F i r s t , I wanted t o d i r e c t some a t t e n t i o n t o t h e f a c t t h a t c e r t a i n r e -

--

search t o o l s t h a t a r e common f o r t h e c o g n i t i v e p s y c h o l o g i s t such as analyz i n g response l a t e n c i e s -- have n o t been u t i l i z e d t o s u b s t a n t i a t e t h e n o t i o n o f judgmental h e u r i s t i c s . Second, t h e r e a r e t o o few o v e r t c o n t r o v e r s i e s i n t h e f i e l d o f person memory and judgment, i n t h e l i g h t o f t h e s e r i o u s problems a c r i t i c a l - m i n d e d researcher must r e a l i z e i n e x i s t i n g t h e o r i e s . Controversies r e f l e c t t h e growth o f s c i e n t i f i c know1edge, T h i r d and f i n a l l y , t h e m o n i s t i c search f o r a v a i l a b i l i t y phenomena, t h a t i s , f o r r e t r i e v a l e f f e c t s on judgments, may have d i s t r a c t e d researchers from an e q u a l l y promising approach, namely, t o i n v e s t i g a t e t h e r o l e o f encoding e f f e c t s i n judgment formation.

References Genesis o f popular b u t erroneous psychoChapman, L.J. & Chapman, J.P. d i a g n o s t i c observations. Journal o f Abnormal Psychology, 1967, 3, 193-204.

TESTABILITY OF THE AVAILABILITY HEURISTIC

Chapman, L.J. & Chapman, J.P. Illusory correlations as an obstacle to the use of valid psychodiagnostic signs. Journal of Abnormal Psychology, 1969, 74, 271-280. Higgins, E.T. & King, G. Accessibility o f social constructs: Informationprocessing consequences o f individual and contextual variability. In N. Cantor & J.F. Kihlstrom (Eds.) Personality, cognition, and social interaction. Hillsdale, N.J.: Erlbaum, 1981. LinQle, - - J.H. & Ostrom, T.M. Retrieval selectivitv in memorv-based impression judgments. Journal of Personality and Social bsychology, 1979, 37, 180-194.

ROSS, M.

& Sicoly, F. Egocentric biases in availability and attribution. Journal o f Personality and Social Psychology, 1979, 37, 322-336.

Thompson, S.C. & Kelley, H.H. Judgments of responsibility for activities in close relationships. Journal of Personality and Social Psychology, 1981 , 41 , 469-477.

Tversky, A. & Kahneman, 0. Availability: A heuristic for judging frequency and probability. Cognitive Psychology, 1973, 5, 207-232. Wyer, R.S.,Jr. , & Srull , T.K. Category accessibility: Some theoretical and empirical issues concerning the processing of social stimulus information. In E.T. Higgins, C.P. Herman, & M.P. Zanna (Eds.) Social cognition: The Ontario Symposium (Vol. 1). Hillsdale, N.J.: Erlbaum, 1981.

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

121

PROBLEM STRUCTURING CALCULI A N D LEVELS OF KNOWLEDGE REPRESENTATION IN DECISION MAKING

DlNA BERKELEY

PATRICK HUMPHREYS

London School of Economics and Political Science Decision Analysis Unit and Social Psychology Department U.K.

Introduction: D i f f i c u l t i e s with the representation of knowledge i n decision making tasks

The t h e s i s we p r e s e n t i n t h i s paper i s t h a t i n any s t u d y o f i n d i v i d u a l d e c i s i o n making i t i s e s s e n t i a l t o t a k e i n t o account, q u i t e , e x p l i c i t l y , t h e s o c i a l c o n t e x t i n which t h e d e c i s i o n making a c t i v i t i e s t a k e place, and w i t h i n which t h e d e c i s i o n maker c o n c e p t u a l i z e s t h e consequences o f h i s o r h e r a c t ions

.

T h i s t h e s i s i s n o t new, i n f a c t i t was t h e f o u n d a t i o n f o r B e r n o u i l l i ' s (1738) d i s c u s s i o n o f how a ' p r u d e n t ' p e r s o n s h o u l d choose between a l t e r n a t i v e a c t s , i n terms o f t h e impact t h a t t h e p o s s i b l e consequences would have on t h e immediate s o c i a l c o n t e x t ; i f t h e p r u d e n t p e r s o n was a merchant, t h i s would be h i s business, and so on.

However, t h e s e ideas formed p a r t o f n o r m a t i v e

d e c i s i o n t h e o r y , p r e s c r i b i n g t h e f o r m o f 'economic r a t i o n a l i t y ' t h a t s h o u l d c h a r a c t e r i z e a p r u d e n t businessman, o r whatever.

A n a l y s i s o f how t h e s o c i a l

c o n t e x t i s t a k e n i n t o account i n t h e process o f d e c i s i o n making has l a r g e l y been concerned w i t h examining how a c t i o n s chosen and i n f e r e n c e s made b y people i n v o l v e d i n d e c i s i o n making t a s k s

conpare w i t h t h e p r e s c r i p t i o n s o f

t h e r e p r e s e n t a t i o n o f t h e d e c i s i o n making t a s k i n a n o r m a t i v e model o f t h e situation.

122

COGNITIVE DECISION RESEARCH

Proceeding i n t h i s manner has e f f e c t i v e l y excluded f r o m a n a l y s i s t h e two aspects o f d e c i s i o n making w i t h which t h i s paper w i l l be p r i n c i p a l l y concerned.

t h e way t h a t knowledge which t h e d e c i s i o n maker These are, (i)

b e l i e v e s t o be r e l e v a n t i s represented, and, (ii)how, on t h e b a s i s o f t h i s knowledge, t h e d e c i s i o n maker generates a s t r u c t u r e r e p r e s e n t i n g h i s o r h e r v i e w o f t h e problem, t h r o u g h a l i n k a g e between k e y elements ( a c t s , events, consequences) and w i t h i n which c o n t e n t ( b e l i e f s , values, p r e f e r e n c e s ) can be e x p l o r e d and m a n i p u l a t e d i n s e a r c h i n g f o r a p r e s c r i p t i o n f o r a c t i o n . The reason why these aspects have been excluded i s t h a t i n c o n v e n t i o n a l d e c i s i o n making experiments, t h e r e p r e s e n t a t i o n o f t h e d e c i s i o n making t a s k a r r i v e s p r e - s t r u c t u r e d a t t h e s t a r t o f t h e experiment. W i t h i n what Kahneman and Tversky (1982) c a l l e d t h e ' c o n v e r s a t i o n a l p a r a d i g m ' , t h e e x p e r i m e n t e r has i n mind a r e p r e s e n t a t i o n o f t h e t a s k which he o r she conveys t o t h e s u b j e c t t h r o u g h t h e e x p e r i m e n t a l i n s t r u c t i o n s . A l l t h e knowledge r e q u i r e d f o r c o n s t r u c t i n g a ' n o r m a t i v e l y c o r r e c t ' s t r u c t u r e i s presumed t o be c o n t a i n e d i n t h e s e i n s t r u c t i o n s , a l t h o u g h i n some e x p e r i m e n t a l s i t u a t i o n s i t i s assumed q u i t e l i k e l y t h a t t h e s u b j e c t w i l l i g n o r e p a r t o f t h i s i n f o r m a t i o n (e.g. i n f o r m a t i o n about base r a t e s i n a r e f e r e n c e p o p u l a t i o n ) and develop an i n c o q l e t e task s t r u c t u r e (c.f.

Bar-Hi 1 l e l , 1982).

I n ' r e a l l i f e ' d e c i s i o n making, e v e r y t h i n g i s n o t so neat. I n s t r u c t i o n s a r e n o t n e a t l y a v a i l a b l e a t t h e o u t s e t o f each d e c i s i o n t a s k , s p e c i f y i n g t h e r e l e v a n t knowledge and i n d i c a t i n g how t o s t r u c t u r e it. O f t e n s t r u c t u r e s do n o t a r r i v e r e a d y f o r use, b u t have t o be generated i n r e p r e s e n t i n g d e c i s i o n problems, so t h a t p r e f e r e n c e s and u n c e r t a i n t i e s can be e x p l o r e d i n d e c i d i n g what t o do.

I n t h i s w i d e r c o n t e x t i t i s our t h e s i s t h a t t h e s o c i a l w o r l d i n

i t s many m a n i f e s t a t i o n s as c u l t u r e and language i n t h e m a c r o l e v e l and as f a m i l y , schoo1,work environment , p e e r group, etc., i n t h e m i c r o l e v e l n o t o n l y determines t h e t e r r a i n o f d e f i n i t i o n s and t h e range o f u n d e r s t a n d i n g an i n d i v i d u a l member can use and p e r f o r m i n h i s o r h e r d e c i s i o n making a c t i v i t i e s b u t a l s o p r o v i d e s t h e person w i t h t h e arena i n which h i s o r h e r c h o i c e s and a c t i o n s w i l l be t r i e d o u t , j u s t i f i e d , i n t e r p r e t e d by o t h e r s and rewarded o r sanctioned. Under such an u n d e r s t a n d i n g o f t h e r e l a t i o n s h i p o f t h e i n d i v i d u a l t o h i s o r h e r environment, t h e i n d i v i d u a l i s seen n o t as p a s s i v e , a c t e d upon by t h e environment, b u t as a c t i v e , i n t e r p r e t i n g , a c t i n g upon and changing t h a t environment.

CALCULI AND KNOWLEDGE REPRESENTATION

123

Adopting t h i s view has i m p l i c a t i o n s f o r t h e t y p e s o f a n a l y t i c models t h a t may be used i n c h a r a c t e r i z i n g d e c i s i o n making.

Under t h e ' c o n v e r s a t i o n a l

paradigm', w i t h i t s use i n c o n t e x t - i s o l a t e d m a i n l y l a b o r a t o r y based t a s k s , i t seemed q u i t e reasonable t o t a k e o v e r models developed by p s y c h o l o g i s t s i n p e r c e p t u a l judgement t a s k s , and much o f t h e t e r m i n o l o g y used i n p s y c h o l o g i c a l d e c i s i o n making m d e l s , l i k e ' a n c h o r i n g and a d j u s t m e n t ' ( T v e r s k y and Kahneman, 1974) o r ' v e r i d i c a l p e r c e p t i o n ' ( T v e r s k y and Kahneman, 19811, p o i n t s t o these r o o t s (c.f.

Johnson, 1972).

With t h i s view o f judgement came t h e emphasis on ' c h a r a c t e r i s t i c s o f t h e r e c e i v e r ' o f t h e i n f o r m a t i o n s u p p l i e d ( c . f . Swets, Tanner and B i r d s a l l , 1961) i n terms o f l o o k i n g f o r suboptimal performance ( a c c o r d i n g t o t h e n o r m a t i v e s t a n d a r d ) o r ' b i a s ' . T h i s emphasis was p e r v a s i v e ; t h e n o t i o n o f ' h e u r i s t i c s ' was o r i g i n a l l y a s s o c i a t e d w i t h i n n o v a t i v e a c t i v i t y w i t h t h e aim o f i n c r e a s i n g scope (Newell and Simon, 1972) o r e f f i c i e n c y (Thorngate, 1980) i n p r o b l e m s o l v i n g a c t i v i t i e s , s a v i n g energy and t i m e . However, i n accounts o f d e c i s i o n making experiments w i t h i n t h e c o n v e r s a t i o n a l paradigm, i t b e c a w l i n k e d w i t h t h e n o t i o n o f b i a s t o t h e e x t e n t t h a t t h e two terms seem t o have a c q u i r e d n e a r l y e q u i v a l e n t meaning ( c . f .

N i s b e t t and R O S S , 1980; Kahneman, S l o v i c and

Tversky, 1982). I n l a t e r papers f o l l o w i n g t h i s approach (e.g., Kahneman and Tversky, 1979) one can f i n d whole networks o f h e u r i s t i c s b e i n g p i e c e d t o g e t h e r t o account f o r complex p a t t e r n s i n b i a s e s which appear when t h e r e s u l t s o f a number o f experiments are c o n s i d e r e d t o g e t h e r . A l l t h i s assumes t h a t t h e i n s t r u c t i o n s presented t o s u b j e c t s p r o v i d e a

r e p o s i t o r y o f knowledge c o n c e r n i n g t h e t a s k s i t u a t i o n which i s s u f f i c i e n t t o p e r m i t t h e c o n s t r u c t i o n o f a p r o b l e m r e p r e s e n t a t i o n w i t h i n which a s u b j e c t ' s a c t u a l d e c i s i o n making performance can be compared w i t h t h a t which i s c o n s i d e r e d t o be ' n o r m a t i v e l y . c o r r e c t ' . However, i n cases where we have analysed t h e way s u b j e c t s a c t u a l l y s t r u c t u r e d e c i s i o n p r o b l e m on t h e b a s i s of i n s t r u c t i o n s p r e s c r i b e d f o r t y p i c a l t a s k s w i t h i n t h e c o n v e r s a t i o n a l paradigm (e.g. t h o s e c i t e d b y Tversky and Kahneman 1974, 1981; Kahneman and Tversky, 1979) v e r y l a r g e i n d i v i d u a l d i f f e r e n c e s have been found i n t h e way s u b j e c t s s t r u c t u r e t h e t a s k s ( P h i l l i p s , 1983). Moreover, t h e s e d i f f e r e n c e s a r e n o t due t o v a r y i n g degrees o f ' i n c o m p l e t e n e s s ' when compared w i t h t h e s t r u c t u r e c i t e d as b e i n g n o r m a t i v e l y c o r r e c t by t h e w r i t e r o f t h e instructions.

I n s t e a d , t h e y r e s u l t f r o m e l a b o r a t i o n s generated by t h e

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COGNITIVE DECISION RESEARCH

s u b j e c t when f a c e d w i t h under-determined and incomplete p r o b l e m d e s c r i p t i o n s ( t h e u s u a l case). The r e s u l t i n g p r o b l e m r e p r e s e n t a t i o n s a r e n o t n e c e s s a r i l y ' s u b - o p t i m a l ' compared t o t h o s e used by t h e e x p e r i m e n t e r i n comp t i n g t h e ' n o r m a t i v e l y expected' i n f e r e n t i a l values.

Indeed t h e o p p o s i t e

s usually

t h e case: s u b j e c t s were o f t e n found t o c r e a t e b e t t e r e l a b o r a t e d s t r u c t u r e s p r o v i d i n g a more complete d e s c r i p t i o n o f t h e a c t u a l t a s k c o n t e x t w i t h i n which t h e s u b j e c t s l o c a t e d themselves. We do n o t wish t o imply t h e c o n c l u s i o n t h a t " e x p e r i m e n t e r s should t r y t o w r i t e b e t t e r i n s t r u c t i o n s f o r t h e i r s u b j e c t s " . Rather, we would l i k e t o s t r e s s the i m p o s s i b i l i t y o f ensuring t h a t subjects w i t h a v a r i e t y o f d i f f e r e n t g o a l s and d i f f e r e n t h i s t o r i e s o f d e c i s i o n making a c t i v i t i e s w i l l e v e r generate f u l l y e q u i v a l e n t problem r e p r e s e n t a t i o n s f o r any p a r t i c u l a r s e t o f i n s t r u c t i o n s g i v e n t o them i n a l a b o r a t o r y s e t t i n g .

I t i s even l e s s

l i k e l y t h a t such communality c o u l d e v e r be ensured i n any r e a l - l i f e c o n t e x t , i n which each d e c i s i o n maker w i l l b r i n g h i s o r h e r d i f f e r e n t h i s t o r y , and d i f f e r e n t r e f e r e n t s f o r t h e meaning o f consequences, e t c . ( B e r k e l e y , 1982).

I n our o p i n i o n , t h e c r u c i a l p o i n t t o n o t e i n s t u d y i n g t h e e f f e c t s o f h e u r i s t i c s and b i a s e s i n f l u e n c i n g human d e c i s i o n making i s t h a t t h e i n f e r e n c e s made and values r e p o r t e d b y s u b j e c t s a r e c o n d i t i o n a l n o t o n l y on t h e i n f o r m a t i o n p r o c e s s i n g h e u r i s t i c s employed b u t also, and more i n p o r t a n t l y , on t h e f o r m o f t h e p a r t i c u l a r s t r u c t u r e generated by t h e s u b j e c t as h i s o r h e r v i e w o f t h e r e p r e s e n t a t i o n o f t h e problem. Hence i m p r o v i n g d e c i s i o n making, o r t r a i n i n g d e c i s i o n makers, r e q u i r e s f i r s t and f o r e m o s t t h a t we understand and a i d t h e s t r u c t u r i n g process.

Conditional rationality: The importance o f ' . f i ~ n' pstructure

D e c i s i o n t h e o r y has t r a d i t i o n a l l y been s i l e n t about i s s u e s i n v o l v e d i n problem s t r u c t u r i n g .

P a r t o f t h i s reason has been t h e c e n t r a l p o s i t i o n o f

t h e i d e a o f 'economic r a t i o n a l i t y ' i n t h e t h e o r y , and t h e g u i d i n g r o l e i t has p l a y e d s i n c e t h e beginning. When d i s c u s s e d as a concept w i t h i n t h e l i t e r a t u r e , r a t i o n a l i t y i s u s u a l l y t a k e n t o r e f e r t o t h e axioms o f t h e t h e o r y , l i k e coherence, t r a n s i t i v i t y , and adherence t o t h e s u b j e c t i v e

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CAL CULl AND KNOWLEDGE REPRESEN TA TION

expected u t i l i t y p r i n c i p l e .

T h i s d e f i n i t i o n o f t h e concept we s h a l l c a l l t h e

t h e o r e t i c a l d e f i n i t i o n i n t h e sense t h a t i t a r t i c u l a t e s t h o s e which, i n t h e o r y , are t h e c h a r a c t e r i s t i c s o f a r a t i o n a l a c t o r o r a c t i o n . However, t e s t i n g t h e e x i s t e n c e o f t h e s e p r i n c i p l e s o r l a c k t h e r e o f i n any g i v e n a c t o r i s achieved o n l y v i a an o p e r a t i o n a l i z a t i o n o f t h i s d e f i n i t i o n . The o p e r a t i o n a l d e f i n i t i o n o f r a t i o n a l i t y i n v o l v e s t h e employment o f f o r m a l models whose c o n s t r u c t i o n i s founded on these axioms. When o p e r a t i o n a l i z e d , t h e s e f o r m a l models e s s e n t i a l l y c o n s i s t o f c o m p o s i t i o n r u l e s ( K r a n t z e t a l , 1971) o p e r a t i n g w i t h i n an a x i o m a t i s e d decomposition o f t h e phenomena o f i n t e r e s t , which i n d e c i s i o n t h e o r y are immediate courses o f a c t i o n . For exanple, SEU p r o v i d e s an a d d i t i v e c o m p o s i t i o n r u l e p r e s c r i b e d when immediate a c t s are deconposed t h r o u g h a c t - e v e n t sequences i n t o consequences a t t h e d e c i s i o n h o r i z o n which, i n t u r n , a r e assigned u t i l i t i e s ( K r a n t z , 1972). MUT p r o v i d e s a c o n p o s i t i o n r u l e f o r o b t a i n i n g values o f t h e s e u t i l i t i e s when t h e y themselves have been decomposed i n t o r e l a t i v e p r e f e r e n c e values on a t t r i b u t e s c h a r a c t e r i z i n g t h o s e consequences (Humphreys,l977);

and

so on. Typically, empirical t e s t i n g o f a d e c i s i o n t h e o r e t i c composition r u l e i n v o l v e s a c o n p a r i s o n procedure l i k e t h a t shown i n f i g u r e 1. I n cases where a d i s c r e p a n c y i s n o t e d between t h e two o u t p u t s i n f i g u r e 1, t h e u s u a l argument i s e i t h e r t o c r i t i c i z e t h e adequacy o f t h e model, (and maybe d i s c u s s i n g how t o r e p l a c e i t w i t h one whose o u t p u t converges more c l o s e l y w i t h t h a t o b t a i n e d f r o m s u b j e c t s , c . f . , Kahneman and T v e r s k y l s 1979 "Prospect Theory") o r t a k i n g t h e model as s p e c i f y i n g t h e i d e a l type, and i d e n t i f y i n g t h e d i f f e r e n c e s i n o u t p u t s as a ' b i a s ' on t h e p a r t o f t h e s u b j e c t . ' C o r r e c t i v e a c t i o n ' would t h e n t a k e on t h e c h a r a c t e r i s t i c s o f ' d e b i a s i n g ' ( F i s c h h o f f , 1982).

I n e i t h e r case, w h i l e e m p i r i c a l t e s t i n g i s c a r r i e d o u t on t h e o p e r a t i o n a l d e f i n i t i o n o f t h e concept o f r a t i o n a l i t y , t h e r e s u l t s o f t h e t e s t i n g a r e taken t o r e f l e c t r a t i o n a l i t y o r i r r a t i o n a l i t y a t the t h e o r e t i c a l l e v e l . However, such i n f e r e n c e s o n l y have meaning i f a common u n d e r s t a n d i n g e x i s t s between s u b j e c t and e x p e r i m e n t e r about ends, means and r e p r e s e n t a t i o n o f knowledge i n t h e p r o b l e m s t r u c t u r e w i t h i n which t h e comparison o f d a t a

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COGNITIVE DECISION RESEARCH

i n d i c a t e d i n F i g u r e 1 i s t o t a k e p l a c e (Hovland, 1952). I n o t h e r words, comparisons l i k e t h o s e i n d i c a t e d i n f i g u r e 1 p o l i c e

conditional rationality. That i s , r a t i o n a l i t y can o n l y be i n v e s t i g a t e d c o n d i t i o n a l on a c h i e v i n g a common u n d e r s t a n d i n g o f t h e f o r m o f t h e problem s t r u c t u r e w i t h i n which content i s m a n i p u l a t e d , whether a c c o r d i n g t o t h e subjects' i n t u i t i v e calculus, o r according t o a t h e o r e t i c formal c a l c u l u s a x i o m a t i z e d on t h e b a s i s o f ' r a t i o n a l ' assunptions. F i g u r e 1:

Comparison procedure used i n g e n e r a t i n g d a t a f o r e m p i r i c a l t e s t i n g o f a decision t h e o r e t i c composition r u l e

Small world constituting problem universe accessed i n comolz by intuitive system (i.e., subject) and normative system

Data D

I x t u i t i v e model used by s u b jec t

Normutiue mode2 (involving the operation of e.g. Bayes' rule or SEU composition rule)

(involving the operation of heuristics)

t -

t

Hypothesis H (input as question) (output resulting from question)

(output resulting from question)

( s o u r c e : B e r k e l e y and Humphreys 1982, F i a u r e 2 ) T h i s common u n d e r s t a n d i n g serves t o t e n p o r a r i l y ' f i x ' t h e f o r m o f t h e d e c i s i o n problem r e p r e s e n t a t i o n w i t h i n which i n f e r e n c e s w i l l be drawn and p r e f e r e n c e s r e v e a l e d b y a l l t h e p a r t i e s i n v o l v e d ( s u b j e c t and experimenter;

CA LCULl AND KNOWLEDGE REPRESENTATION

127

opposing p a r t i e s i n t r e a t y n e g o t i a t i o n s , i n t e r e s t groups i n s o c i a l d e c i s i o n making, e t c . ) .

B e r k e l e y and Hunphreys (1982) d i s c u s s how ' f i x i n g o f t h e

f o r m ' o f a d e c i s i o n problem may i n v o l v e t h e r e s o l u t i o n o f no l e s s than seven d i f f e r e n t types o f uncertainty. Two o f these concerned f i r s t , u n c e r t a i n t y about how t h e d e c i s i o n maker w i l l f e e l , and wish t o act, h a v i n g a r r i v e d a t a subsequent a c t c h o i c e p o i n t a f t e r i n t e r v e n i n g events have u n f o l d e d i n r e a l t i m e ( E l s t e r , 1979) ; and second, u n c e r t a i n t y about t h e e x t e n t t o which one possesses agency f o r i n d u c i n g changes i n t h e p r o b a b i l i t i e s o f subsequent events t h r o u g h b e i n g a b l e t o a l t e r r e l a t i o n s between s t a t e s o f t h e w o r l d (Savage, 1954; V a r i and Vecsenyi, 1983).

Handling these two o f t h e seven t y p e s o f u n c e r t a i n t y a t an i n d i v i d u a l

l e v e l i n v o ves r e s o l v i n g e x i s t e n t i a l problems c o n c e r n i n g t h e p o s i t i o n of o n e ' s s e l f o r t h e o t h e r i n t h e s o c i a l c o n t e x t and l i e o u t s i d e t h e scope o f t h i s paper (Parkes, 1971, discusses i s s u e s o f t h i s t y p e i n terms o f I

psychosoc a1 t r a n s i t i o n s I 1.

Four o f t h e r e m a i n i n g types o f u n c e r t a i n t y can be t a k e n i n t o account i n representation o f decision p r o b l e m , given the c u r r e n t s t a t e o f decision t h e o r y . These r e l a t e t o (i)t h e p r o b a b i l i t i e s o f outcomes o f subsequent events, c o n d i t i o n a l on what has preceded them i n t h e a c t - e v e n t sequence , (ii)t h e p r o b a b i l i t i e s o f outcomes o f subsequent e v e n t s c o n d i t i o n a l on o t h e r extraneous events , (iii)how t o i n c o r p o r a t e p r i o r i n f o r m a t i o n , and ( i v ) how t o c o n c e p t u a l i z e t h e worth o f consequences and, when more t h a n one c r i t e r i o n o f ' w o r t h ' i s i n v o l v e d , how t o combine t h e s e c r i t e r i a . B e r k e l e y and Humphreys (1982) d i s c u s s how d e c i s i o n t h e o r y p r o v i d e s t h e b a s i s f o r r e p r e s e n t i n g t h e way i n which t h e s e f o u r t y p e s o f u n c e r t a i n t y were r e s o l v e d i n r e p r e s e n t i n g any p a r t i c u l a r d e c i s i o n problem t h r o u g h f o u r d i f f e r e n t types o f s t r u c t u r e , each h a n d l i n g one t y p e o f u n c e r t a i n t y . These a r e shown i n t h e boxes i n f i g u r e 2, t o g e t h e r w i t h t h e i n t e r f a c e s between them which may be i n v o l v e d i n d e v e l o p i n g a complete p r o b l e m r e p r e s e n t a t i o n . F i g u r e 2 r e p r e s e n t s t h e c u r r e n t s t a t e of development o f t h e I s u b j e c t i v e ' v i e w of d e c i s i o n t h e o r y p i o n e e r e d b y Edwards (1954; Edwards, Lindman and P h i l l i p s , 1965).

A l l t h e s e systems a r e c o n s i d e r e d t o be i n t e r f a c e d d i r e c t l y w i t h t h e

d e c i s i o n maker's semantic memory. D e c i s i o n p r o b l e m r e p r e s e n t a t i o n s b u i It

COGNITIVE DECISION RESEARCH

128

Figure 2 :

Systems f o r m a l i z e d w i t h i n D e c i s i o n Theory The f o u r systems ( c o r e system, e v e n t b u f f e r system, u t i l i t y b u f f e r system and p r o b a b i l i t y b u f f e r system) a r e o f d i f f e r e n t s t r u c t u r a l types, b u t t h e i n t e r f a c e s between them a r e f u l l y s p e c i f i e d w i t h i n t h e d e c i s i o n t h e o r e t i c framework.

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L~Z.C'SSP&t i O ? l L J

f o r inmediate acts

(source: B e r k e l e y and Humphreys 1982, F i g u r e 1) t h r o u g h t h e use o f t h e s e systems a r e c o n s t r u c t e d on t h e b a s i s o f knowledge r e p r e s e n t e d i n memory b u t do n o t i n themselves d e s c r i b e knowledge s t r u c t u r e s w i t h i n memory. N e i t h e r , i t should be p o i n t e d o u t , do t h e y d e s c r i b e o b j e c t s i n t h e p h y s i c a l world.

' S t a t e s o f t h e w o r l d ' may be named a t nodes l o c a t e d

w i t h i n d e c i s i o n t h e o r e t i c s t r u c t u r e s , b u t what i s r e p r e s e n t e d i s t h e d e c i , s i o n maker's b e l i e f c o n c e r n i n g t h e s u b j e c t i v e p r o b a b i l i t i e s o f these s t a t e s o f t h e

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CALCULI AND KNOWLEDGE REPRESENTATION

w o r l d ' s occurrence under some scenario o f t h e f u t u r e , mediated through h i s o r her knowledge s t r u c t u r e s i n semantic memory. T h i s r a i s e s the fundamental problem o f how t o r e s o l v e t h e seventh, and perhaps most fundamental type o f u n c e r t a i n t y , which Hogarth, Michaud and k r y (1980) c a l l e d procedural u n c e r t a i n t y , and described as " u n c e r t a i n t y concerning means t o handle o r process t h e decision, e.g.

specifying relevant

u n c e r t a i n t i e s , what i n f o r m a t i o n t o seek, and where, how t o i n v e n t a l t e r n a t i v e s and assess consequences, etc." Decision theory,

in i t s

conventional form, i s q u i t e s i l e n t about how t o handle t h i s type o f u n c e r t a i n t y . Yet handling, and r e s o l v i n g procedural m a t e r i a l i s always involved i n t r a n s l a t i n g the generalized p o s s i b i l i t i e s f o r decision theoretic s t r u c t u r e s shown i n f i g u r e 2 i n t o any a c t u a l problem r e p r e s e n t a t i o n which c o u l d be considered t o be ' r e q u i s i t e ' ( P h i l l i p s , 1982) i n r e p r e s e n t i n g and e x p l o r i n g a p a r t i c u l a r problem. Any attempt t o f o r m a l i z e ways i n which procedural u n c e r t a i n t y may be r e s o l v e d i m p l i e s t h e development o f a problem s t r u c t u r ing c a l c u 1us.

Attempts t o build a problem structuring caleulus Berkeley and Hunphreys (1982) i d e n t i f i e d t h r e e d i f f e r e n t approaches t o b u i l d i n g a problem s t r u c t u r i n g c a l c u l u s .

The f i r s t i n v o l v e s c o n s t r u c t i n g a

simple taxonomy o f t a s k s where a n a l y t i c c h a r a c t e r i s t i c s o f a l t e r n a t i v e d e c i s i o n t h e o r e t i c s t r u c t u r e s are matched t o t a s k c h a r a c t e r i s t i c s (Brown and U l v i l a , 1977, Howell and Burnett, 1978).

Two d i f f i c u l t i e s are i n v o l v e d here.

F i r s t , the same problem can be s t r u c t u r e d w i t h i n any one o f a number o f q u i t e d i f f e r e n t a n a l y t i c f r a m w o r k s (Brown, 1978) w i t h no normative way o f d e c i d i n g between them. Secondly, according t o von W i n t e r f e l d t ( l 9 8 0 ) , " s u b s t a n t i v e r a t h e r than a n a l y t i c c h a r a c t e r i s t i c s i d e n t i f y r e a l problems". Von W i n t e r f e l d t (1980) reviewed and r e j e c t e d e x i s t i n g taxonomies, which, f o l l o w i n g t h e f i r s t approach, used m a i n l y a n a l y t i c a l c a t e g o r i e s t o d i s t i n g u i s h problems as being i l l - s u i t e d t o lead a d e c i s i o n maker t o a c l a s s o f problems which had s i m i l a r c h a r a c t e r i s t i c s f o r t h e problem under c o n s i d e r a t i o n . Instead, he f o l l o w e d t h e second approach t o b u i l d i n g a problem s t r u c t u r i n g c a l c u l u s : developing a problem taxonomy, based on

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COGNITIVE DECISION RESEARCH

s u b s t a n t i v e r a t h e r t h a n a n a l y t i c f e a t u r e s o f t a s k s . T h i s taxonomy was designed t o be p r o b l e m - d r i v e n , b u t s t i l l g e n e r a l i z a b l e . For exanple, one c o u l d f i n d a ' p r o t o t y p i c a l s t r u c t u r e ' f o r budget a l l o c a t i o n i n one c e l l of t h e taxonomy, a ' p r o t o t y p i c a l s t r u c t u r e ' f o r f a c i l i t y s i t i n g

i n another

c e l l , and so on. The i d e a b e h i n d s t o r i n g p r o t o t y p i c a l s t r u c t u r e s i n t h e taxonomy i s t h a t t h e s e would s t i l l be g e n e r a l i z a b l e . I n any s p e c i f i c budget a l l o c a t i o n problem, one would e l a b o r a t e t h e a p p r o p r i a t e p r o t o t y p i c a l s t r u c t u r e s u p p l i e d by t h e taxonomy t o meet s p e c i f i c t a s k requirements. The p r o b l e m i s t h a t even when o n l y p r o t o t y p i c a l s t r u c t u r e s are d e f i n e d , b u i l d i n g up a taxonomy o f them according t o s u b s t a n t i v e f e a t u r e s o f problems s t i l l r e q u i r e s a g r e a t number o f c e l l s , each o f which has t o be i n v e s t i g a t e d on an ad-hoc b a s i s . A t h i r d approach, which we s a i d we p r e f e r r e d , was t o employ a c a l c u l u s

c o m p r i s i n g more p o w e r f u l o p e r a t o r s t h a n t h e s i m p l e ' s e a r c h and match' r e q u i r e d i n o p e r a t i n g a taxonomy. We suggested t h a t i t should be p o s s i b l e t o develop a g e n e r a t i v e p r o b l e m s t r u c t u r i n g c a l c u l u s u s i n g p r o d u c t i o n system r u l e s ( c . f . Anderson, 1976, sect.3.3) o p e r a t i n g i n a way analogous t o t h e r e w r i t i n g r u l e s proposed by Chomsky (1957) as t h e b a s i s f o r g e n e r a t i v e grammars: ' r e - w r i t i n g ' s t r u c t u r a l elements i n t o t h e complete s t r u c t u r e w i t h i n which c o n t e n t can t h e n be manipulated. T h i s i d e a was proposed b y Bonczek, H o l s a p p l e and Whinston (1981), as a method f o r g e n e r a t i n g d e c i s i o n s u p p o r t systems, b u t t h e y l i m i t e d t h e i r d i s c u s s i o n t o systems where t h e s t r u c t u r e s were computer programs, and r e - w r i t i n g was p e r f o r m e d on t h e computer language ' s t r u c t u r a l elements' ( i . e . statements w i t h i n t h e program). To us, i t seemed t h a t t h i s i d e a m i g h t w e l l have a much wider a p p l i c a t i o n . Chomsky (1957) and Bonczek e t a1 (1981) had i n d e p e n d e n t l y c o n s i d e r e d two s p e c i f i c , b u t q u i t e d i f f e r e n t s e t s o f grammatical r u l e s ( t h o s e o f E n g l i s h and those o f F o r t r a n ) . F i l l m o r e (1968) , however, has d i s c u s s e d how more g e n e r a l grammars can be c o n s t r u c t e d f o r use i n s u b s t a n t i v e a p p l i c a t i o n s .

The k e y p r o p e r t y o f

F i l l m o r e ' s 'case grammar' approach i s t h a t i n such a grammar, s t r u c t u r a l elements a r e ' t a g g e d ' a c c o r d i n g t o t h e f u n c t i o n a l r e l a t i o n each element p r o v i d e s ( t h e ' c a s e ' i t t a k e s ) when l i n k e d i n t o t h e s t r u c t u r e . Hence, case grammars, u n l i k e Chomsky's p u r e l y s y n t a c t i c , t r a n s f o r m a t i o n a l - g e n e r a t i v e grammar, a r e p r o b l e m d r i v e n and w i t h t h i s comes a l l t h e advantages which von W i n t e r f e l d t (1980) p o i n t e d o u t .

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131

However, t h i s s t i l l leaves one v e r y c r u c i a l problem: how should one s p e c i f y t h e n a t u r e o f t h e s t r u c t u r a l elements t o be m a n i p u l a t e d as c o n t e n t b y casegrammar t y p e p r o d u c t i o n systems i n g e n e r a t i n g d e c i s i o n problem s t r u c t u r e s over t h e enormous range o f a p p l i c a t i o n s and c o n t e x t s where d e c i s i o n making a c t i v i t i e s a r e c a r r i e d o u t ? I f a s i n g l e - l e v e l s t r u c t u r e - g e n e r a t i n g scheme were proposed, t h e number o f elements would be v a s t , as general-purpose ' n a t u r a l ' languages have t o r e l y on v e r y l a r g e v o c a b u l a r i e s . Thus, it would be f u t i l e t o a t t e n p t t o s p e c i f y a l l t h e s t r u c t u r a l elements i n a c a l c u l u s g e n e r a t i n g d e c i s i o n problem r e p r e s e n t a t i o n s a t a s i n g l e l e v e l .

As

an a l t e r n a t i v e t h i s paper i n v e s t i g a t e s what a m u l t i - l e v e l p r o b l e m s t r u c t u r i n g c a l c u l u s m i g h t comprise, and how t h i s l e a d s t o m u l t i p l e l e v e l s o f knowledge r e p r e s e n t a t i o n c o n c e r n i n g d e c i s i o n making tasks.

The p r o b l e m - s t r u c t u r i n g systems we reviewed above a l r e a d y comprise two l e v e l s : a lower l e v e l c a l c u l u s

manipulating content w i t h i n defined s t r u c t u r e (c.f.

f i g u r e 1) and one

operating a t a higher level o f abstraction i n generating t h a t s t r u c t u r e w i t h i n a p a r t i c u l a r (assumed t e n p o r a r i l y f i x e d ) p r o b l e m c o n t e x t . But why stop a t j u s t two l e v e l s o f a b s t r a c t i o n ? To do t h i s would be t o o v e r l o o k t h e i n s i g h t i n P i a g e t ' s (1972, 1977) account o f g e n e t i c e p i s t e m l o g y t h a t , i n a c c o u n t i n g f o r t h e p r o g r e s s o f knowledge, t h e r e i s no reason t o s t o p a t t h e second l e v e l o f a b s t r a c t i o n .

I n a 'two l e v e l ' d e c i s i o n making scheme

what i s r e p r e s e n t e d as f o r m a t t h e f i r s t l e v e l , i s m a n i p u l a t e d as content a t t h e second l e v e l . B u t t h i s r e l a t i o n may be c o n t i n u e d t h r o u g h f u r t h e r l e v e l s of a b s t r a c t i o n , t h a t i s , what i s r e p r e s e n t e d as f o r m a t t h e second l e v e l , can be m a n i p u l a t e d as c o n t e n t a t t h e t h i r d l e v e l , and so on. T h i s i m p l i e s , o f course, t h a t t h e c o n t e n t m a n i p u l a t e d a t each l e v e l i s q u a l i t a t i v e l y d i f f e r e n t , as a r e t h e p r o g r e s s i v e l y m r e p o w e r f u l o p e r a t o r s which become a v a i l a b l e t o a person who can understand t h e p r i n c i p l e s u n d e r l y i n g t h e i r employment w i t h i n t h e c a l c u l u s f o r m a l i z e d a t each p r o g r e s s i v e l e v e l o f a b s t r a c t ion. A d o p t i n g such an approach g r e a t l y s i m p l i f i e s t h e p r o c e s s o f d e c i s i o n problem modeling : a p r o b l e m may be r e p r e s e n t e d a t a number o f l e v e l s o f c o n c e p t u a l i z a t i o n , b u t we may be a b l e t o p r o v i d e a p o w e r f u l and r e l a t i v e l y s i m p l e d e s c r i p t i o n o f t h e p r o b l e m a t each o f t h e l e v e l s i n v o l v e d . The success o f such an e n t e r p r i s e , however, h i n g e s on t h e f a c t t h a t we a r e

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132

p r o p o s i n g a m u l t i - l e v e l scheme f o r c o n c e p t u a l i z i n g d e c i s i o n problems.

Where

i s t h e evidence t h a t d e c i s i o n makers themselves c o n c e p t u a l i z e problems a t a number o f l e v e l s , and how a r e these c o n c e p t u a l i z a t i o n s r e l a t e d t o t h e c o n t e x t o f t h e d e c i s i o n problem? The most comprehensive r e v i e w o f evidence f o r t h e s e p r o p o s i t i o n s t h a t we know o f i s t h a t p r o v i d e d by t h e s t u d i e s o f b u r e a u c r a t i c o r g a n i z a t i o n s c a r r i e d o u t b y E l l i o t t Jaques and h i s c o l l e a g u e s (Jaques 1976, 1982; Jaques e t a l , 1978). W h i l e Jaques' account o f ' l e v e l s o f a b s t r a c t i o n ' w i t h i n t h e h i e r a r c h y o f r o l e s occupied by p e r s o n n e l i n such o r g a n i z a t i o n s does n o t c o n s t i t u t e a microcosm which can s e r v e as a model o f any and a l l s o c i a l c o n t e x t s , i t has t h e advantage t h a t i t a l l o w s us t o examine t h e r e l a t i o n s h i p between l e v e l s o f r e s p o n s i b i l i t y , l e v e l s o f demand c h a r a c t e r i s t i c s o f t a s k s and l e v e l s o f knowledge r e p r e s e n t a t i o n i n s t r u c t u r i n g t h e d e c i s i o n problems i n v o l v e d i n these tasks.

I n o t h e r words, i t p r o v i d e s a c o n c e p t u a l l a b o r a t o r y f o r

e x p l o r i n g p o s s i b l e c h a r a c t e r i s t i c s o f a m u l t i - l e v e l scheme i n a v a r i e t y o f p r a c t i c a l s e t t i n g s i n t h e way we i l l u s t r a t e i n t h e n e x t s e c t i o n .

The paper

concludes by s u g g e s t i n g some i m p l i c a t i o n s f o r t h e s t u d y o f d e c i s i o n making i n g e n e r a l , beyond t h e c o n f i n e s o f a c t i v i t i e s c a r r i e d o u t w i t h i n i d e a l i z e d bureaucracies

.

Levels of abstraction in decision problem representations .The l e f t hand c o l u m i n t a b l e 1 g i v e s a b r i e f summary o f t h e e q u i v a l e n c e e s t a b l i s h e d by Jaques (1976, 1982) between l e v e l s o f a b s t r a c t i o n o f r o l e , r e s p o n s i b i l i t y (expressed i n terms o f ' t i m e - s p a n ' ) and o f demand c h a r a c t e r i s t i c s o f d e c i s i o n makers w i t h i n b u r e a u c r a t i c o r g a n i z a t i o n s . According t o Jaques, t h e qua1 i t a t i v e d i f f e r e n c e s between t h e l e v e l s o f o r g a n i z a t i o n a l r o l e shown i n t h e f i r s t c o l u m can b e u n d e r s t o o d i n terms o f p r o g r e s s i v e l e v e l s o f a b s t r a c t i o n i n t h e symbolic c o n s t r u c t i o n o f a c t i o n s which may be c a r r i e d o u t by e x e c u t i v e s a t each l e v e l . Moreover, t h e s e l e v e l s a r e n o t viewed as a s p e c i f i c p r o d u c t o f o r g a n i z a t i o n a l forms, r a t h e r b u r e a u c r a t i c l e v e l s are p a r a s i t i c o n l e v e l s o f a b s t r a c t i o n o f ( i d e a l i z e d ) tasks w i t h i n organizations,

Table I :

Comparison of demand characteristics of tasks facing personnel having responsibilities at a given organisational level with structuring compatibilities required in representing decision problems at that level.

Level 5

4

Drganisational level in bureaucratic employment hierarchy Corporate subsidiary] enterprise (managing director work system) General management (of e.g.. development production 0~ sales and within work system.

Time span inherent in problem representation at given level

5

2

-

10 years

-5

years

Wmand characteristics of tasks facing personnel with responsibility at given level

Structuring capabilities required in representing decision problems at given level

Problems not dealt with in context set wholly from above can modify boundaries of business within policy or i.e., define work system.

Articulation of principles guiding conditional (goal) closing of an open system, andlor re-opening of a conditionally closed system.

Detachment.from specific cases, seeing them representative examples of issues calling f o r development of a system.

Selectinglinterfacing capability between structural types (DSS generation from scratch or DSS generator construction).

--

-

3

2

I

Department manageriall principal specialist

I

-2

years

Front-line manageriall professional

3 months

Shop and office floor

le6s than 3 months

to I year

-

Control of trend of tasks and problems arising. Extrapolation from trend to ways of formulating problems.

Re-structuring capability within single fixed structural type (e.g., criteria generation in multi-criteria model).

Formal operational, can anticipate changes iu tasks due to any one of: demand, object, p r o G t i o n resource, pathway, or pathway resource.

Manipulation of data on one variable at a time within fixed structure (e.g., sensitivity analysis).

Concrete operational, limited to taks concretely and physically at hand.

Estimation of values at nodes within fixed structure (e.g., informational

.=

c)

C

r-, b

2 0 3 2

0

z

I-

: 0

rn

a rn

a"rn

i 3

a

2

a

W

134

COGNITIVE DECISION RESEARCH

The second column i n t a b l e 1 g i v e s the t y p i c a l time-span i n t h e problem r e p r e s e n t a t i o n which i s r e q u i s i t e f o r a task c a r r i e d o u t by an executive a t each l e v e l . Time span i n d i c a t e s how f a r away t h e d e c i s i o n h o r i z o n (Toda,1976) i s s e t f o r t a s k s f o r which t h e d e c i s i o n maker i s h e l d r e s p o n s i b l e w i t h i n t h e organizational context. For example, a person whose r o l e i n an o r g a n i z a t i o n i s s e t a t l e v e l 1 m i g h t be given t h e r e s p o n s i b i l i t y i n v o l v e d i n minding a machine t o o l . It would m a t t e r f o r h i s career prospects w i t h i n t h e o r g a n i z a t i o n i f he f a i l e d t o decide on a p p r o p r i a t e a c t i o n given concrete evidence o f inpending m a l f u n c t i o n o f t h a t machine. This same person may a l s o g a i n evidence o f o r g a n i z a t i o n a l m a l f u n c t i o n a t a h i g h e r l e v e l (e.g.,

through n o t i c i n g t h a t the manufactured

product i s beginning t o look i n c r e a s i n g l y outmoded, compared w i t h those o f competitors which he o r she has looked a t i n h i g h s t r e e t shop windows). I n t h i s case he o r she may d e s c r i b e t o colleagues what "should be done about t h i s " , o r even submit t h i s i n f o r m a t i o n f o r c o n s i d e r a t i o n a t a h i g h e r l e v e l through a 'suggestion box". However, he o r she would n o t be expected t o decide on t h e proposed a c t i o n , o r be held r e s p o n s i b l e f o r i t s consequences. The t h i r d column i n t a b l e 1 summarizes the d e s c r i p t i o n o f the demand c h a r a c t e r i s t i c s o f t h e tasks f a c i n g personnel w i t h r e s p o n s i b i l i t y a t a g i v e n l e v e l i n an o r g a n i z a t i o n g i v e n by Jaques (1982,

pp 80-85).

Jaques was

p r i m a r i l y concerned w i t h b u r e a u c r a t i c o r g a n i z a t i o n s comprising i d e a l i z e d h i e r a r c h i e s o f work s t r a t a , and i n any a c t u a l o r g a n i z a t i o n a l c o n t e x t we may f i n d personnel a t p a r t i c u l a r o r g a n i z a t i o n a l l e v e l s a l s o r e s p o n s i b l e f o r c a r r y i n g o u t tasks a t lower l e v e l s ( r a t h e r than d e l e g a t i n g them t o subordinates) o r t a k i n g i n i t i a t i v e s a t more than one l e v e l where t h e r o l e s t r u c t u r e p e r m i t s (e.g.,

as i n t h e case o f c o n s u l t a n t s o r "problem f i x e r s " ) .

However, t h e p r e c i s i o n o f t h e equivalence between o r g a n i z a t i o n a l r o l e s and task c h a r a c t e r i s t i c s i s n o t our major concern here. Instead, we were p a r t i c u l a r l y i n t e r e s t e d i n the way Jaques described t h e q u a l i t a t i v e d i f f e r e n c e s between tasks a t t h e v a r i o u s l e v e l s o f a b s t r a c t i o n ( ' q u a l i t a t i v e ' differences

are always i m p l i c i t i n t h e use o f t h e term ' l e v e l ' ;

see Bunge,

1969). I t suggested t o us t h a t we might f i n d p a r a l l e l l e v e l s o f a b s t r a c t i o n i n s t r u c t u r i n g c a p a b i l i t i e s r e q u i r e d i n generating r e p r e s e n t a t i o n s o f d e c i s i o n

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135

problems. The r e s u l t s o f our p r e l i m i n a r y i n v e s t i g a t i o n s i n t h i s r e g a r d a r e o u t l i n e d i n column 4 o f t a b l e 1, and are d i s c u s s e d w i t h some b r i e f examples below. LeveZ 1

S t a r t i n g a t t h e b o t t o m o f t a b l e 1, we f i n d problem p r o c e s s i n g b e i n g handled a t t h e c o n c r e t e o p e r a t i o n a l l e v e l ( l e v e l 1). P i a g e t (1972) d e s c r i b e s k e y p r i n c i p l e s which have t o be understood i n c o n t r o l l i n g o p e r a t i o n s a t t h i s l e v e l as those i n v o l v e d i n s e r i a t i o n (ACE; B
Brown and L i n d l e y ,

1982). However t h e o p e r a t i o n s a c t u a l i z e d a t t h i s l e v e l are l i m i t e d t o p r o v i d i n g b e s t assessments o f q u a n t i t i e s (e.g., p o s i t i o n on a p r o b a b i l i t y o r on a p r e f e r e n c e s c a l e ) t o be r e p r e s e n t e d as c o n t e n t a t a d e f i n e d node i n a f i x e d problem s t r u c t u r e . I n o t h e r words, t h e t y p e o f q u e s t i o n addressed h e r e i s what i s t h e b e s t assessllent o f a p r o b a b i l i t y o f a p a r t i c u l a r e v e n t i n t h e c u r r e n t c o n t e x t ? A good example o f d e c i s i o n making a c t i v i t i e s l o c a t e d a t t h i s l e v e l i s Zimmer's (1983; t h i s volume) account o f p r o c e s s i n g o f s u b j e c t i v e p r o b a b i l i t y assessments ( f r o m t h e s u b j e c t ' s r a t h e r t h a n t h e e x p e r i m e n t e r ' s v i e w p o i n t ) . E f f o r t s t o i n p r o v e t h e q u a l i t y o f d e c i s i o n making l o c a t e d e x c l u s i v e l y a t t h i s l e v e l have c e n t e r e d around a t t e m p t s t o ensure t h a t d e c i s i o n makers a r e " w e l l - c a l i b r a t e d " ( L i c h t e n s t e i n , F i s c h h o f f and P h i l l i p s , 1982). I n each case t h e problem i s seen as how t o ensure t h a t t h e answer g i v e n r e p r e s e n t s t h e d e c i s i o n maker's b e s t a s s e s s m n t s based on h i s o r h e r c u r r e n t knowledge o f s t a t e s of t h e world.

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Level 2 Moving up t o l e v e l 2 i n v o l v e s understanding t h e p r i n c i p l e s u n d e r l y i n g what P i a g e t (1972) c a l l e d f o r m a l o p e r a t i o n a l thought. P i a g e t describes t h e fundamental c h a r a c t e r i s t i c o f c o g n i t i v e a b i l i t i e s a t t h i s l e v e l as t h e " c a p a c i t y t o deal w i t h hypotheses i n s t e a d o f s i m p l y o b j e c t s " , hypotheses which are expressed as p r o p o s i t i o n s r a t h e r than f a c t s . The q u a l i t a t i v e d i f f e r e n c e

between adjacent l e v e l s i s always t h a t what i s

represented as f o r m a t the lower l e v e l i s manipulated as c o n t e n t a t t h e higher level.

P i a g e t (1971; 1972) describes t h e key formal o p e r a t i o n s a t

l e v e l 2 as comprising t h e INRC group: o p e r a t i o n a l i z i n g p r i n c i p l e s r e l a t i n g t o I n v e r s i o n , Negation, R e c i p r o c i t y and C o r r e l a t i o n .

While each o f these types

o f operations may be involved i n a d e c i s i o n maker's attempt t o p r o v i d e ' b e s t assessmentsa i n l e v e l 1 problem s o l v i n g ,

i t i s a t l e v e l 2 t h a t these

p r o p e r t i e s as a group are f i r s t understood and e x p l o i t e d i n e x p l o r i n g aspects o f a ( p r e - s t r u c t u r e d ) r e p r e s e n t a t i o n o f a d e c i s i o n problem. I n t e x t books on d e c i s i o n a n a l y s i s (e.g.

Brown, Kahr and Peterson, 1974) t h i s

t y p e o f e x p l o r a t i o n o f aspects o f a problem i s g e n e r a l l y r e f e r r e d t o as 'I wh a t i f 1" quest ion s about c h an g in g va 1ue s I' sen s it iv it y an a 1ys is " : exp 1o r ing a t nodes i n t h e s t r u c t u r e (e.g.,

t h e p r o b a b i l i t y o f an event i n

an act-event

sequence) t o see what e f f e c t s are propagated throughout t h e s t r u c t u r e . P h i l l i p s (1982) describes a case study showing t h e inportance o f such a c t i v i t i e s i n " r e q u i s i t e " d e c i s i o n modeling i n e n a b l i n g an i n d u s t r i a l conpany's board t o agree on t h e d e c i s i o n s i n v o l v e d i n t h e development and launching o f a new product. Hence, a t l e v e l 2 t h e content manipulated w i t h i n t h e s t r u c t u r e i s n o t " f a c t s " b u t hypotheses (opinions,

views, etc.).

I t i s e x p l i c i t l y recognized t h a t

probabi 1i t i e s can v a r y according t o d i f f e r e n t scenarios o f f u t u r e s t a t e s o f t h e world and t h a t u t i l i t i e s vary i n r e f l e c t i n g t h e range o f s p e c i f i c p a r t i c i p a n t s ' i n t e r e s t s and preferences i n group d e c i s i o n making. A d e c i s i o n maker's knowledge base s u p p o r t i n g h i s o r her problem processing a t t h i s l e v e l has t o comprise procedures Supporting t h e m a n i p u l a t i o n o f content which can be matched t o s p e c i f i c "what i f " e n q u i r i e s .

t 37

CALCULI AND KNOWLEDGE REPRESENTATION

The p o s s i b i l i t i e s f o r r e - s t r u c t u r i n g a d e c i s i o n p r o b l e m r e p r e s e n t a t i o n a t l e v e l 2 a r e n e v e r t h e l e s s e x t r e m e l y l i m i t e d . Jaques'(1982)

account o f demand

c h a r a c t e r i s t i c s o f problems which can be handled by p e r s o n n e l working a t t h i s l e v e l s t r e s s e s t h a t t h e y are l i m i t e d t o those problems where changes i n circumstances w i l l r e q u i r e t h a t changes i n o n l y one major parameter need be i n v e s t i g a t e d a t a t i m e ( s e e t a b l e 1). When changes i n o n l y one a r e i n v e s t i g a t e d , t h e r e s t o f t h e p r o b l e m s t r u c t u r e remains u n d i s t u r b e d , a c t i n g as t h e frame w i t h i n which these changes may be e x p l o r e d . I n s e n s i t i v i t y analyses w i t h i n a d e c i s i o n t r e e , f o r example, changes i n p r o b a b i l i t i e s a t e v e n t nodes are "framed" b y t h e c u r r e n t a c t - e v e n t s t r u c t u r e c o n n e c t i n g t h e s e nodes t o consequences w i t h u t i l i t i e s which remain f i x e d w i t h i n t h e a n a l y s i s , t h a t t h e impact o f these changes on expected u t i l i t y c a l c u l a t i o n s can be assessed. SO

Level 3 More fundamental r e - s t r u c t u r i n g i n v o l v i n g changes i n r e l a t i o n s between two o r more o f t h e major parameters handled i n i s o l a t i o n a t l e v e l 2 r e q u i r e s p r o b l e m p r o c e s s i n g a t l e v e l 3 o r above. A t l e v e l 3, we m v e beyond P i a g e t ' s account o f INRC o p e r a t i o n s d i c t a t i n g t h e f o r m o f t h e c u r r e n t p r o b l e m r e p r e s e n t a t i o n . They now become c o n t e n t m a n i p u l a t e d w i t h i n o p e r a t i o n s aimed a t r e - s t r u c t u r i n g t h e problem under t h e c o n s t r a i n t t h a t t h e type o f s t r u c t u r e used t o r e p r e s e n t t h e problem remains t h e same. The meaning o f ' t y p e ' h e r e can be i n t e r p r e t e d by r e f e r e n c e t o f i g u r e 2, above. Each box i d e n t i f y i n g a d e c i s i o n - t h e o r e t i c subsystem i n t h a t f i g u r e marks o u t t e c h n i q u e s f o r g e n e r a t i n g problem s t r u c t u r e s o f one p a r t i c u l a r t y p e ( a c t - e v e n t l i n k a g e , m u l t i - a t t r i b u t e u t i l i t y decomposition, and so on). R e - s t r u c t u r i n g a c t i v i t i e s which a r e s u p e r o r d i n a t e t o r e o r d e r i n g w i t h i n a. s i n g l e s t r u c t u r a l t y p e are handled a t l e v e l 4 and above i n ways we s h a l l o u t l i n e l a t e r . For an exanple o f d e c i s i o n making a c t i v i t i e s a t t h i s l e v e l , f i g u r e 3 g i v e s an e x t r a c t f r o m a summary o f an i n t e r a c t i v e s e s s i o n w i t h a d e c i s i o n a i d i n g system c a l l e d MAUD4 (Humphreys and Wisudha, 19821, which possesses problem s t r u c t u r i n g c a p a b i l i t i e s a t l e v e l 3. I n t h i s instance, t h e d e c i s i o n maker was p r i m a r i l y s e e k i n g a s s i s t a n c e i n u n d e r s t a n d i n g an aspect o f t h e structure o f h i s problem, which was t o choose

COGNlTl VE DECISION RESEARCH

138

between t h e f o u r j o b o f f e r s l i s t e d a t t h e t o p o f f i g u r e 3. The p r o b l e m was t o i d e n t i f y t h e c r i t e r i a which r e a l l y mattered i n choosing between them. I n w o r k i n g w i t h MAUD4, t h e d e c i s i o n maker i d e n t i f i e d n i n e c r i t e r i a , o f which t h r e e were r e t a i n e d i n t h e f i n a l s t r u c t u r e ( t h e f i f t h , e i g h t h and n i n t h i n o r d e r o f i d e n t i f i c a t i o n ) , Making a c h o i c e between a l t e r n a t i v e j o b o f f e r s i m p l i e s f o r most people choosing between a l t e r n a t i v e f u t u r e s a f f e c t i n g a l a r g e p a r t o f t h e i r l i v e s . Consequently t h e number o f f a c t s which a r e p o t e n t i a l l y r e l e v a n t i n i d e n t i f y i n g c r i t e r i a i s v e r y l a r g e , and so no a t t e m p t was made i n t h i s case t o b u i l d a machine-based knowledge system t o p r o v i d e

F i g u r e 3:

E x t r a c t f r o m t h e summary produced by MAUD4 o f an i n t e r a c t i v e s e s s i o n w i t h a person f a c i n g a j o b d e c i s i o n problem

m

y CF SESIOU DAVID

w s so mn:

Current order of preference of JOBS fran test to wxst (preference v a l w s a r e given in brackets) MIMMIHITY l V (0.781
CRITIC (0.001 a S T

Ratings of JDDS on the scales you are currently using C O M

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(REATIVE(11toKXIffi(91

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CALCULIAND KNOWLEDGE REPRESENTATION

*

Ihe following infonration represents your preEerences f o r the JOUS under consideraticn, on each of the scales. 1.00 represents the test JDB and 0.00 represents t h e worst JCB on each scale.

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If you wish to change anything, you m y d o so a t the end of this surmary. C O M M U

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N4luRRL to MLM EFfCHT lhis scale was cancelled bemuse oE its s i m i l a r i t y w i t h scale nunher 2 (ZOSE m FAMILY to TS?W\EGD rn MLY ?his scale ms m n c e l l e d because of its s i m i l a r i t y w i t h scale nunher 1 muwaxrm to DEWPMNATICN m DPSPAIR ?his scale was cancelled hecause of its s i m i l a r i t y witti scale nunher 4 FEELIffi AT HDME to EEELING ALCNE lhis scale uas cancelled because of its s i m i l a r i t y with scale nunter 3

6 ) MVlWllJRWS to RJDEPE2Wm MR'IURIIY niis scale was c a n c e l l d tecalse of its s i m i l a r i t y w i t h s c a l e nunber 7 ( 7 ) UVJJJSGIN;DIWlCULTIES to ESCAPE FROM REIATICNS ?his scale m s cancelled bemuse OF its s i m i l a r i t y w i t t i s c a l e nrrmher 6 PD OF SWV+RY. (

----------------

t h e i n f o r m a t i o n which would i d e n t i f y t h e c r i t e r i a .

I n s t e a d , t h e knowledge

system w i t h i n MAUD4 o r g a n i z e s procedures f o r e l i c i t i n g , i n v e s t i g a t i n g , and r e o r d e r i n g c r i t e r i a , t h a t i s , s t r u c t u r a l elements r a t h e r t h a n f a c t s about s t a t e s o f t h e world. The domain o f t h e knowledge system i s t h u s b r o u g h t w i t h i n t i g h t bounds, b u t t h e knowledge i t o r g a n i z e s i s r e p r e s e n t e d a t a q u i t e d i f f e r e n t l e v e l f r o m t h e ' f a c t s ' t y p i c a l l y o r g a n i z e d w i t h i n automated d e c i s i o n s u p p o r t systems ( c . f . Bonczek , H o l s a p p l e and Whinston, 1981; McCosh and S c o t t MOrton, 1978).

140

COGNITIVE DECISION RESEARCH

I n o u r view, t h e g e n e r a l c h a r a c t e r i s t i c o f l e v e l 3 p r o b l e m s t r u c t u r i n g c a l c u l i i s t h a t t h e y m a n i p u l a t e as c o n t e n t procedures which serve t o d e f i n e form a t l e v e l 2 , j u s t as l e v e l 2 s t r u c t u r i n g c a l c u l i m a n i p u l a t e p r o p o s i t i o n s t h a t d e f i n e f o r m a t l e v e l 1. MAUD4 r e p r e s e n t s a s p e c i f i c i m p l e m e n t a t i o n

Of

such c a l c u l u s , o p e r a t i n g a t l e v e l 3 i n i n t e r a c t i o n w i t h t h e d e c i s i o n maker, and m a n i p u l a t i n g l e v e l 2 procedures l i k e those l i s t e d above i n g e n e r a t i n g a s t r u c t u r e t o r e p r e s e n t t h e d e c i s i o n problem. However, w i t h i n MAUD4 t h i s s t r u c t u r e i s always expressed i n terms o f a s e t o f a t t r i b u t e s on which p a r t worths (Kneppreth e t a l , 1974) o f t h e s e t o f o p t i o n s c u r r e n t l y under c o n s i d e r a t i o n may be r e p r e s e n t e d . Thus, t h e c o n s t r a i n t under which MAUD4 operates i s t h a t t h i s i s the r e q u i s i t e type o f s t r u c t u r e f o r handling the c u r r e n t d e c i s i o n problem.

Here, t h i s c o n s t r a i n t was acceptable, as t h e

d e c i s i o n maker was f a c i n g a s i t u a t i o n where h i s o p t i o n s were a l r e a d y d e f i n e d i n terms o f a s e t o f immediate a c t s (acceptance o f j o b o f f e r s ) , each of which was d i r e c t l y l i n k e d t o a s i n g l e consequence (consequent c a r e e r ) whose (complex) t o t a l worth had t o be assessed r e l a t i v e t o t h e o t h e r s . F i e l d t e s t s w i t h v a r i o u s v e r s i o n s o f MAUD (Hunphreys and McFadden, 1980, Bronner and de Hoog, 1983) i n d i c a t e d t h a t i n cases where MAUD was s u c c e s s f u l i n a i d i n g d e c i s i o n making, t h i s was due t o i t s a b i l i t y t o a i d t h e d e c i s i o n maker's attempts t o r e s t r u c t u r e t h e problem i n such a way t h a t t h e g o a l s on which t h e s t r u c t u r e i s p r e d i c a t e d a r e c l a r i f i e d , so t h a t a c o h e r e n t c o m p o s i t i o n r u l e can be a p p l i e d i n e x p l o r i n g t h e r e l a t i v e d e s i r a b i l i t y o f t h e I n o t h e r words, t h e k e y t o success l a y i n t h e

o p t i o n s under c o n s i d e r a t i o n .

adequacy o f MAUD's c a p a b i l i t i e s a t l e v e l 3. I n some cases, though, p r o b l e m s t r u c t u r i n g c o n s t r a i n e d t o l e v e l 3 w i l l n o t be acceptable, as when t h e immediate problem i s t o d e c i d e on what t y p e o f s t r u c t u r e t o adopt i n d e v e l o p i n g t h e p r o b l e m r e p r e s e n t a t i o n . I n d e c i d i n g upon a new c a r e e r , such a problem c o u l d a r i s e when, one wished n o t o n l y

t o be a b l e

t o e v a l u a t e c u r r e n t o p t i o n s ( i n v o l v i n g one t y p e o f p r o b l e m r e p r e s e n t a t i o n ) , b u t a l s o t o be a b l e t o develop new o p t i o n s (e.g, t h r o u g h g e n e r a t i n g new a c t event-consequence s c e n a r i o s

-

l i k e " f o r m one's own conpany

.....") .

Choosing

t h e t y p e o f p r o b l e m r e p r e s e n t a t i o n which i s r e q u i s i t e a t any p o i n t i n a d e c i s i o n making process i s i n f a c t a l e v e l 4 p r o b l e m (see below). A t l e v e l 3, o n l y t h e r e s u l t s o f t h i s process are e v i d e n t , where t h e y s e r v e t o d e f i n e t h e t y p e o f p r o b l e m s t r u c t u r e w i t h i n which l e v e l 3, l e v e l 2 , and l e v e l 1

141

CA LCULl AND KNOWLEDGE REPRESENTATION

a c t i v i t i e s are c a r r i e d o u t as, f o r example, i n MAUD4.

LeveZ 4 Level much

4 i s concerned w i t h t h e development o f d e c i s i o n making a c t i v i t i e s scope. Table 1 i n d i c a t e s t h a t t h e demand c h a r a c t e r i s t i c s

wider

problems

r e p r e s e n t e d a t t h i s l e v e l i n v o l v e detatchment f r o m s p e c i f i c

of of

cases ,

seeing them r e p r e s e n t a t i v e o f i s s u e s c a l l i n g f o r t h e development o f a system. Whereas a specific case may be handled t h r o u g h s p e c i f i c problem r e p r e s e n t a t i o n s developed a t l e v e l 3,

structuring capability at level 4

i n v o l v e s t h e a r t i c u l a t i o n o f p r i n c i p l e s which m a n i p u l a t e l e v e l 3 p r o b l e m s t r u c t u r i n g modules as c o n t e n t . There i s no f o r m a l language w i t h i n d e c i s i o n t h e o r y which a r t i c u l a t e s these p r i n c i p l e s , as t h e y a r e s u p e r o r d i n a t e t o t h e forms addressed b y d e c i s i o n t h e o r y , shown i n t h e boxes i n f i g u r e 2. Hence i n working a t l e v e l 4 a d e c i s i o n maker has e i t h e r t o a r t i c u l a t e these p r i n c i p l e s w i t h i n h i s o r h e r own n a t u r a l language o r t o l e a r n a new language e n a b l i n g t h e e f f i c i e n t and c o h e r e n t g e n e r a t i o n o f systems l i n k i n g l e v e l 3 ( s u b ) p r o b l e m r e p r e s e n t a t i o n s i n t o a s t r u c t u r e s p e c i f i c a l l y addressed t o modeling t h e whole range of aspects o f t h e p r o b l e m under c o n s i d e r a t i o n . Hence t h e language employed by t h e d e c i s i o n maker serves as a g e n e r a t i v e problem s t r u c t u r i n g calculus a t t h i s level. F i g u r e 4 shows t h e r e s u l t s o f t h e use o f a 'language c a l c u l u s ' o f t h i s t y p e by t h e manager o f t h e marine c l a i m s department o f a l a r g e insurance company working i n i n t e r a c t i o n w i t h a d e c i s i o n analyst.

The aim was t o g e n e r a t e a

system which c o u l d p r o v i d e f o r d e c i s i o n making c o n c e r n i n g t h e expected s i z e o f a marine c l a i m , f o l l o w i n g n o t i f i c a t i o n a t any t i m e i n t h e f u t u r e o f an i n c i d e n t i n v o l v i n g any one o f a number o f l a r g e s h i p s i n s u r e d w i t h t h e company ( t h i s i s c a l l e d a " h u l l c l a i m " ) . D e t a i l s o f t h e use o f t h i s c a l c u l u s and t h e l e v e l 3 modules i t m a n i p u l a t e s were g i v e n i n Hunphreys, Wooler and P h i l l i p s (1980, s e c t i o n 6). Here we s h a l l o u t l i n e how i t f i t s i n t o o u r d i s c u s s i o n o f d e c i s i o n problem r e p r e s e n t a t i o n a t l e v e l 4. U n l i k e our l e v e l 3 ' M U D ' exanple here t h e problem p r o c e s s i n g system was n o t automated: i t o r i g i n a t e d and remained i n t h e mind o f t h e d e c i s i o n maker who used h i s own ' n a t u r a l ' language i n d e v e l o p i n g t h e problem r e p r e s e n t a t i o n

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COGNITIVE DECISION RESEARCH

shown i n f i g u r e 4. H i s g o a l was t o be a b l e t o assess a s i n g l e complex consequence ( e v e n t u a l s i z e o f s e t t l e m e n t ) whatever t h e circumstances. From t h e way he discussed t h i s consequence it was e v i d e n t t h a t it c o u l d be " r e w r i t t e n " i n terms o f t h e n e t s i z e o f any c l a i m , i n f l a t e d by t h e a p p r o p r i a t e "image f a c t o r " , " r e l a t i o n s h i p f a c t o r ' ' and " h a n d l i n g f a c t o r " . These f a c t o r s sound l i k e j a r g o n t o one o u t s i d e t h e i n s u r a n c e f i e l d . They do n o t e x i s t w i t h i n d e c i s i o n t h e o r e t i c t e r m i n o l o g y , because t h e y a r e names which i n d e x l e v e l 3 d e c i s i o n t h e o r e t i c problem r e p r e s e n t a t i o n s : h a n d l i n g how t o t a k e i n t o account t h e u n c e r t a i n t y about ( i l w h e t h e r t h e way t h e c l a i m i s s e t t l e d w i l l a f f e c t t h e company's image adversely, (ii)whether i t w i l l lead t o d i f f i c u l t i e s i n r e l a t i o n s h i p s w i t h o t h e r c l i e n t s , and ( i i i l w h e t h e r t h e process of h a n d l i n g t h e c l a i m w i l l be t r o u b l e d o r c l e a r . The d e c i s i o n a n a l y s t worked w i t h t h e d e c i s i o n maker i n d e v e l o p i n g t h e s t r u c t u r e o f t h e problem a t l e v e l 4. However, t h e a i d t h a t t h e a n a l y s t c o u l d p r o v i d e a t t h i s l e v e l was n o t i n t e a c h i n g t h e d e c i s i o n maker how t o use a problem s t r u c t u r i n g c a l c u l u s .

I n f a c t t h e o p p o s i t e was t h e case: he had t o

l e a r n t h e c a l c u l u s ( i n c l u d i n g t h e language ' j a r g o n ' ) which t h e d e c i s i o n maker was u s i n g i n t u i t i v e l y .

However, t h e a n a l y s t was a b l e t o c a p i t a l i z e on t h e

r e l a t i o n s h i p between r e - w r i t i n g r u l e s ( i n t h i s case t h o s e i d e n t i f i e d i n t h e d e c i s i o n m a k e r ' s use o f language) and p r o d u c t i o n systems i n f o r m a l i z i n g t h e r e p r e s e n t a t i o n of t h e s t r u c t u r e t h e d e c i s i o n maker was g e n e r a t i n g , and t h u s a i d h i s a t t e m p t s t o check i t s coherence. I n f i g u r e 4 you can see how t h e d e c i s i o n maker's p r o d u c t i o n systems we discussed e a r l i e r , e n a b l e t h e d e c i s i o n m a k e r ' s r e w r i t i n g o f " e v e n t u a l s i z e o f s e t t l e m e n t " t o be expressed as p r o d u c t i o n P l y and f o r t h e t h r e e ' f a c t o r s ' t o be expressed as p r o d u c t i o n s P3, P4 and P5, each o f whose r i g h t hand s i d e i s a t e r m i n a l node i n t h e l e v e l 4 system: a l e v e l 3 module. The d e c i s i o n a n a l y s t was a b l e t o develop t h e l e v e l t h r e e models on t h e b a s i s o f s t r u c t u r e s d e s c r i b e d w i t h i n t h e p r o v i n c e o f d e c i s i o n t h e o r y shown i n f i g u r e 2.

At this

lower l e v e l , i t was t h e a n a l y s t r a t h e r t h a n t h e d e c i s i o n maker who p r o v i d e d t h e language a l t h o u g h t h e d e c i s i o n maker was p r i m a r i l y i n t e r e s t e d t h a t h i s s u b o r d i n a t e s i n t h e a p p r o p r i a t e company departments should understand t h e p r i n c i p l e s a t t h i s l e v e l : t h e y would need t o a r t i c u l a t e t h e s e s t r u c t u r e s i n any i n s t a n c e , and r e p o r t t h e r e s u l t s t o him. Support f o r these l e v e l 3

1

CALCULI AND KNOWLEDGE R EPRESENTA TlON

I

x 1

+

143

144

COGNl TI VE DECISION RESEARCH

s t r u c t u r i n g a c t i v i t i e s i s , o f course, r e q u i r e d a t l e v e l s 2 and 1 i n ways analogous t o t h o s e we d e s c r i b e d f o r t h e l e v e l 3 system MAUD4 above. U n l i k e P3, P 4 and P5, t h e r i g h t hand s i d e o f P 2 i s n o t a t e r m i n a l node i n t h e l e v e l 4 system. When t h e d e c i s i o n maker was asked about t h e n e t s i z e o f t h e c l a i m , he s a i d i t was a f u n c t i o n o f t h e " t o t a l c o s t " minus t h e " n e t c o s t s r e c o v e r a b l e f r o m a t h i r d p a r t y " , each o f which he wished t o express i n h i s

own n a t u r a l language, r a t h e r t h a n as d e c i s i o n t h e o r e t i c modules ( " t o t a l c o s t " i s t h e " s i z e . o f t h e l o s s " minus any " r e d u c t i o n t h a t can be o b t a i n e d f o r n e g l i g e n c e " p l u s "any a d d i t i o n a l payments which have t o be made t o o t h e r p a r t i e s " ) . Hence t h e r i g h t hand s i d e s o f P 6 and P 7 a r e a l s o n o n - t e r m i n a l nodes, as a t these nodes t h e d e c i s i o n maker wished t o c o n t i n u e s t r u c t u r i n g t h e problem a t l e v e l 4. A s c e p t i c m i g h t ask here: why does t h e h u l l c l a i m s problem have t o be expressed a t l e v e l 41 I n t h e case where a l l t h e modules are o f t h e same

s t r u c t u r a l type, i t should be p o s s i b l e t o d e s i g n a system d i r e c t l y l i n k i n g them t o g e t h e r , and even i f t h e y are n o t , one c o u l d implement t h e i n t e r f a c e s shown i n f i g u r e 2 between modules o f d i f f e r e n t s t r u c t u r a l types, so why do we need t h i s f a n c y l e v e l 4 p r o d u c t i o n system? The answer i s as f o l l o w s : f i r s t , w h i l e i t would be q u i t e easy t o d e s i g n a system i n t h e way j u s t d e s c r i b e d g i v e n t h e s t r u c t u r a l r e l a t i o n s o u t l i n e d i n f i g u r e 5, it would n o t have been p o s s i b l e t o g e n e r a t e t h i s system f r o m s c r a t c h w i t h o u t a l e v e l 4 a n a l y s i s o f t h e problem: one would be w i t h o u t t h e e s s e n t i a l i n f o r m a t i o n o f how t h e subproblem r e p r e s e n t a t i o n s should be 1 i n k e d t o g e t h e r . I n t h e h u l l c l a i m s d e c i s i o n making problem, o n l y t h e d e c i s i o n maker ( t h e marine c l a i m s manager) u n d e r s t o o d the p r i n c i p l e s r e q u i r e d i n d e c i d i n g how t o do t h i s , even though h i s u n d e r s t a n d i n g was o n l y i n t u i t i v e , and expressed i n n a t u r a l language. A t l e v e l 4, t h e d e c i s i o n a n a l y s t c o u l d o n l y h e l p h i m f o r m a l i z e t h i s understanding. A t l e v e l 3, t h e d e c i s i o n a n a l y s t ' s t r a i n i n g p r o v i d e d h i m w i t h an i n t u i t i v e u n d e r s t a n d i n g o f how t o g e n e r a t e e v e n t t r e e s s t r u c t u r i n g r e l a t i o n s between u n c e r t a i n s t a t e s o f t h e world, and so he c o u l d t a k e a much more a c t i v e r o l e i n d e v e l o p i n g t h e s t r u c t u r e s a t t h i s l e v e l ( t h e f o r m o f each module), and e x p l a i n i n g t h e i r o p e r a t i o n t o t h e d e c i s i o n maker. I n o t h e r words, t h e knowledge r e p r e s e n t a t i o n s a t l e v e l 4 were accessed f r o m t h e d e c i s i o n maker's semantic memory, b u t t h o s e a t l e v e l 3 were

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145

accessed f r o m t h e d e c i s i o n a n a l y s t ' s semantic memory. A t l e v e l 2 t h e knowledge r e p r e s e n t a t i o n s i n t h e d e c i s i o n a n a l y s t ' s memory were accessed ( h e developed t h e s e n s i t i v i t y analyses

used t o "prune" t h e modules so j u s t t h e

k e y u n c e r t a i n t i e s would be r e p r e s e n t e d ) , b u t a t l e v e l 1, t h e d e c i s i o n maker p r o v i d e d t h e e s t i m a t e s o f t h e p r o b a b i l i t i e s a t a l l t h e e v e n t nodes. Thus, i n g e n e r a t i n g t h e system f r o m s c r a t c h , e i t h e r t h e d e c i s i o n maker o r t h e d e c i s i o n a n a l y s t w o r k i n g w i t h h i m had t o understand t h e p r i n c i p l e s u n d e r l y i n g t h e o p e r a t i o n s , and p r o v i d e t h e knowledge base a t each o f t h e f o u r l e v e l s . However, once t h e system had been generated, i t s use i n e v e n t u a l s i z e o f s e t t l e m e n t s no l o n g e r r e q u i r e d l e v e l 4 d e c i s i o n making. The o u t p u t f r o m t h e l e v e l 4 p r o b l e m p r o c e s s i n g system codifies t h e c o n t e x t w i t h i n which each l e v e l 3 module i s addressed t o a subproblem, and can p r o v i d e t h e b r i e f f o r a manager a t l e v e l 3 i n t h e o r g a n i z a t i o n a l h i e r a r c h y o f t h e company ( s e e T a b l e 1, column 1 ) on t h e i n f o r m a t i o n h i s department needs t o p r o v i d e . S i m i l a r l y the event t r e e s t r u c t u r e s r e s u l t i n g from l e v e l 3 problem s t r u c t u r i n g w i t h i n each module can c o d i f y t h e framework w i t h i n which i n f o r m a t i o n i s t o be accessed ( a s b e s t e s t i m a t e s o f t h e p r o b a b i l i t i e s which a p p l y i n a s p e c i f i c case) a t l e v e l 1 and s u b j e c t e d t o s e n s i t i v i t y a n a l y s i s a t l e v e l 2 . The i d e a behind c o d i f i c a t i o n i n t h i s way has a long h i s t o r y . Examples a r e t h e book o f L e v i t i c u s i n t h e B i b l e ( c . f . Leach, 19641, and t a c t i c a l handbooks f o r m i l i t a r y f i e l d commanders. Understanding how t o c o n s t r u c t such books r e q u i r e s s t r u c t u r i n g knowledge a t a r e l a t i v e l y h i g h l e v e l , which t h e n p r o v i d e s t h e s t r u c t u r a l c o n s t r a i n t s d i c t a t i n g d e c i s i o n making a t a lower l e v e l . C o d i f i c a t i o n , however, i s p r e d i c a t e d on t h e a s s u n p t i o n t h a t t h e r e i s no change i n t h e c i r c u m s t a n c e s addressed a t t h e h i g h e r , c o d i f i e d l e v e l s across d e c i s i o n making a p p l i c a t i o n s i n v o l v i n g t h e use o f t h e codebook o r manual. Chaos may r e s u l t i n cases where t h i s assumption i s n o t met, as t h e p r i n c i p l e s i n v o l v e d i n d e c i s i o n making a t t h e h i g h e r l e v e l s have been e x p l i c i t l y excluded f r o m c o n s i d e r a t i o n t h r o u g h t h e c o d i f i c a t i o n process. T h i s serves t o ' n a t u r a l i z e ' (Barthes,l973)

t h e r e s u l t s o f t h e use o f h i g h e r l e v e l p r i n c i p l e s

as codes o f a c t i o n . These codes appear, a t t h e lower l e v e l s t o be s i n p l y t h e ' n a t u r a l ' way t o approach t h e d e c i s i o n problem and t h e i r h i s t o r y ( t h e p r i n c i p l e s used t o generate them) i s l o s t f r o m view.

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COGNITIVE DECISION R €SEA RCH

Level 5 By analogy w i t h r e l a t i o n s between l o w e r l e v e l s i n our scheme, t h e c o n t e n t

m a n i p u l a t e d a t l e v e l 5 comprises complete l e v e l 4 systems. What i s t h e form though o f t h e c o n t e x t w i t h i n which t h e y a r e l o c a t e d and may be m a n i p u l a t e d by a d e c i s i o n maker who understands t h e p r i n c i p l e s u n d e r l y i n g a l e v e l 5 p r o b l e m s t r u c t u r i n g c a l c u l u s ? A c l u e can be gained f r o m t h e d e s c r i p t i o n g i v e n i n t a b l e 1 o f t h e demand c h a r a c t e r i s t i c s o f t a s k s l o c a t e d a t t h i s l e v e l i n a b u r e a u c r a t i c o r g a n i z a t i o n : problems are n o t d e a l t w i t h i n a c o n t e x t w h o l l y s e t f r o m above and one can m o d i f y t h e boundaries o f o n e ' s b u s i n e s s w i t h i n c o r p o r a t e p o l i c y . Here we have a l i n k w i t h Savage's (1954) n o t i o n o f t h e s m a l l w o r l d w i t h i n whose boundaries p r o b l e m s t r u c t u r i n g a c t i v i t i e s and c u r r e n t knowledge r e p r e s e n t a t i o n s are l o c a t e d . Toda(1976) has d i s c u s s e d t h e n a t u r e o f t h e s e s m a l l worlds, and how t h e i r boundaries are c l o s e d a c c o r d i n g t o t h e g o a l s o f t h e d e c i s i o n maker. T h i s 'boundary s e t t i n g ' f o r a d e c i s i o n problem changes w i t h changes i n m o t i v a t i o n , as does t h e n a t u r e o f t h e s t r u c t u r e s t h e d e c i s i o n maker w i l l c o n s i d e r r e q u i s i t e i n h a n d l i n g any decision problem (c.f.

Jungermann, von U l a r d t and Hausman, 1983 and Hunphreys

e t a l , 1983). Development o f a s t r u c t u r i n g c a l c u l u s a t t h i s l e v e l would mean t h e a r t i c u l a t i o n o f p r i n c i p l e s g u i d i n g t h e c o n d i t i o n a l c l o s i n g o f an open system ( t h e unbounded "grand w o r l d " o f p o s s i b l e views o f t h e f u t u r e ) i n t o t h e s m a l l w o r l d w i t h i n which an i n d i v i d u a l ' s p r o b l e m s t r u c t u r i n g a c t i v i t i e s t a k e p l a c e .

To o u r knowlege no exanple of such a c a l c u l u s has y e t been f o r m a l i z e d .

Higher levels of abstraction Above l e v e l 5 we m v e o u t s i d e t h e scope o f i n d i v i d u a l d e c i s i o n making as a t l e v e l 6 t h e " s m a l l w o r l d s " which d e f i n e f o r m a t l e v e l 5 a r e themselves p l a c e d as c o n t e n t w i t h i n s t r u c t u r e s generated a t h i g h e r l e v e l s . We can understand t h e n a t u r e o f knowledge r e p r e s e n t a t i o n s a t l e v e l s 6 t o 10 by analogy w i t h l e v e l s 1 t o 5 . A t l e v e l 6, p r i n c i p l e s l i k e s e r i a t i o n , t r a n s i t i v i t y and conservation

a r e i n p o r t a n t i n c o n t r o l l i n g o p e r a t i o n s , as a t l e v e l 1, b u t t h e

m a t e r i a l on which t h e y o p e r a t e a r e complete s m a l l worlds, r a t h e r t h a n unitary "facts".

Also, l e v e l 6 d e c i s i o n making r e q u i r e s s u p p o r t a t l e v e l s 1

t o 5 i n d e v e l o p i n g r e q u i s i t e c o n c e p t u a l i z a t i o n s o f t h e s e s m a l l worlds. A

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s i m i l a r c o n p a r i s o n may be made between l e v e l 7 and l e v e l 2, and so on. The reason why we s t a t e t h a t d e c i s i o n making a t t h e s e h i g h e r l e v e l s o f a b s t r a c t i o n must o f n e c e s s i t y be s o c i a l r a t h e r t h a n p e r s o n a l i s t h a t each s m a l l world m a n i p u l a t e d as c o n t e n t a t t h e s e l e v e l s marks f o r t h e i n d i v i d u a l l o c a t e d w i t h i n i t t h e boundary o f t h e ways i n which he o r she can c o n c e p t u a l i z e s c e n a r i o s o f f u t u r e p e r s o n a l e x p e r i e n c e s consequent on a c t i o n s he o r she might t a k e . D e c i s i o n making above l e v e l 5 i s concerned w i t h choosing between courses o f a c t i o n which, when supported a t a l l lower l e v e l s ,

w i l l lead t o

t h e g e n e r a t i o n and r e - o r d e r i n g o f t h e c u l t u r a l s t r u c t u r e s

w i t h i n which i n d i v i d u a l s f i n d t h e i r i d e n t i t y . Few d e c i s i o n makers occupy r o l e s w i t h t h e scope and l e v e l s o f o r g a n i z a t i o n a l s u p p o r t which p e r m i t d e c i s i o n making t a s k s t o be handled a t these l e v e l s o f a b s t r a c t i o n . However, Jaques(1982) d e s c r i b e s o r g a n i z a t i o n a l r o l e s and demand c h a r a c t e r i s t i c s a t l e v e l s 6 and 7.

He c l a i m s t o have i d e n t i f i e d a v e r y few d e c i s i o n makers w i t h

t h e c a p a b i l i t y and o r g a n i z a t i o n a l s u p p o r t a1 l o w i n g them t o o p e r a t e a t l e v e l 8, b u t none above t h i s .

ImpZications for the investigation of individuaZ decision making Given o u r a n a l y s i s o f l e v e l s o f a b s t r a c t i o n i n t h e r e p r e s e n t a t i o n o f d e c i s i o n problems, we conclude b y examining i t s i m p l i c a t i o n s f o r t h e s t u d y o f i n d i v i d u a l d e c i s i o n making under u n c e r t a i n t y t h r o u g h experiments conducted on i n t u i t i v e i n f e r e n c e and d e c i s i o n making w i t h i n t h e " c o n v e r s a t i o n a l paradigm" (see Kahneman and Tversky, 1982). I n a n a l y s i n g t h e r e s u l t s o f such experiments, f i g u r e 1 may be a p p l i e d i n s t u d y i n g d i f f e r e n c e s between s u b j e c t s ' responses and t h e o u t p u t o f a n o r m a t i v e model a t any p a r t i c u l a r l e v e l o f a b s t r a c t i o n .

However, f o r t h e

comparison procedure t o have any v a l i d i t y , two c o n d i t i o n s must be met: ( i ) c o n p a r i s o n s must be made a t one, and o n l y one, p r e - d e f i n e d l e v e l o f a b s t r a c t i o n a t a t i m e , and ( i i ) f o r a comparison t o be made a t any p a r t i c u l a r l e v e l , t h e r e must be a commn u n d e r s t a n d i n g about how t h e s t r u c t u r e o f t h e p r o b l e m r e p r e s e n t a t i o n i s t o be f i x e d a t a l l higher l e v e l s ( t h a t i s , i f t h e procedure i n f i g u r e 1 were t o be a p p l i e d a t these h i g h e r l e v e l s , t h e r e would be no d i f f e r e n c e i n t h e two o u t p u t s ) .

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These two c o n d i t i o n s t o g e t h e r imply t h a t we should s t a r t a t t h e h i g h e s t l e v e l a t which t h e r e i s a p o s s i b i l i t y o f t h e e x i s t e n c e o f d i f f e r e n c e s , and t h e n work down, l e v e l b y l e v e l , i n t e r p r e t i n g t h e d i f f e r e n c e s i n o u t p u t comparisons shown i n f i g u r e 1 a t t h e h i g h e s t l e v e l a t which t h e y occur. The procedure i n v o l v e d i s g i v e n i n f i g u r e 5.

As t h i s paper has been

concerned w i t h s t u d y i n g d i f f e r e n c e s i n i n d i v i d u a l d e c i s i o n making, t h e h i g h e s t l e v e l which i s c o n s i d e r e d e x p l i c i t l y i s l e v e l 5. S t a r t i n g a t l e v e l 5 assumes, however, t h a t t h e r e i s a commn u n d e r s t a n d i n g a t l e v e l s 6 and above. These l e v e l s a r e p r i m a r i l y concerned w i t h s t r u c t u r i n g t h e c u l t u r a l c o n t e x t w i t h i n which i n d i v i d u a l s l o c a t e themselves,

and, indeed, d e f i n e t h e i r

i d e n t i t i e s , t o p i c s o u t s i d e t h e scope o f t h i s paper.

T h e r e f o r e , we s t a r t h e r e

by assuming a commn u n d e r s t a n d i n g about t h e shared c u l t u r a l c o n t e x t which i s l i k e l y t o e x i s t i f a l l concerned ( e x p e r i m e n t e r s , s u b j e c t s , e t c . ) come f r o m s i m i l a r c u l t u r a l backgrounds.

A t l e v e l 5 we a r e concerned w i t h p o s s i b l e d i f f e r e n c e s i n t h e way i n which p e o p l e s e t t h e bounds on t h e small w o r l d s w i t h i n which t h e y may c o n c e p t u a l i z e t h e a l t e r n a t i v e f u t u r e s t h a t may f o l l o w f r o m d e c i d i n g upon any o f t h e o p t i o n s under c o n s i d e r a t i o n .

Hunphreys(1982) has d e s c r i b e d how t h e way i n which

i n d i v i d u a l s s e t t h e s e bounds d e f i n e s t h e l i m i t s o f t h e 'background o f s a f e t y ' (Sandler and Sandler, 1978) w i t h i n which s c e n a r i o s o f such p o s s i b l e f u t u r e s a r e c o n s t r u c t e d and e x p l o r e d .

W i t h i n these bounds, t h e way s c e n a r i o s a r e

a r t i c u l a t e d w i l l depend upon t h e g o a l s o f t h e i n d i v i d u a l i n d e f i n i n g c h a r a c t e r i s t i c s o f t h e ( s m a l l ) w o r l d he o r she would, o r would n o t , l i k e t o l i v e in. An example o f an area where d i f f e r e n c e s a t t h i s l e v e l come t o t h e f o r e i's p u b l i c debates c o n c e r n i n g c o n f l i c t s o v e r n u c l e a r power. debate (e.g.

One s i d e i n such a

a t a p u b l i c i n q u i r y ) may d e f i n e t h e ' a c c e p t a b i l i t y ' o f n u c l e a r

power i n terms o f i t s s a f e t y where ' s a f e t y ' i s d e f i n e d i n terms o f expected number o f f a t a l i t i e s d i r e c t l y a t t r i b u t a b l e t o t h e r e a c t o r process i n o p e r a t i o n , whereas t h e o t h e r s i d e may be p r i n c i p a l l y concerned w i t h ( u n ) a c c e p t a b i l i t y i n terms o f e x p l o r i n g s c e n a r i o s c o n c e r n i n g i n c r e a s e d c e n t r a l i z a t i o n o f s t a t e power and s e c u r i t y , and problems o f f u t u r e hazards f r o m r a d i o a c t i v e waste ( c . f .

Gyorgi, 1981; Watson, 1981; von W i n t e r f e l d t and

CALCULI AND KNOWLEDGE REPRESENTATION

Figure 5:

149

Multi-level comparison procedure for investigating differences between "intuitive" and "normative" problem representations

Explore crossc u l t u r a l differences a t l e v e l 6 and above

NO model of the decision problem share a

I t i s p o s s i b l e t o study d i f f e r e n c e s i n individual decision making within comon c u l t u r a l context. 5) S t a r t t h i s study a t l e v e l 5 ( s e t n

-

L I Apply comparison shown i n f i g u r e I on representations of problem a t level n.

r

YES

Try t o resolve differences i n problem representation a t l e v e l n.

Explain d i f f e r e n c e s between s u b j e c t s and experimenter's view of problem r e p r e s e n t a t i o n (or t h a t of various groups of s u b j e c t s a t l e v e l n.

I

i

150

COGNITIVE DECISION RESEARCH

R i o s , 1980).

I n t h i s case, seeking evidence which assumes t h a t t h e d i f f e r e n c e l i e s a t l e v e l 1 (e.g. " i t i s i n p o r t a n t t o a s c e r t a i n how s a f e a

p l a n t w i l l b e i n terms o f t h e b e s t e s t i m a t e o f expected number o f f a t a l i t i e s p e r y e a r " ) w i l l never r e s o l v e t h e d i f f e r e n c e s between t h e p a r t i e s .

It cannot

e v e r address them, as t h e r e i s no commn agreement about t h e s c e n a r i o s o f t h e f u t u r e which need t o be e x p l o r e d i n a s c e r t a i n i n g t h e ' a c c e p t a b i l i t y ' o f what i s t h u s c o n c e p t u a l i z e d and which w i l l d e f i n e t h e themes t o be s t r u c t u r e d a t l e v e l 4 and below. I f however, consensus can be achieved a t l e v e l 5, a commn u n d e r s t a n d i n g c o n c e r n i n g t h e bounds o f t h e p r o b l e m t o be s t r u c t u r e d a t t h a t l e v e l may be d e f i n e d and t h e comparisons shown i n f i g u r e 1 can t h e n be made a t l e v e l 4

,

where we a r e i n t e r e s t e d i n how i n d i v i d u a l s g e n e r a t e t h e s t r u c t u r e s which d e s c r i b e t h e i r v i e w o f t h e problem. I f agreement i s n o t reached on t h e

structural types t o be used i n r e p r e s e n t i n g t h e problem, and how t h e y a r e t o be l i n k e d t o g e t h e r ( c . f . f i g u r e 41, t h e n i t i s necessary t o analyse t h e d i f f e r e n t l i n k a g e s a t t h i s l e v e l . F o r exanple, c o n s i d e r t h e f o l l o w i n g q u e s t i o n s , posed by P h i l l i p s and B o x a l l (1983) t o s u b j e c t s i n an experiment w i t h i n t h e ' c o n v e r s a t i o n a l paradigm' (Kahneman and Tversky, 1982). "Peter i s c o n s i d e r i n g i n s t a l l i n g a b u r g l a r a l a r m t o p r o t e c t h i s home. Which i s more l i k e l y : (i)t h a t P e t e r ' s home w i l l be broken i n t o b e f o r e t h e end o f n e x t year i f he i n s t a l l s a b u r g l a r a l a r m b e f o r e t h e end o f n e x t year, o r (ii)t h a t P e t e r ' s home w i l l be broken i n t o b e f o r e t h e end o f n e x t year i f he does n o t i n s t a l l a b u r g l a r a l a r m b e f o r e t h e end o f n e x t y e a r 1" P h i l l i p s and B o x a l l (1983) r e p o r t an a n a l y s i s o f t h e problem r e p r e s e n t a t i o n s a c t u a l l y used b y s u b j e c t s ( o b t a i n e d t h r o u g h a ' t h i n k a l o u d ' technique, f o l l o w i n g Feldman, 1963 and Svenson, 1974) i n d e c i d i n g how t o answer t h i s and e l e v e n o t h e r q u e s t i o n s developed f r o m those o r i g i n a l l y posed t o s u b j e c t s b y T v e r s k y and Kahneman. They f o u n d t h a t 58% o f t h e s u b j e c t s e l a b o r a t e d t h e q u e s t i o n i n a d i f f e r e n t way f r o m t h e l i n k a g e o f one b i n a r y event f o l l o w e d b y another assumed b y T v e r s k y and Kahneman(l980) i n d i s c u s s i n g t h e e x t e r n a l ' d i a g n o s t i c ' p e r s p e c t i v e s u b j e c t s may use i n a r r i v i n g a t an answer. I n some cases, s u b j e c t s r o l e - p l a y e d P e t e r o r t h e b u r g l a r , t a k i n g an i n t e r n a l

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CALCULI AND KNOWLEDGE REPRESEN TA TION

p e r s p e c t i v e when t h e p r o b l e m i s r e p r e s e n t e d t h r o u g h an a c t - e v e n t

sequence,

i n s t e a d o f an event-event sequence. Accounting f o r d i f f e r e n c e s between s u b j e c t s who adopted an ' e x t e r n a l p e r s p e c t i v e , and those who adopted an i n t e r n a l ' p e r s p e c t i v e t h u s r e q u i r e d an e x p l a n a t i o n c e n t r e d on a l t e r n a t i v e l e v e l 4 problem r e p r e s e n t a t i o n s . However, n o t a l l t h e q u e s t i o n s s t u d i e d b y P h i l l i p s and B o x a l l l e d t o d i f f e r e n c e s which had t o be accounted f o r a t l e v e l 4.

For some, t h e i n f o r m a t i o n p r o v i d e d t o

s u b j e c t s was found t o be s u f f i c i e n t t o ensure a 'common u n d e r s t a n d i n g ' o f t h e problem r e p r e s e n t a t i o n a t t h i s l e v e l . Achievement o f a common u n d e r s t a n d i n g a t l e v e l 4 a l l o w s examination o f d i f f e r e n c e s i n t h e way i n which i n d i v i d u a l s e l a b o r a t e t h e i r r e p r e s e n t a t i o n s o f a decision problem w i t h i n a p a r t i c u l a r s t r u c t u r a l type represented a t l e v e l 3. ' S o c i a l d e c i s i o n making' schemes l o c a t e d a t t h i s l e v e l have been d e s c r i b e d b y Edwards (1977) and von W i n t e r f e l d t and R i o s (19801, among others.

Each o f these r e p o r t s d e s c r i b e s cases where consensus between

v a r i o u s i n d i v i d u a l s was assumed o r e n g i n e e r e d a t l e v e l s 4 and 5, and differences displayed w i t h i n a problem representation o f a s i n g l e s t r u c t u r a l t y p e : a m u l t i a t t r i b u t e u t i l i t y d e c o n p o s i t i o n c o r r p r i s i n g a l l a t t r i b u t e s which any o f t h e p a r t i e s c o n s i d e r e d r e l e v a n t . D i f f e r e n c e s between p a r t i e s can t h e n be expressed b y d i f f e r e n t v e c t o r s o f weights a p p l i e d w i t h i n t h e o v e r a l l

decomposition ( a s s i g n i n g a zero w e i g h t t o any p a r t o f t h e decomposition e f f e c t i v e l y d e l e t e s i t f r o m c o n s i d e r a t i o n as would be t h e case where .the a t t r i b u t e s i n t h e ' s o c i a l d e c i s i o n making' d e c o m p o s i t i o n were n o t p r e s e n t i n a p a r t i c u l a r individual's m u l t i a t t r i b u t e u t i l i t y structure, characterizing h i s o r her l e v e l 3 view o f t h e problem). B e r k e l e y and Hunphreys (1982) and P h i l l i p s and B o x a l l (1983) d e s c r i b e some a l t e r n a t i v e e l a b o r a t i o n s which may b e used by s u b j e c t s p r e s e n t e d w i t h d e c i s i o n problems by e x p e r i m e n t e r s u s i n g t h e ' c o n v e r s a t i o n a l paradigm'. In t h e s c h e m we d e s c r i b e here, t h e s e a l t e r n a t i v e e l a b o r a t i o n s r e p r e s e n t d i f f e r e n c e s i n ways of c o n c e p t u a l i z i n g t h e d e c i s i o n problems a t l e v e l 3. Here, t h e a p p r o p r i a t e e x p l a n a t i o n has t o be i n terms o f a c c o u n t i n g f o r t h e d i f f e r e n c e s i n elements i n t h e s t r u c t u r e (e.g. i n t e r v e n i n g a c t s , e v e n t s o r a t t r i b u t e s o f consequences) and i n how t h e y are l i n k e d t o g e t h e r .

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When, f i n a l l y , we can ensure t h a t t h e r e i s a commn understanding between subjects and experimenter concerning the s t r u c t u r e o f the problem representation a t l e v e l 3, it then becomes possible t o make the comparisons show i n f i g u r e 1 a t l e v e l 2. Now we can look, f o r example, f o r evidence whether subjects are 'normatively c o n s i s t e n t ' o r biased towards causal reasoning (c.f.

Tversky and Kahneman, 1980) i n the way i n which they

i n t e r p r e t the diagnostic s i g n i f i c a n c e o f evidence presented i n sequences 1i k e t h a t given i n the ' b u r g l a r alarm' question we discussed above. I n sumnary, we share w i t h Amarel (1968) the view t h a t "The act o f choosing a representation f o r a problem involves t h e s p e c i f i c a t i o n of a space where the search f o r s o l u t i o n can take place.

-

-

Such a s p e c i f i c a t i o n involves the choice o f a language and i t s use f o r expressing problem conditions, p r o p e r t i e s o f solutions, and knowledge o f r e g u l a r i t i e s i n the search space. I n addition, i t involves t h e formulation o f a schema according t o which problem conditions and knowledge about p r o p e r t i e s o f the space can be used e f f e c t i v e l y i n t h e s o l u t i o n seeking procedures. Such a schema can be as s i n p l e as a s e t o f l o c a l moves o r as elaborate as a global plan..''

(Amare1,1968;

p.94).

I n t h i s paper we have attempted t o explore t h i s space, d e f i n i n g general conditions f o r the formulation o f such a schema and i l l u s t r a t i n g how a t 'least f i v e q u a l i t a t i v e l y d i f f e r e n t l e v e l s o f a b s t r a c t i o n must be encompassed i n t h i s enterprise.

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the

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i r r a t i o n a l i t y . London, Cambridge U n i v e r s i t y Press. Feldman, J.: 1963. S i m u l a t i o n o f b e h a v i o u r i n t h e b i n a r y c h o i c e experiment. I n : Feldman, J. & Fergenbaum ( e d s ) , Computers and t h o u g h t . New York. McGraw H i l l . F i l l m o r e , C.J.: 1968. The case f o r case. I n : Bach, E. & Harms, R.T. ( e d s ) : U n i v e r s a l s i n l i n g u i s t i c t h e o r y . New York. H o l t , R i n e h a r t & Winston. ( p p .1-88 1 F i s c h h o f f , B.: 1982. A.(eds), o p . c i t .

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S l o v i c , P. & Tversky,

Gyorgy, A. & f r i e n d s : 1979. No Nukes: Everyone's guide t o N u c l e a r Power. Boston, Mass. South End Press. Hogarth, R.M., Michaud, C. & Mery, J.L.: 1980. D e c i s i o n b e h a v i o r i n urban development: A m e t h o d o l o g i c a l approach and s u b s t a n t i v e c o n s i d e r a t i o n s . A c t a P s y c h o l o g i c a , 45, 95-117. 1978. U n c e r t a i n t y measurement : a c o g n i t i v e Howell, W.C. & B u r n e t t , S.A.: taxonomy. O r g a n i z a t i o n a l Behavior and Human Performance, 22, 45-68. Hovland, C.I.: 1952. A 'communication a n a l y s i s ' o f concept l e a r n i n g . P s y c h o l o g i c a l review, 59, 461-472. Hunphreys, P.C.: 1977. A p p l i c a t i o n s o f M u l t i a t t r i b u t e U t i l i t y Theory. I n : Jungermann, H. & de Zeeuw G.(eds), D e c i s i o n making and change i n human a f f a i r s . Dordrecht. Reidel. Hunphreys, P.C.: 1982. Value s t r u c t u r e s u n d e r l y i n g r i s k assessments. I n : Kunreuther, H. (ed.). R i s k : a seminar s e r i e s . Laxemburg, A u s t r i a . I n t e r n a t i o n a l I n s t i t u t e o f A p p l i e d Systems A n a l y s i s . Hunphreys, P.C., L a r i c h e v , O.I., V a r i , A. & Vecsesnyi, J.: 1983. C o n p a r a t i v e a n a l y s i s o f use o f d e c i s i o n s u p p o r t systems i n R&D d e c i s i o n s . I n : S o l , H. G. ( e d ) : Processes and t o o l s f o r d e c i s i o n s u p p o r t . Amsterdam. N o r t h Holland. Hunphreys, P.C. & McFadden, W.: 1980. Experiences w i t h MAUD: a i d i n g d e c i s i o n s t r u c t u r i n g v e r s u s b o o t s t r a p p i n g t h e decision-maker. A c t a P s y c h o l o g i c a , 45, 51-69. Svenson, 0. & V a r i , A. ( e d s ) : 1983. A n a l y s i n g and a i d i n g Hunphreys, P.C., d e c i s i o n processes. Amsterdam. N o r t h Holland. Hunphreys, P.C. & Wisudha, A,: 1982. MAUD4. D e c i s i o n A n a l y s i s U n i t T e c h n i c a l r e p o r t 82-5. London. London School o f Economics and P o l i t i c a l Science.

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Jaques, E.:

1976. A g e n e r a l t h e o r y o f Bureaucracy.

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1982. Free e n t e r p r i s e , f a i r employment. London. Heinemann.

Jaques, E., Gibson, R.O. & Isaac, D.J.: 1978. L e v e l s o f a b s t r a c t i o n 'in l o g i c and human a c t i o n . London. Heinemann.

1972. A s y s t e m a t i c i n t r o d u c t i o n t o t h e psychology o f t h i n k i n g . Johnson, D.M.: New York. Harper & Row. Jungermann, H., von U l a r d t , I. & Hausman, L.: g e n e r a t i n g a c t i o n s . I n : Hunphreys, P.C., op. c i t .

1983. The r o l e of g o a l i n Svenson, 0. & V a r i , A.(eds),

Kahneman, O., S l o v i c , P. & Tversky, A. ( e d s ) : 1982. Judgement under u n c e r t a i n t y : H e u r i s t i c s and Biases. New York: Cambridge U n i v e r s i t y Press. Kahneman, D. & Tversky, A.: 1979. P r o s p e c t t h e o r y : an a n a l y s i s o f d e c i s i o n under r i s k . Econometrica, 47, 263-291. Kahneman, 0. & Tversky, A.: 1982. On t h e s t u d y o f s t a t i s t i c a l i n t u t i o n s . C o g n i t i o n , 11, 123-141.

1974. Kneppreth,N.P., Gustafson, D.H. , L e i f e r , R.P. 81 Johnson, E.M.: Techniques f o r t h e assessment o f worth. T e c h n i c a l paper 254 (AD784629). A r l i n g t o n , Va.. Army Research I n s t i t u t e f o r t h e B e h a v i o r a l and S o c i a l Sciences. K r a n t z , D.H., Luce, R.D., Suppes, P. & Tversky, A.: measurement ( V o l 1). New York. Academic Press.

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D.H. : 1972. Measurement s t r u c t u r e s and p s y c h o l o g i c a l laws. Science, 175, 1427-1435.

Krantz,

Leach, E.R.: 1964. A n t h r o p o l o g i c a l aspects o f language. I n : Lenneberg, E.H.(ed). New d i r e c t i o n s i n t h e s t u d y o f language. Boston, Mass. I n s t i t u t e o f Technology Press, 23-63. L i c h t e n s t e i n , S.C., F i s c h h o f f , B. & P h i l l i p s , L.D.: 1982. C a l i b r a t i o n o f p r o b a b i l i t i e s : t h e s t a t e o f t h e a r t t o 1980. I n : Kahneman, D . , S l o v i c , P. & Tversky, A.(eds), o p . c i t . McCosh, A.M. & S c o t t Morton, M.S.: London. MacMi 11an.

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P h i l l i p s , L.D.: 1983. A t h e o r e t i c a l p e r s p e c t i v e on h e u r i s t i c s and biases i n Svenson, 0. & V a r i , A. p r o b a b i l i s t i c t h i n k i n g . In: Hunphreys, P.C., (eds), o p . c i t . P h i l l i p s , L.D. & B o x a l l , 3.: 1983. A problem s t r u c t u r i n g a n a l y s i s o f h e u r i s t i c s and biases. D e c i s i o n A n a l y s i s U n i t . Technical Report 83-1. London. London School o f Economics and P o l i t i c a l Science. P i a g e t , 3.:

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Piaget, J.: 1972. The p r i n c i p l e s o f genetic epistemology. and Kegan Paul.

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1980. E f f i c i e n t d e c i s i o n h e u r i s t i c s . Behavioral Science, 25,

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VERBAL VS. NUMERICAL PROCESSING OF SUBJECTIVE PROBABILITIES

ALF C. ZIMMER Universitat Miinster Psychologisches lnstitut F. R. Germany

The problem o f d e c i d i n g between d i f f e r e n t a l t e r n a t i v e s w i t h e q u a l l y p o s i t i v e outcomes has an a p p a r e n t l y t r i v i a l s o l u t i o n : t a k e t h e one where t h e p o s i t i v e outcome i s most p r o b a b l e . That t h i s s o l u t i o n i s n o t as t r i v i a l as i t seems t o be can be seen i n t h e f a c t t h a t i t was n o t u n t i l 1660 f o r t h e concept o f p r o b a b i l i t y t o be t r e a t e d m a t h e m a t i c a l l y (HACKING, 1975), t h e r e b y p r o v i d i n g s o l u t i o n s f o r how t o decide i n t h e case o f u n c e r t a i n events. As e a r l y as 1713 JACQUES BERNOULLI has made t h e d i s t i n c t i o n between p r o b a b i l i ties

-

l a t e r d e f i n e d by LAPLACE as t h e r e l a t i v e f r e q u e n c i e s o f successes

and s u b j e c t i v e " d e g r e e s o f confidence!

-

By means o f t h i s d i s t i n c t i o n i t became

p o s s i b l e t o account f o r i n c o n s i s t e n c i e s o r i r r a t i o n a l i t i e s i n human d e c i s i o n s . The course o f a c t i o n t a k e n was assumed t o depend on t h e s e s u b j e c t i v e "degrees o f c o n f i d e n c e " i n s t e a d o f t h e o b j e c t i v e p r o b a b i l i t i e s as d e r i v e d f r o m l o n g r u n experience. P r a c t i c a l as w e l l as p h i l o s o p h i c a l c o n s i d e r a t i o n s t h e n l e d t o v a r i o u s a t t e m p t s t o d e f i n e s u b j e c t i v e p r o b a b i l i t y f o r m a l l y ( e t g . DE FINETTI, 1937) and t o p r e s c r i b e procedures o f d e t e r m i n i n g k h e t h e r numerical e s t i m a t e s o f p r o b a b i l i t i e s g i v e n by human r a t e r s f u l f i l t h e axioms o f s u b j e c t i v e p r o b a b i l i t i e s (The system o f axioms most o f t e n i n v e s t i g a t e d i s t h e one due t o SAVAGE, 1954). A t h o r o u g h c r i t i q u e o f t h i s approach can be found i n SLOVIC & TVERSKY (1974) and SUPPES (1974). LINDLEY (1974) adds a f u r t h e r d i s c u s s i o n f r o m t h e p o i n t o f view o f Bayesian s t a t i s t i c s . The c o r e o f b o t h SUPPES' and TVERSKY's c r i t i q u e i s t h a t human d e c i s i o n makers and p r o b a b i l i t y r a t e r s m a r k e d l y and s y s t e m a t i c a l l y d e r i v a t e f r o m what i s p r e s c r i b e d by SAVAGE'S theory

, e s p e c i a l l y c o n c e r n i n g t h e axiom o f i n d i f f e r e n c e .

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Froni t h e p o i n t o f view o f human i n f o r r a t i o n o r o c e s s i n g t h e framework f o r i n v e s t i g a t i n g t h i s problem i s g i v e n i n t a b l e 1. Table 1 iuman i n f o r m a t i o n p r o c e s s i n g depends on:

I. Modes o f r e p r e s e n t a t i o n verbal ( i ) propositional

numerical

g e n e r a l l y symbolic

imagi n a l ( ii) analogue a u t o m a t i c frequency m o n i t o r i n g these modes d i f f e r i n :

-

t h e t r a n s f o r m a t i o n s w h i c h can be a ? p l i e d .

The c o n s t r a i n t s on t h e s e r e p r e s e n t a t i o n s can be "strict" o r "elastic".

-

t h e mental work t h e y impose on t h e i n f o r m a t i o n processing capacity

?.A v a i l a b l e

procedures

( i ) r u l e s (e.g. grammars, a r i t h m e t i c s , " G e s t a l t laws" i n p e r c e p t i o n e t c . ) t h e s e i n t e r a c t w i t h t h e above mentioned a p p l i c a b l e t r a n s f o r m a t i o n s and a r e t h e r e f o r e c o n t e x t s p e c i f i c . Less c o n t e x t s p e c i f i c a r e ( i i ) h e u r i s t i c s , w h i c h can be regarded as t o o l s f o r n a r r o w i n g down t h e number o f p o s s i b l e c a n d i d a t e s among t h e t r a n s f o r m a t i o n s i n t h e case o f i n s u f f i c i e n t i n f o r m a t i o n L e a s t c o n t e x t s p e c i f i c and t h e r e f o r e a p p l i c a b l e i n s i t u a t i o n s o f i n f o r m a t i o n overload are ( i i i ) r u l e s o f thumb, which can be a p p l i e d a l m o s t w i t h o u t any cons t r a i n t s . They p r o v i d e " q u i c k and d i r t y " procedures f o r t h e r e d u c t i o n o f i n f o r m a t i o n so t h a t t h e p r o c e s s i n g o v e r l o a d i s r e l i e v e d . (e.g."Take

whatever comes t o y o u r mind f i r s t " . )

PROCESSING OF SUBJECTIVE PROBABILITIES

161

Most i n v e s t i g a t i o n s o f s u b j e c t i v e p r o b a b i l i t i e s have been much more concerned w i t h t h e u n d e r l y i n g procedures ( see bottom h a l f o f t a b l e 1 ) t h a n w i t h t h e modes o f r e p r e s e n t a t i o n (see upper h a l f o f t a b l e 1 ) . These i n v e s t i g a t o r s have imp1 i c i t l y assumed t h a t i n f o r m a t i o n i s s t o r e d p r o p o s i t i o n a l l y and t h a t a f t e r r e t r i e v a l i t can be s t a t e d n u m e r i c a l l y . Any i n c o n s i s t e n c y between i n f o r m a t i o n i n t a k e ( t h e o b j e c t i v e s i d e ) and t h e n u m e r i c a l l y expressed subj e c t i v e p r o b a b i l i t y ( t h e s u b j e c t i v e s i d e ) i s t h e n a s c r i b e d t o t h e procedures. That i s , t h e s u b j e c t i s presumed t o have chosen an i n a p p r o p r i a t e a l g o r i t h m . Examples f o r such proposed p r o c e d u r a l m i s t a k e s a r e o v e r c o n f i d e n c e ( s e e F i g u r e l ) , conservatism (i.e.

s t i c k i n g t o an i n i t i a l a p p r a i s a l o f a s i t u a t i o n i n

s p i t e o f new i n f o r m a t i o n a v a i l a b l e f o r r e v i s i o n ) , and n e g l i g e n c e o f t h e r e g r e s s i o n e f f e c t ( i . e . t h e i m p l i c i t assumption o f a p e r f e c t c o r r e l a t i o n between t h e p r e d i c t o r v a r i a b l e and t h e c r i t e r i o n ) . Theapproach t a k e n h e r e begins by a s k i n g how u n c e r t a i n e v e n t s m i g h t be r e presented i n t e r n a l l y . T h a t i s , a r e t h e y r e p r e s e n t e d i n ( i ) a v e r b a l propos i t i o n a l mode, ( i i ) a numerical p r o p o s i t i o n a l mode, o r ( i i i ) i n an analogue mode o f a u t o m a t i c frequency m o n i t o r i n g ( see t h e n e x t c h a p t e r ), Thus, what i s q u e s t i o n e d h e r e i s n o t t h e adequacy o f human j u d m e n t s c o n c e r n i n g u n c e r t a i n t y , b u t r a t h e r t h e i n d i v i d u a l s ’ a b i l i t y t o express n u m e r i c a l l y what i s i n t e r n a l l y represented. Coping w i t h u n c e r t a i n t y i s u b i q u i t o u s i n humankind ( WRIGHT and PHILLIPS, 1980) and e x p r e s s i o n s f o r d i f f e r e n t degrees o f c e r t a i n t y can be f o u n d i n most languages. Yet, as r e p o r t e d above i t t o o k u n t i l t h e 1 7 t h c e n t u r y t o develop these concepts m a t h e m a t i c a l l y . Even t h e n i t t o o k p l a c e o n l y i n t h e European c u l t u r e where i t was m o t i v a t e d by p r a c t i c a l problems; such as gambl i n a and insurance. I t seems u n l i k e l y t h a t t h e m a t h e m a t i c a l l y a p p r o p r i a t e procedures w i t h numerical e s t i m a t e s o f u n c e r t a i n t y have become a u t o m a t i z e d since then. I t i s more l i k e l y t h a t p e o p l e h a n d l e u n c e r t a i n t y by customary v e r b a l e x p r e s s i o n s and t h e i m p l i c i t and e x p l i c i t r u l e s o f c o n v e r s a t i o n conn e c t e d w i t h them. I n t h e r e s e a r c h r e p o r t e d h e r e I t h e r e f o r e s t a r t f r o m t h e a n a l y s i s o f meaning o f common v e r b a l e x p r e s s i o n s f o r u n c e r t a i n events. These e x p r e s s i o n s a r e i n t e r p r e t e d as p o s s i b i l i t y f u n c t i o n s (ZADEH, 1978) and t h e procedures a p p l i c a b l e t o them ( r u l e s as w e l l as h e u r i s t i c s and b i a s e s ) a r e m o d e l l e d i n t h e framework o f p o s s i b i l i t y t h e o r y . S i n c e t h i s t h e o r y a l l o w s

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1.0 [

1

&--

------

I

I

I

Hazard 8, Peterson, 1973: Probabilities

.......... Hazard 8 Peterson, 1973: Odds

0..

n.-.-.-.-

- -0

.-.5

Philips 8 Wright, 1977 Lichtenstein (unpublished)

.6

.7 .8 .9 Subjects' Responses

1.0

Figure 1 . Calibration curves adapted from LICHTENSTEIN, FISCHHOFF, and PHILLIPS (1982) Overconfidence is shown by the fact that in the upper part of the graph the subjective probability estimates are significantly higher than the objective relative frequencies ( probabilities). for a numerical interpretation by means o f determining the elastic constraints on the usage of such expressions, the results gained by interpreting verbal expressions of uncertainty as possibility functions can be compared to the results of the above mentioned studies, where subjects provide the numerical expressions themselves.

163

PROCESSING OF SUBJECTIVE PROBABILITIES

The Automatic Monitoring o f Frequencies Before developing t h i s theory i n d e t a i l f o r the purposes o f our study i t seems t o be worthwhile t o ask how people gather the knowledge which i s r e f l e c t e d i n the verbal expressions f o r the p r o b a b i l i t i e s o f u n c e r t a i n events. One can assume t h a t f o r r e p e t e t i v e events e.g. the outcomes o f ballgames o r the d a i l y weather, they m n i t o r the frequency o f outcomes automatically and t h a t thev r e v i s e t h e i r knowledge accordingly. HINTZMAN, NOZAWA, and IRMSCHER (1982) have studied t h i s automatic monitoring o f frequencies and they r e p o r t t h a t i t i s n o t i n t e r f e r e d w i t h by numerical processing. From t h e i r r e s u l t they conclude t h a t t h i s information i s stored i n a non-numerical analogue mode (see item 1 ( i i ) i n Table 1 ) . Such automatic monitoring o f frequencies seems t o be a p l a u s i b l e candidate f o r the mode o f representation underlying the generation o f judgments concerning subjective p r o b a b i l i t i e s . I n an experiment I have t r i e d t o determine whether the frequencies o f more than one unattended stimulus a t t r i b u t e can be monitored i n order t o determine the degree o f a u t o m a t i c i t y and mutual i n t e r f e r e n c e o f m u l t i p l e frequency judgments. I n the experiment 150 c o l o r s l i d e s d e p i c t i n g e i t h e r landscapes o r b u i l d i n g s were presented t o 80 I n t r o d u c t o r y Psychology students i n a probabil i t y learning task, i n which they had t o p r e d i e t the content landscape

-

-

b u i l d i n g vs.

o f each upcoming s l i d e . Afterwards they were asked f o r the r e -

l a t i v e frequencies o f two presumably attended features o f the s l i d e s , o r i e n t a t i o n and form, which were uncorrelated w i t h the content o f the s l i d e s . The question format was e i t h e r a numerical scale, on which subjects had t o mark t h e i r frequency estimate, o r a sequence o f verbal expressions f o r r e l a t i v e frequencies, o u t o f which they had t o p i c k the best f i t t i n g one. For the exact experimental design and the r e s u l t s (mean squared e r r o r s ) see Table 2 . The r e s u l t s show ( i ) t h e unattended features d i f f e r e d i n saliency ( o r i e n t a t i o n was easier t o monitor than form), ( i i ) t h e verbal judgments were s l i g h t l y b e t t e r than the numerical ones, ( i i i ) i n a l l cases the second f r e quency judgments were worse than the f i r s t ones, b u t more important, the impairment was l e s s i n a l l cases where one o f the judgments was verbal and i t was l e a s t when both were verbal. From these r e s u l t s i t seems p l a u s i b l e t o conclude, f i r s t , t h a t more than one unattended v a r i a b l e can be automaticall y monitored even i f they are c l o s e l y related, b u t t h e mode o f probing t h i s

COGNITIVE DECISION RESEARCH

164

Table 2 The v e r b a l judgments have been t r a n s f o r m e d i n t o n u m e r i c a l v a l u e s by assuming t h a t t h e l a b e l s p a r t i t i o n t h e range (0%

mode o f

mean

-

100%) e q u a l l y .

udgment

squared

verbal

numerical

errors

other position

other position

v e r b a l judgm.

I numerical j

v e r b a l judgm.

I

numerical j

posi t i o i

1

E :1

positioi

2

posit i o i E

1

0

.r

c, IIJ

CI

Z

positioi

A

2

.r

knowledge i n f l u e n c e s t h e p r e c i s i o n o f judgments. Second, i f more t h a n one judgment has t o be made t h e r e i s i n f e r e n c e between them, b u t t h e amount o f i n f e r e n c e depends on t h e modes o f judgments. T h i s a p p a r e n t s u p e r i o r i t y o f t h e v e r b a l mode l e a d s t o t h e t e n t a t i v e i n t e r p r e t a t i o n t h a t t h e v e r b a l mode f o r r e p r e s e n t i n g knowledge i s a b l e t o process i n f o r m a t i o n more e f f e c t i v e l y t h a n t h e numerical mode, because t h e r u l e s a p p l i c a b l e t o p r o c e s s i n g v e r b a l p r o p o s i t i o n s have been a c q u i r e d e a r l i e r and a r e t h e r e f o r e more a u t o m a t i z e d . T h i s would e x p l a i n why t h e y r e q u i r e l e s s mental p r o c e s s i n g c a p a c i t y and consequentl y a r e l e s s prone t o i n t e r f e r e n c e f r o m simultaneous processes. Ift h e o v e r -

PROCESSING OF SUBJECTIVE PROBABILITIES

165

l o a d o f t h e mental p r o c e s s i n g c a p a c i t y n e c e s s i t a t e s t h e a p p l i c a t i o n o f heur i s t i c s , and i f i n t u r n b i a s e s i n human judgment can be t r a c e d back t o m i s t a k e n a p p l i c a t i o n s o f h e u r i s t i c s ( f o r an o v e r v i e w see NISBETT and ROSS, 1980), t h e f o l l o w i n g c o n j e c t u r e seems t o be p l a u s i b l e : Any mode o f judgment i m p o s i n g l e s s mental work l o a d s h o u l d be more v a l i d t h a n one r e q u i r i n g more mental p r o c e s s i n g c a p a c i t y . T h i s c o n j e c t u r e m o t i v a t e d t h e f o l l o w i n g e x p e r i ments, i n which I have l o o k e d f o r p o s s i b l e ways o f making t h e i n f o r m a t i o n p r o c e s s i n g e a s i e r i n t a s k s which i m p l y t h e a n a l y s i s o f u n c e r t a i n events. I n o r d e r t o a n a l y z e t h e e f f i c i e n c y o f d i f f e r e n t modes o f r e p r e s e n t a t i o n i n i n f o r m a t i o n p r o c e s s i n g i t i s necessary t o d e v i s e an i n t e r p r e t a t i o n f o r t h e meaning o f v e r b a l e x p r e s s i o n s which can be t r a n s l a t e d i n t o numerical v a l u e s . Fuzzy s e t t h e o r y p r o v i d e s a t o o l f o r such i n t e r p r e t a t i o n b y means o f num e r i c a l l y s t a t e d e l a s t i c c o n s t r a i n t s f o r t h e a p p l i c a b i l i t y o f v e r b a l exp r e s s i ons ,

A Fuzzy Set T h e o r e t i c a l Approach t o S u b j e c t i v e P r o b a b i l i t i e s According t o f u z z y s e t t h e o r y t h e meaning o f concept

"5'' i n a u n i v e r s e o f

d i s c o u r s e "U" can be modelled by t h e p o s s i b i l i t y f u n c t i o n f o r

"J,"

i n "U",

which i n d i c a t e s f o r which s t a t e s i n "U" t h e concept 'I&" f i t s , f o r which i t can be p o s s i b l y a p p l i e d , and f o r which i t does n o t f i t a t a l l . F i g u r e 2 d e p i c t s t h e p o s s i b i l i t y f u n c t i o n s f o r two concepts as w e l l as t h e b a s i c two b i n a r y operations :

( i i ) the disjunction

The v e r s i o n o f f u z z y s e t t h e o r y a p p l i e d here, e s p e c i a l l y c o n c e r n i n g t h e f o r m o f p o s s i b i l i t y functions, deviates

f r o m approaches where t h e e x a c t f o r m o f t h e s e f u n c t i o n s i s determined e i t h e r t h e o r e t i c a l l y ( f o r an o v e r v i e w see KAUF-

MA",

1975, o r DUBOIS and PRADE, 1980) o r e m p i r i c a l l y (e.g.

HERSH and CARA-

MAZZA, 1976; ZIMMER, 1980, 1982 a ) . The a p p r o p r i a t e n e s s o f t h e s e s p e c i f i c membership f u n c t i o n s as p a r t s o f t h e mental r e p r e s e n t a t i o n o f vague concepts

166

COGNl TI VE D EClSlO N RESEARCH

9 4 n ecessar

B

/ -

possible

F i g u r e 2. P o s s i b i l i t y d i s t r i b u t i o n s and b i n a r y o p e r a t i o n s upon them

i s a t l e a s t d o u b t f u l . FREKSA (1981, 1982) has shown t h a t f o r p r a c t i c a l p u r poses such s p e c i f i c a t i o n i s unnecessary. By r e s t r i c t i n g t h e p o s s i b i l i t y t o t h e values: "necessary", " p o s s i b l e " , and " i m p o s s i b l e " , one can a v o i d t h e deb a t e d q u e s t i o n o f how e x a c t t h e i n f o r m a t i o n a b o u t one's own knowledge can be ( c f . NISBETT and WILSON

1977) w i t h o u t l o s i n g t h e amount o f s p e c i f i c i t y ne-

cessary t o a p p l y f u z z y s e t t h e o r y t o v e r b a l concepts. The meanings o f t h e s e concepts i s t h e n g i v e n by t h e e l a s t i c c o n s t r a i n t s imposed upon them by t h e p o s s i b i l i t y functions. M o d e l l i n g t h e meanings o f v e r b a l l a b e l s f o r r e l a t i v e f r e q u e n c i e s i s s t r a i g h t f o r w a r d i n t h i s framework. The u n i v e r s e o f d i s c o u r s e i s t h e u n i t i n t e r v a l and t h e r e g i o n s of a p p l i c a b i l i t y , p o s s i b l e a p p l i c a b i l i t y , and i n a p p l i c a b i l i t y can t h e n be determined e m p i r i c a l l y . The r e s u l t i n g p o s s i b i l i t y f u n c t i o n s f o r a g i v e n s e t o f v e r b a l e x p r e s s i o n s i s d e p i c t e d i n F i g u r e 3. The s p a c i n g of t h e p o s s i b i l i t y f u n c t i o n s i n t h e u n i t i n t e r v a l i s r e m i n i s c e n t o f WITTE's (1960) t r e a t m e n t o f v e r b a l judgments s t a b i l i z e d i n memory o v e r t i m e . I ( Z I M MER, 1980) have shown t h a t such e q u i d i s t a n t and e q u a l l y shaped c a t e g o r i e s a r e o p t i m a l f r o m t h e p o i n t o f v i e w o f language p r a g m a t i c s , b u t i t has t o be k e p t

PROCESSING OF SUBJECTIVE PROBABILITIES

167

100 frequ. F i g u r e 3. P o s s i b i l i t y d i s t r i b u t i o n s f o r t h e meaning of v e r b a l e x p r e s s i o n s f o r uncertainty f o r a subject using 5 labels i n mind t h a t t h i s c o m m u n i c a b i l i t y c o n s t r a i n t i s a s i m p l i f y i n g assumption, which i s n o t a p r e r e q u i s i t e f o r t h i s k i n d o f m o d e l l i n g . WITTE's approach, however, i m p l i e s two q u e s t i o n a b l e assumptions: t h a t t h e v e r b a l l a b e l s a r e u n i q u e l y ordered f o r a l l s u b j e c t s and t h a t a l l s u b j e c t s a r e e q u a l l y f a m i l i a r w i t h t h e l a b e l s p r o v i d e d by t h e e x p e r i m e n t e r .

E m p i r i c a l D e t e r m i n a t i o n o f t h e Fuzzy Meanings o f Verbal Expressions I n t h e p r e l i m i n a r y s t e p f o r my experiment I observed t h e s u b j e c t ' s usage o f e x p r e s s i o n s f o r u n c e r t a i n events and asked them a f t e r w a r d s t o o r d e r t h e used l a b e l s a c c o r d i n g t o frequency. I t t u r n e d o u t t h a t t h e s u b j e c t s d i f f e r e d markedl y i n v e r b a l l a b e l s used and moreover i n t h e s e r i a l o r d e r o f t h e same l a b e l s

used by d i f f e r e n t s u b j e c t s . F o r i n s t a n c e some s u b j e c t s r e p o r t e d t h e i n c r e a s i n g s e r i a l o r d e r " p o s s i b l e , 1 i k e l y , p r o b a b l e " whereas o t h e r s i n t e r c h a n g e d "poss i b l e " and "1 ik e l y " o r " 1 ike1y"and "probable" (WALLSTEN, n o t e 1, r e p o r t s s i m i l a r r e s u l t s ) . I n t h e f o l l o w i n g i t w i l l be assumed t h a t s u b j e c t s used t h e s e e x p r e s s i o n s f n t h e way t h e y r e p o r t e d . As i t w i l l t u r n o u t l a t e r t h i s assumption i s i n l i n e w i t h t h e e m p i r i c a l r e s u l t s . The e l i c i t a t i o n o f t h e v e r b a l l a b e l s t h a t s u b j e c t s were most f a m i l i a r w i t h and used most f r e q u e n t l y , r e v e a l e d t h a t most s u b j e c t s used m e r e l y f i v e o r s i x d i f f e r e n t e x p r e s s i o n s ( see Table 3 ) . A t o t a l of t w e l v e d i f f e r e n t l a b e l s was

168

COGNITIVE DECISION RESEARCH

Table 3

number o f 1abel s f o r uncertain

4

5

6

7

a

2

a7

55

5

1

events

number o f s u b j e c t s

found. The e m p i r i c a l d e f i n i t i o n s o f t h e meanings had t o be done i n d i v i d u a l l y because of t h e i n t e r i n d i v i d u a l d i f f e r e n c e s . From t h e p o i n t o f v i e w o f language pragmatics t h e s e i n d i v i d u a l d i f f e r e n c e s i n u s i n g t h e same e x p r e s s i o n s a r e a p u z z l i n g a s p e c t because t h e y c o n t r a d i c t t h e assumed c o m m u n i c a b i l i t y c o n s t r a i n t and c a s t i n t o doubt t h e p r a g m a t i c v a l u e o f such v e r b a l judgments. From t h e interviews w i t h the subjects i n t h e d e b r i e f i n g sessions I learned t h a t these marked i n d i v i d u a l d i f f e r e n c e s were p a r t i a l l y due t o t h e c o m p o s i t i o n s o f our sample o f s u b j e c t s : 30 s t u d e n t s i n psychology, 30 s t u d e n t s i n b u s i n e s s and economy, and 90 d r a f t e d s o l d i e r s o f t h e German Bundeswehr. Whereas t h e f i r s t two groups had f i n i s h e d courses i n s t a t i s t i c s and p r o b a b i l i t y t h e o r y , i n t h e l a s t group o n l y a s m a l l p r o p o r t i o n had s t u d i e d p r o b a b i l i t y theovy i n h i g h school. When asked what t h e y would do i f t h e y r e a l i z e d t h a t somebody e l s e m i g h t misunderstand t h e i r v e r b a l f r e q u e n c y e x p r e s s i o n s , most s u b j e c t s r e p o r t e d t h a t i n such cases t h e y would use t h e r e f e r e n c e t o g e n e r a l l y known s t a n d a r d s i t u a t i o n s as a means t o r e s o l v e t h i s m i s u n d e r s t a n d i n g w i t h an anal o g y . F o r i n s t a n c e t h e y m i g h t p o i n t o u t t h a t a c e r t a i n e v e n t i s as p r o b a b l e as w i n n i n g i n t h e German L o t t e r y . The e m p i r i c a l d e t e r m i n a t i o n a f t h e meanings f o r t h e i n d i v i d u a l l y used v e r b a l e x p r e s s i o n s i s done i n t h e f o l l o w i n g s t e p s : ( i ) survey o f r e c u r r e n t l y used l a b e l s f o r r e l a t i v e f r e q u e n c i e s ( i i ) t e s t o f e q u i v a l e n c e f o r l a b e l s (Do some o f t h e words mean t h e same f o r you?)

PROCESSING OF SUBJECTIVE PROBABILITIES

169

( i i i ) d e t e r m i n a t i o n o f t h e ranges f o r t h e l a b e l s , s ( k ) , assuming t h a t t h e l a b e l s p a r t i t i o n t h e o v e r a l l range e q u a l l y ( i v ) s e q u e n t i a l e s t i m a t i o n o f t h e e m p i r i c a l meanings f o r t h e l a b e l s used by t h e s u b j e c t s under examination: The f i r s t s t e p i n t h e e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e meanings f o r t h e v e r b a l e x p r e s s i o n s used by t h e s u b j e c t s c o n s i s t e d i n d e t e r m i n i n g t h e s t a r t i n g p o i n t f o r t h e e s t i m a t i o n procedure. T h i s was done by t a k i n g t h e m i d p o i n t o f t h e range i n case o f an uneven number o f t h e v e r b a l l a b e l s , t h a t i s , 50%, o r i n t h e case o f an even number o f c a t e g o r i e s a s t a r t i n g p o s i t i o n was t a k e n s l i g h t l y moved t o t h e l o w e r o r upper p a r t o f t h e range, t h a t i s , e i t h e r 33% o r 67%. I n o r d e r t o r e p r e s e n t o b j e c t i v e r e l a t i v e f r e q u e n c i e s f o r t h e c a l i b r a t i o n o f t h e i n d i v i d u a l l y used v e r b a l expressions, I showed a window on a c o m p u t e r - d r i v e n t v - s c r e e n which cont a i n t e d up t o 6144 d o t s ( 1 0 0 % ) . The number o f t h e w h i t e d o t s spaced r a n domly i n t h e window i n r e l a t i o n t o t h e s i z e o f t h e screen r e p r e s e n t e d the o b j e c t i v e r e l a t i v e frequencies. A f t e r the s t a r t i n g c o n f i g u r a t i o n had been shown t h e s u b j e c t had t o g i v e a v e r b a l e x p r e s s i o n f o r t h e chance t h a t any p o i n t i n s i d e o f t h e window would be covered b y a w h i t e d o t , when t h e same frequency of d o t s were p r e s e n t e d a g a i n . A f t e r t h e s u b j e c t s had g i v e n t h e i r answers t h e n e x t r e l a t i v e frequency o f w h i t e d o t s was determined a c c o r d i n g t o a m o d i f i e d RCBBINS-MONROE procedure w i t h an e v e r d e c r e a s i n g s t e p s i z e as l o n g as t h e s u b j e c t s gave t h e same v e r b a l l a b e l . When t h e s u b j e c t changed t h e l a b e l t h e d i r e c t i o n o f change f o r t h e r e l a t i v e frequencies was changed, t o o . A f t e r more t h a n 4 s w i t c h e s i n d i r e c t i o n , a new s t a r t i n g p o s i t i o n was t a k e n i n a n o t h e r p a r t o f t h e range a c c o r d i n g t o t h e number of v e r b a l e x p r e s s i o n s used by t h e s u b j e c t

i n q u e s t i o n . T h i s r e c u r s i v e procedure was r e p e a t e d u n t i l t h e b o u n d a r i e s f o r each v e r b a l l a b e l had beet1 determined. The boundaries, t h a t i s , t h e areas where s u b j e c t s s w i t c h e d f r o t r one l a b e l t o another, were i n t e r p r e t e d as t h e e l a s t i c c o n t r a i n t s c h a r a c t e r i z i n g t h e medning o f t h e v e r b a l expression. A t y p i c a l r e s u l t o f t h i s e x p e r i m e n t a l procedure i s shown i n F i g u r e 3. The r e s u l t s (e.g. F i g u r e 3, l a b e l " l i k e l y " ) can be i n t e r p r e t e d i n t h e f o l l o w i n g way: f o r t h e s u b j e c t under c o n s i d e r a t i o n t h e l a b e l " l i k e l y " f i t s f o r

170

COGNITIVE DECISION RESEARCH

events w i t h t h e r e l a t i v e f r e q u e n c y o f occurence between .65 and .75. I t m i g h t be a p p l i e d t o o i n t h e i n t e r v a l s .55

-

.65 and .75

-

.85 ( p o s s i b l e a p p l i c a -

b i l i t y ) , and i t i s n o t a p e r m i s s i b l e e x p r e s s i o n f o r e v e n t s o u t s i d e o f t h i s interval. The M o d e l l i n g o f t h e A v a i l a b i l i t y B i a s i n t h i s Model

I have argued elsewhere (ZIMMER, 1982 b ) t h a t t h e c o n t e x t s p e c i f i c meanings o f

q u a n t i f i e r s i n c o l l o q u i a l E n g l i s h c a n be decomposed i n t o knowledge a b o u t t h e occurence o f q u a n t i f i e d statements i n t h e s e c o n t e x t s and i n t o c o n t e x t f r e e meanings o f q u a n t i f i e r s . T h i s s e p a r a b i l i t y o f c o n t e x t u a l and g e n e r a l meaning m i g h t e x p l a i n why my s u b j e c t s a r e a p p a r e n t l y a b l e t o d i s t i n g u i s h between a t l e a s t f o u r q u a n t i f i e r s , whereas BEGG (1982), who does n o t t a k e i n t o account t h e c o n t e x t , concludes f r o m h i s d a t a t h a t normal language r e l i e s on o n l y t h r e e q u a n t i f i e r s . T h i s s e p a r a b i l i t y can a l s o be used t o model t h e a v a i l a b i l i t y b i a s i n t h e model developed i n t h i s paper. It has been shown

( f o r an o v e r v i e w see TVERSKY and KAHNEMAN, 1973 ) t h a t j u d g i n g t h e r e l a t i v e frequency o f a c l a s s o f events, such as t h e p r o b a b i l i t y o f d y i n g due t o an a i r c r a f t accident, s u b j e c t s t a k e i n t o account n o t o n l y t h e occurence o f t h i s e v e n t b u t a l s o t h e d i f f i c u l t y o r ease t h e y have i n coming up w i t h a t y p i c a l example f o r t h i s c l a s s o f events; t h e more a v a i l a b l e i n f o r m a t i o n a b o u t t h e e v e n t t h e r e i s , t h e more t h e p r o b a b i l i t y o f occurence i s o v e r e s t i m a t e d . T h i s a v a i l a b i l i t y b i a s can be accounted f o r i n o u r model b y t h e c o n j u n c t i o n o f t h e f u z z y meaning o f each v e r b a l e x p r e s s i o n under c o n s i d e r a t i o n and a scope f u n c t i o n r e p r e s e n t i n g t h e i n f l u e n c e o f a v a i l a b i l i t y . I n F i g u r e s 4, 5, and 6 scope f u n c t i o n s f o r h i g h , moderate, and l o w a v a i l a b i l i t y a r e shown. F o r t h e s e scope f u n c t i o n s o n l y m o n o t o n i c i t y i s assumed; t h e e x a c t shape i s t o be det e r m i n e d e m p i r i c a l l y , e.g. b y measuring t h e a s s o c i a t i o n speed o r a s i m i l a r v a r i a b l e . The c o n j u n c t i o n o f t h e s e scope f u n c t i o n s w i t h t h e p o s s i b i l i t y f u n c t i o n s f o r t h e meanings o f t h e v e r b a l l a b e l s l e a d s , a f t e r r e n o r m a l i z a t i o n , t o p o s s i b i l i t y f u n c t i o n s o f t h e b i a s e d judgments ( see f i g u r e s 7, 8, and 9 ) . The comparison o f t h e b i a s e d w i t h t h e c o n t e x t f r e e p o s s i b i l i t y judgments r e v e a l s t h a t t h e model p r e d i c t s an upward s h i f t o f t h e h i g h u n c e r t a i n t y l a b e l s i n t h e case o f h i g h a v a i l a b i l i t y and a downward s h i f t o f t h e l o w u n c e r t a i n t y l a b e l s i n t h e case o f l o w a v a i l a b i l i t y , whereas t h e l a b e l s on t h e o t h e r end of t h e range remain v i r t u a l l y unchanged. T h i s e f f e c t i s i n l i n e w i t h e x p e r i -

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mental r e s u l t s (e.g. t h o s e o f SLOVIC, FISCHHOFF, and LICHTENSTEIN, 1979). I f t h e p o s t u l a t e d s e p a r a b i l i t y o f t h e f u z z y meanings and t h e a v a i l a b i l i t y b i a s proves t o be e m p i r i c a l l y v a l i d , t h i s model can have f r u i t f u l a p p l i c a t i o n s i n methods f o r e s t i m a t i n g s u b j e c t i v e p r o b a b i l i t i e s , because i t p r o v i d e s a p r o cedure f o r d e b i a s i n g ( f o r d i f f e r e n t approaches t o d e b i a s i n g see FISCHHOFF, 1982).

Scope functions

1.00 F i g u r e 4. Scope f u n c t i o n f o r a " h i g h a v a i l a b i l i t y b i a s "

F i g u r e 5. Scope f u n c t i o n f o r a "moderate a v a i l a b i l i t y b i a s "

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1.00 F i g u r e 6. Scope f u n c t i o n f o r a

"low a v a i l a b i l i t y b i a s "

Influence of availability on categories -----_-- X A bias normalized axn bias'

F i g u r e 7. I n f l u e n c e o f h i g h a v a i l a b i l i t y on t h e meaning o f " h i g h l y improbable" and " l i k e l y " i n F i g u r e 3

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173

F i g u r e 8. I n f l u e n c e o f moderate a v a i l a b i l i t y on t h e meaning o f " h i g h l y i m probable" and " l i k e l y " i n F i g u r e 3

F i g u r e 9 . I n f l u e n c e o f low a v a i l a b i l i t y and " v e r y l i k e l y " i n F i g u r e 3

on

t h e meaning o f " h i g h l y improbable"

Up t o t h i s p o i n t I have i m p l i c i t l y assumed t h a t t h e p o s s i b i l i t y f u n c t i o n s f o r the verbal l a b e l s a r e w e l l c a l i b r a t e d , which i s t a u t o l o g i c a l l y t r u e f o r t h e r e l a t i v e frequencies o f w h i t e d o t s on a t v screen. I t can a l s o be shown f o r tasks u s u a l l y undertaken t o t e s t c a l i b r a t i o n , where s u b j e c t s answer q u e s t i o n s about almanac knowledge and e s t i m a t e immediately a f t e r w a r d s the s u b j e c t i v e p r o b a b i l i t y o f t h e i r answer b e i n g c o r r e c t ( f o r an overview see LICHTEMSTEIN, FISCHHOFF, and PHILLIPS, 1982).

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C a l i b r a t i o n o f Verbal Expressions f o r Probabi 1it i e s

I n o r d e r t o s e p a r a t e t h e v a r i a b l e s " d i f f i c u l t y o f t h e i t e m " and " o f t h e s u b j e c t " , b o t h o f which i n f l u e n c e t h e p r o b a b i l i t i e s o f c o r r e c t answers, t h e c o n j o i n t measurement approach o f RASCH (1966) i s a p p l i e d , which p r o v i d e s i n dependant e s t i m a t e s o f t h e a b i l i t y o f a s u b j e c t , 5 ( i ) , and t h e d i f f i c u l t y o f an i t e m j , 6 ( j ) . These e s t i m a t e s i n t u r n a l l o w us t o assess t h e p r o b a b i l -

i t y t h a t a s u b j e c t i s o l v e s i t e m j, a c c o r d i n g t o t h e f o l l o w i n g f o r m u l a :

e

l+e

ti- 6j

MAY ( n o t e 2) has suggested t h i s approach t o c a l i b r a t i o n w h i c h t a k e s i n t o account i n d i v i d u a l d i f f e r e n c e s . The s p e c i f i c d i f f i c u l t y o f a g i v e n i t e m j f o r a s u b j e c t i i s t h e d i f f e r e n c e between t h e d i f f i c u l t y o f t h e i t e m and t h e a b i l i t y o f t h e s u b j e c t . On t h i s s c a l e t h e r e l a t i v e f r e q u e n c i e s o f t h e l a b e l s

which t h e s u b j e c t s use t o c h a r a c t e r i z e t h e i r p r o b a b i l i t y o f success, can be compared w i t h t h e e s t i m a t e d o b j e c t i v e p r o b a b i l i t i e s . A t e s t c o n s i s t i n g o f 150 p o l i t i c a l knowledge i t e m s was a d m i n i s t e r e d t o 90

d r a f t e d s o l d i e r s o f t h e German Bundeswehr. The i t e m s were i n an open response f o r m a t w i t h s u b j e c t s f i r s t answering t h e i t e m and i m m e d i a t e l y a f t e r w a r d s g i v i n g t h e i r v e r b a l e x p r e s s i o n s f o r t h e p r o b a b i l i t y o f a c o r r e c t answer. F i g u r e 10 r e p r e s e n t s t h e r e s u l t s o f a s i n g l e s u b j e c t , who a p p l i e d f i v e d i f f e r e n t e x p r e s s i o n s f o r h i s s u b j e c t i v e p r o b a b i l i t i e s . F i g u r e 11 g i v e s t h e d i f f i c u l t y - b y - l a b e l c u r v e s f o r a l l s u b j e c t s u s i n g f i v e l a b e l s . The curves a r e f a i r l y p a r a l l e l and t h e median values o f t h e p o s s i b i l i t y f u n c t i o n s ( r a n g e s o f a p p l i c a b i l i t y f o r t h e e x p r e s s i o n s ) a r e i n good agreement w i t h t h e p r o b a b i l i t i e s e s t i m a t e d f r o m t h e model. I n o r d e r t o make t h e s e r e s u l t s more comparable t o t h e ones d e p i c t e d i n F i g u r e

10 I have computed t h e p r o b a b i l i t i e s o f successes f o r g i v e n a b i l i t y l e v e l s and g i v e n d i f f i c u l t y l e v e l s c o n d i t i o n e d on t h e l a b e l s a p p l i e d . These "obj e c t i v e " p r o b a b i l i t i e s a r e p l o t t e d a g a i n s t t h e median v a l u e s o f t h e p o s s i b i l i t y f u n c t i o n s r e p r e s e n t i n g t h e meaning o f t h e v e r b a l e x p r e s s i o n s i n F i g u r e 12.

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F i g u r e 10. The c u m u l a t i v e f r e q u e n c i e s f o r t h e v e r b a l l a b e l s o f one s u b j e c t . The curves a r e drawn f o r i t e m s f r o m seven c a t e g o r i e s o f d i f f i c u l t y f o r t h i s s u b j e c t , which determine t h e i n d i v i d u a l probabil i t i e s o f successes a c c o r d i n g t o t h e Rasch model. The curves a r e l a b e l l e d w i t h t h e mean f r e q u e n c i e s o f t h e second experiment (see F i g u r e 3). The s u b j e c t would be p e r f e c t l y c a l i b r a t e d i f t h e l a b e l i n g o f t h e curves and t h e c u t t i n g p o i n t s w i t h t h e dashed l i n e were t h e same. T h i s s u b j e c t i s s l i g h t l y u n d e r c o n f i d e n t f o r h a r d it e ms b u t w e l l c a l i b r a t e d f o r t h e easy ones.

I

A 0 - b y - label curves

0

A

8.

F i g u r e 11. The average c u m u l a t i v e d i f f i c u l t y by l a b e l curves f o r a l l s u b j e c t s applying f i v e expressions f o r t h e i r s u b j e c t i v e p r o b a b i l i t i e s o f successes

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.15

.33

.51

.67

.86 Su bj. resp. ( r i k I

Figure 1 2 . The p r o b a b i l i t i e s o f successes f o r given a b i l i t y and d i f f i c u l t y l e v e l s conditioned on the applied verbal expressions p l o t t e d against the median values o f t h e - p o s s i b i l i t y f u n c t i o n s f o r these verbal expressions.

Although t h e r e are marked variances, the r e s u l t s i n d i c a t e t h a t on t h e whole 1 t h i s group o f subjects i s w e l l c a l i b r a t e d .

SCHUETT (1981) reports t h a t a f t e r s u f f i c i e n t t r a i n i n g and optimal support by a computerized information processing system h i s subjects were able t o estimate s u b j e c t i v e p r o b a b i l i t i e s , which d i d n o t v i o l a t e S A V A G E ' S (1954) axioms. This r e s u l t does n o t c o n t r a d i c t the conclusions I have drawn from my experiments; t h a t i s , humans u s u a l l y express t h e i r confidence v e r b a l l y and they are able t o perform by means o f verbal arguments a task which i s as complex as S A V A G E ' S system o f axioms. Furthermore t h e r e s u l t s reported here i n d i c a t e t h a t by using t h i s k i n d of i n f o r m a t i o n processing people are able t o g i v e v e r i d i c a l judgments w i t h o u t t h e amount o f external support provided i n SCHUETT's (1981) experiment.

PROCESSING OF SUBJECTIVE PROBABILITIES

177

F u r t h e r A p p l i c a t i o n s o f t h i s Model As p o i n t e d o u t i n t h e b e g i n n i n g I assume t h a t any mode o f p r o c e s s i n g i n f o r m a t i o n which imposes h i g h e r mental work l o a d i s bound t o r e n d e r any c o n s e c u t i v e r a t i n g more b i a s e d t h a n a mode imposing a l o w e r mental work l o a d . Examples f o r t h e n e g a t i v e e f f e c t o f a h i g h mental work l o a d can be found i n t h e s t u d i e s done on c o n s e r v a t i s m i n i n f o r m a t i o n p r o c e s s i n g and on t h e n e g l i g e n c e o f t h e r e g r e s s i o n e f f e c t i n numerical p r e d i c t i o n t a s k s . I n o r d e r t o f i n d o u t i f t h e a p p l i c a t i o n o f verbal expressions i n these tasks r e l i e v e s t h e b i a s i n g e f f e c t , I have a p p l i e d my model t o two p r o t o t y p i c a l experiments f r o m t h i s area.

Conservatism A t y p i c a l example f o r t h e suboptimal i n f o r m a t i o n p r o c e s s i n g has been demonstrated i n experiments on conservatism, where s u b j e c t s t e n d t o s t i c k t o

t h e i r i n i t i a l assumptions c o n c e r n i n g t h e p r o b a b i l i t i e s o f events d e s p i t e t h e f a c t t h a t i n t h e l i g h t o f new i n f o r m a t i o n t h e y s h o u l d r e v i s e t h e s e assumptions. The o p t i m a l r e v i s i o n s t r a t e g y f o r assumptions i n t h i s t a s k i s t h e a p p l i c a t i o n of Bayes’theorem, which t h e r e f o r e can be used as a n o r m a t i v e s t a n d a r d f o r t h e s u b j e c t s ’ performance. I n a s e r i e s o f experiments PHILLIPS and EDWARDS (1966) have i n v e s t i g a t e d t h i s phenomenon w i t h t h e r e s u l t t h a t i n a l l cases c o n s e r v a t i s m occured. T h i s e f f e c t c o u l d be s l i g h t l y reduced by r e 1 i e v i n g t h e memory l o a d o f t h e s u b j e c t s and i n one experiment by p e r m i t t i n g them t o answer v e r b a l l y . 1 have r e p l i c a t e d p a r t o f t h e i r experiment I11 w i t h 30 s u b j e c t s (undergraduates i n economy). S u b j e c t s were shown two i d e n t i c a l b i n s p i c t u r e d s i d e by s i d e on a t v screen. They were t o l d ( A ) c o n t a i n e d 70% b r i c k s and 30% b a l l s and t h e o t h e r (B) 30% b r i c k s and 70% b a l l s . On each t r i a l s u b j e c t s pushed a b u t t o n and t h e r e b y drew a sample o f one either a brick or a ball

-

-

f r o m each o f t h e b i n s . S u b j e c t s were asked t o d e s c r i b e v e r b a l l y t h e chance o f g e t t i n g a b r i c k o r a b a l l t h e n e x t t i m e f r o m t h e b i n . On a second s c r e e n a t a l l y was k e p t o f t h e numbers o f b r i c k s and b a l l s drawn f r o m each b i n , t o g e t h e r w i t h t h e l a s t v e r b a l e x p r e s s i o n t h e s u b j e c t had g i v e n f o r t h e chances. The r e s u l t s p l u s t h e ones f r o m PHILLIPS and EDIdARDS (1966, experiment 111) a r e shown i n F i g u r e 13. These r e s u l t s c l e a r l y i n d i c a t e t h a t v e r b a l responses t o g e t h e r w i t h a v i s i b l e r e c o r d induces near o p t i m a l , o r , Bayesian performance i n t h e s u b j e c t s . I n t h e l i g h t o f t h i s r e s u l t s one can assume t h a t t h e so

t 78

COGNITIVE DECISION RESEARCH

Conservatism

2.0

Figure 13. Comparison of the r e s u l t s from PHILLIPS and EDWARD'S experiment I11 w i t h the ones from our experiment w i t h verbal expressions f o r chances (labelled ".70 (verb.)"

called conservatism in information processing i s mainly due t o a conservative handling of numbers and computation and i s not necessarily due t o a biased processing procedure. Prediction In t h e i r 1973-study KAHNEMANN and TVERSKY have t r i e d t o determine which kinds of heuristics and biases influence subjects' performance in predicting future events. They studied cateqorical as well as numerical predictions and found in a l l cases that subjects tended t o neglect the base rates and exhibited mistaken intuitions a b o u t regression. For instance, they acted as if: in a l l cases the correlation between the predictor variable and the c r i t e r i o n were perfect and as i f the predictor variablie were measured with perfect r e l i a b i l i t y . In order t o determine i f these mistaken intuitions are a t l e a s t p a r t i a l l y caused by the apparent d i f f i c u l t i e s subjects have in processing

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179

numerical v a r i a b l e s , I have undertaken t h e f o l l o w i n g experiment: s u b j e c t s ( 3 0 undergraduate s t u d e n t s i n psychology who had n o t a t t e n d e d courses i n

s t a t i s t i c s ) were asked t o p r e d i c t t h e success o f s t u d e n t s i n t h e u n i v e r s i t y f r o m t h e i r performance a t h i g h s c h o o l . S u b j e c t s were g i v e n a samnle o f 75 cases which i n c l u d e s t h e grades t h e s t u d e n t s r e c e i v e d a t h i g h s c h o o l and i n t h e g r a d u a t i n g exam a t t h e u n i v e r s i t y . The r e s u l t s were v e r y s i m i l a r t o t h e ones o b t a i n e d by KAHMEMAM and TVERSKY (1973) b u t when t h e s u b j e c t s were asked how c e r t a i n t h e y were about t h e c o r r e c t n e s s o f t h e i r p r e d i c t i o n s , t h e y gave q u i t e l o w s u b j e c t i v e degrees o f c o n f i d e n c e . When probed f u r t h e r a b o u t t h e d i r e c t i o n o f t h e i r p r o b a b l e e r r o r o f p r e d i c t i o n 27 s u b j e c t s i n d i c a t e d c o r r e c t l y t h a t t h e t r u e v a l u e would p r o b a b l y be c l o s e r t o t h e mean p e r f o r mance t h a n t h e y had p r e d i c t e d . Only 3 s u b j e c t s answered t h a t t h e y c o u l d n o t tell. T h i s r e s u l t as w e l l as those r e p o r t e d above seem t o i n d i c a t e t h a t s u b j e c t s a r e b e t t e r a b l e t o t a k e i n t o account complex dependencies by means o f v e r b a l p r o c e s s i n g t h a n t h e y a r e i f t h e y a r e f o r c e d t o process t h e same amount o f i n f o r m a t i o n n u m e r i c a l l y . As GREGORY (1982) p o i n t e d o u t , numbers and comp u t a t i o n form

a . more r e c e n t t o o l o f mind t h a n language and t h e r e f o r e t h e

numerical i n f o r m a t i o n p r o c e s s i n g i s l e s s a u t o m a t i c .

A f u r t h e r a s p e c t o f t h e v e r b a l s u p e r i o r i t y i s i l l u s t r a t e d by t h e outcome o f an experiment f r o m a d i f f e r e n t s t u d y . I n t h i s experiment2 s u b j e c t s (24 bank c l e r k s r e s p o n s i b l e f o r f o r e i g n exchange) were asked t o p r e d i c t what t h e exchange r a t e between t h e US D o l l a r and t h e Deutschmark would be f o u r weeks l a t e r . 12 S u b j e c t s had t o g i v e t h e p r e d i c t i o n s " i n t h e i r own words as t h e y would t a l k t o a c l i e n t " , whereas t h e o t h e r t w e l v e were asked t o g i v e numer i c a l e s t i m a t e s i n percentage o f change. B o t h groups were asked t o v e r b a l i z e t h e s t e p s t h e y t o o k i n o r d e r t o come up w i t h t h e p r e d i c t i o n . A f t e r t h e p r e d i c i t i o n s o f t h e f i r s t group had been c a l i b r a t e d w i t h a t e c h n i q u e s i m i l a r t o t h e one o f o u r second experiment, t h e y were compared t o t h e one made b y t h e numerical f o r e c a s t i n g group. I t t u r n e d o u t t h a t t h e f i r s t group was more

*

I n a d i f f e r e n t c o n t e x t I have r e p o r t e d o t h e r aspects o f t h e r e s u l t s o f t h i s experiment.

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c o r r e c t and more i n t e r n a l l y c o n s i s t e n t . W h i l e t h i s i s i n t e r e s t i n g i n i t s e l f , a n o t h e r p o i n t i s more i m p o r t a n t : The s l i g h t d i f f e r e n c e i n t h e i n s t r u c t i o n s g i v e n caused marked d i f f e r e n c e s i n t h e way t h e s u b j e c t s performed t h e i r t a s k as r e v e a l e d by v e r b a l i z i n g . The v e r b a l p r e d i c t i o n group used q u a n t i t a t i v e v a r i a b l e s (e.g. t h e GNP i n c r e a s e i n p e r c e n t ) as w e l l as q u a l i t a t i v e v a r i a b l e s (e.g. t h e s t a b i l i t y o f t h e German government) f o r d e r i v i n g t h e i r p r e d i c t i o n s , whereas t h e o t h e r group m e r e l y t o o k i n t o account t h o s e v a r i a b l e s which were u s u a l l y expressed n u m e r i c a l l y . From t h i s i t seems p l a u s i b l e t o assume t h a t t h e reason f o r t h e s u p e r i o r i t y i n t h e v e r b a l f o r e c a s t i n g c o n d i t i o n i s t h e f a c t t h a t t h e knowledge base on which these s u b j e c t s r e l i e d was b r o a d e r and a l l o w e d f o r more e l a b o r a t i o n .

An a l t e r n a t i v e i n t e r p r e t a t i o n o f t h o s e r e s u l t s i s p o s s i b l e t o o : I t m i g h t be t h a t t h e t h i n k i n g - a l o u d i n t e r f e r e d s p e c i f i c l y w i t h t h e numerical r e a s o n i n g .

General C o n c l u s i o n A1 though t h e s t u d i e s r e p o r t e d h e r e have o r i g i n a t e d f r o m a p e r s p e c t i v e o f a p p l i e d psychology, a i m i n g a t b e t t e r ways t o implement vague knowledge i n t o d e c i s i o n processes, t h e r e may be consequences o f t h i s k i n d o f r e s e a r c h f o r c o g n i t i v e psychology i n g e n e r a l . Nhen analyzed by f o r m a l means i t t u r n s o u t t h a t d e c i s i o n making and f o r e c a s t i n g on t h e one hand and r e a s o n i n g and problem s o l v i n g on t h e o t h e r hand a r e s t r u c t u r a l l y i d e n t i c a l . T h e r e f o r e i t can be assumed t h a t t h e dependency o f c o g n i t i v e processes on t h e u n d e r l y i n g mental r e p r e s e n t a t i o n s m i g h t be a q u i t e g e n e r a l problem. The e x p e r i m e n t a l r e s u l t s p r e s e n t e d h e r e i n d i c a t e t h a t : f i r s t , t h e a v a i l a b l e procedures f o r human i n f o r m a t i o n p r o c e s s i n g a r e s t r o n g l y dependent on t h e modes o f mental r e p r e s e n t a t i o n s , i n which t h e y a r e a p p l i e d ; and second, human s u b j e c t s a r e more e f f e c t i v e i n r e a s o n i n g w i t h v e r b a l e x p r e s s i o n s t h a n w i t h numerical expressions, even i f t h e t a s k s performed r e l y on f r e q u e n c y i n f o r mation. Reference Notes 1 . p e r s o n a l communication

PROCESSING OF SUBJECTIVE PROBABILITIES

181

2. personal communication (ZIMMER, note 3 ) 3. ZIMHER, A.

A model f o r t h e i n t e r p r e t a t i o n o f v e r b a l p r e d i c t i o n s , manus c r i p t , Stanford, 1983, (submitted f o r p u b l i c a t i o n )

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DE FINETTI, B. (1937) La p r e v i s i o n : des logiques, ses sources s u b j e c t i v e . Annales de 1 ' I n s t i t u t e Henri Poincar6, 7, 1-68

DUBOIS, D. & PRADE, H. (1980). Fuzzy Sets and Systems: Theory and A p p l i c a t i o n . New York: Academic Press FERRELL, #.F. & MC GOEY, D.J. (1980). A model o f c a l i b r a t i o n f o r s u b j e c t i v e p r o b a b i l i t i e s . Organisational Behavior and Human Performance, 26, 32-53 FREKSA, C. (1981) L i n g u i s t i c P a t t e r n Recognition.Ph.

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FREKSA, C. (1982) L i n g u i s t i c d e s c r i p t i o n o f human judgments i n e x p e r t systems and i n " s o f t " sciences. I n : L.TRONCALE (ed.) Systems Science and Science. L o u i s v i l l e , Ky.:Society f o r General Systems Research GREGORY, R.L.

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HACKING, I. (1975) The Emergence o f P r o b a b i l i t y . Cambridge: Cambridge U n i v e r s i t y Press HERSH, H.M. & CARAMAZZA, A. (1976) A f u z z y s e t approach t o m o d i f i e r s and vagueness i n n a t u r a l language. Journal o f Experimental Psychology, General,l05, 254-276

HINTZMAN, D.L., NOZAWA, G. & IRMSCHER, M. (1982) Frequency as a nonpropos i t i o n a l a t t r i b u t e o f memory. Journal o f Verbal Learning and Verbal Beh a v i o r , 21, 127-141 KAHNEMAN, D. & TVERSKY, A. (1973) On t h e psychology o f p r e d i c t i o n . Psychol o g i c a l Review, 80, 237-251 KAUFMANN, A. (1975) Theory o f Fuzzy Subsets. Vol

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LICHTENSTEIN, S., FISCHHOFF, B. & PHILLIPS, L.D. (1982) C a l i b r a t i o n o f p r o b a b i l i t i e s : The s t a t e o f t h e a r t t o 1980. I n : KAHNEMAN, D., SLOVIC, P. , PVERSKY, A. (eds.) Judgment under u n c e r t a i n t y . H e u r i s t i c s and Biases. Cambridge: Cambridge U n i v e r s i t y Press

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LINDLEY (1974) D i s c u s s i o n on t h e papers by P r o f e s s o r T v e r s k y and P r o f e s s o r Suppes. J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B. 36,181-182 NISBETT, R.E. & ROSS, L. (1980) Human I n f e r e n c e : S t r a t e g i e s and Shortcomings o f S o c i a l Judgment. Englewood C l i f f s , N.J.: P r e n t i c e H a l l NISBETT, R.E. & WILSON, T.D. (1977) T e l l i n g more t h a n we know: Verbal r e p o r t s on mental processes. Psychological Review, 84, 231-259 PHILLIPS, L. & EDWARDS, W. (1966) Conservatism i n a s i m p l e p r o b a b i l i t y i n f e r e n c e t a s k . J o u r n a l o f Experimental Psychology, 72, 346-354 RASCH, G. (1966) An i t e m a n a l y s i s which t a k e s i n d i v i d u a l d i f f e r e n c e s i n t o account. B r i t i s h J o u r n a l o f Mathematical and S t a t i s t i c a l Psychology, 19, 49-57 SAVAGE, L.J.

(1954) The Foundations o f S t a t i s t i c s . New York: W i l e y

SCHOTT, K. P. (1981) Wahrscheinlichkeitsschatzungen i m Computer-Dialog. S t u t t g a r t : Poeschel V e r l a g SLOVIC, P., FISCHHOFF, B. & LICHTENSTEIN, S. (1979) Images o f d e s a s t e r : P e r c e p t i o n and acceptance of r i s k s from n u c l e a r power. I n : G.T. GOODMAN & W.D. ROYCE (eds.) Energy R i s k Management. London: Academic Ppess SLOVIC, P. & TVERSKY, A. (1974) Who a c c e p t s Savage’s axiom? B e h a v i o r a l Science, 19, 368-373 SUPPES, P. (1974) The measurement o f b e l i e f . J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B, 36, 160-175 TVERSKY, A. (1974) Assessing u n c e r t a i n t y . J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B, 36, 148-159 WITTE, W. (1960) S t r u k t u r , Dynamik und Genese von Bezugssystemen. Psycholo-g i s c h e B e i t r a g e , 4, 218-252 WITTGENSTEIN, L. (1969) P h i l o s o p h i s c h e Untersuchungen. F r a n k f u r t : Suhrkamp WRIGHT, G.N. & PHILLIPS, L.D. (1980) C u l t u r a l v a r i a t i o n i n p r o b a b i l i s t i c t h i n k i n g : a l t e r n a t i v e ways o f d e a l i n g w i t h u n c e r t a i n t y . I n t e r n a t i o n a l J o u r n a l o f Psychology, 15, 239-257 ZADEH, L.A. (1978) Fuzzy s e t s as a b a s i s f o r a t h e o r y o f p o s s i b i l i t y . Fuzzy Sets and Systems, 1 , 3-38

ZIMMER, A. (1980) E i n e Formal i s i e r u n g m n e s t i s c h s t a b i l i s i e r t e r Bezugssysteme a u f d e r Grundlage von Toleranzmengen. In : A. THOMAS & R. BRACKHANE (eds.) b!ahrnehmen Urteilen Handeln. Bern: Huber

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ZIMMER, A . (1982 a ) What r e a l l y i s t u r q u o i s e ? A n o t e on t h e e v o l u t i o n o f c o l o r terms. P s y c h o l o g i c a l Research, 44, 213-230 ZIMMER, A. (1982 b) Some experiments c o n c e r n i n g t h e f u z z y meaning o f l o g i c a l q u a n t i f i e r s . 1n:L. TRONCALE(ed.) Systems Science and Science. L o u i s v i l l e , K .: S o c i e t y f o r General Systems Science

DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

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SUBJECTIVE EXPECTED UTILITY: A MODEL OF DECISION-MAKING11

BARUCH FISCHHOFF ZUR SHAPIRA BERNARD GOlTElN Decision Research A Branch of Perceptronics USA

A sim p le and comprehensive r u l e f o r making d e c i s i o n s i s t h e f o l l o w i n a : L i s t a l l f e a s i b l e courses o f a c t i o n . F o r each a c t i o n , enumerate a l l p o s s i b l e consequences. F o r each consequence, assess t h e a t t r a c t i v e n e s s o r aversiveness o f i t s occurrence, as w e l l as t h e p r o b a b i l i t y t h a t i t w i l l be i n c u r r e d should t h e a c t i o n be taken. Compute t h e expected w o r t h o f each consequence by m u l t i p l y i n g i t s w o r t h by i t s p r o b a b i l i t y o f occurrence. The expected w o r t h o f an a c t i o n i s t h e sum o f t h e expected w o r t h o f a l l p o s s i b l e consequences. Once t h e c a l c u l a t i o n s a r e completed, choose t h e a c t i o n w i t h t h e g r e a t e s t expected worth. S t u d y i n g t h e p r e s c r i p t i v e and d e s c r i p t i v e v a l i d i t y o f t h i s model i s t h e preoccupat io n o f res ea r c h e r s i n a d i f f u s e f i e l d known as b e h a v i o r a l d e c i s i o n t h e o r y (BDT). Some l o o k a t t h e model as a whole, o t h e r s c o n c e n t r a t e on i t s components (e.g. , assessments o f w o r t h o r p r o b a b i l i t y ) , and s t i l l o t h e r s concern themselves w i t h s p e c i a l cases. The most general f orm i s known as t h e s u b j e c t i v e expected u t i l i t y (SEU) model. I n it, p r o b a b i l i t i e s a r e t r e a t e d as b e i n g s u b j e c t i v e , w h i l e w o r t h i s expressed i n u t i l i t y , a g e n e r a l i z e d measure

Adapted f r o m a paper by t h e same a u t h o r s e n t i t l e d "The Experienced U t i l i t y o f Expected U t i l i t y Approaches." I n : N.T. F eat her (ed.), Expect at ions and A c t io ns . H i l l s d a l e , NJ: Erlbaum Associates, 1981.

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o f ( u n ) d e s i r a b i l i t y . The "S" i s dropped i f p r o b a b i l i t i e s are considered t o be o b j e c t i v e (a d i s t i n c t i o n c l a r i f i e d below). The "U" becomes a "V" ( f o r "value") i f an absolute standard o f worth, l i k e d o l l a r s , i s used. The E i s dropped i f p r o b a b i l i t i e s are ignored. F i n a l l y , the various a t t r i b u t e s o f each consequence may o r may n o t be considered e x p l i c i t l y . When they are, the r u b r i c " M u l t i a t t r i b u t e " i s invoked. This essay traces t h e h i s t o r y o f SEU r e search, w i t h p a r t i c u l a r emphasis on i t s v a r i e d i n t e l l e c t u a l roots and the i n s i g h t s and b l i n d spots they have e n t a i l e d . One p o i n t o f reference f o r t h e review w i l l be the concurrent, b u t r e l a t i v e l y independent, work done under the r u b r i c o f Expectancy-Valence (E-V)

, research

[ l l . Although they have used SEU models, E-V i n v e s t i g a t o r s have been l e s s i n terested i n understanding the general p r o p e r t i e s o f the model than i n using i t t o p r e d i c t and abet human performance i n s p e c i f i c a p p l i e d settings, such as career counseling o r employee management. The experiences and b e l i e f s o f E-V and BDT researchers provide i n t e r e s t i n g counterpoints. The great advantage o f E-V studies i s the realism o f t h e i r s e t t i n g s ; t h e i r great disadvantage l i e s i n the complexity o f those settings. Many o f the conclusions o f behavioral decision theory regarding the i n t r i c a c i e s o f decision behavior and i t s study were discovered and documented slowly and p a i n s t a k i n g l y i n h i g h l y constrained experimental s i t u a t i o n s . Such s u b t l e signals might never have been detected i n the f l e s h . Conversely, f i e l d research o f t h e E-V type reminds us o f the key variables t h a t should be presented i n l a b o r a t o r y work.

Fo&Z

Models of Decision-Making

Attempts t o describe how people make decisions took t h e i r i n i t i a l marching orders from p r e s c r i p t i v e models, t e l l i n g how people should make decisions. The appeal o f SEU models i n t h i s r e p o r t i s obvious. They reduce the universe o f decisions t o a common s e t o f p r i m i t i v e s (options, p r o b a b i l i t i e s , u t i l i t i e s ) about which one could hope t o derive universal t r u t h s (e.g., place s i m i l a r u t i l i t i e s on losses and gains o f the same absolute magnitude) i n due time. To some extent, t h i s promise has been borne o u t by r e a l i t y . Three hundred a r t i c l e s w r i t t e n y e a r l y on behavioral decision theory probably produce much more cumulative knowledge than three thousand p u b l i c a t i o n s on p e r s o n a l i t y [2]. A second enticement i s the presenceof an axiomatic basis f o r these

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models. Any i n d i v i d u a l who s u b s c r i b e s t o a s m a l l s e t o f reasonable r u l e s (e.g.,

t r a n s i t i v i t y o f p r e f e r e n c e s ) s h o u l d behave a c c o r d i n g t o t h e model.

P r e s c r i p t i v e reasonableness i s presumably a necessary c o n d i t i o n f o r t h e des c r i p t i v e v a l i d i t y o f a t h e o r y o f conscious b e h a v i o r ( i . e . , people a r e supposed t o be f o l l o w i n g

-

whatever r u l e

according t o one's theory

-

should

make sense t o them). I n mainstream American economics, t h e i d e a t h a t p e o p l e maximize V o r U o r

E-V o r SEU o r m u l t i a t t r i b u t e SEU has t h e s t a t u s o f a metatheory; i t goes w i t h o u t s a y i n g t h a t such m a x i m i z a t i o n i s what people t r y and o f t e n manage t o do. The goal o f t h e t h e o r i s t i s n o t t o t e s t t h i s assumption, b u t t o d i v i n e j u s t what i t i s t h a t people a r e maximizing. With s u f f i c i e n t i n g e n u i t y , one always f i n d s something t h a t a p a r t i c u l a r decision-maker has maximized i n a p a r t i c u l a r s i t u a t i o n . L i k e o t h e r m e t a t h e o r i e s , SEU i s n o t f a l s i f i a b l e and t h u s n o t a t h e o r y a t a l l , b u t r a t h e r a r u l e i n t h i s game. The s t o r y o f b e h a v i o r a l d e c i s i o n t h e o r y has, however, been t h e g r o w i n g r e a l i z a t i o n t h a t SEU o f t e n does n o t d e s c r i b e t h e d e c i s i o n - m a k i n g process e i t h e r as i t i s designed o r executed. The d r a m a t i c t e n s i o n has been p r o v i d e d by SEUls remarkable a b i l i t y t o hang on d e s p i t e mounting doubts about i t s des c r i p t i v e competence. The c l i m a x o f t h i s y e t i n c o m p l e t e t a l e seems t o have something t o do w i t h r e a l i z i n g t h a t a c l e a r l y erroneous model s t i l l may s e r v e sow u s e f u l r o l e . The denouncement ( o r perhaps sequel ) may i n v o l v e e l a b o r a t i n g and c i r c u m s c r i b i n g t h a t r o l e . The r e t r o s p e c t i v e l o o k which f o l l o w s suggests t h a t t h e s e c r e c y o f i t s l o n g e v i t y l i e s i n t h e f a c t t h a t t h e model ( a ) expresses some fundamental t r u t h i n some s i t u a t i o n s , ( b ) has a remarkable a b i l i t y t o be c l o s e t o t h e mark i n s i t u a t i o n s where i t s u n d e r l y i n g l o g i c i s c l e a r l y i n a p p r o p r i a t e , and ( c ) has proponents o f h e r o i c t e n a c i t y .

Model Fitting The t y p i c a l way t o t e s t t h e d e s c r i p t i v e v a l i d i t y o f t h e SEU model i s t o see i f i t can " e x p l a i n " p a s t d e c i s i o n s . A s e t o f s i m i l a r d e c i s i o n s i s c o l l e c t e d and t h e r e s e a r c h e r s e t s about i d e n t i f y i n g a s e t o f p r e d i c t o r s ( p r o b a b i l i t i e s and u t i l i t i e s ) t h a t , when combined a c c o r d i n g t o t h e SEU r u l e , w i l l account f o r those decisions.

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COGNlTl V E DECISION RESEARCH

U n d e r l y i n g t h i s approach i s a f a i r l y reasonable r e s e a r c h s t r a t e g y . I t i s ext r a o r d i n a r i l y d i f f i c u l t t o study simultaneously the form o f t h e decision r u l e t h a t people use and t h e s u b s t a n t i v e c o n s i d e r a t i o n s i n c o r p o r a t e d i n them; t h e r e f o r e , assume t h e t r u t h o f a g e n e r a l model and c o n c e n t r a t e on i d e n t i f y i n g t h e i n p u t s t h a t w i l l make i t work. S p e c i f i c a l l y , assume t h a t p e o p l e use t h e SEU model and f i n d what t h e i r p r o b a b i l i t i e s and u t i l i t i e s a r e . W i t h such a s t r a t e g y , i t i s h a r d t o l o s e i n t h e s h o r t run. The c l e v e r r e s e a r c h e r can u s u a l l y f i n d some s e t s o f values i n a p a r t i c u l a r s i t u a t i o n t h a t w i l l g i v e good enough p r e d i c t i o n s o r p o s t d i c t i o n s t o m a i n t a i n f a i t h i n t h e model. Moreover, any d e s c r i p t i v e f a i l u r e by an SEU model can be e x p l a i n e d away as a f a i l u r e i n measurement r a t h e r t h a n t h e use o f t h e wrong c o m b i n a t i o n r u l e . The wrong consequences were i n c l u d e d ; t h e i r w o r t h was assessed i n an u n r e l i a b l e o r c o n f u s i n g manner; u n c e r t a i n t i e s were i g n o r e d o r e l i c i t e d i m p r o p e r l y and so on. With t h e complementary m o d e l - f i t t i n g s t r a t e g y one must assume t h a t one knows t h e p r o b a b i l i t i e s and u t i l i t i e s t o which people a t t e n d and t h e n must e x p e r i ment w i t h d i f f e r e n t c o m b i n a t i o n r u l e s u n t i l one i s f o u n d t h a t works. The p r i c e p a i d f o r s t r i c t adherence t o e i t h e r s t r a t e g y i s i n t h e l o n g run. As t h e number o f p r e d i c t o r s ( d i f f e r e n t v a r i a b l e s , measurement techniques, d a t a t r a n s f o r m a t i o n s , e t c . ) increases, t h e model becomes i n c r e a s i n g l y vague and i m p r e c i s e . Given a s u f f i c i e n t l y l a r g e s e t o f p r o b a b i l i t i e s and u t i l i t i e s , one can, o f course, d e v i s e a r u l e " p r e d i c t i n g " p a s t d e c i s i o n s t o any d e s i r e d l e v e l o f p r o f i c i e n c y . I n r e g r e s s i o n terms, by expanding t h e s e t o f independ e n t v a r i a b l e s one can always f i n d a s e t o f p r e d i c t o r s ( o r even one p r e d i c t o r ) w i t h any d e s i r e d c o r r e l a t i o n w i t h t h e dependent v a r i a b l e . The p r i c e one pays f o r o v e r f i t t i n g i s shrinkage, f a i l u r e o f t h e r u l e t o "work" on a new sample o f cases. With w e l l d e f i n e d problems, t h e p r e d i c t o r / s a m p l e r a t i o a l l o w s one t o " c o r r e c t f o r s h r i n k a g e " and e s t i m a t e t h e p r e d i c t i v e v a l i d i t y o f an SEU model f o r f u t u r e d e c i s i o n s . Such c o r r e c t i o n s a r e i m p o s s i b l e when i t i s u n c l e a r j u s t how many v a r i a n t s o f t h e model have been t r i e d and d i s -

carded b e f o r e t h e b e s t - f i t t i n g one was i d e n t i f i e d . Thus, when an SEU model "works" on a s e t o f data, i t i s d i f f i c u l t t o know t h a t t h a t proves. Have we proven t h a t t h e decision-makers we a r e s t u d y i n g a r e u s i n g SEU, and h e r e a r e t h e i r w e i g h t ? O r has o u r f i s h i n g e x p e d i t i o n f i n a l l y stumbled upon an SEUl i k e model t h a t mimics t h e i r a c t u a l b e h a v i o r ?

SUBJECT1VE EXPECTED UTILITY

Simple Models or Simple Process?

Growing unease w i t h the elusiveness o f the proper SEU model was compounded by r e s u l t s emerging from t h e study o f c l i n i c a l judgment, another i n t e l l e c t u a l t r a d i t i o n upon which behavioral decision theory draws. C l i n i c a l judgment i s exercised by a r a d i o l o g i s t who s o r t s x rays o f u l c e r s i n t o "benign" and "malignant", by a personnel o f f i c e r who chooses the best applicants from a s e t o f candidates, by a c r i s i s center counselor who decides which c a l l e r s threatening s u i c i d e are serious. I n each o f these examples, the diagnosis i n volves making a decision on t h e basis o f a s e t o f cues o r a t t r i b u t e s . When, as i n these examples, the decision i s r e p e t i t i v e and a l l cases can be chara c t e r i z e d by the same cues, i t i s possible t o model the judges' decisionmaking p o l i c y s t a t i s t i c a l l y . Following t h e l o g i c o f the SEU modeling described i n the previous section, a s e t o f decisions are gathered and m u l t i p l e regression i s used t o determine the cue-weighting scheme t h a t best accounts f o r the decisions a c t u a l l y made by the judge. Two decades o f such poticycapturing studies p e r s i s t e n t l y produced a conclusion: Simple l i n e a r models, using a weighted sum o f the cues, d i d an e x c e l l e n t j o b o f p o s t d i c t i n g judges' decisions, despite the judges' c l a i m t h a t they were using much more complicated s t r a t e g i e s [3-51. A commonly asserted form o f complexity i s c a l l e d " c o n f i g u r a l " judgment, i n which the diagnostic meaning o f one cue depends on the meaning o f other cues (e.g., " t h a t tone o f v o i c e makes me t h i n k ' n o t s u i c i d a l ' unless the c a l l comes a t midday"). Two reasons f o r t h i s c o n t r a s t have emerged, each w i t h negative i m p l i c a t i o n s f o r the d e s c r i p t i v e v a l i d i t y o f SEU. One involves the growing r e a l i z a t i o n t h a t combining l a r g e amounts o f i n f o r m a t i o n i n one's head overwhelms the comp u t a t i o n a l capacity o f anyone b u t an i d i o t suvunt[6,71. Thus, a judge t r y i n g t o implement a complex strategy simply would n o t be able t o do so w i t h great consistency. Since the SEU model i s such a r u l e , these r e s u l t s suggest t h a t people could n o t use i t even i f they t r i e d . The second r e a l i z a t i o n t h a t emerges from c l i n i c a l judgment research i s t h a t simple l i n e a r models are e x t r a o r d i n a r i l y powerful p r e d i c t o r s . I f one can i d e n t i f y and measure t h e a t t r i b u t e s t h a t decision-makers consider, one can mimic t h e i r decisions t o a l a r g e degree w i t h simple models bearing no resem-

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b l a n c e t o t h e i r c o g n i t i v e processes. T h a t i s , one can m i s s p e c i f y w e i g h t s and even c o m b i n a t i o n r u l e s and s t i l l do a p r e t t y good j o b o f p r e d i c t i n g d e c i s i o n s under v e r y g e n e r a l c o n d i t i o n s [8]. Thus, whatever p e o p l e a r e d o i n g w i l l l o o k t h e a p p l i c a t i o n o f an a d d i t i v e l i n e a r model (such as SEU). I n Hoffman's [91 terms, such models a r e parmorphic i n t h a t t h e y can r e p l i c a t e t h e i n p u t - o u t p u t r e l a t i o n s h i p s o f phenomena w i t h o u t any guarantee o f f i d e l i t y t o t h e underl y i ng processes.

The response o f t h e r e s e a r c h community t o t h e s e problems was a p a r t i n g o f ways between two camps which m i g h t be d e s c r i b e d as i n v o l v i n g c o g n i t i v e a p p l i e d and a p p l i e d c o g n i t i v e p s y c h o l o g i s t s [10,11]. The l a t t e r were i n t e r e s t e d i n how people t h i n k i n t h e g e n e r a l a p p l i e d c o n t e x t o f decision-making; t h e i r p r o g r e s s i s d e s c r i b e d i n subsequent s e c t i o n s . The f o r m e r were i n t e r e s t e d i n s o l v i n g s p e c i f i c a p p l i e d problems i n v o l v i n g c o g n i t i v e decision-making. F o r them, t h e p r e d i c t i v e power o f t h e s e models made them a h i g h l y u s e f u l t o o l . O f t e n one does n o t c a r e how d e c i s i o n s a r e made, as l o n g as one can p r e d i c t t h e i r outcome. I f i t works, even an a d m i t t e d l y paramorphic model may be good enough f o r d e s i g n i n g an e f f e c t i v e m a r k e t i n g

campaign o r remuneration scheme. Indeed such a model n i g h t be used t o r e p l a c e c l i n i c a l judges w i t h f o r m u l a s m o d e l i n g t h e i r b e h a v i o r . C a l l e d " b o o t s t r a p p i n g " , t h i s approach has decision-makers i d e n t i f y t h e v a r i a b l e s upon which t h e y base t h e i r d e c i s i o n s and t h e n " c a p t u r e s " t h e p o l i c y t h e y use i n a s e t o f t r i a l cases. T h a t p o l i c y , as embodied i n a f o r m u l a , i s t h e n used i n p l a c e o f t h e judges. Even i f i t does n o t embody t h e i r t h o u g h t processes, a f o r m u l a may mimic t h e i r d e c i s i o n s w i t h g r e a t e r r e l i a b i l i t y (and hence v a l i d i t y ) t h a n t h e judges themselves. Formulas n e v e r have o f f - d a y s o r s u f f e r f r o m f a t i q u e o r d i s t r a c t i o n b y i r r e l e v a n t cues 151. C l e a r l y , p e o p l e know more t h a n i s i n c l u d e d i n such f o r m u l a s . However, t h e r e i s n o e v i d e n c e t o d a t e i n d i c a t i n g t h a t t h e y can c o n v e r t t h i s s e n s i t i v i t y i n t o s u p e r i o r p r e d i c t i o n s . E m p i r i c a l l y d i s c o v e r i n g an a n a l y t i c a l r e s u l t b y W i l k s [ E l , Dawes and C o r r i gan 1131 showed t h a t c o n s i d e r a b l e p r e d i c t i v e success i s p o s s i b l e w i t h even minimal modeling. A l l one has t o do i s i d e n t i f y t h e v a r i a b l e s ( o r a t t r i b u t e s ) t o which a decision-maker a t t e n d s and d e c i d e whether t h e y a r e r e l a t e d p o s i t i v e l y o r n e g a t i v e l y t o the decision c r i t e r i o n (i.e.,

whether t h e y a r e good

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o r bad signs). I f these variables are expressed i n standard u n i t s , they can be given u n i t weights (+1o r -1, as appropriate). Such a unit-weighted model w i l l , under very general conditions, p r e d i c t decisions as w e l l as a weightedsum model using regression weights and i s a n a l y t i c a l l y q u i t e s i m i l a r . Thus, a simple substantive theory i n d i c a t i n g what v a r i a b l e s people care about when making decisions may be a l l one needs t o make p r e t t y good p r e d i c t i o n s o f t h e i r behavior. I f they l i k e more o f good things and l e s s o f bad things, j u s t count up the number o f good things an o p t i o n leads t o and s u b t r a c t the number o f bad things and you have a good idea o f how favorably i t w i l l be viewed. The p r e d i c t i v e power o f such a simple model provides a baseline against which more sophisticated models must be compared. Obviously, some goods ( p o s i t i v e a t t r i b u t e s ) are more important than others. Therefore, a model using importance weights should, i n p r i n c i p l e , p r e d i c t b e t t e r than one using u n i t weights. S i m i l a r l y , goods obtained w i t h h i g h p r o b a b i l i t y should be valued more than those obtained w i t h low p r o b a b i l i t y . Therefore, a model using p r o b a b i l i t y weights i n p r i n c i p l e should p r e d i c t b e t t e r than one using u n i t weights. However, any u n r e l i a b i l i t y o r misspecif i c a t i o n o f those weights reduces t h e i r c o n t r i b u t i o n very r a p i d l y . I n t h e extreme, models using poorly conceived o r executed weighting schemes may succeed i n s p i t e o f r a t h e r than because o f t h e i r d i f f e r e n c e s from t h e u n i t weight model.

Divide and Conquer Feeling t h a t the SEU model was impregnable and impenetrable when d e a l t w i t h as a whole, the a p p l i e d c o g n i t i v e psychologists among behavioral d e c i s i o n t h e o r i s t s have l a r g e l y turned t o an examination o f the model's p a r t s and t h e i r v a l i d i t y . The f o l l o w i n g sections describe the answers found by asking: (1) Do people accept t h e axioms upon which the model i s based? ( 2 ) Are they

capable o f p r o v i d i n g the inputs which i t requires? ( 3 ) I n s i m p l i f i e d s i t u ations, which r e l i e v e t h e i r computational load, do they e x h i b i t SEU-like behavior?

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Prescriptive VaZ i d i t y One great a t t r a c t i o n o f SEU approaches i s t h a t they represent reasonable dec i s i o n rules, ways i n which people would want t o make decisions. The SEU model states t h a t options become more a t t r a c t i v e as t h e i r p o s i t i v e consequences become more appealing o r more l i k e l y , o r as t h e i r l e s s appealing consequences become l e s s l i k e l y . Since the expected u t i l i t i e s o f various p o s s i b l e consequences are added, one consequence's low o r negative u t i l i t y can, i n p r i n c i p l e , be compensated f o r by s u f f i c i e n t l y high u t i l i t y on others. I f people endorse these rules, then the task o f d e s c r i p t i v e research i s j u s t studying how ( w e l l ) they are c a r r i e d out. When might people r e j e c t these r u l e s ? (a) When p r o b a b i l i t i e s and values are n o t independent. One form o f dependence i s when the p r o b a b i l i t y o f r e c e i v i n g a consequence a f f e c t s the value ascribed t o i t . For example, people w i l l sometimes forego an increment i n p r o b a b i l i t y o f success i n order t o be challenged by a s i t u a t i o n , o r t o have

some suspense associated w i t h i t s outcome, o r t o l e a r n something from how i t turns out [141. One might p r e f e r a class i n which t h e t o p 80 % receive A's t o one i n which A's are automatically given t o everyone. Assigning i n t r i n s i c values t o p r o b a b i l i t i e s o f success v i o l a t e s the SEU model's axioms. However, i t i s possible t o redefine the consequences so as t o save the model. Thus, the "cinch A" and the "earned A" might be considered d i f f e r e n t consequences

whose u t i l i t i e s d i f f e r enough t h a t a 1.0 chance o f the former i s l e s s a t t r a c t i v e than a 0.8 chance o f the l a t t e r . Influence i n the opposite d i r e c t i o n i s a l s o possible. P r o b a b i l i t i e s may be influenced by u t i l i t i e s , as when o p t i m i s t s exaggerate the p r o b a b i l i t y o f good things happening. For example, Weinstein [15] has found t h a t people t y p i c a l l y b e l i e v e t h a t they are more l i k e l y than t h e average i n d i v i d u a l t o have good things happen and l e s s l i k e l y t o s u f f e r various maladies. I t i s more d i f f i c u l t t o rescue them from these deviations. ( b ) When options are evaluated by noncompensatory c r i t e r i a . Two p o s s i b l e noncompensatory r u l e s are the conjunctive and d i s j u n c t i v e . By the conjunctive

SUBJEC Ti VE EXPECTED U TiL I T Y

191

r u l e , an o p t i o n has t o s c o r e f a i r l y h i g h on each consequence t o be considered. F o r example, a v a c a t i o n o p t i o n must be r e a s o n a b l y p r i c e d , a v a i l a b l e when needed, s u i t a b l y sunny and f a i r l y q u i e t . I f an o p t i o n f a i l s t o pass m u s t e r on any one o f these a t t r i b u t e s , i t s r a t i n g on t h e o t h e r s would be i m m a t e r i a l , e.g., n o amount o f sun w i l l compensate f o r a l o t o f n o i s e . Accordirig t o t h e d i s j u n c t i v e r u l e , an o p t i o n t h a t i s adequate on any one a t t r i b u t e i s a c c e p t a b l e . F o r example, an i n v e s t m e n t o p p o r t u n i t y m i g h t be chosen i f i t were good enough as a s p e c u l a t i o n , t a x s h e l t e r , o r hedge a g a i n s t i n f l a t i o n , no m a t t e r how badl y i t r a t e d on t h e o t h e r dimensions. Investment p o r t f o l i o s o f t e n i n c l u d e

v a r i e d i t e m s chosen by t h e same d i s j u n c t i v e r u l e . Compound s t r a t e g i e s a r e a l s o p o s s i b l e , such as u s i n g a d i s j u n c t i v e or c o n j u n c t i v e r u l e t o reduce a l a r g e s e t o f o p t i o n s t o a s m a l l e r one t o which a compensatory r u l e i s t h e n a p p l i e d t 161. A1 though any r e 1 i a n c e on noncompensatory r u l e s would t h r e a t e n t h e p r e s c r i p t i v e v a l i d i t y o f SEU models, E i n h o r n [ 1 7 ] f o u n d l i n e a r models t o be so p o w e r f u l t h a t t h e y s t i l l p r o v i d e d r e l a t i v e l y good p r e d i c t i o n s o f t h e b e h a v i o r o f p e o p l e c l e a r l y u s i n g noncompensatory models. Choice t h e o r y [18]

,

however, o f f e r s a v a r i e t y o f o t h e r noncompensatory d e c i s i o n r u l e s whose use has been observed i n one s i t u a t i o n o r another. Attempts t o p r e d i c t t h e usage o f such s t r a t e g i e s have a d i s t i n c t l y ad hoe c h a r a c t e r . (c)

When consequences a r e n o t e v a l u a t e d i n d e p e n d e n t l y . L i t e r a l usage o f an

SEU model r e q u i r e s one t o e v a l u a t e each consequence o f each o p t i o n s e p a r a t e l y and t h e n combine t h o s e assessments. I n some s i t u a t i o n s , however, t h e d e c i sion-maker m i g h t want t o e v a l u a t e a p a r t i c u l a r consequence d i f f e r e n t l y dependi n g on t h e v a l u e o f o t h e r consequences. F o r example, one m i g h t l i k e e i t h e r a s l i g h t l y u n c o m f o r t a b l e c h a i r o r a moderate l e v e l o f ambient n o i s e when d e c i d i n g where t o s i t i n a l e c t u r e , knowing t h a t e i t h e r w i l l h e l p one s t a y awake. However, a s e a t t h a t i s b o t h h a r d and l o c a t e d i n a n o i s y a r e a w i l l be u n d e s i r a b l e s i n c e i t d i v e r t s t o o much a t t e n t i o n . U s i n g t h e language o f judgment r e search, i t r e f l e c t s c o n f i g u r a l judgment. I n o t h e r words, w o r t h i s an i n t e r a c t i v e f u n c t i o n o f t h e a t t r i b u t e s . Such s i t u a t i o n s can, i n p r i n c i p l e , be handled by e v a l u a t i n g a t t r i b u t e s i n p a i r s o r t r i p l e t s r a t h e r t h a n s e p a r a t e l y ( i . e . , one c o u l d e v a l u a t e a s o f t s e a t i n a n o i s y area, a h a r d s e a t i n a n o i s y area, a s o f t s e a t i n a q u i e t area, and h a r d s e a t i n a q u i e t a r e a ) . With a l l b u t t h e s i m p l e s t o f a t t r i b u t e - a t t r i b u t e i n t e r a c t i o n s , however, t h i s would prove q u i t e laborious.

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( d ) When o p t i o n s a r e n o t e v a l u a t e d s i m u l t a n e o u s l y . Simon's [7] n o t i o n o f s a t i s f i c i n g grew o u t o f o b s e r v i n g t h e p r e d i c t i v e f a i l u r e s o f t h e SEU-like models o f c l a s s i c a l economics. S a t i s f i c e r s l o o k f o r any d e c i s i o n a l t e r n a t i v e t h a t i s "good enough". T h e i r search t e r m i n a t e s when a s a t i s f a c t o r y o p t i o n has been i d e n t i f i e d and evaluated. T h i s o p t i o n may, however, be i n f e r i o r ( i n an SEU sense) t o o t h e r o p t i o n s t h a t have n o t even been considered, because t h e s e r i a l e v a l u a t i o n process was t r u n c a t e d p r e m a t u r e l y . ( e ) When people e x p l i c i t l y r e j e c t t h e SEU model's axioms. One way t o see i f p e o p l e accept t h e axioms i s t o ask them. I n 1969, Tversky [191 f o u n d t h a t a l t h o u g h p e o p l e l i k e t o be t r a n s i t i v e i n t h e i r c h o i c e s ( t h u s f u l f i l l i n g one axiom), one can d e s i g n s i t u a t i o n s i n which t h e y a r e b o t h i n t r a n s i t i v e and una b l e t o r e s o l v e t h e i n c o n s i s t e n c y i n t h e i r choices. He s p e c u l a t e d t h a t mark e t i n g approaches may be designed t o e x p l d i t such d i f f i c u l t i e s . The A l l a i s and E l l s b e r g paradoxes [20,21]

a r e two demonstrations o f p e o p l e r e j e c t i n g

Savage's independence axiom, a c c o r d i n g t o which p r e f e r e n c e s between a1 t e r n a t i v e s s h o u l d be independent o f any consequences t h e y have i n common. S l o v i c and Tversky [221

p r e s e n t e d s u b j e c t s w i t h arguments f o r and a g a i n s t t h i s

axiom and found p e r s i s t e n t v i o l a t i o n s , even a f t e r t h e axiom was p r e s e n t e d i n a c l e a r and presumably c o m p e l l i n g f a s h i o n . The problem w i t h such demonstrat i o n s i s t h a t t h e y do n o t i n d i c a t e t h e p r e v a l e n c e and d i s t r i b u t i o n o f v i o l a t i o n s . Are t h e y j u s t concocted c u r i o s i t i e s ? O r do t h e y r e p r e s e n t modal beh a v i o r i n some i m p o r t a n t realms? Kahneman and T v e r s k y ' s [231 " p r o s p e c t t h e o r y " a t t e m p t s t o l e n d system t o v i o l a t i o n s o f t h e SEU axioms b y e x p l a i n i n g them as r e s u l t s o f c o n s i s t e n t b e h a v i o r a l t e n d e n c i e s . One such tendency, t h e " c e r t a i n t y e f f e c t " , causes consequences t h a t a r e o b t a i n e d w i t h c e r t a i n t y t o be g i v e n more w e i g h t t h a n u n c e r t a i n consequences. The second, l a b e l e d t h e " r e f e r e n c e e f f e c t " , l e a d s p e o p l e t o e v a l u a t e a l t e r n a t i v e s r e l a t i v e t o a r e f e r e n c e p o i n t w h i c h may c o r r e spond t o t h e i r p r e s e n t s t a t u s , t o some e x p e c t a t i o n a b o u t f u t u r e s t a t u s , o r t o some a d a p t a t i o n l e v e l [241. By a l t e r i n g t h e r e f e r e n c e p o i n t , a l t e r n a t i v e v e r s i o n s o f t h e same d e c i s i o n problem t h a t a r e f o r m a l l y e q u i v a l e n t i n terms o f t h e SEU model may e l i c i t d i f f e r e n t p r e f e r e n c e s . Once a d e c i s i o n problem has been posed i n a p a r t i c u l a r way, Kahneman and T v e r s k y ' s t h e o r y makes p o w e r f u l p r e d i c t i o n s about responses t o it, p r e d i c t i o n s t h a t a r e o f t e n con-

SUBJECTIVE EXPECTED UTILITY

193

t r a r y t o p r e d i c t i o n s o f the SEU model. It needs, however, t o be complemented by a substantive theory regarding the way i n which decision questions a r e posed by nature and i n t e r p r e t e d by decision-makers.

AvaiZabiZity of Inputs People who wish t o engage i n SEU-like behavior must provide a l i s t o f a l l f e a s i b l e options and possible consequences, along w i t h assessments o f t h e i r p r o b a b i l i t y and u t i l i t y . One need n o t perform these tasks w e l l i n order t o attempt SEU-like behavior. Poor performance w i l l , however, lead t o suboptimal decisions, r e l a t i v e t o what could be obtained by the best possible usage o f the SEU model. Not making the best decisions need n o t mean t h a t people are n o t t r y i n g t o use the SEU model. Nonetheless, acute i n a b i l i t y t o provide the i n p u t s required t o use the SEU model w e l l must cast some doubts on i t s popul a r i t y , f o r why would people p e r s i s t i n doing something they do q u i t e p o o r l y ? Whether i t i s reasonable t o expect people t o p e r s i s t a t suboptimal SEU be; havior, r a t h e r than o p t i n g f o r some simpler r u l e , requires consideration o f three questions: ( a ) J u s t how good ( o r bad) are they? (b) How l i k e l y are they t o l e a r n about t h e i r mistakes? ( c ) What are t h e i r options? I f people could n o t perform these tasks, r e a l i z e t h e i r l i m i t a t i o n s , have v i a b l e options, t h e SEU model would be a l e s s l i k e l y candidate f o r how people t r y t o make deci-

sions. What e l s e could people do

, besides

t r y t o l i s t a l l courses o f a c t i o n open

t o them and assess t h e i r expected consequences? One o p t i o n i s t r y n o t t o t h i n k a t a l l , b u t r e l y on nonanalytical decision rules, such as " t h i s i s (most l i k e ) what I ' v e always done" o r " t h i s i s what my (most expert) f r i e n d s t e l l me t o do" o r " t h i s i s what everyone else i s doing". Another o p t i o n i s refuse t o make a ( f i n a l ) decision, p r e f e r r i n g t o muddle through by t r i a l and e r r o r , making small incremental decisions t h a t a f f o r d an opportunity t o change courses i f things are n o t going w e l l [251. A t h i r d option i s t o r e c r u i t a number o f people t o t h i n k about issues too complex and poorly understood t o analyze alone and r e l y on whatever conclusion these chaotic group processes produce. Such r e l i a n c e on group r a t h e r than i n d i v i d u a l " r a t i o n a l i t y " i s common i n work s i t u a t i o n s 126,271, w i t h perhaps the best known example being the emergence o f group production norms i n defiance o f piecework

pay schemes.

194

COGNITIVE DECISION RESEARCH

Problem Structuring L i k e o t h e r a n a l y t i c approaches, SEU b e g i n s w i t h a s t r u c t u r i n g o f t h e problem, namely l i s t i n g a l l r e l e v a n t o p t i o n s and consequences. The obvious performance s t a n d a r d h e r e i s completeness. A modest amount o f s y s t e m a t i c and a n e c d o t a l evidence suggests t h a t p e o p l e have d i f f i c u l t y i n d e p e n d e n t l y p r o d u c i n g comp l e t e problem r e p r e s e n t a t i o n s . When asked t o j u d g e completeness o f represent a t i o n s generated b y themselves o r o t h e r s , however, t h e y o f t e n seem t o be v e r y i n s e n s i t i v e t o t h e s e inadequacies. What i s o u t o f s i g h t e f f e c t i v e l y i s o u t o f mind [28,291.

Thus i n t h e s h o r t r u n a t l e a s t , p e o p l e may n o t r e a l i z e

t h i s l i m i t on t h e i r a n a l y t i c a l d e c i s i o n - m a k i n g a b i l i t y . I n t h e l o n g r u n , o m i t t i n g o p t i o n s and consequences s h o u l d l e a d t o p o o r d e c i s i o n s . However, l i f e seldom sends us l a r g e enough batches o f unambiguous s i g n a l s t o make i t c l e a r j u s t how good o u r decision-making i s and where t h e problems l i e [30,311.

Assessing Probabilities A c o r n e r s t o n e o f SEU t h i n k i n g i s t h a t we l i v e i n an u n c e r t a i n w o r l d . Accord-

i n g t o t h e Bayesian o r s u b j e c t i v i s t p o s i t i o n t h a t u n c e r t a i n t y r e f l e c t s t h e l i m i t s of o u r understanding. From t h i s p o s i t i o n , a l l statements a r e q u a l i f i e d i m p l i c i t l y o r e x p l i c i t l y by o u r degree o f b e l i e f i n them. Degrees o f b e l i e f a r e expressed n u m e r i c a l l y i n s u b j e c t i v e p r o b a b i l i t i e s , i . e .

, expressions

t h e f o r m "My p e r s o n a l p r o b a b i l i t y t h a t s t a t e m e n t A i s t r u e i s .XX.".

with

Such

p r o b a b i l i t i e s a r e e n t i r e l y i n t h e eye o f t h e b e h o l d e r and a r e n o t p r o p e r t i e s o f t h e w o r l d . Two o b s e r v e r s o f a s i t u a t i o n i n p r i n c i p l e can a s s i g n d i f f e r e n t p r o b a b i l i t i e s o f b e i n g t r u e t o t h e same s t a t e m e n t about i t , e i t h e r because t h e y i n t e r p r e t what t h e y see d i f f e r e n t l y o r have d i f f e r e n t background i n f o r ma t i on. I n f a c t , s u b j e c t i v i s t s would argue t h a t t h e whole n o t i o n o f " o b j e c t i v e proba b i l i t y " i s meaningless. A s s i g n i n g any p r o b a b i l i t y r e q u i r e s some i n t e r p r e t a t i o n o f t h e observed s i t u a t i o n and t h a t a c t o f i n t e r p r e t a t i o n imputes a subj e c t i v e element. What a r e commonly t h o u g h t o f as o b j e c t i v e p r o b a b i l i t i e s m e r e l y r e f e r t o s i t u a t i o n s i n w h i c h t h e r e i s consensus among r e a s o n a b l e obs e r v e r s about how t o i n t e r p r e t e x i s t i n g evidence. The p r o b a b i l i t y o f a f a i r c o i n f a l l i n g on "heads" on i t s n e x t f l i p i s commonly h e l d t o be 0.5,

but

195

SUBJECT1VE EXPECTED UTlLITY

t h a t i s o n l y because observers agree on how t o e v a l u a t e p a s t performance and r e l a t e i t t o subsequent performance (e.g.

, what

c o n s t i t u t e s an adequate s e r i e

o f independent, e q u i v a l e n t t r i a l s ) . Bayesians do n o t t a l k about estimating p r o b a b i l i t i e s which i m p l i e s t h a t t h e r e i s something " o u t t h e r e " i n t h e w o r l d t o b e appraised, b u t o f assessing p r o b a b i i t i e s , s i g n i f y i n g t h e e v a l u a t i o n o f an i n t e r n a l s t a t e . Assessing peopl e l s performances i n assess ng p r o b a b i l i t i e s poses some problems. I n d i v i d u a l p r o b a b i l i t i e s can a l m o s t n e v e r be e v a l u a t e d ( e x c e p t i n t h e case where one says "There i s absoZuteZy no chance o f t h i s s t a t e m e n t b e i n g wrong" r e g a r d i n g a statement t h a t i s , i n f a c t , wrong). Sets o f s u b j e c t i v e p r o b a b i l i t i e s , however, may be e v a l u a t e d a c c o r d i n g t o two c r i t e r i a . One i s c o n s i s t e n c y w i t h t h e laws o f p r o b a b i l i t y o r ( i n t e r n a l ) coherence [32]. F o r example, t h e p r o b a b i l i t i e s assigned t o a s t a t e m e n t and i t s complement s h o u l d t o t a l 1.0. The second i s " c a l i b r a t i o n " , a p p l i c a b l e when t h e t r u t h o f uncert a i n statements e v e n t u a l l y becomes known. The we1 1 c a l ib r a t e d p r o b a b i 1 it y assessor s h o u l d t e n d t o have more t r u e statements a s s o c i a t e d w i t h h i g h t h a n w i t h l o w p r o b a b i l i t i e s . S p e c i f i c a l l y , XX % o f t h e s t a t e m e n t s assigned proba b i l i t y .XX s h o u l d be t r u e . T h a t i s , i f I am w e l l c a l i b r a t e d , 70 % o f t h e t i m e when I s t a t e " t h e r e i s a 0.7 chance o f t h i s s t a t e m e n t b e i n g t r u e " , s h o u l d be t r u e .

it

The e a r l i e s t s t u d i e s o f p r o b a b i l i t y assessment used r e p e t i t i v e s e r i e s o f events ( l i k e drawing r e d o r b l u e b a l l s f r o m an u r n ) and d e f i n e d p r o b a b i l i t y as r e l a t i v e f r e q u e n c y o f occurrence. Research showed t h a t people were q u i t e good a t a p p r a i s i n g such f r e q u e n t i s t i c p r o b a b i l i t i e s [33]. More r e c e n t l y , t h e f o c u s o f r e s e a r c h has t u r n e d t o how p e o p l e a s s i g n proba b i l i t i e s t o statements, i n c l u d i n g u n i q u e events (e.g.

, Carter

w i l l be r e -

e l e c t e d ) , where r e l a t i v e frequency has no meaning. Here, performance has been l e s s t h a n o u t s t a n d i n g . S t u d i e s u s i n g a v a r i e t y o f t a s k s , response modes, and s u b j e c t s have shown t h a t p r o b a b i l i t y assessments t e n d t o be c a l i b r a t e d p o o r l y [341. The most common e r r o r i s overconfidence, i . e . ,

people t h i n k

t h a t t h e y know more t h a n i s a c t u a l l y t h e case [351. Where observed, t h i s b i a s seems t o be so r o b u s t t h a t p e o p l e a r e w i l l i n g t o engage i n h i g h l y disadvantageous b e t s based on t h e i r c o n f i d e n c e judgments [28].

196

COGNITIVE DECISION RESEARCH

Various forms o f incoherence have a l s o been observed, alt hough w i t h o u t any c l e a r i n d i c a t i o n o f t h e i r prevalence. F o r example, Wyer [ 36] f ound t h a t people t e nd t o exaggerate t h e p r o b a b i l i t y o f t h e c o n j u n c t i o n o f two events, r e l a t i v e t o t h e p r o b a b i l i t i e s assigned t o t h e two i n d i v i d u a l l y . Kahneman and Tversky [371 f ound t h a t r a t h e r t h a n combinding base-rat e ( i . e . , background) i n f o r m a t i o n w i t h i n f o r m a t i o n r e g a r d i n g a p a r t i c u l a r case, people t e n d t o i g n o r e b as e-rat e p r o b a b i l i t i e s . Slovici, F i s c h h o f f , and L i c h t e n s t e i n [38] demo n s t r a t e d s i t u a t i o n s i n which t h e j u d g e d p r o b a b i l i t y o f a compound event was l a r g e r t h an t h e p r o b a b i l i t i e s o f i t s c o n s t i t u e n t events. A more fundamental l e v e l o f c o n f u s i o n m i g h t be f o u n d i n t h e f o l k l o r e o f survey research which h o l d s t h a t one cannot even ask f o r numerical p r o b a b i l i t y assessments f rom a random sample o f t h e p o p u l a t i o n w i t h o u t g e t t i n g d i s t u r b i n g l y h i g h r a t e s ( i . e . , o v e r 20 % ) of nonresponse [391. On t h e o t h e r hhnd, people seem t o have a good a p p r e c i a t i o n o f t h e meaning o f p r o b a b i l i s t i c weather f o r e c a s t s [401. Perhaps t h e most d i s t u r b i n g a s p e c t o f t h e s e r e s u l t s f o r a proponent o f SEU as a d e s c r i p t i v e model i s n o t t h e presence o f biases, b u t t h e t h o u g h t processes t h a t seem t o u n d e r l i e b o t h erroneous and a c c u r a t e p r o b a b i l i t y assess ments. Tversky and Kahneman [411

have argued t h a t many p r o b a b i l i s t i c judg-

ments a r e produced by u s i n g r u l e s o f thumb o r h e u r i s t i c s whose i n t e r n a l l o g bears l i t t l e resemblance t o t h e r u l e s o f p r o b a b i l i t y . These a r e g e n e r a l l y e f f e c t i v e ways o f c o p i n g w i t h an u n c e r t a i n environment t h a t b o t h deny i t s proba b i 1i s t i c c h a r a c t e r and p r e c l u d e l e a r n i n g a b out some b a s i c phenomena ( l i k e r e g r e s s i o n t o t h e mean). O f t e n t h e y l e a d t o s u b s t a n t i a l l y b i a s e d judgments and t o d e c i s i o n s n o t i n t h e i n d i v i d u a l ' s b e s t i n t e r e s t . To t h e e x t e n t t h a t these h e u r i s t i c s c a p t u r e p e o p l e ' s t h o u g h t processes, t h e y argue ( a ) t h a t we a r e n o t e f f i c i e n t p r o b a b i l i s t i c f u n c t i o n a l i s t s , and ( b ) t h a t even when people do i n c o r p o r a t e u n c e r t a i n t y i n t h e i r decision-making, t h e y do so

i n terms q u i t e d i f f e r e n t f r o m t h o s e o f t h e p r o b a b i l i t y c a l c u l u s o f t h e SEU model s.

Assessing Values The st udy o f v a l u e judgments and t h e i r v a l i d i t y has languished r e l a t i v e t o t h e st udy of p r o b a b i l i t y judgments. The reason f o r t h i s apparent d i s i n t e r e s t i s pro bably t h e absence o f an a c c e p t a b l e c r i t e r i o n by which t o e v a l u a t e v a l u e

SUBJECT1VE EXPECTED UTILITY

197

judgments. Such judgments would seem t o be t h e l a s t r e d o u b t o f u n a i d e d int u i t i o n . Who knows b e t t e r t h a n a respondent what he o r she p r e f e r s ? Recent r e s e a r c h has, however, r e v i v e d and e l a b o r a t e d a l e s s o n l o n g known t o a t t i t u d e r e s e a r c h e r s . S u b t l e changes i n how v a l u e q u e s t i o n s a r e phrased and responses a r e e l i c i t e d can have marked e f f e c t s on t h e p r e f e r e n c e s p e o p l e express. T h i s l a b i l i t y i n v a l u e judgments seems t o have i m p o r t a n t i m p l i c a t i o n s f o r b o t h how v a l u e s a r e c o n c e p t u a l i z e d and how t h e y a r e s t u d i e d . An a r t i c l e o f f a i t h among s t u d e n t s o f v a l u e , choice, and a t t i t u d e judgments i s t h a t people have r e a s o n a b l y w e l l d e f i n e d o p i n i o n s r e g a r d i n g t h e d e s i r a b i l i t y o f d i f f e r e n t e v e n t s . Although t h e s e o p i n i o n s may n o t be f o r m u l a t e d i n t u i t i v e l y i n numerical ( o r even v e r b a l ) form, c a r e f u l q u e s t i o n i n g can e l i c i t judgments r e p r e s e n t i n g p e o p l e ' s u n d e r l y i n g values. From t h i s stance, e l i c i t a t i o n procedures a r e n e u t r a l t o o l s , b i a s - f r e e channels which t r a n s l a t e subj e c t i v e f e e l i n g s i n t o s c i e n t i f i c a l l y u s a b l e e x p r e s s i o n s . They impose no views

on respondents, beyond f o c u s s i n g a t t e n t i o n on t h o s e v a l u e i s s u e s o f i n t e r e s t t o the investigator. What happens i n cases where p e o p l e do n o t know, o r have d i f f i c u l t y a p p r a i s i n g , what t h e y want? Under such circumstances, e l i c i t a t i o n procedures used i n t h e r e s e a r c h may become m a j o r f o r c e s i n shaping t h e v a l u e s expressed, o r apparentl y expressed, i n t h e judgments t h e y e l i c i t . Experimenters can i n d u c e random

e r r o r (by c o n f u s i n g t h e respondent), s y s t e m a t i c e r r o r ( b y h i n t i n g a t what t h e " c o r r e c t " response i s ) , o r u n d u l y extreme judgments ( b y s u g g e s t i n g c l a r i t y and coherence o f o p i n i o n t h a t a r e n o t w a r r a n t e d ) . I n such cases, t h e method becomes t h e message, a r e f l e c t i o n o f t h e r e s e a r c h e r ' s t o o l s r a t h e r t h a n t h e respondent I s wishes, People a r e most l i k e l y t o have c l e a r p r e f e r e n c e s r e g a r d i n g i s s u e s t h a t a r e f a m i l i a r , simple, and experienced d i r e c t l y . Each o f t h e s e p r o p e r t i e s i s assoc i a t e d w i t h o p p o r t u n i t i e s f o r t r i a l - a n d - e r r o r l e a r n i n g , such as, m i g h t be summarized i n r e a d i l y a p p l i c a b l e r u l e s , t h a t p r o v i d e s t e r e o t y p i c , e a s i l y j u s t i f i a b l e responses t o f u t u r e q u e s t i o n s o f values. When adopted b y i n d i v i d u a l s , t h e y may be seen as h a b i t s ; when adopted b y groups, t h e y c o n s t i t u t e traditions.

198

COGNITIVE DECISION RESEARCH

The power o f t h e s e r u l e s o f thumb f o r a s s e s s i n g v a l u e s comes f r o m t h e i r development and a p p l i c a t i o n t o s i m p l e and r e p e t i t i v e problems. T h e i r v i a b i l i t y becomes suspect when t h e i s s u e s a r e u n f a m i l i a r and complex, t h e o l d

i n t u i t i o n s i r r e l e v a n t , t h e o l d r u l e s untested. Unfortunately, these a r e p r e c i s e l y t h o s e s i t u a t i o n s i n which values need s t u d y i n g . I n them, however, we may n e v e r have c o n s i d e r e d t h e i m p l i c a t i o n s o f t h e v a l u e s and b e l i e f s acq u i r e d i n s i m p l e r s e t t i n g s . As a r e s u l t , we may have no a r t i c u l a t e d p r e f e r ences. I n some fundamental sense, o u r v a l u e s may be i n c o h e r e n t , n o t t h o u g h t through. I n t h i n k i n g about what a r e a c c e p t a b l e l e v e l s o f r i s k , f o r example, we may be u n f a m i l i a r w i t h t h e terms i n which i s s u e s a r e f o r m u l a t e d (e.g., s o c i a l d i s c o u n t r a t e s , minuscule p r o b a b i l i t i e s , o r megadeaths). We may have c o n t r a d i c t o r y v a l u e s (e.g.,

a strong aversion t o catastrophic losses o f l i f e

and a r e a l i z a t i o n t h a t we a r e n o t more moved by a p l a n e

c r a s h w i t h 500 f a -

t a l i t i e s t h a n one w i t h 300). We may occupy d i f f e r e n t r o l e s i n l i f e ( p a r e n t s , workers, c h i l d r e n ) which produce c l e a r c u t , b u t i n c o n s i s t e n t values. We may v a c i l l a t e between compatible, b u t s t r o n g l y held, p o s i t i o n s (e.g.

, freedom

of

speech i s i n v i o l a t e , b u t s h o u l d be d e n i e d t o a u t h o r i t a r i a n movements). Gle may n o t even know how t o b e g i n t h i n k i n g about some i s s u e s (e.g.,

t h e appro-

p r i a t e t r a d e o f f between t h e o p p o r t u n i t y t o dye o n e ' s h a i r and a vague, m i n u t e i n c r e a s e i n t h e p r o b a b i l i t y o f cancer 20 y e a r s f r o m now). Our views may undergo changes o v e r t i m e (say, as we n e a r t h e hour o f d e c i s i o n o r t h e consequence i t s e l f ) and we may n o t know w h i c h v i e w s h o u l d f o r m t h e b i a s o f o u r d e c i s i o n . Such i n a r t i c u l a t e d p r e f e r e n c e s a r e h a r d l y c o m p a t i b l e w i t h t h e s o r t o f r i g o r ous s y s t e m a t i c t h i n k i n g r e q u i r e d b y SEU decision-making. Furthermore, t h e y g i v e pause f o r t h o u g h t a b o u t j u s t what i t i s s u b j e c t s a r e g i v i n g us when we e l i c i t t h e v a l u e s needed t o t e s t t h e c o n f o r m i t y o f t h e i r b e h a v i o r w i t h t h e SEU model. L i s t i n g a few s p e c i f i c e f f e c t s may i n d i c a t e t h e power a n e l i c i t o r may d e l i b e r a t e l y , or i n a d v e r t e n t l y , w i e l d i n shaping, d i s t o r t i n g , o r even c r e a t i n g expressed p r e f e r e n c e s . One such e f f e c t mentioned e a r l i e r was t h e ease w i t h which one can a l t e r t h e d e s i r a b i l i t y o f outcomes b y s h i f t i n g t h e r e f e r e n c e p o i n t i n t h e l i g h t o f which t h e y a r e e v a l u a t e d [371. The same s a f e t y program m i g h t l o o k q u i t e d i f f e r e n t when c o n c e p t u a l i z e d i n terms o f " l i v e s saved" o r " l i v e s l o s t " , w i t h t h e respective reference p o i n t s o f t h e c u r r e n t s i t u a t i o n

199

SUBJECTIVE EXPECTED UTILITY

o r an i d e a l one. A f a m i l i a r e f f e c t f r o m t h e s t u d y o f m o t i v a t i o n i s t h a t as one g e t s c l o s e r t o an e v e n t w i t h mixed consequences, t h e aversiveness o f i t s n e g a t i v e aspects in c r e a s e s more r a p i d l y t h a n t h e a t t r a c t i v e n e s s o f i t s p o s i t i v e aspects, making i t appear, on t h e whole, l e s s d e s i r a b l e t han i t d i d f r o m a d i s t a n c e [42]. People may have o p p o s i t e o r d e r s o f p r e f e r e n c e f o r gambles when asked which t h e y p r e f e r (which focusses t h e i r a t t e n t i o n on how l i k e l y t h e y a r e t o w i n ) and when asked how much t h e y would pay t o p l a y each (which h i g h l i g h t s t h e amount t o w i n ) 1431. People may p r e f e r t o t a k e a chance a t l o s i n g a l a r g e sum o f money r a t h e r t h a n absorb a s m a l l e r loss, b u t change t h e i r mind when t h e " s u r e l o s s " i s r e l a b e l e d as an "insurance premium" 1441. A r e l a t i v e l y u nimp or t a n t a t t r i b u t e may become t h e d e c i s i v e f a c t o r i n choosing between a s e t o f o p t i o n s i f t h e y a r e p r e s e n t e d i n such a way t h a t t h e a t t r i b u t e a f f o r d s t h e e a s i e s t comparison between them [45]

.

Three i m p o r t a n t f e a t u r e s o f these s h i f t i n g judgments a r e ( a ) people a r e t y p i c a l l y unaware o f t h e potency o f such s h i f t s i n t h e i r p e r s p e c t i v e , ( b ) t h e y o f t e n have no g u i d e l i n e s as t o which p e r s p e c t i v e i s t h e a p p r o p r i a t e one, and ( c ) even when t h e r e a r e g u i d e l i n e s , p e o p l e may n o t want t o g i v e up t h e i r own i n c o n s i s t e n c y 1441. E l i c i t o r s must decide a t some p o i n t whether t h e y have capt ured t h e i r respond e n t ' s values adequately. The u s u a l c r i t e r i a a r e r e l i a b i l i t y and i n t e r n a l consist enc y (e.g., t r a n s i t i v i t y ) . However, where t h e t a s k i s p o o r l y unders t o o d because t h e c o m p l e x i t y o r u n f a m i l i a r i t y , c o n s i s t e n c y o f response w i t h i n a g i v e n e l i c i t a t i o n mode may o n l y r e f l e c t t h e power o f t h a t mode t o impose a p a r t i c u l a r p e r s p e c t i v e . G r e a t e r i n s i g h t i n t o values may come f r o m p o s i n g d i v e r s e q u e s t i o n s i n t h e hope o f e l i c i t i n g i n c o n s i s t e n t responses, i n o r d e r t o see how c o h e r e n t t h e u n d e r l y i n g values a r e and what changes make a difference. I n t h i s l i g h t , t h e r i g h t way t o ask a q u e s t i o n depends on one's purpose i n asking. I f one i s i n t e r e s t e d i n how p e o p l e express t h e i r values i n t h e r e a l world, one q u e s t i o n may be enough. Th a t w o r l d o f t e n asks o n l y one q u e s t i o n (e.g., i n a b a l l o t measure). A c a r e f u l a n a l y s i s o f how an i s s u e i s posed may a l l o w one t o i d e n t i f y t h a t q u e s t i o n and p r e d i c t responses a c c u r a t e l y . I f one i s i n t e r e s t e d i n what people r e a l l y f e e l a b o ut a v a l u e issue, t h e r e may

200

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be no s u b s t i t u t e f o r an i n t e r a c t i v e e l i c i t a t i o n procedure, one t h a t acknowledges t h e e l i c i t o r ' s r o l e i n h e l p i n g t h e respondent t o c r e a t e and e n u n c i a t e values. That h e l p would i n c l u d e a conceptual a n a l y s i s o f t h e problem and o f t h e personal, s o c i a l , e t h i c a l p e r s p e c t i v e s f r o m w h i c h t h e respondent m i g h t w i s h t o view i t .

Decision Contexts Although b e h a v i o r a l d e c i s i o n t h e o r i s t s have n o t developed a t h e o r y o f d e c i s i o n s i t u a t i o n s t o complement t h e i r e v o l v i n g t h e o r y o f t h e i n d i v i d u a l d e c i sion-maker upon o c c a s i o n t h e y have a t t e m p t e d t o r e p l i c a t e t h e i r l a b o r a t o r y s t u d i e s i n t h e r e a l w o r l d , These experiments a t r o u g h i n g i t have been prompte d by ( a ) t h e f e e l i n g t h a t d e c i s i o n t h e o r y l i k e o t h e r areas i n a p p l i e d cogn i t i v e psychology s h o u l d work i n t h e w o r l d ; ( b ) t h e r e f u s a l o f economists t o a c c e p t ( o r even l o o k a t ) l a b o r a t o r y data; ( c ) t h e hope o f b e i n g a b l e t o e l a b o r a t e t h e i r t h e o r i e s by embedding them i n a b r o a d e r c o n t e x t ; and ( d ) c o n t i n u e d i n t e r a c t i o n w i t h d e c i s i o n a n a l y s t s , t h o s e p u r v e y o r s o f SEU as a n o r m a t i v e g u i d e t o d e c i s i o n - m a k i n g who a r e a l s o p a r t i c u l a r l y a t t u n e d t o t h e s u b j e c t i v e q u a l i t y o f judgments o f f a c t and value. One obvious s t e p i n t h e d i r e c t i o n o f r e a l i s m i s t o use r e a

r a t h e r t h a n hy-

p o t h e t i c a l s t a k e s i n s t u d i e s o f decision-making. A second

s t o use e x p e r t s r a t h e r t h a n n a i v e s u b j e c t s . A t h i r d i s u s i n g more r e a l i s t i c s t i m u l i . A l t h o u g h

a v a i l a b l e evidence i s modest, none o f t h e s e m a n i p u l a t i o n s has p r o v i d e d s u f f i c i e n t l y d r a m a t i c r e s u l t s t o c a s t s e r i o u s doubt on most o f t h e r e s e a r c h conc l u s i o n s c i t e d above 135,461. None o f t h e s e m a n i p u l a t i o n s , however, l e a d s t o t h e s t u d y o f a c t u a l judgments

in situ. P r o b a b l y t h e most e l a b o r a t e e x c u r s i o n o f b e h a v i o r a l d e c i s i o n t h e o r y i n t o t h e r e a l w o r l d has been a s t u d y by Kunreuther e t a l . 1471, designed t o understand t h e f a i l u r e o f p e o p l e t o buy f e d e r a l l y s u b s i d i z e d f l o o d i n s u r a n c e . T h a t i n s u r a n c e i s o f f e r e d a t h i g h l y a t t r a c t i v e r a t e s , where " h i g h l y a t t r a c t i v e " was determined by p l a n n e r s who assumed t h a t r e s i d e n t s had a c c u r a t e r i s k and c o s t d a t a and combined them a c c o r d i n g t o SEU p r i n c i p l e s . A n a t i o n a l s u r vey o f f l o o d - p r o n e areas discovered, however, t h a t r e s i d e n t s ' judgments o f r i s k s and c o s t s were h i g h l y d i v e r g e n t f r o m t h o s e o f t h e e x p e r t s . O f t e n t h e s e

201

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misjudgments seemed t o have t h e same r o o t s as t h e judgmental b i a s e s observed i n l a b o r a t o r y s t u d i e s . Obviously, even i f r e s i d e n t s were SEU decision-makers, t h e y c o u l d n o t be expected t o behave i n t h e p r e d i c t e d way ( i . e . ,

purchase

i n s u r a n c e ) i f t h e y d i d n o t s h a r e t h e p l a n n e r s ' i n f o r m a t i o n base. The d e s i g n o f t h e Kunreuther e t a l . s t u d y a l s o a l l o w e d an a p p r a i s a l o f how f a v o r a b l e i n s u r a n c e s h o u l d be t o r e s i d e n t s g i v e n what t h e y t h o u g h t t o be t h e f a c t s o f t h e m a t t e r and assuming t h a t t h e y were SEU decision-makers. A s t r a i g h t f o r w a r d d e r i v a t i o n t r a n s l a t e d a t t r a c t i v e n e s s o f i n f o r m a t i o n t o an i n d e x R , w i t h h i g h e r R v a l u e s b e i n g a s s o c i a t e d w i t h more a t t r a c t i v e i n s u r a n c e s i t u a t i o n s . Comparison o f i n d i v i d u a l s who had and had n o t purchased i n s u r a n c e r e v e a l e d l i t t l e d i f f e r e n c e i n t h e two d i s t r i b u t i o n s o f R values. Thus, whatever g u i d e d i n s u r a n c e purchase d e c i s i o n s , i t was n o t an SEU a n a l y s i s based on e i t h e r an " o b j e c t i v e " ( e x p e r t s ' ) o r " s u b j e c t i v e " ( r e s i d e n t s ' ) view o f t h e s i t u a t i o n . What d i d g u i d e d e c i s i o n s ? K u n r e u t h e r ' s b e s t guess was t h a t people r e l i e d on a s e r i e s o f i n f o r m a l d e c i s i o n r u l e s such as t h e answers t o t h e q u e s t i o n s : Are my n e i g h b o r s and r e l a t i v e s b u y i n g i t ? What does my i n s u r a n c e a g e n t say?

Do I have t h e c a p a c i t y t o w o r r y a b o u t one more t h i n g ? The a p p l i c a t i o n o f expectancy v a l u e t h e o r y t o p r e d i c t i n g w o r k e r s ' e f f o r t and performance on t h e i r j o b s r e p r e s e n t s a p r o l o n g e d e x t e n s i o n o f SEU t h e o r y i n t o t h e r e a l world. Reviewing t h i s e f f o r t , Schwab, O l i a n - G o t t l i e b , and Heneman

1481 found t h a t on t h e average o n l y 9 % o f t h e v a r i a n c e i n e f f o r t o r performance i s e x p l a i n e d by E-V models. A l t h o u g h t h e i r r e v i e w c o n s i d e r e d o n l y between-subject a p p l i c a t i o n s o f t h i s w i t h i n - p e r s o n t h e o r y o f c h o i c e ( t h e r e b y i n c r e a s i n g t h e amount o f e r r o r v a r i a n c e ) , analogous w i t h i n - p e r s o n s t u d i e s r a r e l y e x p l a i n more t h a n 25 % o f t h e v a r i a n c e . Furthermore, w i t h one except i o n , t h e few f a c t o r s t h a t Schwab, O l i a n - G o t t l i e b ,

and Heneman i d e n t i f y as

c o n t r i b u t i n g t o b e t t e r p r e d i c t i o n a r e e i t h e r o m i t t e d f r o m SEU t h e o r y o r a r e a n t i t h e t i c a l t o it. They conclude: " t h e r e i s a n a g g i n g s u s p i c i o n t h a t expectancy t h e o r y o v e r i n t e l l e c t u a l i z e s t h e c o g n i t i v e processes p e o p l e go t h r o u g h when choosing a l t e r n a t i v e a c t i o n s " [48, P. 1461. More e x p l i c i t l y p e s s i m i s t i c , Staw 1491 suggests t h a t o r d i n a r y c h o i c e b e h a v i o r i s b e t t e r modeled as " i m p u l s e b u y i n g " d e c i s i o n s o f consumers t h a n on t h e l i n e s o f SEU t h e o r y . Perhaps as r e v e a l i n g as t h e c o n t e x t t o which l a b o r a t o r y r e s u l t s have been

202

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extended are the contexts t h a t have been avoided. Researchers' assumptions about s i t u a t i o n s t o which t h e i r r e s u l t s w i l l n o t generalize provide an i m p l i c i t p a r t i a l theory o f decision-making environments. One such 1 i m i t i n g f e a t u r e o f most behavioral decision theory experiments i s t h a t subjects are rewarded f o r responding as honestly as they can. However, t h e r e obviously are s i t u a t i o n s where i t pays t o l i e (e.g., when t h e boss asks about work environment) and they may engender r a t h e r d i f f e r e n t behavior. A second l i m i t i n g f e a t u r e i s the absence o f time and emotional pressure t h a t would create t h e s o r t o f "hot cognition" studied by Janis and Mann [501. A t h i r d feature i s i s o l a t i o n o f the i n d i v i d u a l from s o c i a l i n t e r a c t i o n s , which may both f o s t e r i n d i v i d u a l i s t i c behavtor and e l i m i n a t e a common decision-making a i d

- t a l k i n g t o others.

A f o u r t h c h a r a c t e r i s t i c o f most studies i s the preclusion o f a no-choice o r p r o c r a s t i n a t i o n o p t i o n [51]. Each o f these c h a r a c t e r i s t i c s might represent an environmental v a r i a b l e r e s t r i c t i n g t h e v a l i d i t y o f behavioral d e c i s i o n theory r e s u l t s o r p e r t u r b a t i o n as minor as the s h i f t from hypothetical t o r e a l stakes o r from naive t o expert subjects. Cone lusion

The s t o r y o f SEU research has i n some sense been a t a l e o f deadends and hard-earned lessons. An enormous amount o f e f f o r t was devoted t o "capturing" decision s t r a t e g i e s before i t was r e a l i z e d t h a t the power o f l i n e a r models v i r t u a l l y precluded l e a r n i n g what weights i n d i v i d u a l s were g i v i n g t o d i f f e r e n t a t t r i b u t e s o r i f they were using an SEU decision r u l e a t a l l . Many s t u dies o f how people estimate r e l a t i v e frequencies were conducted before r e searchers r e a l i z e d t h a t such tasks f a i l e d t o capture t h e s o r t o f u n c e r t a i n t y found i n most decision s i t u a t i o n s . For years, researchers derived s a t i s f a c t i o n from the e l i c i t a t i o n o f consistent, r e l i a b l e value judgments, before beginning t o worry t h a t such o r d e r l i n e s was a product o f t h e i r methods. Having disabused ourselves o f these p a r t i c u l a r i l l u s i o n s , where do we go from here, i n a d d i t i o n t o having b e t t e r respect f o r the power o f our methods t o produce t i d y , b u t misleading r e s u l t s ? Substantively, we should begin t o develop an understanding o f decision environments t h a t w i l l enable us t o understand how general l a b o r a t o r y r e s u l t s are and how far-reaching the e f f e c t s they produce. One key component o f t h i s understanding w i l l be an

203

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e r r o r theory e x p l i c a t i n g the consequences o f poor performance i n s t r u c t u r i n g decisions and i n assessing and combining p r o b a b i l i t i e s and values. The sensit i v i t y o f our decisions t o such deviations f r o m o p t i m a l i t y w i l l c l a r i f y which p r i n c i p l e s o f decision-making need t o be and can be learned from experience. Theoretically, we need t o go beyond simply t r y i n g t o f a l s i f y SEU theory, which, i f o n l y because o f the power o f l i n e a r models, w i l l almost always provide a t l e a s t mediocre predictions. Instead, we need t o move on t o sophist i c a t e d f a l s i f i c a t i o n [521, f i n d i n g t h e o r i e s t h a t do what SEU does and a t l e a s t a l i t t l e more. The shape o f those theories might be new c a l c u l i , such as Prospect Theory, using d i f f e r e n t p r i m i t i v e s and combination rules, o r process-tracing models which attempt t o t r e a t n o t o n l y input-output r e l a t i o n ships, but a l s o the i n t e r v e n i n g thought processes [16,531, o r contingencybased models t h a t preserve some o f t h e SEU l o g i c but a l s o incorporate a theory of the environment and the way i n which i t i s s e q u e n t i a l l y decomposed

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1979. Prospect theory. I n : Econometrica

47, 263-291 1241

Helson, H.: 1959. Adaptation l e v e l theory. I n : S. Koch (ed.), Psychology: A study o f a science, Vol. 1. New York: McGraw-Hill

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March, J.G.: 1978. Bounded r a t i o n a l i t y , ambiguity and the engineering o f choice. I n : B e l l Journal o f Economics 9, 587-608

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Salancik, G. & P f e f f e r , J.: 1978. A s o c i a l i n f o r m a t i o n processing approach t o j o b a t t i t u d e s and task design. I n : Administrative Science Q u a r t e r l y 23, 224-253 White, S.E. & M i t c h e l l , T.R.: 1979. Job enrichment versus s o c i a l cues: A comparison and competitive t e s t . I n : Journal o f Applied Psychology 64, 1-9 Fischhoff, B./Slovic, P. & Lichtenstein, S.: 1977. Knowing w i t h cert a i n t v : The aooroori ateness o f extreme confidence. I n : Journal o f Eiperimentai Psychology: Human Perception and Performance 3, 552-564 Fischhoff, B./Slovic, P. & Lichtenstein, S.: 1978. F a u l t t r e e s : Sensit i v i t y o f estimated f a i l u r e p r o b a b i l i t i e s t o problem representat i o n . I n : Journal o f Experimental Psychology: Human Perception and Performance 4, 342-355 Fischhoff, B.: 1975. Hindsight # f o r e s i g h t : The e f f e c t o f outcome knowledge on judgement uncertainty. I n : Journal o f Experimental Psychology: Human Perception and Performance 1, 288-299 Einhorn, H.J. & Hogarth, R.: 1978. Confidence i n judgement: Persistence o f the i l l u s i o n o f v a l i d i t y . I n : Psychological Review 85, 395-416 Kyburg, H.E. J r . & Smokler, H.E.: a b i l i t y . New York: Wiley

1964. Studies i n s u b j e c t i v e prob-

Peterson, C.R. & Beach, L.R.: 1967. Man as an i n t u i t i v e s t a t i s t i c i a n . I n : Psychological B u l l e t i n 68, 29-46 Lichtenstein, S./Fischhoff, B. & P h i l l i p s . L.D.: 1977. C a l i b r a t i o n o f p r o b a b i l i t i e s : The s t a t e o f the a r t . I n : H. Jungermann & G. de Zeeuw (eds.), Decision making and change i n human a f f a i r s . Amsterdam: Reidel Fischhoff, B.: Debiasing. I n : D. Kahneman/P. S l o v i c & A. Tversky (eds.), Judgement under uncertainty: H e u r i s t i c s and biases. New York: Cambridge U.P., i n press Wyer, R.S. : 1974. Cognitive organization and change: An i n f o r m a t i o n processing approach. Potomac, MD: Lawrence Erlbaum Associates Kahneman, D. & Tversky, A.: 1973. On t h e psychology o f p r e d i c t i o n . In: Psychological Review 80, 237-251 Slovic, P./Fischhoff, B. & Lichtenstein, S. : 1976. Cognitive processes and s o c i e t a l r i s k taking. I n : J.S. C a r r o l l & J.W. Payne (eds.), Cognition and s o c i a l behavior. Potomac, MD: Lawrence Erlbaum Associates Quinn, R.P. & Staines, G.L.: 1978. The 1977 q u a l i t y o f employment survey. Ann Arbor, M I : I n s t i t u t e f o r Social Research, U n i v e r s i t y o f Michigan

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Murphy, A./Lichtenstein, S./Fischhoff, 6. & Winkler, R.L. : 1980. Misinterpretations of precipitation probability forecasts. In: Bulletin of the American Meteorological Society 61, 695-701

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1541

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SOCIAL INTERACTION

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

21 1

CONDITIONS 0F VI 0LATI NG INDIVI DUAL RAT IONA LITY

WERNER H. TACK Universitat des Saarlandes Psychologisches lnstitut F. R. Germany

Introduction

This paper deals w i t h a very special type of group decision making, namely decisions to s e t t l e conflicts of i n t e r e s t between a t l e a s t three partners. The exclusion of the two-partner-case imp1 ies that (1) there i s always room f o r non-trivial coalition formation, and ( 2 ) there are always reasons for- a t l e a s t partial - cooperation due t o the f a c t that a perfectly competitive situation f o r every pair a t the same time i s impossible. Furthermore, the situations regarded here are cooperative i n the sense t h a t the partners involved have the possi bi 1 i ty to communicate w i t h each other and a r e allowed to enter binding agreements on how t o solve the conflict a t hand. Formally, any of these situations can be represented by a s e t N of partners (persons o r groups) - called "players" i n game-theoretic one terminology -, a s e r i e s of s e t s A1, A2,... An of feasible options s e t f o r each player -, and a s e r i e s of preference relations Pi - again one f o r each player - on the s e t of outcomes resulting from combinations

-

212

SOCIAL INTERACTION

o f chosen options. I n t r o d u c i n g the a d d i t i o n a l assumption t h a t a l l i n d i vidual preferences f o l l o w the r u l e o f negative t r a n s i t i v i t y and t h a t

-

in

-

case o f an i n f i n i t e outcome s e t the outcome s e t contains a denumerable order-dense subset, i n d i v i d u a l preferences can be represented by payoff functions xi such t h a t f o r every outcome p a i r a,b a Pi b i f f xi(a)

> xi(b)

Thus, a l l s i t u a t i o n s regarded i n t h i s paper are representable by n-persongames i n normal form. I n t h i s context, the term "cooperative" simply r e f e r s t o t h e f a c t t h a t there i s some k i n d o f communication between the players involved, c o n s t i t u i n g a bargaining process before f i n a l decisions, and t h a t binding agreements between some o r a l l o f these players on which options t o choose are a possible r e s u l t o f such a bargaining process. As a matter o f course, the p o s s i b i l i t y o f binding agreements does n o t imply t h a t any

player must e n t e r such an agreement. Thus, he can take the value o f an outcome he can reach w i t h o u t a c t u a l l y cooperating w i t h someone e l s e as a y a r d s t i c k f o r deciding which possible agreements are too low i n value according t o h i s own preference system t o be acceptable. Such

-

-

a behavior follows t h e simple p r i n c i p l e not t o accept an o f f e r o r t o e n t e r an agreement i f there i s a p r e f e r r e d outcome which can be reached by appropriate a c t i o n without c o l l a b o r a t i o n w i t h any o f t h e other players involved i n t h e given c o n f l i c t o f i n t e r e s t . I t i s e x a c t l y t h i s simple concept which i s termed individua2 r a t i o n a l i t y . ExverimentaZ Games The experimental games used i n our former research (KAUFMANN & TACK, 1975, LICHTENBERGER, 1975, TACK, 1980, 1981) can be taken as examples o f the cooperative s i t u a t i o n s t o be analyzed here. They are much simpler and more transparent than usual real l i f e s i t u a t i o n s ; we know a l o t about t h e i r i n t e r n a l s t r u c t u r e , and we have some experiences on t h e behavior o f experimental subjects i n these types o f s i t u a t i o n s .

VIOLATING INDIVIDUAL RATIONALITY

213

The f i r s t example is the "real e s t a t e brokers problem" which has been introduced as an experimental paradigm a couple of years ago (KAUFMANN & TACK, 1975). Imagine three real e s t a t e brokers - called Adams, Brown, and Clay. Each of them wants t o s e l l a piece of land. The three estates are neighbouring; they a l l have pairwise common borders, and the whole situation looks l i k e t h a t depicted in figure 1. There are seven prospective buyers on the market. One of them wants t o buy Adams' e s t a t e (A) and offers 7 units (maybe 700.000 German Marks o r whatever one likes t o use as the monetary equivalent of a u n i t ) . The second offers 2 units f o r Brown's e s t a t e ( B ) , and the third 3 units f o r Clay's one ( C ) . The next three prospective buyers are interested in pairs of estates only. One offers 10 units f o r A t B , one 7 units f o r B t C , and one 1 2 units for A+C. Finally, there is a company who wants t o buy the whole area; they are w i l l i n g t o pay 14 units f o r a l l three pieces of land together.

Figure 1: !epresentation of a sample situation from experiments using the real e s t a t e brokers problem".

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SOCIAL INTERACTION

These values c o n s t i t u t e a f u n c t i o n from the s e t o f p o s s i b l e coaZitions S o f brokers i n t o t h e reals, i f

-

as usual i n t h i s context

-

the term

-

" c o a l i t i o n " i s used f o r any nonempty subset o f players, i n c l u d i n g f o r i s o l a t e d players as "one person c o a l i t i o n s " the sake o f s i m p l i c i t y and

-

-

a t the o t h e r extreme

-

the "grand c o a l i t i o n " comprising a l l

partners. Such a f u n c t i o n i s c a l l e d a characteristic function and i t i s u s u a l l y taken t o be superadditive which means t h a t f o r every p a i r o f d i s j o i n t c o a l i t i o n s the f u n c t i o n value o f t h e i r union i s a t l e a s t as great as the sum o f the values o f the o r i g i n a l c o a l l t i o n s ; f o r m a l l y : f o r every S,T c_ N w i t h SnT =

0:

v(SuT) _> v(S)tv(T). I n the brokers s i t u a t i o n , superaddi t i v i ty i s a necessary requirement t o make things p l a u s i b l e . I f there i s

-

f o r instance

- an o f f e r o f 10

u n i t s f o r the areas A and B together, and another o f 3 u n i t s f o r C alone, somebody i n t e r e s t e d i n the whole area must o f f e r a t l e a s t the sum o f

-

these values nameily 13 as a possible c l i e n t .

-

i n order t o be s e r i o u s l y taken i n t o account

The brokers are t h e players engaged i n a c o n f l i c t o f i n t e r e s t . They have t o solve t h e problem, t o whom t o s e l l and how t o share t h e r e s u l t i n g gain. Introducing students as experimental subjects playing the r o l e s o f the brokers gives an example o f one o f the experimental s i t u a t i o n s used t o c o l l e c t data on bargaining behavior, c o a l i t i o n formation, and c o n f l i c t r e s o l u t i o n . Experiments o f t h i s type are r a t h e r usual ; except t h a t o t h e r researchers mostly g i v e t o t h e i r subjects the a b s t r a c t values o f a c h a r a c t e r i s t i c f u n c t i o n w i t h o u t such a n i c e cover s t o r y , and w i t h a second d e v i a t i o n from t h e paradigm introduced here which w i l l be mentioned l a t e r i n another context. For t h i s type o f s i t u a t i o n , t h e concept o f individwlZ rationatittl i s extremely simple: Nobody should agree on a s o l u t i o n which gives him l e s s than t h e amount o f u n i t s he can gain by s e l l i n g h i s r e a l e s t a t e alone t o

215

VIOLATING INDIVIDUAL RATIONALITY

t h e one person who i s i n t e r e s t e d i n t h i s and o n l y t h i s area. Things are a b i t more complicated i n another class o f experimental s i t u a t i o n s , t h e so-called "three-person-matrix-game".

Three subjects are

seated a t a round t a b l e . Each o f them faces a d i s p l a y showing three t i l t e d 3x3-matrices l i k e those depicted i n f i g u r e 2. Each o f t h e 27 c e l l s o f these matrices represents a possible outcome o f the s i t u a t i o n which i s symbolized by t h r e e numbers. These numbers s p e c i f y amounts o f money. An outcome i s r e a l i z e d by paying t h e amount given by t h e top number t o the subject f a c i n g t h i s display, w h i l e t h e amounts given by t h e bottoml e f t and bottom-right numbers are payed t o the players s i t t i n g on h i s l e f t - and on h i s right-hand-side r e s p e c t i v e l y .

t

r

tt

Finure 2: Example o f t h e representation o f an outcome c o n f i g u r a t i o n as displayed t o an experimental subject i n a matrix-game situation.

216

SOCIAL INTERACTION

The f e a s i b l e options o f each p l a y e r are the t h r e e matrices on h i s d i s p l a y (marked "r", "s", and "t" on f i g u r e 2 ) . He can choose one o f them, thereby r e s t r i c t i n g the s e t o f possible outcomes t o those given by t h e chosen m a t r i x . Information on t h e d i s p l a y faced by the other subjects i s arranged such t h a t an o p t i o n o f t h e p a r t n e r s i t t i n g on the l e f t - h a n d s i d e corresponds t o one o f the rows from u p p e r - l e f t t o lower-right (marked 'lull, 'Iv", and I'w"), w h i l e the right-hand p l a y e r can r e s t r i c t p o s s i b l e outcomes t o those i n one o f t h e rows going from upper-right t o l o w e r - l e f t (marked and "z"). Imagine

-

f o r example

-

I'X",

t h a t the p l a y e r f a c i n g the l a y o u t given i n f i g u r e 2

and h i s right-hand partner agree t o choose t h e middle m a t r i x ( s ) and the p a r t n e r ' s d i s p l a y

-

"y",

-

on

a m a t r i x corresponding t o the upper row from upper-

r i g h t t o l o w e r - l e f t ( x ) . Then, o n l y t h e three outcomes i n the upper row from upper-right t o l o w e r - l e f t i n t h e middle m a t r i x ( t h e s-x combination) are s t i l l possible. I f t h e t h i r d player decides t o choose on h i s d i s p l a y a m a t r i x which i s represented here as the lower row from u p p e r - l e f t t o lower-right (u), s-u-x marks the p o s i t i o n o f t h e f i n a l outcome. The person f a c i n g the l a y o u t depicted i n f i g u r e 2 has earned 8 u n i t s w h i l e t h e o t h e r two players gain 3 and 6 u n i t s .

rationu2 Again, an outcome i n t h i s type o f s i t u a t i o n s i s c a l l e d ird~~v.2'dv.aZh i f nobody gains l e s s than the amount he can assure f o r himself by choosing

appropriate actions w i t h o u t c o l l a b o r a t i o n w i t h any o t h e r partner. But how

-

-

can a person confronted w i t h a s i t u a t i o n o f t h i s t y p e enforce something? There are a t l e a s t two answers t o t h i s question, corresponding t o the d i s t i n c t i o n between alpha- and b e t a - r a t i o n a l i t y introduced by AUMANN and PELEG (1960) more than 20 years ago. T h i s d i s t i n c t i o n can be embedded i n t o a more general concept o f s i t u a t i o n a l I' c o n t r o l " ; t h i s approach

w i l l be used here t o c l a r i f y the d i s t i n c t i o n between d i f f e r e n t kinds o f individual rationality.

ControZ Systems I t i s a c e n t r a l aspect of a l l the s o c i a l s i t u a t i o n s we a r e regarding i n t h i s context, t h a t every p a r t n e r engaged i n such a s i t u a t i o n and every has p o s s i b i l i t i e s t o r e s t r i c t by appropriate actions t h e c o a l i t i o n too

-

-

VIOLATING INDIVIDUAL RATIONALITY

-

217

-

s e t of feasible outcomes; he has a t l e a s t partial control. The system o f outcome subsets generated by the different p o s s i b i l i t i e s of r e s t r i c t i o n which are available t o a partner i s called a controZ system Ci o f the partner i E N. T h u s

where P(A) denotes the power s e t of the unrestricted outcome s e t A. I t i s necessary t o introduce some additional properties of control systems. (1) I t should aZwuys be possible t o refrain from any action which makes some outcomes unreachable; thus - there i s always a possibility t o force no r e s t r i c t i o n a t a l l . T h i s means: A

E

Ci for every i

E

N.

( 2 ) I t should never be possible to make a l l outcomes unreachable, o r - i n other words t o force the situation into an eternal s t a t e where i t never can end. Formally:

-

0

4 Ci

f o r every i

E

N.

( 3 ) I t should always be possible t o r e s t r i c t the s e t of available outcomerestricting actions , which means t h a t

i f E,F E C i , then E u F E Ci f o r every i

E

N.

In f a c t , control systems should be introduced by s t a t i n g some more properties than these three, b u t the others have t o do w i t h systems of coalitions consisting o f more than one partner only and are therefore - of no special i n t e r e s t i n the relevant context of the considerations following in this paper.

-

Given a control system Ci of an individual player i , the best value i can enforce equals the maximum o f the minima of outcome-values from the several

218

SOCIAL INTERACTION

outcome sets belonging t o the sytem. By r e s t r i c t i n g the s e t o f f e a s i b l e outcomes t o a s p e c i f i c subset B belonging t o Cia he can guarantee t o reach the minimum value over a l l outcomes i n B. Choosing f o r r e s t r i c t i o n t h a t subset B ' which gives t h e best guarantee i n t h e sense t h a t t h e r e i s no other subset w i t h a higher minimum value, the maximum o f the minima becomes a lower bound f o r the values o f a l l the outcomes which are s t i l l f e a s i b l e . Formally, i can enforce Max {Min {xi

(b)

I

b

E

BI

I

B

E

Ci).

Thus, outcome b i s individuaZZy rationu2 i f and o n l y i f f o r every i r N : xi(b)

,Max

{Min {xi(a)

I

a

E

BI

I

B

E

Cil.

Going back t o the m a t r i x s i t u a t i o n , one can introduce a t l e a s t two d i f f e r e n t c o n t r o l systems. A very simple type o f c o n t r o l consists o f r e s t r i c t i n g the own s e t o f options. I f a player f a c i n g the l a y o u t given i n f i g u r e 2 decides n o t t o take t h e r i g h t matrix, he r e s t r i c t s the s e t o f f e a s i b l e outcomes t o those given by t h e two other matrices. Thus, the set o f these outcomes i s an element o f a c o n t r o l system c a l l e d alpha-control system. By deciding t o take the lower l e f t matrix, he can enforce t o gain a t l e a s t

1 u n i t ( i n t h e o t h e r matrices, he can - p o s s i b l y - get nothing). Thus, 1 u n i t i s the value he can enforce, and - i n order t o be i n d i v i d u a l l y alphar a t i o n a l f o r him - an outcome must g i v e him a t l e a s t 1 u n i t . Beta-control i s another s t o r y . I n h i s case, t h e person f a c i n g the l a y o u t given i n f i g u r e 2' decides on which m a t r i x t o choose dependent on t h e choices o f h i s partners. He reacts t o t h e decisions o f t h e others by choosing a f u n c t i o n from the s e t o f combinations o f t h e i r options i n t o h i s own s e t o f a v a i l a b l e a l t e r n a t i v e s . One such f u n c t i o n determines the s e t o f f e a s i b l e options depicted i n f i g u r e 3 , which guarantees an outcome w i t h a t l e a s t 3 u n i t s f o r t h e person f a c i n g t h i s layout. Thus, under betacontroZ 3 u n i t s i s a value he can enforce, and - i n order t o be i n d i v i d u a l l y beta-rational f o r him - an outcome must give him a t l e a s t 3 u n i t s .

VIOLATING INDl VIDUAL RATIONALITY

f

r

f

t

F l y r e 3: Outcome configuration of a matrix-game showing a set of outcomes (unhatched) belonging t o the beta-control system o f the player facing this layout. The problem with beta-rationality i s t h a t the realization of beta-control presupposes a fixed time order of final decisions and f u l l information about the decisions of the other players. In the experimental games r u n a t our laboratory (LICHTENBERGER, 1975; TACK, 19812, 1981), there was no fixed time order, b u t d u r i n g bargaining we heard very often sentences l i k e " I f you do t h i s , I ' l l do t h a t , otherwise...''

I f people accepb such announcements as serious ones, they behave as i f a l l of them could simultaneously exert beta-control , which is factually impossible. AUMANN and PELEG (1960) suspect that while alpha-control i s obviously more plausible i t may turn out t h a t beta-control i s much more e have some empirical evidence important f o r real cooperative behavior. W

219

220

SOCIAL INTERACTION

supporting t h i s conjecture. ResuZts

Looking a t experiments using characteristic function situations - l i k e the "real e s t a t e brokers problem", b u t usually without such a nice cover story - the relevant l i t e r a t u r e reveals a rather strange situation. I t seems t o be usually taken f o r granted that subjects i n experimental games agree on individually rational outcomes - maybe w i t h a very small number of exceptions -, b u t nearly a l l experiments (e.g. RIKER, 1967; LAING & MORRISON, 1973; RAPOPORT & KAHAN, 1976; ALBERS, 1978; RAPOPORT & KAHAN, 1979) have a 0-percent violation o f individual r a t i o n a l i t y due to the f a c t that the experimenters d o n ' t give t h e i r subjects any chance t o violate this principle. The characteristic functions have zero-values f o r the 'one-person coalitions' , which simply means t h a t there is no possible outcome over which an individually enforcable outcome is preferred. Looking a t experiments using the real-estate-brokers paradigm, there are up t o now - data from 228 single experiments u s i n g 25 different characteristic functions. Most of these data a r e published (KAUFMANN & TACK, 1975; TACK, 1980), some of them .have been collected by students as p a r t of t h e i r seminar work (reference note). The proportion of experimental bargaining situations with final agreements on individually non-rational outcomes is between 2 and 3 percent only. T h u s , violations of individual rationality seem t o be a very r a r e event. This can lead t o the conclusion t h a t such a spurious phenomenon does not deserve any special i n t e r e s t , t h a t i t can be p u t aside as one more manifestation of simple f a u l t s and errors which occur i n every context where f a l l i b l e human subjects a r e involved. B u t , when looking a t the distribution of the violations of individual rationality over the different characteristic functions, t h i n g s a r e starting to become interesting. All violations are concentrated a t about 10 percent of a1 characteristic functions. Taking alone these functions into account the proportion o f violations goes u p to 19 percent. First consequence:

-

Wheter violations of individual r a t i o n a l i t y are a rare event o r not

22 1

VIOLA TING INDl VIDUAL RA TIONAL l TY

depends on t h e choice o f c h a r a c t e r i s t i c functions used f o r experiment a t i on. Asking f o r t h e special p r o p e r t i e s o f c h a r a c t e r i s t i c f u n c t i o n s which seemingly a t t r a c t v i o l a t i o n s o f i n d i v i d u a l r a t i o n a l i t y shows t h a t they are marked by l a r g e d i f f e r e n c e s between t h e i n d i v i d u a l l y enforcable values , combined w i t h a small differencebetween t h e t o t a l amount a v a i l a b l e f o r d i s t r i b u t i o n i n t h e grand c o a l i t i o n and t h e sum o f t h e i n d i v i d u a l l y enforcable values. I n terms o f our real-estate-brokers paradigm: The probab i l i t y o f v i o l a t i o n s o f i n d i v i d u a l r a t i o n a l i t y goes up i f there are marked differences i n the p r i c e s o f f e r e d f o r s i n g l e pieces o f l a n d and a very small a d d i t i o n a l gain which can be earned by s e l l i n g the whole area t o the one prospective buyer who wants t o have a l l t h r e e pieces together. I n such a s i t u a t i o n , t h e r e i s a very small monetary i n c e n t i v e f o r t h e formation o f a grand c o a l i t i o n ; the grand c o a l i t i o n i s r a t h e r worthless under payoff maximizing considerations. Thus, i f

-

nevertheless

-

the

grand c o a l i t i o n i s formed, there must be another reason t o do t h i s . One possible explanation i s t h a t

-

i n t h i s special t y p e o f s i t u a t i o n

-

sub-

j e c t s use t h e grand c o a l i t i o n t o diminish i n t e r i n d i v i d u a l diflferences i n enforcable r e s u l t s . I t i s a w e l l known f a c t , t h a t e q u i t y considerations p l a y an important r o l e when people have t o share a c e r t a i n amount o f money, and i f t h e r e i s no power o r e n f o r c a b i l i t y enabling s t r a t e g i c considerations o f t h e k i n d regarded i n t h i s paper. I f c o a l i t i o n s are r a t h e r useless under s t r a t e g i c aspects, they probably become useful f o r o t h e r purposes, e.g. f o r "greater e q u a l i t y " i n the sense o f diminishin? differences. On the basis o f t h i s i n t e r p r e t a t i o n , t h e connection between enforcable values i s a k i n d o f secondary e f f e c t . The greater the variance o f i n d i v i d u a l l y enforcable r e s u l t s , t h e higher t h e p r o b a b i l i t y t h a t tendencies i n d i r e c t i o n t o an equal share a u t o m a t i c a l l y v i o l a t e i n d i v i d u a l r a t i o n a l i t y . For b e t t e r i n f o r m a t i o n on t h i s phenomenon, i t i s necessary t o design an experiment c o n t r o l l i n g as an experimental v a r i a b l e t h e number o f players f o r whom an equal share o f the g r a n d - c o a l i t i o n value i s individually rational. Data on v i o l a t i o n s o f i n d i v i d u a l r a t i o n a l i t y i n m a t r i x game s i t u a t i o n s

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are - for many reasons - much more informative than those from brokers problem sessions. Mainly in connection with a project run by LICHTENBERGER (1975) data have been collected from 390 bargaining sessions using 32 different outcome configurations. This data file contains 23 violations of individual rationality. Before calculating any relative frequencies, one has to subtract 85 experiments with payoff configurations in which every outcome is individually both alpha- and beta-rational , What remains are data from 305 experiments showing 22 violations of alpha-rationality (that is 7 percent) and 23 violations of beta-rationality (that is about the same percentage). Since every matrix situation has a finite number of outcomes, one can count the different numbers of available outcomes meeting certain criteria. In the whole set of outcome configurations - excluding those in which every outcome is alpha- as well as beta-rational - 41 percent of all the available outcomes are individually alpha-rational while 33 percent are beta-rational The extremely small difference between the frequencies of alpha- and beta-rationality among the chosen outcomes is not deducible from a similarly small difference in the set of avaiZabZe ones. Betarationality implies alpha-rationality; nearly all - with one exception only - chosen individually rational outcomes comply with the stricter laws of beta-rationality. This is one of the findings supporting the conjecture that, despite the greater pl ausi bi 1 i ty of a1 pha-control , beta-control has a higher explicative value for real bargaining behavior and its results. In our data files, there are some outcome configurations in which individual alpha- and beta-rationali ty cannot be differentiated each alpha-rational outcome is beta-rational too and some other configurations in which every available outcome i s individually alpharational , such that only the a d d i t b a a 2 requests of the beta version specify beta-rational ity. In both subgroups of experiments, the percentage of violations of individual beta-rationality is the same. This is another evidence for the fact, that it is the concept of beta-rationality which really counts if we are dealing with empirical data. One of the reasons for not recognizing this earlier could be that nearly a1 1 experimentation has used side payment situations in which alpha- and beta-control cannot be distinguished from each other.

.

-

-

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VIOLA TING INDl VIDUA L RA TIONAL I TY

I f one takes a c l o s e r l o o k a t t h e v i o l a t i o n s o f i n d i v i d u a l r a t i o n a l i t y

neglecting the alpha-beta-distinction which i s r e l e v a n t f o r one s i n g l e case only, there i s a c l e a r and eye-catching d i f f e r e n c e between two kinds o f v i o l a t i o n s . The f i r s t k i n d i s marked by t h e f a c t t h a t t h e p l a y e r f o r whom the outcome i s n o t i n d i v i d u a l l y r a t i o n a l i s always i s o l a t e d a t t h e end o f the bargaining process; he i s never a member o f a l a r g e r

coalition.

Further, v i o l a t i o n s o f t h i s type are scattered about t h e d i f f e r e n t outcome configurations, showing no recognizable c o r r e l a t i o n w i t h any property o f the configurations. T h i s i n d i c a t e s t h a t t h e r e i s no consensus t o s a c r i f i c e i n d i v i d u a l r a t i o n a l i t y i n order t o reach some o t h e r common goal; i t i s a s i n g l e person's d e c i s i o n which leads t o a v i o l a t i o n o f i n d i v i d u a l r a t i o n a l i t y . I t i s probably n o t too complicated t o construct a number o f h e u r i s t i c decision r u l e s l e a d i n g t o e x a c t l y t h i s k i n d o f phenomenon. One could s t a r t by s t a t i n g assumptions

on a person's assump-

-

t i o n s concerning h i s partners decision behavior o r as another p o s s i b i l i t y by i n t r o d u c i n g an appropriate mechanism which overlooks c e r t a i n combi-

-

nations o f the o t h e r persons' options o r c e r t a i n a v a i l a b l e outcomes. F u r t h e r p u r s u i t o f t h i s type o f i n t e r p r e t a t i o n presupposes a g r e a t e r number o f r e a l i z e d v i o l a t i o n s o f t h i s kind; thus, i t i s probably advisable t o postpone such an approach u n t i l t h e r e i s some more i n f o r m a t i o n on t h i s k i n d o f violations. The second type o f v i o l a t i o n s looks t o t a l l y d i f f e r e n t . Here, I t i s an agreement o f a l l persons engaged i n t h e s i t u a t i o n which leads t o t h e v i o l a t i o n o f i n d i v i d u a l r a t i o n a l i t y ; i t i s never a s i n g l e person's b u t always a common d e c i s i o n t o choose an outcome which i s n o t i n d i v i d u a l l y r a t i o n a l . V i o l a t i o n s o f t h i s k i n d are no longer unsystematically scattered over t h e whole set o f outcome configurations; t h e i r d i s t r i b u t i o n

i s concentrated a t some very few configurations only. Non-rational outcomes o f t h i s type always show d i f f e r e n c e s between the amounts payed t o t h e partners which are d e f i n i t e l y smaller than t h e d i f f e r e n c e s between t h e amounts enforcable by i n d i v i d u a l c o n t r o l . The c o n f i g u r a t i o n s w i t h a high r a t e o f agreements on i n d i v i d u a l l y non-rational outcomes o f t h i s t y p e show d i s t i n c t differences between t h e i n d i v i d u a l l y enforcable values combined w i t h small o r even

no advantages of e n t e r i n g some c o a l i t i o n .

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The properties o f configurations which seemingly a t t r a c t t h i s t y p e of i n d i v i d u a l l y non-rational outcomes can be i l l u s t r a t e d by a c o n f i g u r a t i o n which i s r a t h e r extreme i n a l l these respects. The person f a c i n g t h e l a y o u t given i n f i g u r e 4 can decide t o take t h e l e f t m a t r i x and enforce t o

tt

t

r

Figure 4:

Special case o f an outcome c o n f i g u r a t i o n leading t o an e x c e p t i o n a l l y high p r o p o r t i o n o f v i o l a t i o n s o f i n d i v i d u a l r a t i o n a l it y

.

reach 5 un t s by excerting alpha-control. The maximum enforcable by each o f h i s par ners i s 1 u n i t only. Beta-control does n o t c r e a t e any advantage t o anybody. Since 5 u n i t s i s the maximum f o r t h e player f a c i n g t h i s layout, t h e r e i s no i n c e n t i v e t o enter a c o a l i t i o n . Each of t h e others can gain from a grand c o a l i t i o n only, b u t has t o take i n t o account t h a t the p a r t n e r f a c i n g t h e l a y o u t from f i g u r e 4 has no reason t o e n t e r

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VIOLA TING INDl VIDUA L RA TIONALl TY

such an agreement. Under gain maximizing considerations, one expects that the bargaining ends with three isolated partners, each of them getting what he can enforce and t h u s - contributing to an individually rational outcome. In 10 out of 15 experiments we run using this configuration, something t o t a l l y different happened.The partners entered a grand coalition and agreed on an outcome being not individually rational b u t dininishino the differences between enforcable values. In f a c t the grand coalition's choice was t o give 4 units to two of them and 3 u n i t s t o the third or - b u t i n one case only to realize the non-Pareto-optimal equal share solution which gives 3 units t o each of them.

-

-

Comparing these results w i t h those from the brokers problem experiments shows t h a t the second k i n d of violations encountered when r u n n i n g matrix experiments is again a grand c o a l i t i o n ' s decision i n sibuations w i t h marked interindividual differences which serves the purpose t o d i m i n i s h these differences and t o create a solution closer to an equal share. The main results of the analysis of violations of individual rationality are: (1) There i s evidence t h a t the s t r i c t e r properties of beta-rationality are more useful in explaining behavior i n experimental games than those stemming from alpha-control considerations only. T h i s is not observable in side-payment situations which do not allow to d i f f e r e n t i a t e between a1 pha- and heta-control

.

(2) There are two clearly differentiable types of violations of individual rationality. One type consists of violations due t o - in the sense o f rationality considerations - un-adequate decisions of a single and isolated partner. These violations show no recognizable dependencies on properties of the given outcome configuration; they are not observable i n side payment situations i n which an isolated partner automatically gains his individually enforcable value. (3) The second type of violations - observable in side payment situations as well as i n experimental games without side payments is baded on

-

grand coalition decisions. I t i s prominent i n such situations where the individually enforcable values are very different and where the

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s t r a t e g i c i n c e n t i v e o f e n t e r i n g a c o a l i t i o n i s r a t h e r small. These v i o l a t i o n s always d i m i n i s h the i n t e r i n d i v i d u a l d i f f e r e n c e s leading t o outcomes which are n o t i n d i v i d u a l l y r a t i o n a l f o r those partners being s t r o n g according t o what they can enforce w i t h o u t a c t u a l l y cooperation. Looking a t o t h e r kinds o f "wrong" o r " f a l s e " human decisions, i t i s probably i n f o r m a t i v e t h a t our research shows t h e f r u i t f u l n e s s o f considering systematic differences between several I types' o f e r r o r s which n o t o n l y show d i f f e r e n t dependencies on s i t u a t i o n a l v a r i a b l e s b u t serve d i f f e r e n t purposes too. Looking a t f u t u r e research on v i o l a t i o n s o f i n d i v i d u a l r a t i o n a l i t y , i t i s probably one o f the most important r e s u l t s o f our analysis t h a t we now know how t o c o n s t r u c t s i t u a t i o n s l e a d i n g t o a h i g h e r number o f v i o l a t i o n s . This allows t h e design o f experiments being more i n f o r m a t i v e w i t h respect t o these v i o l a t i o n s and t o leave the f i e l d o f r a t h e r generous hypotheses formation based on extremely small numbers o f cases observed i n experiments designed f o r o t h e r purposes.

Reference note Anstadt, T./Niessner, M./Schaefer, H.J. & Thraem, S . : 1980. Entscheidungsverhal t e n bei 3-Personen-Spielen m i t Seitenzahlungen. Fachrichtung Psychologie der U n i v e r s i t d t des Saarlandes, Saarbrucken

References Albers, W.: 1978. Bloc forming tendencies as c h a r a c t e r i s t i c s o f t h e bargaini n g behavior i n d i f f e r e n t versions o f the Apex game. I n : E.H. Sauermann (ed.), C o a l i t i o n forming behavior. Tibingen: Mohr Aumann, R.J. & Peleg, B.: 1960. Von Neumann-Morgenstern s o l u t i o n s t o coope r a t i v e games w i t h o u t s i d e payments. I n : B u l l e t i n o f t h e American Mathematical Society 66, 173-179 Kaufmann, M.A. & Tack, W.H.: 1975. K o a l i t i o n s b i l d u n g und Gewinnaufteilung b e i s t r a t e g i s c h Bquivalenten 3-Personen-Spielen. I n : Z e i t s c h r i f t fir Sozialpsychologie 6, 227-245 Laing, J.D. & Morrison, R.J.: 1973. C o a l i t i o n s and payoffs i n three-person sequential games. In: Journal o f Mathematical Sociology 3, 3-25 Lichtenberger, J. : 1975. Drei-Personen-Spiele m i t Seitenzahlungen. Doctora d i s s e r t a t i o n , U n i v e r s i t a t des Saarlandes, Saarbrucken

VIOLA TlNG INDI VIDUAL RA TlONALl TY

Rapoport, A. & Kahan, J.P.: 1976. When t h r e e i s n o t always two against one: Coal i t i o n s i n experimental three-person cooperative games. I n : Journal o f Experimental Social Psychology 12, 253-273 Rapoport, A. & Kahan, J.P.: 1979. C o a l i t i o n a l power and p a y o f f disbursement i n three-person cooperative games. Chapel H i l l , N.C. : L.L. Thurstone Laboratory, U n i v e r s i t y o f North Carolina, Report No. 159 Riker, W.H.: 1967. Bargaining i n a three-person game. I n : The American Poli t i c a l Science Review 61, 642-656 Tack, W.H.: 1980. Cooperative c o n f l i c t r e s o l u t i o n . A formal approach and some empirical r e s u l t s . I n : E.D. Lantermann & H. Feger (eds.), S i m i l a r i t y and choice: Papers i n honour o f Clyde Coombs. Bern: Huber, 290-326 Tack, W.H. : 1981. Domination i n experiments on cooperative games w i t h o u t s i d e payments. I n : G.E. Lasker (ed.), Applied systems and cybernetics, Vol. 2. New York: Pergamon Press, 728-732

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DECISION MAKING UNDER UNCERTAINTY

R.W. Scholz (editor) 229

0 Elsevier Science Publishers B.V. (North-Holland), 1983

BIASES IN GROUP DECISION MAKING

HELMUT W. CROTT

JOHANNES A. ZUBER

Universitat Freiburg Psychologisches lnstitut F. R. Germany

1. I

Introduction

The following study intends t o e x p l a i n PoZarization i n group decision making by group decision rules, which take i n t o account the rankorder o f i n d i v i d u a l preferences over a1 ternatives. The research on Group Choice S h i f t o r Group PoZarization ( a l i a s r i s k y s h i f t / cautious s h i f t ) was i n i t i a t e d by STONER (1961), who compared the r i s k t a k i n g o f i n d i v i d u a l s and groups. I n h i s experimental s e t t i n g he asked t h e p a r t i c i pants f i r s t t o respond i n d i v i d u a l l y t o a series o f s t o r y problems. Each problem described a r i s k i n v o l v i n g dilemma s i t u a t i o n . I n a second phase p a r t i cipants assembled as groups and discussed the problem u n t i l they came t o an agreement. STONER'S s t r i k i n g f i n d i n g was t h a t the group decisions were u s u a l l y more r i s k y than the average i n d i v i d u a l decision. I n addition, i t was found t h a t a b r i e f p e r i o d o f discussion, followed by an i n d i v i d u a l decisionmaking w i l l a l s o cause a s h i f t i n the group average. Although the average s h i f t i n p r o b a b i l i t y was o n l y about .10 or less, t h i s f i n d i n g , which was unexpected according t o the e x i s t i n g theories, has r a i s e d i n t e r e s t among s o c i a l and organizational psychologists. Besides i t s r o l e i n the understanding o f human behavior i n i n t e r a c t i o n s i t u a t i o n s , t h e ' r i s k y s h i f t ' appears o f importance f o r many organizational s i t u a t i o n s i n which decisions are made by teams. It i s easy t o imagine t h e severe r e s u l t s t h a t may ensue when business committees, m i l i t a r y decision groups, counselling

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groups and j u r i e s f i r s t h o l d a group discussion and then make a j o i n t decision. Therefore, i t i s n o t s u r p r i s i n g t h a t STONER'S i n i t i a l study was followed by a wave o f i n v e s t i g a t i o n s o f group r i s k - t a k i n g . The t h e o r e t i c a l explanations f o r t h a t phenomenon based on experimental research can o n l y be mentioned b r i e f l y i n the given t e x t ( f o r more d e t a i l e d i n f o r m a t i o n see summaries by LAMM & MYERS 1978,and MYERS 1982). The f o l l o w i n g f a c t o r s have been taken i n t o account: 1.

I n groups, the responsibiZity f o r a decision i s distributed among t h e p a r t i c i p a n t s , thus the d e c i s i o n may tend t o be more r i s k y .

2. A t the o u t s e t

Ss are not familiar with the material and thus tend t o be

very cautious. During the discussion Ss exchange information about the matter and i n so doing a r e able t o perceive the s i t u a t i o n more r e a l i s -

3. 4.

t i c a l l y , which u s u a l l y leads t o a higher r i s k taking. The opinion leaders i n the group are uswrZZy more r i s k y . The group atmosphere i s comparably more p o s i t i v e and thus creates optimism.

5. Risk is a vaZue i n o u r society, therefore, t h e m a j o r i t y w i l l already lean towards being more r i s k y and s o c i a l comparison w i l l cause t h e cautious m i n o r i t y t o j o i n the m a j o r i t y . Some evidence has been provided f o r each o f these hypotheses. However, most o f these explanations can be found f a u l t y , since i t has been observed t h a t some material creates more conservative than r i s k y s h i f t s (as was already found w i t h items i n STONER'S study). On t h e o t h e r hand, a t l e a s t statements 3. and 5 . can be adapted i n order t o account f o r conservative s h i f t s as w e l l . This can be seen i n t h a t the leader u s u a l l y makes extreme decisions, e i t h e r i n the d i r e c t i o n o f being conservative o r r i s k y and i n the f a c t t h a t cautiousness, as w e l l as r i s k i n e s s , i s a highly-esteemed value i n our society. I t was a l s o found t h a t group s h i f t s appeared w i t h regard t o nonr i s k s i t u a t i o n s . Therefore, i n t h e l a t e s i x t i e s the term r i s k y s h i f t was subs t it ut e d by ' group- ( response )-s h i f t ' and I pol a r i za t ion ' ( MOSCOVI C I & ZAVALLONI 1969). Together w i t h the i n t r o d u c t i o n o f these more general terms,

BIASES IN GROUP DECISIONS

231

the t h e o r e t i c a l explanations had t o be reformulated i n order t o cover s h i f t s i n a t t i t u d i n a l , value-related and judgmental items. 1.2

Definition of S h i f t and Polarization

MYERS (1982, p. 127) describes group p o l a r i z a t i o n as the f a c t t h a t " t h e average post-discussion response (and group decision) w i l l tend t o be more extreme i n t h e same d i r e c t i o n as the average o f the pregroup responses". The question here i s how the direction i s determined w i t h regard t o the midpoint o f the scale o r w i t h regard t o the average i n d i v i d u a l response o f a l l Ss t a k i n g p a r t i n t h e experiment? I n experimental research both c r i t e r i a a r e used t o determine the d i r e c t i o n o f s h i f t . These two c r i t e r i a f o r p o l a r i z a t i o n have, however, c e r t a i n shortcomings. The midpoint o f the scale does n o t a l ways represent the psychological midpoint and i t o f t e n does not even e x i s t (e.g.,

i n estimation problems which have a natural zero p o i n t b u t n o t a l -

ways a n a t u r a l maximum).

In regards t o the second c r i t e r i a the average f i r s t

i n d i v i d u a l preference i s dependent on the population and thus, i t i s a f f e c t e d by sampling biases. As a t h i r d method, e s p e c i a l l y s u i t a b l e t o t e s t t h e degree o f p o l a r i z a t i o n n o t caused by t h e Ss' tendency t o decide according t o a m a j o r i t y r u l e , MYERS (1982, p. 139) suggests t o compare the median o f the i n d i v i d u a l responses w i t h the group decision o r t h e post discussion responses. This method corresponds b e t t e r t o t h e o r i g i n a l d e f i n i t i o n o f s h i f t . The a r i t h m e t i c mean w i t h i n the group i s taken as c r i t e r i o n and the l o c a t i o n o f the median as an i n d i c a t o r o f the skewness o f the d i s t r i b u t i o n . 1.3

Social Decision Rules

I n the discussion above, i t was already assumed t h a t t h e i m p l i c i t r u l e which groups apply i n order t o reach a decision might cause t h e d i f f e r e n c e between t h e a r i t h m e t i c mean and the group decision. The importance o f decision r u l e s (amalgamation r u l e s , s o c i a l decision schemes) was f i r s t pointed o u t by economists i n discussing problems o f s o c i a l welfare (see e.g. ARROW 1951, BLACK 1948). Social psychologists became aware o f the problem through the research on Social Decision Schemes (see DAVIS 1973). Theories on s o c i a l decision making and r e l e v a n t research w i l l n o t be d e t a i l e d i n t h i s paper. W i t h i n t h e given framework, some features o f s o c i a l decision r u l e s which are important

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f o r the following analysis of polarization and s h i f t will be discussed. Decision rules usually predict the outcome of a group decision on the basis of the individual preferences of the participants by integrating the individual preferences into a collective decision according t o the given rule. Most decision rules investigated empirically thus f a r , take into account only the f i r s t preferences of the individuals. Some decision rules predict a d i s t r i bution of group decisions. Others make exact point predictions o r specify the range in which the group decision i s most likely t o f a l l . Examples for distributional group decision rules are as follows:

-

ProportionaZity ruZe

- The distribution of group decisions i s proportional

to the distribution of the f i r s t preferences of the individuals. - Any alternative preferred by a t l e a s t one indivi-

- EquiprobabiZity ruZe

dual will have the same probability of becoming the group decision. The following three are examples f o r point/range predictions:

- Average of a l l f i r s t preferences of the individuals. Majority Decision - The absolute majority f o r one of the alternatives. PZuraZity ruZe - An alternative will r e s u l t from a group decision which

- Arithmetic Mean

-

has the relative majority, i.e. the modal alternative. Naturally, i t depends on the task i f a model makes sense as a group decision predictor. Whereas the Proportionality rule and the Equiprobability rule might apply for any problem, rules l i k e the Arithmetic Mean and others make sense only i f the object of decision i s quantifiable on the level of an i n terval scale. Some decision rules do n o t lead t o unique decisions. In these cases, e i t h e r the range of outcomes i s predicted o r a secondary decision ruZe i s introduced. Consequently, sometimes in small groups there is more than one Plurality ( i . e . , two o r more modal values in the distribution of f i r s t preferences). When t h i s occurs, one might predict a s e t of outcomes ( a l l alternatives with a plurality are equally l i k e l y ) or one might predict with another secondary decision rule t h a t the outcome will be the median plurality o r will be located in the middle between opposing p l u r a l i t i e s .

BIASES IN GROUP DECISIONS

1.4

233

The relation between Social Decision Rules and Polarization

Researbh on s h i f t and p o l a r i z a t i o n has i m p l i c i t l y taken t h e a r i t h m e t i c mean o f the i n d i v i d u a l responses as the natural candidate f o r t h e group decision. The f a c t t h a t the actual group decision deviated from t h e a r i t h m e t i c mean was therefore i n t e r p r e t e d by psychological e f f e c t s 1i k e t h e e f f e c t s o f group atmosphere, leader c h a r a c t e r i s t i c s , information exchange e f f e c t s and s o c i e t a l values (as described i n t h e i n t r o d u c t a r y section). However, i t i s easy t o v e r i f y t h a t most decision r u l e s w i l l lead t o a s h i f t i n r e l a t i o n t o t h e a r i t h metic mean, when t h e d i s t r i b u t i o n o f the o r i g i n a l i n d i v i d u a l preferences w i t h i n the group i s n o t completely symmetric. Moreover t a k i n g srrall groups w i t h i n a population w i l l u s u a l l y lead t o asymmetric d i s t r i b u t i o n s , even i f t h e popul a t i o n ' s d i s t r i b u t i o n i s symmetric. So p o l a r i z a t i o n i s l i k e l y t o occur as a consequence o f t h e decision r u l e . Taking thab i n t o account the c e n t r a l question f o r t h e f o l l o w i n g empirical i n v e s t i g a t i o n s can be formulated:

'Do

s o c i a l decision r u l e s account f o r a l l the p o l a r i z a t i o n ' o r a l t e r n a t i v e l y

'Is t h e r e substantial p o l a r i z a t i o n l e f t a f t e r the a p p l i c a t i o n o f a decision rule?'. 1.5

Some remarks regarding the concept of the following empim'cal studies

The empirical studies reported i n the f o l l o w i n g s e c t i o n were o r i g i n a l l y designed t o analyse which decision schemes f i t best on c o l l e c t i v e decision. However, f o r t h e reasons discussed above, the studies w i l l s u i t a r e a n a l y s i s i n terms o f the p o l a r i z a t i o n e f f e c t s . Ad hoc groups have u s u a l l y been taken as the basis o f analysis f o r research on decision schemes. However, t h e d i s t r i b u t i o n o f i n d i v i d u a l preferences i n ad hoc groups does n o t very o f t e n a l l o w f o r d i f f e r e n t i a t i o n o f d i f f e r e n t s o c i a l decision rules. I t may happen w i t h ad hoc groups t h a t according t o the o v e r a l l s t a t i s t i c s a l l decision schemes p r e d i c t e q u a l l y w e l l , although i n any s i n g l e group the decision can be best explained by a c e r t a i n p o i n t p r e d i c t i o n model. Because o f t h e d i f f i c u l t y t o d i f f e r e n t i a t e between c e r t a i n models by using ad hoc groups i t i s more powerful t o c o n s t r u c t groups i n a way t h a t the i n d i v i d u a l preferences f u l f i l l c e r t a i n d i s t r i b u t i o n a l charact e r i s t i c s . This has been done i n study I 1 and 111. That would n o t have been

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SOCIAL INTERACTION

necessary f o r the analysis o f the p o l a r i z a t i o n e f f e c t , b u t t h i s does not, however, a f f e c t the v a l i d i t y o f t h a t analysis. I n a l l t h r e e studies hypotheses about the best f i t t i n g s o c i a l decision scheme were formulated i n advance, i.e.,

depending on the task, v a r i a t i o n s o f t h e m a j o r i t y d e c i s i o n r u l e

were assumed t o be best f i t t i n g . The basic idea behind the m a j o r i t y decision r u l e s applied i n the three studies i s the assumption t h a t group decisions w i l l f a l l on the f i r s t preference o f t h e median i n d i v i d u a l i n tasks w i t h underlying scales o f ( a t l e a s t ) o r d i n a l character. As BLACK (1948) has proved, t h a t i s i n f a c t the case when t h e preference curves o f the Ss over the scale p o i n t s are s i n g l e peaked, and when the decision r u l e i s the p a i r w i s e comparison o f a l t e r n a t i v e s by m a j o r i t y voting. S i n g l e peaked means t h a t the i n d i v i d u a l has a preference p o i n t ( b l i s s point, i d e a l p o i n t ) on t h e scale, and t h a t h i s degree o f s a t i s f a c t i o n w i t h o t h e r scale p o i n t s decreases cont i n i o u s l y l e f t and r i g h t o f t h a t preference p o i n t . As w i l l be demonstrated l a t e r t h i s assumption i s not t r i v i a l , e s p e c i a l l y n o t f o r r i s k - t a k i n g s i t u a t i o n s . I f the s i n g l e peakedness c o n d i t i o n does n o t h o l d then, a m a j o r i t y v o t i n g over the a1 t e r n a t i v e decision might l e a d t o t h e group decision d i f f e r e n t from the median. This d i f f e r s from most o t h e r decision schemes i n t h a t those aggregation r u l e s have t o take i n t o account n o t only the f i r s t preferences b u t the complete preference rankorders. The decision r u l e s pred i c t e d as best f i t t i n g are given i n a b r i e f d e s c r i p t i o n o f the experiments i n the next sections. I n order t o analyse the p o l a r i z a t i o n e f f e c t i n terms o f soc i a l decision schemes, we have t o compare actual p o l a r i z a t i o n and the polar i z a t i o n predicted by the s o c i a l decision r u l e . As a basis we take the decis i o n schemes t h a t proved as best f i t t i n g , assuming t h a t those r u l e s approximate best what Ss r e a l l y used as a c o l l e c t i v e decision device. The decision r u l e s assumed as best f i t t i n g i n advance indeed showed the smallest d e v i a t i o n from t h e actual group decision. I n a d d i t i o n we analysed various o t h e r p r e d i c t i o n schemes, l i k e geometric mean, a r i t h m e t i c mean, p l u r a l i t y , p r o p o r t i o n a l i t y and equiprobabil it y (CROTT 1965, CROTT/SCHERMER/KNOLL & THOENE, i n prep. ,and CROTT/SCHERMER & ZUBER, i n prep.). 1.6

General Hypotheses f o r Experiments I t o IIIandtechnical remarks regarding the analysis and presentation o f data

Our general idea says t h a t i n cases where the decision a l t e r n a t i v e s can be

BIASES IN GROUP DECISIONS

235

rank ordered a t l e a s t on an ordinal scale the collective decision i s reached by some kind of majority voting. If there i s an absolute majority f o r one alternative (more than 50 % of group members give the f i r s t rank t o the same alternative) then t h i s alternative will result as a group decision. In the more c r i t i c a l cases, mainly investigated in the following experiments in which no absolute majority exists f o r one single alternative, the r e s u l t of the majority voting depends on the structure of the individual preference orders: 1. I f a l l individual preference orders a r e single peaked, the r e s u l t of the majority voting will be the alternative most preferred by the median individual. If there are two median individuals a secondary decision rule has t o be applied (e.g., midpoint of the range of the two median individuals). 2.

In cases i n which the single peakedness condition i s not f u l f i l l e d f o r a t least one individual in the group a paired c o ~ m i s o nby majority voting over a l l pairs of alternatives might lead to a result differing from the median. Hence, when single peakedness of individual preference curves can't be assumed t o hold generally, we have t o compute the pairwise majority voting result f o r each groups of Ss separately. - In some cases t h i s procedure would r e s u l t i n i n t r a n s i t i v i t i e s i n the group's rankorder of the alternatives. Then the group could n o t be able t o decide f o r one of the alternatives as a group decision. Again f o r those cases we have t o establish a secondary decision rule.

Experiment I i s a reanalysis of a study by CROTT (1964/65) w i t h a judgmental task (estimation of the number of points on a s l i d e a f t e r a short exponation). In Experiment I1 and I11 (CROTT e t a l . , in prep. a,b) Ss have t o decide on bets with equal expected value, varying in the degree of risk. I t i s the aim of a l l three analyses t o demonstrate t h a t there exists s h i f t and polarization when conventional measure, namely the deoiation of the arithmetic mean, i s taken. A t the same time we claim t h a t s h i f t and polarization reduce considerably when an appropriate decision rule i s applied. In order t o save space we only give a brief description of the experimental method. More detailed information i s (or will be) given in the original a r t i c l e s . For the same

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SOCIAL INTERACTION

reason we mainly present the more general p o l a r i z a t i o n data. The main resul t s regarding l e f t / r i g h t s h i f t o r risky/conservative s h i f t are mentioned i n footnotes under the p o l a r i z a t i o n tables. The model t e s t i n g approach a p p l i e d here, i m p l i e s maintaining the model, when deviations from t h e pred i c t i o n are i n s i g n i f i c a n t . Following a convention, t h e 20 % l e v e l o f s i g n i ficance w i l l be chosen when s i g n i f i c a n t r e s u l t s a r e n o t expected according t o our hypotheses. Otherwise the 5 % l e v e l i s applied. This approach i s cons e r v a t i v e i n making i t d i f f i c u l t t o g e t supportive evidence i n f a v o r o f t h e hypotheses. 2.

Experiment I :

Group Decision under different motivational conditions

(CROTT 1964, 1965) 2.1

Brief description of the experimental conditions

Subjects and Conditions of participants: Volunteering 72 male and 72 female students i n 36 groups o f four Ss. A l l members o f a group were o f same sex. Tasks: a) I n d i v i d u a l estimation o f randomly d i s t r i b u t e d p o i n t s on a s l i d e a f t e r a short exponation (10 sec. ). b) Giving a group judgment on t h e number o f points. Procedure: Ss denoted t h e i r i n d i v i d u a l estimations, then t h e experimenter announced them i n a random order. Then Ss wrote down new estimations w i t h o u t seeing t h e s l i d e again. This was repeated u n t i l a l l estimations agreed w i t h i n a range o f 4 points. Motivational conditions: Motivation A: We want t o know i f groups reach b e t t e r estimations o f the number o f p o i n t s than i n d i v i d u a l s do on the average. Theref o r e please be cooperative t o come t o a good group r e s u l t . Motivation B: This experiment has t o do w i t h t h e r e l a t i o n s h i p between i n t e l l i g e n c e and t h e a b i l i t y t o judge f a c t s c o r r e c t l y . We want t o know i f i n d i v i d u a l s w i t h the b e t t e r estimations i n f l u e n c e the group decision process more than i n d i v i d u a l s w i t h worse estimations . Under both conditions the experimenter announced t h a t he would n o t be able t o t e l l them the r e a l number o f p o i n t s i n order t o prevent t h a t Ss would t a l k about i t t o l a t e r coming Ss. Experimental factors: Sex and motivational conditions. Social Decision Schemes applied: Primary SDS: Median Secondary SDS: Cooperative Motivation: Midpoint o f the median range (Median/Mi dpoi n t )

BIASES IN GROUP DECISIONS

237

Competitive Motivation: A r i t h m e t i c mean i f w i t h i n t h e median range o r t h a t bound o f the median range which i s c l o s e r t o the a r i t h metic mean (Median/Mi n. Var. D i stance). Relevant Hypothesea: I n a paired comparison over the o r i g i n a l i n d i v i d u a l estimations t h e majori t y r u l e could l e a d t o a group decision w i t h i n the range of the two median S s , regardless o f the motivational conditions. The secondary decision scheme d i f f e r s depending on the motivational c o n d i t i o n (as described above). We expect t h a t no p o l a r i z a t i o n w i l l e x i s t a f t e r a p p l i c a t i o n o f the approp r i a t e decision scheme.

2.2

Results and discussion for Experiment I

Since the Ss' estimations are given i n numbers here, the s i n g l e peakedness o f the i n d i v i d u a l preference orders appears t o be a n a t u r a l property o f t h e task. With f o u r Ss i n a group there e x i s t two median points. Therefore a secondary decision r u l e has t o be defined. Under t h e cooperative m o t i v a t i o n we postulate t h a t Ss w i l l soon agree on the midpoint o f the median range as an estimate. The secondary decision r u l e f o r t h e competitive m o t i v a t i o n f o l lows a hypothesis by HOFFMAN, FESTINGER and LAWRENCE (1954). It s t a t e s t h a t because o f s o c i a l comparison processes Ss w i l l tend t o minimize variance between the p a r t i c i p a n t s . Therefore we assume t h a t Ss w i l l agree on the a r i t h m e t i c mean when i t l i e s w i t h i n t h e median range or, otherwise, on t h a t bound o f the median range t h a t i s c l o s e r t o the a r i t h m e t i c mean. The s o c i a l decision r u l e might t h e r e f o r e s t a t e 1. t h e median range 2. w i t h i n the median range, the midpoint o f the two extreme Ss o r t h a t bound o f the median range which i s c l o s e r t o i t . We w i l l however n o t present the data f o r t h a t analysis here, since differences t o the decision scheme suggested by s o c i a l comparison theory are n e g l i g i b l e . The r e s u l t s i n t a b l e I/1 and I / 2 show t h a t i n the given judgmental context

A p l a u s i b l e a l t e r n a t i v e secondary group decision r u l e would l e a d t o a l most i d e n t i c a l r e s u l t s , namely t h a t t h e two extreme Ss bargain much tougher f o r t h e group r e s u l t than the middle Ss which a r e already close t o t h e i r o r i g i n a l estimation by t h e primary d e c i s i o n r u l e .

230

SOCIAL INTERACTION

Table I/1:

Frequency analysis o f Polarization (from the A r i t h m e t i c Mean i n d i r e c t i o n t o midpoint o f t h e Median range) ') Po 1 a r iza t i on

Polarization

A r i t h m e t i c Mean

Median/Midpoint

t

Cooperative Motivation

I

15 (13.5)

-

3 (4.5)

+

-

I

11

7

Median/Min.Var.Dist. Competitive Motivation

-

12 (13.5) 27

6 (4.5) 9

7

11

18

18

No differences i n frequencies between male and female groups.

Table I/2:

r

Deviation analysis o f P o l a r i z a t i o n (from t h e Arithme i c ean i n t o the d i r e c t i o n o f t h e midpoint o f the median range) + tt

Y

A r it h me t ic Mean Cooperative Motivation

ji

Competitive Motivation

ji

I

Sums o v e r both conditions

t

79.18 147.28 2,28*05

t

34.63 57.62 2. 5 5 ' 0 5

t

56.91 111.83 3.05*01

'1

A p p r o p r i a t e SDS: Med ia n / M i d p o i n t

8.82 30.85 1 . 2 1 - 30 Median/Min. Var. Distance -5.04 46.25 -.46'70 1.89 39.31 .2gs70

No differences between means f o r male and female S s . Regarding l e f t and r i g h t s h i f t s t h e r e i s l e f t s h i f t from the A r i t h m e t i c Mean f o r m o t i v a t i o n A and B together, w i t h t = 1.62, d f = 35, p < .20 and almost no s h i f t f o r the appropriate SDSs.

BIASES IN GROUP DECISIONS

239

t h e r e e x i s t strong p o l a r i z a t i o n e f f e c t s according t o the a r i t h m e t i c mean. I n t h a t context l e f t / r i g h t p o l a r i z a t i o n from the a r i t h m e t i c mean can h a r d l y be accounted f o r by t h e usual psychological explanations provided f o r r i s k and a t t i t u d i n a l changes since i t i s j u s t a change i n the e s t i m a t i o n o f t h e

num-

bers o f points. The p o l a r i z a t i o n i s seen i n the frequency t a b l e I/1 as w e l l as i n the analysis o f deviations i n t a b l e I / 2 . P o l a r i z a t i o n e f f e c t s can, however, be reduced t o almost zero when the hypothesized decision r u l e s are applied.

3.

3.1

Experiment 11: Social Decision Scheme and Group Polarization i n Group Decisions among b e t s (CRGTT, SCHEMER, KNOLL & THOENE, i n prep.) Brief description of the experimental conditions

Subjects and conditions of participation: 260 male students o f economy and law. Each S g o t 5 DM f o r p a r t i c i p a t i o n . Ss were informed, t h a t they could p o s s i b l y win a d d i t i o n a l l y a considerable amount o f money. For each session between 15 and 25 Ss were i n v i t e d . Instructions, Tasks, and Procedure: a ) Explanation o f p r o b a b i l i t i e s i n terms o f r e l a t i v e frequencies and the expected value as a long run expectation. Explanation o f the meaning o f i n d i v i d u a l preferences over bets w i t h equal expected values. I t was already mentioned t o the Ss here, t h a t they afterwards had t o discuss t h e i r preferences w i t h others t o come t o a group decision. b ) Measurement o f i n d i v i d u a l preferences f o r t h e f o l l o w i n g f o u r bets:

The Payoff conditions served as two d i f f e r e n t experimental conditions. c ) Arrangement o f Ss i n t o groups which f i t i n t o one o f the f o l l o w i n g d i s t r i b u t i o n s o f occupancy numbers:

d) Group discussion and decision: Ss had t o come t o a group decision among the bets ( a f t e r a t most 15 minutes o f discussion). e) Second measurement o f i n d i v i d u a l preferences. f ) Playing o u t o f the bets: The chosen b e t was played o u t once f o r t h e whole group and the amount o f money o f the b e t o r nothing was p a i d t o the Ss depending on the r e s u l t .

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SOCIAL INTERACTION

Experimental Factors: The two payoff conditions under section I n s t r u c t i o n s b ) and the f o u r d i s t r i b u t i o n s o f occupancy numbers under section I n s t r u c t i o n s c ) . Social Decision Schemes applied: Primary S D S : Median Secondary SDS: Any o f the two w i t h equal p r o b a b i l i t y , i f there are two 'median' bets. The group decision w i l l f a l l on the median b e t ( o r one o f the median bets f o r 4-person groups). Relevant Hypotheses: The SDS Median, as described above, w i l l p r e d i c t group decisions c o r r e c t l y . No diffefences are expected between payoff conditions.

Results and discussion for Experiment 11

3.2

I n accord w i t h the r e s u l t s o f Experiment I we expected the Median d e c i s i o n r u l e t o apply f o r bets w i t h equal expected value (see hypotheses i n box

3.1). Table

II/l:

Frequency analysis o f PoZariz t i o i n d i r e c t i o n o f t h e Median)

+? ttr

positive

O

(from t h e Arithmetic Mean

negative

Sign t e s t f o r Arithm. Mean, 42 : 16 cases, p < .01 Sign t e s t f o r Median, +)

4 : 16 cases, p < .01

The analysis f o r t h i s experiment i s o n l y conducted f o r frequencies. An analysis i n terms o f deviations does n o t provide more information since 56 o u t o f 58 s h i f t s are e x a c t l y one u n i t .

++) Since most s h i f t s were r i s k y , r e s u l t s f o r a r i s k y s h i f t analysis a r e al'most i d e n t i c a l w i t h the p o l a r i z a t i o n analysis (43 : 15 and 18 : 2 r e s p e c t i v e l y ).

BIASES IN GROUP DECISIONS

241

Again there was a s t r o n g p o l a r i z a t i o n (and s h i f t a l s o ) from t h e a r i t h m e t i c mean (see t a b l e II/1) b u t contrary t o our foregoing r e s u l t s there was s t i l l p o l a r i z a t i o n a f t e r the a p p l i c a t i o n o f the Median d e c i s i o n r u l e . Although 47 o f 58 group decisions f a l l e x a c t l y on ( o r one o f ) the median bet(s), p o l a r i z a t i o n i s s i g n i f i c a n t ; i n d i c a t i n g t h a t there i s a systematic d e v i a t i o n from the median. A c l o s e r look a t t h e data reveals t h a t the negative p o l a r i z a t i o n from t h e median i s p r i m a r i l y due t o r i s k s h i f t (18 : 2). The p o l a r i z a t i o n i s p o s i t i v e from the a r i t h m e t i c mean (42 : 16) and negative from the median (4 : 16). This outcome i s due t o the f a c t t h a t when a r i s k s h i f t was expected from t h e a r i t h m e t i c mean i n d i r e c t i o n o f the median i t occurred (group types I1 and 111), whereas when a conservative s h i f t was expected from the a r i t h e m t i c

mean i n d i r e c t i o n o f the median i t d i d n o t occur (group types I and I V ) . For the a r i t h m e t i c mean taken as standard, t h i s r e s u l t s i n a p o s i t i v e p o l a r i z a t i o n , f o r t h e median as standard i t r e s u l t s i n a negative p o l a r i z a t i o n . I n t r y i n g t o e x p l a i n these r e s u l t s , we suggest two a l t e r n a t i v e hypotheses:

1. Money ScaZe Hypothesis: When deciding on t h e bets, Ss a r e o r i e n t e d on t h e money scale r a t h e r than t h e p r o b a b i l i t y scale. I n addition, the assumption underlying the median model o f BLACK (1948) and COOMBS (1954) ( t h a t o n l y the rankorder o f a l t e r n a t i v e s i s important f o r group decision, n o t the distances) i s i n c o r r e c t . I n t h e case o f Ss deciding p r i m a r i l y on the money scale w i t h distances between a l t e r n a t i v e s as an important fact o r , i t might be expected t h a t

-

because o f t h e r a p i d increase i n money

-

value t h e arguments f o r bets w i t h higher money values (and lower prob a b i l i t i e s ) would be more convincing i n the group decision (see Box 11, I n s t r u c t i o n s , Tasks, and Procedures , b ) ) . 2.

VioZation of SingZe Peakedness Hypothesis: The basic s o c i a l decision r u l e ( m a j o r i t y v o t i n g over a l t e r n a t i v e s ) i s c o r r e c t , b u t the c o n d i t i o n o f s i n g l e peakedness o f i n d i v i d u a l preference d i s t r i b u t i o n s i s v i o l a t e d . A d d i t i o n a l l y , we have t o assume t h a t t h e v i o l a t i o n s do n o t wash o u t on the average b u t cause systematic deviations from t h e median d e c i s i o n rule.

242

SOCIAL INTERACTION

Although we found i n our studies t h a t 86 % (263 o f 305 cases) o f t h e i n d i vidual preference orders were s i n g l e peaked, we have t o expect t h e o r e t i c a l l y t h a t 48 % o f the groups would have a t l e a s t one i n d i v i d u a l v i o l a t i o n o f the c o n d i t i o n o f s i n g l e peakedness ( t h a t would amount t o 28 out o f 58 cases). I f the s i n g l e peakedness had a systematic e f f e c t on t h e group decisions

(e.g.,

towards r i s k ) , then 48 % o f the groups were p o t e n t i a l l y a f f e c t e d .

Both a l t e r n a t i v e explanations s h a l l be t e s t e d i n the f o l l o w i n g experiment.

4.

4. I

Experiment 111: Group Decision among bets o f equaZ expected vatue, with two different types of bets (CRO!l'T, SCHEMER, ZUBER, i n prep. I Brief description of the experimentat conditions

Subjects and conditions of participation: Ss were 280 male students o f law and economy. Each S g o t 5 DM f o r p a r t i c i pation. Ss were informed t h a t they could p o s s i b l y a d d i t i o n a l l y win a considerable amount o f money. For each session, between 15 and 25 Ss were i n vited. Instructions and Tasks: a) Explanation o f the meaning o f p r o b h b i l i t i e s i n terms o f r e l a t i v e f r e quencies and the meaning o f expected value as a l o n g run expectation. Explanation o f the meaning o f i n d i v i d u a l preferences among bets w i t h equal expected value ( f o r the bets see t a b l e I I I / l ) . Ss were informed i n advance t h a t they have t o discuss t h e i r preferences w i t h others and have t o come t o a c o l l e c t i v e decision. b) Measurement o f i n i t i a l i n d i v i d u a l preferences by rankordering t h e s i x bets.

Procedure: The Ss were assigned t o 5-person groups w i t h one o f t h e f o l l o w i n g basic patterns o f occupancy numbers: aa) 2 1 1 1 ba) 2 2 1 c ) 2 1 2 bb) 1 2 2 ( m i r r o r ab) 1 1 1 2 ( m i r r o r image) image) Examples : Bet.No. 1 2 3 4 5 6

Bet.No. 1 2 3 4 5 6

1

Bet.No. 2 3 4

5

6

0

0

2

0

2

1

2

0

2

0

1

0

c ) Group discussion and decision ( a f t e r 15 minutes). d) Second measurement o f i n d i v i d u a l preferences: A f t e r t h e group decision,

BIASES IN GROUP DECISIONS

243

i n d i v i d u a l preferences were measured a second time and t h e b e t was played once. Repetitions (two times) o f the procedure w i t h s e t I 1 and I11 (see t a b l e III/l). Playing o u t ' t h e bets: The chosen b e t was played o u t once f o r t h e whole group and the amount o f money o f the b e t o r nothing was paid t o the Ss depending on t h e r e s u l t s o f the l o t t e r y .

Experimental Factors: a) The f i v e types o f d i s t r i b u t i o n s o f occupancy numbers. b ) Conditions P and DM f o r the s t r u c t u r e o f bets. Under c o n d i t i o n P bets have equal p r o b a b i l i t y (and consequently unequal money distances). , Under c o n d i t i o n DM bets have equal money distances ( b u t unequal p r o b a b i l i t y distances). Social Decision Schemes applied: For both conditions, P and DM, we t r y o u t the f o l l o w i n g SDS: Primary SDS:

Secondary SDS:

1. 2. 3. 4.

Median b e t i n the range i n case o f intransitivities. Median b e t i n the range i n case of intransitivities.

A r i t h m e t i c Mean Median F u l l Paired Comparison Reduced Paired Comparison

Example f o r SDS 3. and 4.: Priniary SDS: > 2 <3 > 4 <

1

5>

6

Secondary SDS: 3

Relevant Hypotheses: There i s no p o l a r i z a t i o n i n any o f t h e conditions a f t e r a p p l i c a t i o n o f t h e appropriate Social Decision Scheme.

4.2

Results and discussion for Experiment 111

I n order t o make t h e i n v e s t i g a t i o n o f the two hypotheses possible, t h e bets f o r Experiment I11 .under c o n d i t i o n P were constructed w i t h equal p r o b a b i l i t y distances (as before i n Experiment 11) and w i t h equal money distances under c o n d i t i o n DM (see t a b l e III/l). I n addition, the complete rankorder o f i n d i vidual preferences over t h e a l t e r n a t i v e s was taken from each i n d i v i d u a l before and a f t e r the group decision. The s o c i a l decision scheme assumed t o be c o r r e c t according t o hypothesis 2 f o r groups without an 'Absolute M a j o r i t y ' (see r e s u l t s and discussion f o r Experiment 11) was the m a j o r i t y decision i n paired comparison o f a l t e r n a t i v e s . Besides the A r i t h m e t i c Mean and the Median, two v a r i a n t s o f the M a j o r i t y Decision by Paired Comparison o f a l t e r n a t i v e s have been investigated. The

244

SOCIAL INTERACTION

Table 11111:

The three s e t s of bets used i n the experiment under c o n d i t i o n P and DM S e t 11:

I

S e t 111:

S e t I: 0.25/18.00 0.351 12.80 0.451 9.90 0 . 5 5 1 8.20 0.651 6.90 0.751 6.00

0 . 2 5 1 1 2 .oo 0,351 8.60 0.451 6.70 0.551 5.50 0.651 4.60 0.751 4.00

0 . 2 5 1 6 .oo 0.3514.30 0.45l3.30 0.5512.70 0,6512.30 0 . 7 5 1 2 .OO

0.24l18.50 0.28116.00 DM-con- 0 . 3 3 / 1 3 . 5 0 d i t i o n 0 . 4 1 1 11.00 0.531 8.50 0.751 6.00

0.26l11.50 0.30/10.00 0.351 8.50 0.431 7.00 0.551 5.50 0.751 4.00

0.3314.50 0 . 3 8 1 4 .OO o .43/3.50 0.50/3.00 0.60/2.50 0 . 7 5 / 2 .OO

P-condition

Table 11112:

1

Decision Scheme 'Absolute M a j o r i t y ' f o r groups i n Set I11

I

Absol Ute M a j o r i t y Predicsion c o r r e c t

I

I

Absolute M a j o r i t y Prediction incorrect

I

i

I

40

I 1 and

12 +)

+) It doesn't make sense here t o analyse the 12 cases according t o r i s k y

s h i f t and p o l a r i z a t i o n . Only i n two o f the 12 cases r i s k y s h i f t andposip o s i t i v e p o l a r i z a t i o n are possible. The remaining 10 cases a l l o w f o r cautious s h i f t o r negative p o l a r i z a t i o n only, because t h e 'Absolute M a j o r i t y ' already p r e d i c t s the extremest bet. f i r s t one i s c a l l e d ' F u l l Paired Comparison/Median Bet' and the second one ' Reduced Pai red Compari son1Medi an Bet ' (see box). F u l l Paired Comparison means t h a t a l l a l t e r n a t i v e s are compared i n order t o p r e d i c t the group decision. For the Reduced Paired Comparison only t h e neighboured bets are compared successively, t h a t i s b e t 1 vs. 2 , bet 2 vs. 3 and so on. I n both bases (but e s p e c i a l l y w i t h t h e Reduced Paired Comparison) the r e s u l t i n g d i s t r i b u t i o n o f the group preference order might n o t be t r a n s i t i v e . As a secondary decision r u l e we take t h e median b e t o f an i n t r a n s i t i v e sequence as t h e best estimation (see example i n box). I f hypothesis 1 was true, we

BIASES IN GROUP DECISIONS

245

should get differences in the P and DM condition even a f t e r applying the decision rule considered appropriate; according t o hypothesis 2 ( i . e . , under the P condition) there should remain polarization, b u t not under the DM condi t i on. Table 111/3:

Frequencies of groups without violations o f single peakedness (or irrelevant violations) and groups containing a t l e a s t one member w i t h violations of single peakedness f o r condition P and DM separately.

Groups with relevant

Groups without violations of single peakedness (or irrelevant violations)

I I p

I

22

53 X’

(26.5)

violations o f single peakedness

I

1

20

(15.5)

142

I

31

4.14, df = 1, p < .05, descriptive +)

I1 and I 1 1 which consist of same Ss as groups from s e t I are included in the table, so frequencies are partly dependent.

+) 29 groups o f s e t

In table III/3 we can compute t h a t the percentage of groups w i t h a t l e a s t one member with a t l e a s t one relevant violation of the single peakedness i s 37 %. A violation i s considered relevant when the Full Paired Comparison of a l l alternatives by majority voting leads t o predicted group decision different from the median. We can also see from table III/3 t h a t there are more groups with relevant violations under the p than under the DM condition. Polarization and s h i f t have f i r s t been analysed i n terms o f frequencies (comparison of number of cases with positive vs. number of cases w i t h negat i v e polarization and number of cases w i t h risky or with conservative s h i f t ) . Results of t h a t analysis are given i n tables III/3 and III/4. Table III/5 shows the polarization results of the analysis of deviations.

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SOCIAL INTERACTION

p o s i t i v e no

2 negative z - t e s t (p:)

20

0

9

.10

Median

6

14

9

.70

F u l l Paired Comp.

8

15

6

.80

11

14

4

* 20

A r i t h m e t i c Mean

Reduced Paired Comp.

%Yncies of

PoZarization ( t o the Median) under c o n d i t i o n

.90

Reduced Paired Comp.

10

22

10

.99

Regarding s h i f t t h e r e are s i g n i f i c a n t r e s u l t s f o r a l l f o u r models (.0001, .01, .05, .20) under the P c o n d i t i o n and f o r t h e ' A r i t h m e t i c Mean' under the DM c o n d i t i o n (.05). To provide an overview i n t a b l e I I I / 6 t h e remaining p r o b a b i l i t i e s f o r a l l s t a t i s t i c a l t e s t s are collected. The values above the diagonal r e f e r t o the frequency analyses the values below t o the d e v i a t i o n analyses. I t can be seen from column 1 t h a t the group d e c i s i o n i s s i g n i f i c a n t l y d i f f e r e n t from the Arithmetic Mean, t h i s i s t r u e f o r frequency- and f o r d e v i a t i o n analysis, f o r r i s k y s h i f t and p o l a r i z a t i o n , f o r c o n d i t i o n P and DM and f o r both condit i o n s together. Among the 12 values t h e r e are o n l y two exceptions f o r p o l a r -

-

i z a t i o n deviations under c o n d i t i o n

P ( w i t h p < .10 and p

< .30). ') The

Regard t h a t t h e A r i t h m e t i c Mean and the Median a r e n o t the hypothesized Decision Schemes, so we take t h e 5 % l e v e l o f s i g n i f i c a n c e i n these cases, and the 20 % l e v e l f o r t h e f o l l o w i n g t e s t s o f t h e F u l l Paired Comparison and t h e Reduced Paired Comparison.

BIASES IN GROUP DECISIONS

247

Deviation A n a Z p i ~of PoZarization ( t o the Median): Deviation

Table 11115:

from t h e Social Decision Scheme p r e d i c t i o n (group de i s i n minus SDS p r e d i c t i o n ) Set I, 11, and I 1 1 'Average' ++?

+F

Arithmetic

lSt SDS Mean __--___.----------

Median

Median Bet

2nd SDS

-n / d f

25/24

X

0.027

P s t

n/df X

DM s t

-n / d f P+ DM t

+++e

Full Pair d

---------_ -E:rlleL ---_--_

25/24 -0.016 0.121 0.112 1 . 2 1 . ~ ~-0.66' 60

25/24 -0.009 0.146 -0.31' 80

27/26 0.032 0.045 3.68."

27/26 -0.002 0.036 -0.24' 90

27/26 0.002 0.046 0.2 7' 8o

52/50 0.030 0.084 2.58' O2

52,150 -0.009 0.088 -0.74' 50

52/50 -0.003 0.106 -0.20' 90

25/24 0.020 0.118 0.85.~' 27/26 -0.001 0.054 -0.08' 90

52/50 0.009 0.091 D. 71-50

i)e

Ss o f the groups i n Set I 1 and I 1 1 which correspond t o t h e occupancy numbers as described i n Box 4.1. are i d e n t i c a l w i t h t h e Ss i n Set I, we averaged t h e deviations from the p r e d i c t e d values f o r Set I, 11, and 1 1 1 i n cases i n which t h e r e was more than one s e t w i t h c r i t i c a l occupation numbers.

++)

Considering o n l y r i s k y s h i f t ' A r i t h m e t i c Mean' shows s i g n i f i c a n t dev i a t i o n s f o r P, DM and P + DM ( w i t h p < .0001, < .01, < .001, respecti v e l y ) . The median model shows s i g n i f i c a n t d e v i a t i o n s f o r P and P + DM (< .01, < .01) whereas f o r DM there i s no r i s k y s h i f t . For the model ' F u l l Paired Comparison' the p r o b a b i l i t y i s below .20 only f o r P + DM together. For t h e 'Reduced Paired Comparison' model the p r o b a b i l i t i e s do n o t go below .20.

+++I As ' F u l l

Paired Comparison' and 'Reduced Paired Comparison' sometimes p r e d i c t t h e most extreme bet, r i s k y s h i f t and p o s i t i v e p o l a r i z a t i o n were n o t always possible under those models. Therefore a conservative c o r r e c t i o n f o r f l o o r / c e i l i n g e f f e c t s was a p p l i e d by e n l a r g i n g the r i s k y s h i f t and p o s i t i v e p o l a r i z a t i o n one standard deviation. However, t h i s procedure had very l i t t l e e f f e c t s on the and t - t e s t values, so t h a t f l o o r l c e i l i n g e f f e c t s apparently can be neglected i n t h e given experimental m a t e r i a l .

248

SOCIAL INTERACTION

Table 111/6: Summary t a b l e o f s i g n i f i c a n c e s o f frequency- and d e v i a t i o n analysis f o r r i s k y s h i f t and p o l a r i z a t i o n (see a l s o t a b l e s III/4 and 111/5). Frequency s i g n i f i c a n c e s above, d e v i a t i o n significances be1ow diagonal

Median (column 2 ) shows some s i g n i f i c a n t deviations regarding r i s k y s h i f t e s p e c i a l l y under t h e P condition. P o l a r i z a t i o n from the Median i s n o t s i g n i f i c a n t i n any case. Also f o r the F u l l Paired Comparison/Median Bet t h e r e are s t i l l some s i g n i f i c a n t r e s u l t s f o r r i s k y s h i f t under the P c o n d i t i o n and both conditions taken together (p < .05 and < . l o ) . For the Reduced Paired Comparison/Median Bet t h e r e are o n l y marginal s i g n i f i c a n c e s l e f t f o r r i s k y s h i f t and p o l a r i z a t i o n i n frequency a n a l y s i s under P, t h a t vanish when P and DM a r e taken together. These r e s u l t s demonstrate t h a t t h e v i o l a t i o n o f s i n g l e peakedness had a systematic e f f e c t on t h e group decisions. When t a k i n g i n d i v i d u a l character-

249

BIASES IN GROUP DECISIONS

i s t i c s i n t o ac c ount by u s i n g t h e P a i r e d Comparison Model , however, t h e r i s k y s h i f t and p o l a r i z a t i o n reduce c o n s i d e r a b l y under any c o n d i t i o n , P o r DM. R e s ult s a r e a l s o summarized i n F i g u r e III/1 which g i v e s a g r a p h i c a l r e p r e s e n t a t i o n o f t h e r e d u c t i o n s i n r i s k y s h i f t and p o l a r i z a t i o n by a p p l i c a t i o n o f a p p r o p r i a t e d e c i s i o n r u l e s . By t h e Reduced P a i r e d Comparison/Median Bet r i s k y s h i f t i s reduced t o about 17 % o f t h e o r i g i n a l amount, p o l a r i z a t i o n t o ap p ro x ima t e ly 30 %. Risky s h i f t

Polarization

-05

.05

.04

.04

I

+

.03

.02

t

.ol

A r i t h m . Median Mean

t

F.P.C. median bet

B.P.C. median bet

indicates positive polarization,

Figure I I I / l :

Arithm. Median Mean

-

F.P.C. median bet

R.P.C. median bet

negative polarization.

G ra p h i c a l r e p r e s e n t a t i o n o f r i s k y s h i f t and p o l a r i z a t i o n e f f e c t s f o r P and DM

We conclude r i s k y s h i f t and p o l a r i z a t i o n can be e x p l a i n e d by t h e e f f e c t o f t h e d e c i s i o n r u l e . A d m i t t i n g t h a t t h e r e i s s t i l l some r i s k y s h i f t and p o l a r i z a t i o n l e f t (sometime p o s i t i v e , sometime n e g a t i v e ) n e v e r t h e l e s s r i s k y s h i f t and p o l a r i z a t i o n a r e reduced c o n s i d e r a b l y by a p p l y i n g t h e a p p r o p r i a t e decision rule. Hypothesis 2 i s s upp o r t e d r a t h e r t h a n h y p o t h e s i s 1. T h i s corresponds w e l l w i t h Experiment I , where t h e r e was no s y s t e m a t i c d e v i a t i o n f rom a p p r o p r i a t e d e c i s i o n r u l e a l t h o u g h t h e d i s t r i b u t i o n o f e s t i m a t i o n s w i t h i n groups tended t o be f a i r l y skewed.

250

5.1

SOCIAL INTERACTION

Swnmary and conclusions f o r Experiments I t o 111

I n a l l t h r e e experiments ( i n t o t a l 178 systematically varied groups) we found s i g n i f i c a n t p o l a r i z a t i o n from the a r i t h m e t i c mean i n d i r e c t i o n o f t h e median. I n experiment I1 and 111 r i s k y s h i f t from t h e a r i t h m e t i c mean was s i g n i f i c a n t as well. I n experiment I Ss had t o estimate a number o f p o i n t s on a s l i d e b r i e f l y exposed t o them. Then they had t o engage i n a c o l l e c t i v e estimation. Since s i n g l e peakedness o f i n d i v i d u a l preference f u n c t i o n s appears t o be a n a t u r a l property o f the task

( i n t h e sense o f BLACK),the median i s t h e appropriate primary s o c i a l decision r u l e . Results show t h a t when depending on the motivational conditions, the two v a r i a n t s o f t h e median model are applied t o the data, p o l a r i z a t i o n vanishes almost complete1Y.

I n experiment 11, i n which Ss had t o decide c o l l e c t i v e l y upon bets o f equal expected value, the median d i d n o t reduce r i s k y s h i f t and p o l a r i z a t i o n t o an i n s i g n i f i c a n t l e v e l . Even though 47 o f 58 decisions were on the ( o r one o f the) mPdian values, the t e s t i n g procedure revealed s i g n i f i c a n t (negative) p o l a r i z a t i o n and r i s k y s h i f t . Regarding t o f o r g o i n g results,experiment I11 was designed t o t e s t w i t h a s i m i l a r task the f o l l o w i n g a l t e r n a t i v e hypotheses:

1.

The assumption t h a t only rankorders are important f o r group decisions (not distances) i s wrong. W.O.W. the assumption underlying t h e BLACK (median) model i s wrong, w i t h regard t o the actual behavior o f i n d i viduals i n a group decision process.

2.

The basic assumption o f the Black model i s c o r r e c t , b u t the formulation o f the decision has t o take i n t o account v i o l a t i o n s o f the s i n g l e peakedness condition. The two decision r u l e s defined i n a d d i t i o n t o t h e median were F u l l Paired Comparison and Reduced Paired Comparison.

Other decision r u l e s l i k e P r o p o r t i o n a l i t y , E q u i p r o b a b i l i t y , and P l u r a l i t y were a l s o tested. The r e s u l t s are n o t reported here, however, since t h e i r p r e d i c t i o n s are l e s s r e l i a b l e than any o f the o t h e r t h r e e decision r u l e s (Median, F u l l Paired Comparison, Reduced Paired Comparison). Moreover, they are sometimes even l e s s precise i n p r e d i c t i o n than the A r i t h m e t i c Mean.

BIASES I N GROUP DECISIONS

251

A p p l i c a t i o n of the decision r u l e s F u l l and Reduced Paired Comparison o f t e n leads t o more r i s k y p r e d i c t i o n s f o r the group decision, i n d i c a t i n g t h a t the v i o l a t i o n s o f the s i n g l e peakedness c o n d i t i o n do n o t wash o u t i n the average regarding r i s k y and conservative s h i f t s . This supports the hypothesis t h a t the observed r i s k y s h i f t t h a t goes beyond t h e median model i s due t o v i o l a t i o n s o f the s i n g l e peakedness condition. The r e s u l t s i n d i c a t e t h a t the basic ideas o f m a j o r i t y v o t i n g over a l t e r n a t i v e s accounts f o r most of the biases c a l l e d risky s h i f t o r poZurizution. We do n o t claim t h a t genuine r i s k y / c a u t i o u s s h i f t and p o l a r i z a t i o n are none x i s t e n t . Nevertheless we i n d i c a t e t h a t a more c a r e f u l analysis o f the c o l l e c t i v e decision processes should precede a s h i f t o r p o l a r i z a t i o n analys i s i n arder t o be able t o t e s t the existence o f t r u e s h i f t / p o l a r i z a b i o n effects. References Arrow, K.J.:

1951. Social choice and i n d i v i d u a l values. New York: Wiley

Black, D.: 1948. On the r a t i o n a l e o f group decision making. I n : Journal o f P o l i t i c a l Economy 56, 23-34 1954. Social choice and strength o f preference. I n : R.N. T h r a l l / Coombs, C.H.: C.H. Coombs & R.L. Davis (eds.), Decision processes. New York, 69-86 C r o t t , H.W.: 1964. Experimentelle Untersuchung eines Modells f u r das Gruppenentscheidungsverhalten i n Verhandlungssituationen. Zulassungsarbeit am Psychologischen I n s t i t u t der Uni v e r s i f a t Hamburg C r o t t , H.W. : 1965. Gruppenentscheidungen unter verschiedenen Motivationsbedingungen. I n : Archiv f u r d i e gesamte Psychologie 117, 113-130 Crott, H.W./Schermer, Th./Knoll, C. & Thoene, B. : Experimental analysis o f s o c i a l d e c i s i o n schemes and r i s k y s h i f t i n group decisions among bets. I n prep. Crott, H.W./Schermer, Th. & Zuber, J.A.: Experimental analysts o f s o c i a l decision schemes i n various group decision tasks. I n prep. Davis, J.H.: 1973. Group decision and s o c i a l i n t e r a c t i o n . A theory o f s o c i a l decision schemes. I n : Psychological Review 80, 97-125 Hoffman, P.J./Festinger, L. & Lawrence, D.R. : 1954. Tendencies toward group comparability i n competitive game behavior. I n : Human Relations 7,

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141-159 Lamm, H. & Myers, D.G.: 1978. Group-induced p o l a r i z a t i o n o f a t t i t u d e s and behavior. I n : L. Berkowitz (ed.), Advances i n experimental s o c i a l psychology, Vol. X I . New York: Academic Press Moscovici, S. & Zavalloni, M.: 1969. The group as a p o l a r i z e r o f a t t i t u d e s . I n : Journal o f Personality and Social Psychology 12, 125-135 Myers, D.G.: 1982. P o l a r i z i n g e f f e c t s o f s o c i a l i n t e r a c t i o n . I n : H. Brands t a t t e r / J . Davis & G. Stocker-Kreichgauer, Group decision making. London: Academic Press Stoner, J.A.F.: 1961. A comparison o f i n d i v i d u a l and group decision i n c l u d i n g r i s k . Unpublished diploma paper, School o f I n d u s t r i a l Management, Mi1.T.

DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

253

THE DEPENDENCE OF COGNITION AND BEHAVIOR IN 2 x 2 MATRICE GAMES

ULRICH SCHULZ

THE0 MAY

Universitat Bielefeld Abteilung fur Angewandte und Experimentelle Psychologie F. R. Germany

In small group research experimental games are studied which have the following simple characteristics: The game i s played by two subjects. Each subject has two alternative choices ( s t r a t e g i e s ) . The game i s played without bargaining o r communication. Therefore each player's decision i s made independently from the other subject. The outcome of the game consists i n a pair of choices ( s t r a t e g i e s ) . Each outcome has a u t i l i t y f o r each subject, usually measured in money (payoff). In most experiments the payoff is displayed i n matrix form:

strategy I

strategy I payoff of S1 f o r outcome (1,I)

strategy I 1 payoff of S1 for outcome (1,II)

subject 1 strategy I 1 payoff of S1 f o r outcome (I1,I)

payoff of S1 f o r outcome (I1,II)

SOClA L INTERACTION

254

Both subjects know t h e i r own and t h e i r p a r t n e r ' s payoff. I n c o n f l i c t games there e x i s t s no s t r i c t competition between the subjects as the p r o f i t t o one subject causes no l o s s f o r t h e other. The p a y o f f s t r u c t u r e i s designed i n such a way t h a t choices, based o n l y on t h e own u t i l i t y , r e s u l t w i t h c e r t a i n t y i n low payoff decisions. I f the o t h e r ' s p a y o f f i s also taken i n t o consideration, t h i s may r e s u l t i n uncertain outcomes: It i s possible t h a t reasonable payoffs f o r both subjects can be reached b u t a low p a y o f f i s a l s o possible. I n such c o n f l i c t s the p r i n c i p l e o f i n d i v i d u a l r a t i o n a l i t y no longer provides reasonable s o l u t i o n s . An.

RAPOPORT argued t h a t the study o f gambling behavior might provide new i n d i c a t i o n s leading t o a s o l u t i o n concept f o r these games. 1.

Prisoner's Dilemma Game and Chicken Came

Two-person games are considered i n which each person i (i= 1,2) has two pure s t r a t e g i e s c a l l e d C and 0. The game i s played w i t h o u t comnunication. Player i ' s payoffs are denoted by ai (X,Y), where X and Y symbolize the respective choices o f the player and o f h i s opponent. P r i s o n e r ' s Dilemma Games (PDG) are defined,

i f we only consider r e s t r i c t e d and symnetric games by the i n e q u a l i t i e s : al

(l)

(D,C)

> al

(C,C) > al

(D,D) >al (C,D)

2al (c,c) > al (D,c) t al (c,D) a2 (X,X) = al (X,X) f o r X = C,D a2 (X,Y) = al

(Y,X) f o r X 4 Y

X,Y

= C,D

The symmetric and r e s t r i c t e d Chicken Game (CG) i s defined by the i n e q u a l i t y

I n the CG the order o f the outcomes D,D and C,D are simply reserved. A l l other d e f i n i t i o n s remain the same. Both games belong t o the class o f c o n f l i c t games. C o n f l i c t arises i n these games from the f a c t t h a t a t l e a s t the p a y o f f o f one p l a y e r f o r e q u i l i b r i u m p o i n t s i s lower than the payoff f o r the s t r a t e g y combination C,C.

2 55

MATRICE GAMES

2.

Psychological Research in Experimental Matrice Games

Psychological researches have mainly focused on the t o p i c s o f c o n f l i c t r e s o l u t i o n , s o c i a l motivation, and i n f o r m a t i o n processing i n i t e r a t e d games (e.g. KRIVOHLAVY, 1974). Beside studying the p l a y e r ' s choice behavior, psychologists soon became very i n t e r e s t e d i n the s u b j e c t ' s p r e d i c t i o n o f the other p l a y e r ' s behavior and the accuracy o f t h i s p r e d i c t i o n . The main r e s u l t s o f t h i s research can be summarized as f o l l o w s : Additional r e g i s t r a t i o n of! P r e d i c t i o n does n o t i n f l u e n c e decnsion and gaming behavior (TEDESCHI e t a l . ,

1968; HALPIN & PILISUK, 1970; SCHULZ & MAY, 1982).

Changes i n p r e d i c t i o n s i n the course o f the gaming sequence show t h a t players process information about the other p l a y e r ' s gaming behavior. (SERMAT, 1964; BIXENSTINE & BLUNDELL, 1966; PYLYSHYN e t a l . , 1966; TEDESCHI e t a1 ., 1968; HALPIN & PILISUK, 1970; REVENSTORFF e t al., 1977; SCHULZ & HESSE, 1978; BLUDSZUWEIT & KAUKE, 1976; SCHULZ & JONAS, 1982). There i s a minor group o f players whose p r e d i c t i o n s become i n c r e a s i n g l y more accurate i n the course of gaming. These subjects are c a l l e d r e a l i s t s . The m a j o r i t y o f subjects i s n o t r e a l i s t i c . (PYLYSHYN e t a l . ,

1966; TEDESCHI e t a l . ,

1968; BLUDSZUWEIT & KAUKE, 1976; BARTH, 1979; FRINDTE, 1979). The combination o f p r e d i c t i o n and s t r a t e g y choice i n each game o f t h e sequence may be i n t e r p r e t e d as a m o t i v a t i o n a l s t a t e (e.g. t r u s t , r i s k e t c . ) . (LOOMIS, 19kj9; DEUTSCH, 1960; SOLOMON, 1960; TERHUNE, 1968; REVENSTORFF e t a1 1977; SCHULZ & HESSE, 1978).

.,

The s u b j e c t ' s information processing causes changes i n the frequencies o f the motivational states i n the gaming sequence. (HALPIN & PILISUK, 1970; SCHULZ & HESSE, 1978).

The m a j o r i t y of r e a l i s t s have a noncooperative motivation. They take t h e i r p r e d i c t i o n s i n t o consideration and t r y t o maximize t h e i r u t i l i t y . (BLUDSZUWEIT & KAUKE, 1976; FRINDTE, 1979; BARTH, 1979). , Because o f these r e s u l t s , and as an attempt t o e x p l a i n the phenomenon o f realism, SCHULZ & MAY (1982) performed PDG-experiments where subjects had t o i n d i c a t e t h e i r p r e d i c t i o n , t h e i r s u b j e c t i v e c e r t a i n t y of the p r e d i c t i o n , and t h e i r choice. The authors argued t h a t for r e a l i s t s , s u b j e c t i v e c e r t a i n t y o f p r e d i c t i o n should h i g h l y c o r r e l a t e w i t h c o r r e c t p r e d i c t i o n s . I n t h e experiments the s u b j e c t ' s s u b j e c t i v e c e r t a i n t y was i n d i c a t e d by b i n a r y forced choice

SOCIAL INTERACTION

256

between the two s t a t e s " c e r t a i n " and "uncertain".

3.

A Model for Choice Behavior, Prediction and Subjective Certainty

SCHULZ & MAY (1982) proposed a model f o r the choice, the p r e d i c t i o n and i t ' s c e r t a i n t y , i n sequences o f

PDG which i s presented here along w i t h a more

s p e c i f i e d psychological i n t e r p r e t a t i o n o f some processes. The subject's process w i t h regard t o the other p l a y e r ' s choice i s the e s t i mation process f o r the unknown p r o b a b i l i t y

IT

t h a t the o t h e r w i l l cooperate.

General experience and informati on gained during the preceding games speci f i es some p r o b a b i l i t y d i s t r i b u t i o n o f the unknown parameter

'TI

as i n d i c a t e d i n

f i g u r e 1. ( S i m i l a r models can be found as p r i o r d i s t r i b u t i o n i n Baysian s t a t i s t i c s (e.g. SCHAEFER, 1976; WINKLER, 1967) .)

I

I I

I

I

01

D'S

D'U

I

C'U

C'S

; > x 1

Fig. 1

Ifwe denote the s u b j e c t ' s p r e d i c t i o n s by C' and D ' and the states "certai.n" by S and "uncertain" by U t h e p a i r i n g s of p r e d i c t i o n and c e r t a i n t y can be i n t e r p r e t e d as l a b e l s f o r ordered i n t e r v a l s on the a x i s o f the parameter TI as i n d i c a t e d i n f i g u r e 1. I f the subject has t o p r e d i c t the others choice, and i t ' s c e r t a i n t y , he/she

2 57

MATRICE GAMES

i s sampling according t o the p r o b a b i l i t y d i s t r i b u t i o n f ( a ) a value o f

a'.

The subject looks f o r the i n t e r v a l which contains a * and r e p o r t s the l a b e l o f t h i s i n t e r v a l . I n t e r v a l bounderies are considered t o be f i x e d . Observable p r o b a b i l i t i e s o f the i n t e r v a l s can be calculated by summation o r i n t e g r a t i o n . The probabl i1ity d i s t r i b u t i o n i s considered t o vary during the sequence o f the games. The p r o b a b i l i t i e s o f the respective p r e d i c t i o n - c e r t a i n t y s t a t e d c o n d i t i o n a l on the gambling h i s t o r y o f the process are denoted by S,n, E,,'~,.,, E ~ E ~ ~ ( n ~= ~ number ~, of, ~ t r i ~a l s~ ) . T h e o r e t i c a l l y these p r o b a b i l i t i e s can be c a l c u l a t e d by i n t e g r a t i o n o f the E

l

actual d i s t r i b u t i o n f ( a ) . The model assumes t h a t f changes i n such a way t h a t observable p r o b a b i l i t i e s E are changed by l e a r n i n g operators: denote by

(3a)

'X

EX,ntl = gX,X(EX,n)

l/ax 'X,n

= l'aX

'X X = C I S , C'U, D'U, ax > 0.

D'S;

EY,ntl = gX,y(Ey,n)

=

'X,n

t

1-

E

X,n

'Y,n

-

EX,n l% 1 X #

Y and

X,Y

= CIS, C'U,

EXyn

D'U, D ' S .

These two equations specify the l e a r n i n g operators a p p l i e d t o the p r o b a b i l i t i e s E.

Figure 2 shows the graph o f the function g x Y x . These operators are simple

one paremetric noncommutative operators. The p l a y e r ' s p r e d i c t i o n , and h i s p a r t n e r ' s choice i n t r i a l n o f the game sequence are i n d i c a t e d i n the f i r s t and second column of t a b l e 1. The t h i r d column i n d i c a t e s which p r o b a b i l i t y w i l l be transformed a f t e r observ a t i o n o f the specified prediction-choice combination of column 1 and 2 by the operator g x Y x . The l a s t column s p e c i f i e s the parameter used i n gx,x. A l l other p r o b a b i l i t i e s E are transformed by gxYy u s i n g the appropriate Y,n

SOClA L lNTERACTlON

258

1.0

0.1

@.I

0.3

0.6

0.5

0 1

1.1

0.1

0.9

1.0

Fig. 2

Table 1 contains the reinforcement schedule of the model : Table 1 (Reinforcement schedule) prediction i n trial n

choice by alter i n trial n

X for which !1earni ng E ~ , ~= g + X~, X ( E n j parameter

C'

C

C'S

aC'S

C'

D

C'U

aC'U

D'

C

D'U

aD'U

D'

D

D'S

"D'S

parameter a . All the information processing assumptions of the model are now specified by equations. For each player we introduced four s t a r t i n g probabi 1i t i e s X = C I S , C'U, D'U, D ' S , three of which have t o be estimated. Further E X,1' parameters are the four learning parameters of table 1, which also have t o be estimated.

MATRICE GAMES

259

The model assumes t h a t i n each game the f i r s t stage i n each p l a y e r ' s d e c i s i o n and choice process i s described sampling o f a p r e d i c t i o n - c e r t a i n t y s t a t e . The model f u r t h e r assumes t h a t given the p r e d i c t i o n - c e r t a i n t y state, players have f i x e d propensities t o choose between C and D. These propensities are independent of the h i s t o r y of the gambling process and o f the number of t r i a l s . They are measured by f o u r c o n d i t i o n a l p r o b a b i l i t i e s :

The simple n o t a t i o n i s j u s t i f i e d by the above mentioned independence assumpt i o n s . The propensities 5 are f o u r additi'onal parameters introduced f o r each player, so t h a t each p l a y e r i s characterized by 11 unknown parameters.

4.

VaZidation of the Mode2 for PDG and CG

aI

Exprimen t s

Experiments were performed w i t h 3 PDG matrices and 3 CG matrices. SCHULZ & MAY (1982) describes the experimental procedure i n d e t a i l . Subjects were male students, b u t n o t psychology students. Subjects were payed f o r t h e i r p a r t i c i p a t i o n . Their renumeration depended on the r e s u l t s of the experimental games. Table 2 indicates the matrices used, the number of t r i a l s i n the sequence and the number o f players.

SOCIAL INTERACTION

260

Table 2 matrix

Code o f game

number o f t r i a1 s

number o f subjects

45

30

50

20

CG2

50

18

PDGl

50

20

PDG2

50

22

PDGO

45

30

CGO

CG 1

bl

c

I

4

I

D

I

9

1

C

l

-21 -8 J

D

l

Parameter Estimation and Model Control

I n xperimental games, research i n t o d e t a i l e d processes o n l y seems t o successful i f in t e r i ndi v i dual differences (due t o d i f f e r e n t motives, d i f f e r e n t experiences and d i f f e r e n t i n f o r m a t i o n processing) are taken i n t o consideration. I n the absence o f b e t t e r methods, the parameters o f i n d i v i d u a l subjects are estimated by Maximum-Li kelihood-Methods using data provided by i n d i v i d u a l p a i r s o f players. I t should be mentioned t h a t the parameters 5 are estimated by c o n d i t i o n a l r e l a t i v e frequencies ( f o r more d e t a i l e d i n f o r m a t i o n see SCHULZ & MAY, 1982). Model t e s t s are a l s o performed f o r i n d i v i d u a l p a i r s o f players. Indices, such as frequency of b i l a t e r a l cooperation i n the f i r s t o r second h a l f o f the

MATRICE GAMES

26 1

sequence o f games, can be computed f r o m e m p i r i c a l data. By s i m u l a t i o n o f t h e model u s i n g t h e e s t i m a t e d parameters i t i s p o s s i b l e t o decide whether t h e observed index v a l u e s a r e e x t r e m e a n d u n l i k e l y i f t h e model i s v a l i d ( a = 0.05). I f t h e i n d i c e s a r e extreme and u n l i k e l y t h e model i s n o t a b l e t o d e s c r i b e t hese aspects o f e m p i r i c a l d a t a . ( F o r more d e t a i l s o f t h e model t e s t s see SCHULZ & MAY, 1982).

As i n SCHULZ & MAY (1982) 48 i n d i c e s p e r p a i r o f p l a y e r s were t e s t e d u s i n g a = 0.05 p e r t e s t . I f t h e model i s v a l i d , two o r t h r e e v i o l a t i o n s

can be expected. T he r e f o r e we decided t o c o n s i d e r t h a t more t han t h r e e model v i o l a t i o n s produce s e r i o u s descrepancy between d a t a and model p r e d i c t i o n . Table 3 shows t h e number o f s e r i o u s model v i o l a t i o n s f o r t h e s i x games.

Table 3

1Game

I

Model r e j e c t i o n s CGO

number o f pairs with 0 model r e j e c t .

CG1

CG2

PDGl

PDG2

1

0

3

1

PDGO 0

F o r PDG, t h e model was r e j e c t e d i n 11 % o f a l l cases. F o r CG, t h e model was r e j e c t e d i n 3 % o f t h e cases. We conclude t h a t t h e model can be used f o r t h e d e s c r i p t i o n o f d e c i s i o n and choice b ehav ior i n s e q u e n t i a l PDG and CG. ( D e t a i l e d r e s u l t s of t h e model t e s t s f o r PDG a r e r e p o r t e d i n SCHULZ & MAY, 1982 , f o r CG i n SCHULZ & MAY, 1983.) The p a i r s of p l a y e r s f o r which t h e model was r e j e c t e d were excluded from f u r t h e r a n a l y s i s .

5.

The InfZuence o f Certainty on Choices

The model s t a t e d an i n f l u e n c e o f s u b j e c t i v e c e r t a i n t y on c h o i c e behavior. E a r l i e r i n v e s t i g a t i o n s of c h o i c e b e h a v i o r i n e x p eriment al games have shown t h a t s imp le models i n v o l v i n g o n l y c h o i c e and p r e d i c t i o n can be used t o des c r i b e t h e main f e a t u r e s o f d e c i s i o n and c h o i c e processes i n PDG (SCHULZ & JONAS, 1982; MAY, 1982). Th e r e f o r e t h e q u e s t i o n a r i s e s as t o whether c e r t a i n t y of p r e d i c t i o n i s necessary f o r t h e d e s c r i p t i o n o f d e c i s i o n and c h o i c e be-

262

SOCIAL INTERACTION

havior. This question can be answered by examining the submodel which i s characterized by

As described i n SCHULZ & MAY (1982)

, the

support o f the submodel by data

r e l a t i v e t o the support o f the general model can be demonstrated by consider i n g the differences o f the doubled l o g - l i kelihood-values o f the submodel and the general model. Because we know nothing about the d i s t r i b u t i o n o f the l i k e l i h o o d r a t i o s we cannot perform the usual chi-square t e s t . Using a c r i t i c a l difference of 9.5 for the double o f l o g - l i k e l i h o o d d i f f e r e n c e s (corresponding t o a = 0.05 i n a chi-squared d i s t r i b u t i o n w i t h 4 degrees o f freedom), the general model was d i s t i n c t l y b e t t e r supported by experimental data than the submodel ( i n 24 % o f the CG-cases and i n 13 % o f PDG-cases). More d e t a i l e d i n s p e c t i o n shows t h a t the i n f l u e n c e o f c e r t a i n t y may depend on the payoff m a t r i x . I n 40 % o f the CGO cases the general model was d i s t i n c t l y b e t t e r supported. Our conclusion i s t h a t t h e submodel should n o t be used f o r CG.

6.

InfZuence of Payoff Matrix on Model Parameters

Descriptive comparison o f behavior i n PDG and CG has been reported by several authors (RAPOPORT & CHAMMAH, 1969; SERMAT, 1967; SCHLENKER & GOLDMAN, 1978). The general r e s u l t i s t h a t the d i f f e r e n t character o f the c o n f l i c t s i n CG and PDG leads t o higher cooperation i n CG. The next step, i n which behavior and experimental games i s analysed by mathemtical models , consists of an attempt t o i n v e s t i g a t e the r e l a t i o n s between model parameters and s i t u a t i o n a l parameters , p a r t i c u l a r l y the p a y o f f m a t r i x o f the game. Our attempt concentrates on t h e parameters 5 . Theoretical and experimental work i n experimental games (e.g. DEUTSCH, 1960; DEUTSCH, 1973; McCLINTOCK, 1972; McCRIMMON & MESSICK, 1976; KELLEY & THIBAULT, 1978) has shown t h a t s o c i a l motives h e a v i l y i n f l u e n c e t h e s u b j e c t ' s e v a l u a t i o n

MATRICE GAMES

263

o f t h e p a y o f f m a t r i x . The m a j o r i t y o f these a u t h ors p o s t u l a t e t h a t t h e subj e c t i v e u t i l i t y o f t h e game i s some s i m p l e a l g e b r a i c f u n c t i o n o f t h e p a y o f f m a t r i x , such as t h e sum o r t h e d i f f e r e n c e o f t h e p a y o f f s . T h i s f u n c t i o n depends on t h e s u b j e c t ' s s o c i a l m o t i v a t i o n . Th i s s u b j e c t i v e t r a n s f o r m a t i o n may l e a d t o a c o n f l i c t s t r u c t u r e o f t h e s u b j e c t i v e u t i l i t i e s t h a t d i f f e r s f rom t h e p r e v i o u s p a y o f f m a t r i x . Th e r e f o r e (when c o n s i d e r i n3 d i f f e p e e t m a t t v a t t o o r ) i t may be d i f f i c u l t t o d e r i v e g e n e r a l r e s u l t s f o r d i f f e r e n t games w i t h o u t

knowledge o f t h e s u b j e c t ' s m o t i v a t i o n a l t e n d e n c i es. Consequently experiments w i t h s p e c i a l p a y o f f m a t r i c e s were designed i n o r d e r t o demonstrate some e f f e c t s o f t h e p a y o f f m a t r i x on t h e p r o p e n s i t i e s o f s u b j e c t s which would be v a l i d under f a i r l y weak assumptions about t h e subj e c t i v e u t i l i t y s t r u c t u r e o f t h e payoff matrices. F or some s t r a t e g y c o m b i n a t i o n of b o t h p l a y e r s Z ( Z = CC, CD, DC, DD),

let

xz denote t h e p l a y e r ' s own p a y o f f and yz t h e p a y o f f o f h i s p a r t n e r . Then (acco rd ing t o LUCE & SUPPES, 1965) we may assume t h e e x i s t e n c e o f a subj e c t i v e s t r e n g t h f o r each p a y o f f c o m b i n a t i o n (x,y) denoted by v(x,y I t i s assumed t h a t s t r e n g t h i s some s t r i c t m o n o t o n i c a l l y i n c r e a s i n g t r a n s f o r m a t i o n o f u t i l i t y . The main p r o p e r t y of s t r e n g t h i s i t ' s p o s i t i v t Y > 0. We assume t h r e e f u r t h e r p o r p e r t i e s o f s t r e n g t h :

v(x,y)

Translation invariance o f ratios: v(x

V (X , Y )

f

d, y

f

d)

Monotoni c i t y f o r equal p a y o f f s : (6)

v(x,x)

> V(Y,Y) f o r x > Y

M o n o t o n i c i t y on c o n s t a n t p a y o f f f o r t h e o t h e r :

(7)

v(x,z)

> v(y,z)

f o r z >.x > y

The n e x t s t e p i s t o f i n d a r e l a t i o n between t h e p r o b a b i l i t i e s 5 and t h e corre s ponding s t r e n g t h s . Going back t o f i g u r e 1 we assume t h a t each i n t e r v a l has a c h a r a c t e r i s t i c v a l u e o f s u b j e c t i v e p r o b a b i l i t y rZ, Z = C I S , C'U, D'U, D'S (e.g. t h e midpoints f o r the i n t e r v a l ) .

204

SOCIAL INTERACTION

'CIS

2

2 'D'U

'c1u

f

? 'D'S

In)

A

,

W

I

,.

I

v

h

"

I

I

"ctl

"D'U

"0's

n

I

"cis

1

I

>*

Fig. 3 We f u r t h e r assume t h a t t h e p r o b a b i l i t i e s of c h o i ces, c o n d i t i o n a l on t h e p r e d i c t i o n - u n c e r t a i n t y s t a t e s , a r e i n f l u e n c e d by t h e c h a r a c t e r i s t i c values nz a c c ord ing t o t h e p r i n c i p l e of s t r i c t expect ed u t i l i t y i n t r o d u c e d by LUCE & SUPPES (1965).

"Z v(xCCs YCC)

sz

' (l-*Z)

V(X('DY YCD) (8)

=

'Z(v(xCCs YCC)

' v(xDCY YDC) ' (I-'Z)

(v(xCD, YCD)

' v(xDD*

YDD))

f o r Z = C I S , C'U, D ' U , D ' S Through t h e assumptions ( 5 ) - ( 7 ) we a r e now a b l e t o d e r i v e some s p e c i a l p r e d i c t i o n s about t h e parameters cz f o r t h e m a t r i c e s d i s p l a y e d i n t a b l e 4. Table 4

CGl

um

I8

10

1.1

.ll

1.97)

1.751

1.0 .I5 I.@)11.0)

1.0 1.891

.Sl

.I6

.I1

LI.01

11.01

11.01

."*.',.,,,,' 11.01

11.01

1.1311.11,

MATRICE GAMES

6.1

265

Comparison of CGO and PDGO

cZ

With assumption (8) parameters

are i n v a r i a n t under t r a n s l a t i o n o f payoffs.

Therefore CGO and the CG have the same propensities w i t h the f o l l o w i n g 2 matrix: C D

For the outcomes, C,C; C,D;

D,C t h i s m a t r i x has the same payoffs as PDGO. Applyi n g assumptions ( 5 ) t o the strength o f outcome DD o f t h i s m a t r i x we conclude:

-> S z , p ~ G o f o r z

eZ,CGO

=

CIS, C ’ U , D ’ U , D ’ S

Since ncIs 1 and n c l u >> 0 i s a p l a u s i b l e a d d i t i o n a l assumption, we may d i f f e r e n t i a t e f u r t h e r : the d i f f e r e n c e between ccs , cGo and ccs , PDGO ~

S

~

‘ ccs,pDGO ~ ,

The d i f f e r e n c e between

~

scu, cGo

~ and

scu,

~ should be moderate b u t the

t o t a l i n f l u e n c e o f the outcome DO w i l l be observed f o r the o t h e r two because IT^ i s then small, almost 0. Therefore we expect

czy

’ %IU,PDGO

%u,

CGO

‘DS,

CGO >

‘DS, PDGO

.

I f we take independent samples o f players from the same population f o r CGO we p r e d i c t s i g n i f i c a n t differences i n the c e n t r a l tendencies o f cDu and sDs f o r both games, and a t most, s l i g h t d i f f e r e n c e s f o r scs between these two games. Mann-Withney-tests f o r the two variables show s i g n i f i c a n t differences only f o r cDu and cDsy and no s i g n i f i c a n t d i f f e rence f o r cCs ( t h e t e s t s were performed on the 5%-ilevel).

6.2

Comparison of CGI, CG2, PDGI, PDGI

CG1 and CG2 d i f f e r o n l y i n the payoffs o f the DD outcome. According t o ( 5 ) and v a r i a t i o n

‘Z,CGl

nZ

‘Z,CG2

we conclude Z = C’S,...,

D’S

266

SOCIAL INTERACTION

‘CIS,

CG1

2

‘CIS,

CG2’ ‘D’U,

CG1 > ‘D’U,

CG2

‘D’S, CG1 > ‘D’S, CG2 Using the same arguments f o r P D G l and PDG2 produces

sZ, PDGl 2 ‘Z, PDG 1

‘C’S,

Z = CIS,

PDG2 ‘C’S,

PDG 2

and

,

.,D’S

SDIu,

PDG 1 > EDIU, PDG 2

‘D’S,

PDG 1 > ‘D’U,

PDG 2

Relations between CG2 and P D G l can o n l y be established f o r small values o f nzy e s p e c i a l l y f o r nDIU and vDlS near 0. I n t h i s case sDIu and sDlS assumptions ( 7 ) and ( 6 ) i t follows: v(-2,9)

v(-2,-2)

> v(-3,9) > v(-3,-3)

We therefore conclude ‘D’U, ‘D‘S

CG2 > ‘D’U,PDGl

,CG2

> ‘D’S

,P D G ~

Ifwe combine a l l separate hypotheses we expect no s i g n i f i c a n t d i f f e r e n c e s Summabetween CG1 and CG2, o r between P D G l and PDG2, i n the v a r i a b l e r i z i n g the i n e q u a l i t i e s f o r the parameters

cDIu

we expect decreasing c e n t r a l

tendencies going from CG1, over CG2 and PDG1, t o PDG2. The same hypotheses can be s t a t e d f o r the v a r i a b l e sDIs. Median values o f the prob a b i l i t i e s are depicted i n f i g u r e 5. The l a s t two hypotheses were t e s t e d by the Kruskal-Wallis-test. There were s i g n i f i c a n t differences i n the variables sDls and sDlu ( a = 0.05). Figure 4 a l s o demonstrates the monotonic decreasing tendency. There was no s i g n i f i c a n t difference between CG1 and CG2 i n the v a r i a b l e

sclsy

b u t there was a s i g n i f i c a n t difference i n t h i s v a r i a b l e between PDGl and PDG2 ( a = 0.05).

267

MATRICE GAMES

7.

Discussion

P r e d i c t i o n o f the o t h e r ' s choice and i t s c e r t a i n t y can be i n t e g r a t e d i n a model, f o r decision and choice behavior i n PDG and CG. The model has two somewhat independent parts: i n f o r m a t i o n processing t h a t o n l y influences p r e d i c t i o n s and c e r t a i n t y , and the constant s t r a t e g i c propensities. The model can be used f o r PDG and CG because o n l y a few model r e j e c t i o n s can be stated. The analysis o f the submodel shows t h a t s u b j e c t i v e u n c e r t a i n t y has t o be considered a t l e a s t i n CG. To resolve t h i s question w i t h i n the system o f c l a s s i c a l s t a t i s t i c s more data should be c o l l e c t e d . The i n t e r p r e t a t i o n o f the f o u r p r e d i c t i o n - c e r t a i n t y - s t a t e s as being ordered i n t e r v a l s w i t h c h a r a c t e r i s t i c s u b j e c t i v e p r o b a b i l i t i e s can be used f o r the d e r i v a t i o n o f p r e d i c t i o n s f o r the propensities 5 , i f weak conditions are f u l f i l l e d by the subjective strength o f game matrices, and i f the model o f s t r i c t expected u t i l i t y i s v a l i d . Though t h i s t h e o r e t i c a l d e r i v a t i o n may n o t be the o n l y explanation o f the r e l a t i o n s between the parameters, the proposed i n t e r p r e t a t i o n o f the decision and choice process i n terms o f t r a d i t i o n a l theory i s able t o show t h a t using f a i r l y weak assumptions, i n t e r e s t i n g aspects o f the parameters can be predicted. Perhaps i t would be possible t o formalize these assumptions and demonstrate t h e i r v a l i d i t y i n more detai l e d experiments. Fig. 4

l

a CGO

I

PDGO

>

268

SOCIAL INTERACTION

Fig. 5

I ; CG1

I

CG2

'

POGl

>

\. PDG2

References:

Barth, W . : 1979. Faktorenstruktur verschiedener Nichtnullsummenspiele unter besonderer Berucksichtigung des PDG. In: Wiss. Ztschr. FriedrichSchiller-Univ. Jena, Ges.- und Sprachwiss., 28, 817-823. Bixenstine, V.E. & Blundell, H.: 1966. Control of choice exerted by structural factors in two-person, non-zero-sum games. In: Journal of Conflict Resolution, 4, 478-487. Bludszuweit, H. & Kauke, M.: 1976. Kooperatives Verhalten und realistische Partnererwartungen im Spielexperiment. In: Zei t s c h r i f t fur Psychologie, 184, 157-170. Deutsch, M . : 1960. The e f f e c t of motivational orientation upon trust and suspicion. In: Human Relations, 13, 123-140. Deutsch, M.: Haven.

1973. The resolution of conflict. Yale University Press, New

Frindte, F.: 1979. Erfahrungen mit einem neuen experimentellen Spiel. In: Wiss. Ztschr. Friedrich-Schiller-Univ. Jena, Ges. u . Sprachwiss., 28, 6. Halpin, S.M. & Pilisuk, M.: 1970. Prediction and choice in prisoner's dilemma. In: Behavioral Science, 15, 141-153. Kelley, H.H. & Thibaut, J.W.: 1978.Interpersonal relations: A theory of i n t e r dependence. Wiley, New York.

MA TRICE GAMES

Krivohlavy, J . : 1974. Zwischenmenschliche Konflikte und experimentelle Spiele. Huber, Bern. Loomis, J.L.: 1959. Communication, the development of trust, and cooperat i v e behavior. In: Human Relations, 12, 305-315. Luce, R.D. & Suppes, P.: 1965. Preference, u t i l i t y and subjective probab i l i t y . In: Luce, R.D., Bush, R . R . & Galanter, E.(Eds.): Handbook of mathematical psychology, Vo. 111, Wiley, New York. May, T.: 1982. EinfluB der Situation auf das Verhalten i n experimentellen Konfliktspielen. Paper presented on the 24. Tagung experimentell arbeitender Psychologen, Trier. McClintock, C.G.: 1972. Social Motivation Behavioral Science, 17, 438-454.

-

a s e t of propositions. In:

McCrimmon, K.R. & Messick, D.M.: 1976. A framework f o r social motives. In: Behavioral Science, 21, 86-100. Pylyshyn, Z. Agnew, N. & Illingworth, J . : 1966. Comparison of individuals and pairs as participants i n a mixed-motive game. In: Journal of Conflict Resolution, 10, 211-220. Rapoport, AN. & Chammah, M.A.: 1965, Prisoner's Dilemma: a study of conf l i c t and cooperation. University of Michigan Press, Ann Arbor. Revenstorff, D., Wegscheider, R., Fitting, U. & Mai, N . : Markov models of gaming behavior in experimental nun-zero-sum games. In: Kempf, W.F. & Repp, B.H.: Mathematical models f o r the social psychology. Huber, Bern. Schaefer, R.E. : 1976. Probabilistische Infomationsverarbei t u n g . Huber, Bern. Schlenker, B.R. & Goldman, H.J.: 1978. Cooperators and Competitors in conflict. A t e s t of the "Triangle Model". In: Journal of Conflict Resolution, 22, 393-410. Schulz, U. & Hesse, W.: 1978. fin Modell fur das Verhalten in Prisoner'sDilemma-Spielen unter Berucksichtigung von Erwartungen. In: Psychologische Bei trage, 20, 551-570. Schulz, U. & Jonas, K.: 1982. Kooperation und Wettbewerb, Tit-for-TatStrategie und Kooperationserhohung auf dem H i ntergrund eines Model 1s fur das Spiel verhal ten in wiederhol ten Prisoner Is-Di lemma-spiel en. Schulz, U. & May, T . : 1982. Ein Modell f u r das Verhalten in Sequenzen von Konfliktspielen unter Berucksichtigung von Antizipationen und subjektiven Sicherheiten. Paper submitted to Zeitschrift fur Sozialpsychologie. Schulz, U. & May, T.: 1983. Verhalten in Sequenzen von "Chicken"-Spielen unter Berucksichti gung von Antizi pationen und subjekti ven Sicherhei ten. Paper presented on the 25. Tagung experimentel 1 arbei tender Psychologen, Hamburg.

269

270

SOCIAL INTERACTION

Sermat, V . : 1967. The e f f e c t o f an i n i t i a l cooperative o r competitive t r e a t ment upon a s u b j e c t ' s response t o c o n d i t i o n a l cooperation. I n : Behavior a l Science, 12, 301-313. Solomon, L.: 1960. The i n f l u e n c e o f some types o f power r e l a t i o n s h i p s and game s t r a t e g i e s upon the development o f interpersonal t r u s t . I n : Journal o f Abnormal and Social Psychology, 2, 223-230. Tedeschi , J.T. , Steele, M.W., Gahagan, J.P. & Aranoff, D.: 1968. I n t e n t i o n s , p r e d i c t i o n and patterns o f s t r a t e g y choice i n a P r i s o n e r ' s Dilemma Game. I n : Journal o f Social Psychology, 75, 199-207. Terhune, K.W.: 1968. Motives, s i t u a t i o n , and interpersonal c o n f l i c t w i t h i n Prisoner's Dilemma. I n : Journal o f Personality and Social Psychology, 8, 1-24. Winkler, R.L.: 1967. The q u a n t i f i c a t i o n o f judgement: Some experimental r e s u l t s . Proceedings o f the American S t a t i s t i c a l Association, 386-395.

DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor)

271

0 Elsevier Science Publishers B.V. (Nonh-Holland), 1983

ON THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

WULF ALBERS

GlSELA ALBERS

Universitat Bielefeld lnstitut fur Mathematische Wirtschaftsforschung F. R. Germany

Summary: This paper gives a f i r s t approach t o a theory o f t h e prominence o f numbers o f the decimal system. Basic components o f t h e theory are the l i m i t e d r a t i o n a l p r i n c i p l e s O f r u l e cons t r u c t i o n by i t e r a t e d a d d i t i o n o r s u b t r a c t i o n o f a given amount, t h e refinement o f a scale by adding the means o f any two neighbours, and i t s coarsening by o m i t t i n g a l l uneven elements. Applying these p r i n c i p l e s t o the powers o f 10 ( i . e . t h e most prominent numbers o f the decimal system) various scales w i t h d i f f e r e n t degrees o f exactness can be constructed. Empirical observations suggest t h a t among d i f f e r e n t scales w i t h about the same exactness generally t h a t one i s p r e f e r r e d which can be constructed i n the e a s i e r way, where the c o n s t r u c t i o n p r i n c i p l e s are those from above and the coarsening procedure i s assumed t o have the same complexity as the twof o l d refinement. This reduces the r e l e v a n t scales t o (z*a-lOn, z € Z ) w i t h o r .21 n E Zand aL(1,.5,.25

.

The prominence o f a number o r a s e t of numbers i s defined as the l e n g t h o f the steps o f the coarsest scale containing the number o r the set. (For empirical a p p l i c a t i o n s i t i s suggested t o permit

25 percent exceptions. )

Empirical r e s u l t s i n d i c a t e t h a t the prominence o f the s e t o f numbers r e s u l t i n g from a s p e c i f i c decision s i t u a t i o n i s i n many cases about one t e n t h o f the numbers i n question.

SOCIAL INTERACTION

272

More precisely the prominence (or exactness of s c a l e ) of a specific decision problem seems t o be selected according t o the rule that i t i s maximal subject t o the condition t h a t the range of reasonable alternatives contains a t l e a s t three prominent numbers. Results of n-person bargaining experiments (with real valued characteristic function games) suggest that the evaluation of outcomes (more exactly, of payoff differences) i s done via the underlying prominence, such that the distance of two outcomes i s evaluated by the number of prominence steps between them. By measureing incentives in t h i s way a general theory of the probability t h a t a "social" coalition (mainly an equal share coalition of a l l players) i s formed could be developed f o r games of apex type, by which the empirical observation that the probabilities can be quite different i f the same game i s played on scales with different prominence can be explained. This result suggests t h a t the prominence of the (spontaneously selected) scale can have basic importance f o r the analysis of individual or group decisions on numerical values. 1.

Prominent p o i n t s of the decimaZ system

The basic principles of constructing scales seem t o be the i t e r a t e d addition (and subtraction) o f a given amount x so t h a t a l l integer multiples of x are obtained ( b ) the refinement of a scale by adding the means of any two neighbours ( c ) the coarsening of a scale by omitting a l l uneven elements.

(a)

In the decimal system the main reference points are the powers lon ( n € Z ) , and according t o principle ( a ) each of these powers produces a scale (z-10n7 z € Z ) which can be refined or coarsened. The f i r s t refinement of such a scale according t o ( b ) gives the numbers (z-.5-lOn, z € Z ) , the second produces (z*.25*1OnYz E Z ) . These types of refinement seem t o be empirically relevant. - A t h i r d application of ( b ) gives (z..125*10n, zcZ) b u t we did not observe s e t s of empirical data ( w i t h the character o f decimals, not multiples of 1/8 ) in which the multiples of .125 were essentially more frequent t h a n other multiples of .025. I t seems t h a t

273

THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

t h e system

( z * . l * l O n y z E Z ) = (z.lOn-',

z€Z)

i s c l e a r l y preferred.

The c oars e ning p r i n c i p l e ( c ) g i v e s t h e systems ( ~ ~ 2 . 1z E 0 B~ ) ~= (zs.2 lont1, z E B ) which can be chosen as an a l t e r n a t i v e o f ( ~ - . 2 5 . 1 0 ~ + ~ ,

-

I t seems z E Z ) , s i n c e b o t h systems have a b o u t t h e same degree o f exactness. t h a t i n a f i x e d s i t u a t i o n each i n d i v i d u a l s e l e c t s one of t hese two scales. T hat

b o t h o f them have some e m p i r i c a l e v i d e n c e can be seen f rom t h e f a c t t h a t some c u r r e n c i e s have p iec e s o f v a l u e .02, .20, 2 , and 20 (as S w i t z e r l a n d ) , w h i l e The twoo t h e r s i n t r o d u c e d .25, 2.5, and 25 i n s t e a d ( as t h e Net herlands).

-

f o l d a p p l i c a t i o n of p r i n c i p l e ( c ) g i v e s ( z . 4 - 1 0 n y z E Z ) which, however, seems never t o be used. The s c a l e ( z 5 lon, z E Z ) w i t h a comparable degree of exactness seems t o be c l e a r l y p r e f e r r e d . The r e s u l t o f t h es e c o n s i d e r a t i o n s a r e t h e s c a l es (z*a,lOn, ~ E Z ) where n E Z and ah(1,.5,.25 The l e n g t h o f t h e s t e p c a l l a r e a l number

A =

lon

o r .21.

o f such a s c a l e i s c a l l e d i t s prominence. We

x prominent with A

i f i t i s c o n t a i n e d i n such a s c a l e

w i t h prominence A ' 2 A. A s e t M o f r e a l numbers i s prominent with A i f a l l ( i n e m p i r i c a l a p p l i c a t i o n s 75 % o f a l l ) elements o f M a r e prominent with

A

and A

i s maximal w i t h t h i s p r o p e r t y .

The i d e a i s t h a t e v e r y d e c i s i o n o f an i n d i v i d u a l f o r a number i s made on one o f the s e s c a les c o n s i d e r i n g o n l y numbers w i t h a g i v e n prominence where t h e prominence seems t o be s e l e c t e d s u f f i c i e n t l y h i g h such t h a t t h e s e t o f a l t e r n a t i v e s which have t o be checked as p o s s i b l e s o l u t i o n s o f t h e d e c i s i o n problem remains s u f f i c i e n t l y low. We mention t h e f o l l o w i n g d e v i a t i o n s f r o m t h e p r i n c i p l e t h a t o n l y s c a l e s o f t h e types above a r e s e l e c t e d . ( 1 ) I n s i t u a t i o n s w i t h more t h a n one d e c i s i o n maker i t i s f r e q u e n t l y c o n s i d e r ed t o s p l i t t h e j o i n t outcome e q u a l l y . T h i s l e a ds t o a d d i t i o n a l prominent numbers which,however,have

n o t been used t o generate s c a l e s a c c o r d i n g t o p r i n c i p l e

274

SOCIAL INTERACTION

( a ) in the cases we observed.

( 2 ) If the decision task i s t o divide a given amount x among different people, i t can happen that prominent divisors o f x a r e selected t o generate scales using principle ( a ) (Some subjects selected a prominence of 30 , when x was 150 - We explain t h i s deviation from the theory by the assumption t h a t the reference number x caused the subjects t o leave the decimal system. )

.

.

(3) If the selected number i s - as in price s e t t i n g - such that others a r e supposed t o react on them, then i t can make sense t o replace the numbers (z*a*lOn,z € Z ) by alternatives with a higher efficiency anticipating the reactions of the others. (Frequently 2.98, 2.78, 2.48, 2.28 are selected instead of 3.00, 2.75, 2.50, 2.25 , such t h a t the prices are kept j u s t below the prominence steps of the buyers. But there are different rules of t h i s type, also depending on the degree o f reflection related with the a c t of buying, and we do not want t o go into these d e t a i l s here. In the followin? we shall only consider examples w i t h an obvious transformation rule. )

2.

The reZative s i z e of the prominence

Empirical results give the impression t h a t sets of data (as prices of comparable goods in the same shop, proposals in similar bargaining situations, e t c . ) quite frequently have a prominence of about 10 % of the numbers and the range i s between 5 % and 25 % . To give an impression of a s e t of possible prices resulting from such a behaviour, we consider as an example the case that each price i s calculated w i t h a relative prominence n o t below 5 % (We suppose t h a t each price iscalculated separately and selected from a separate system (zqa, z € Z ) w i t h a prominence A = a - l O n which i s as low as possible, b u t not below 5 % of the f i n a l l y selected price.) Then the s e l l e r can chose among the following numbers in the range between 1 and 10 : I )

.

l ) We assume t h a t a = .25 i s prefered t o a = .2 . - Possible prices between 1On and lont1 are obtained by multiplication with 10n

.

275

THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

1.01, 1.10, 1.20, 1.25, 1.30, 1.40, 1.50, 1.60, 1.70, 1.75, 1.80, 1.90

(Al)

2.00, 2.25, 2.50, 2.75, 3.00, 3.25, 3.50, 3.75, 4.00, 4.25, 4.50, 4.75 5.00, 5.50, 6.00, 6.50, 7.00, 7.50, 8.00, 8.50,9.00, 9.50

(A2) (A31

1.00 . l o , i . e . f r o m 5 % t o 10 %), a t 2.00 (where t h e prominence changes f r o m .10 t o - 2 5 , i . e . f r o m 5 % t o 12.5 % ) , and a t 5.00 (where t h e prominence changes f r o m .25 t o .50, i . e . f r o m 5 % t o 10 %). So t h e r u l e t h a t no a r t i c l e i s c a l c u l a t e d w i t h a h i g h e r r e l a t i v e prominence t h a n 5 % causes r e l a t i v e prominences between 5 I and 12.5 %. I n t h i s system o f p o s s i b l e p r i c e s t h e r e a r e s t r u c t u r a l breaks a t (where t h e prominence changes f r o m

.05

to

( G e n e r a l l y , a minimal r e l a t i v e prominence o f between

c1

% and a t most

2.5

a %

a

% causes r e l a t i v e prominences

i n t h e c o r r e s p o n d i n g system o f p o s s i b l e

alternatives.) Another example which i s a l s o comparable w i t h observed p r i c e systems i s t h e s e t o f p o s s i b l e a l t e r n a t i v e s between

1 and 10, where each number i s c a l 10 %:

c u l a t e d w i t h a minimal r e l a t i v e prominence o f

1.00, 1.25, 1.50, 1.75, 2.00, 2.25 2.50, 3.00, 3.50, 4.00, 4.50 5.00, 6.00, 7.00, 8.00, 9.00

(B1) (B2) (B3)

1.00, 2.50, and 5.00, where t h e r e l a t i v e prominences change f r o m .10 t o .25 ( i . e . f r o m 10 % t o 25 %), f r o m .25 t o .50 ( i . e . f r o m 10 % t o 20 %), and f r o m .50 t o 1.00 ( i . e . f r o m 10 % t o 20 9 ) . t h e r e t h e s t r u c t u r a l breaks a r e a t

These t h e o r e t i c a l examples i l l u s t r a t e how p r i c e s w i t h a r e l a t i v e prominence between 5 % and 25 % can occur. To u n d e r l i n e t h e e m p i r i c a l e v i d e n c e o f t h e s e c o n s i d e r a t i o n s we g i v e some examples o f observed p r i c e systems. Each o f t h e examples g i v e s a l l observed p r i c e s o f a s e l e c t e d department o f one shop o r a l l p r i c e s o f an a d v e r t i s e m e n t o r l e a f l e t . (We remark t h a t examples have been s e l e c t e d such t h a t t h e prominence s t r u c t u r e can e a s i l y be recognized. )I)

A l l p r i c e s have been observed i n shops a t B i e l e f e l d i n t h e f i r s t months o f 1983.

276

SOCIAL INTERACTION

Example 1: ideal real

slabs o f chocolate (department s t o r e ) l )

.85 .85

.90 .89

.99 .98

f r e q u . l l

-i d e a l real frequ.

1.80 1.78. 1.80 2

1.10 1.08

1.00

2

3

(1.85) 1.85

1.20 1.18

3

1.90 1.90

...

1.40 1.38

1.30 1.28

1

2.00 1.98 2.00 5

1

1

1.25 1.23

5

1.50 1.48

2

...

2.25(?) 2.30

1.60 1.58 1.59 1.60

1.70

-

-

3

-

-

2

2.75(?)2.70

1

1

T h i s p r i c e system f i t s q u i t e w e l l w i t h t h e p r e d i c t i o n s o f row t h a t t h e s t r u c t u r a l break a t

2.00

i s also

1.75

IAl).

Note

according t o ( A l ) .

It i s i n t e r e s t i n g t o n o t e t h a t p r i c e systems o f d i f f e r e n t a r t i c l e s w i t h d i f -

f e r e n t s e l l i n g p r i c e s can have t h e same r e l a t i v e prominence. For example i n two o f t h e department s t o r e s we observed the prominence o f t e l e v i s i o n s e t s and p o r t a b l e s was a l s o comparable w i t h ( A l ) . They were prominent w i t h 100 i n a reange between 800 and 2000, where e x c l u s i v e l y p r i c e s o f t h e t y p e z * 100 t x w i t h z E Z! and x = 99, 98 o r 48 were observed, and i n one store

x = 48

Example 2:

5 out o f

64 cases.

unpacked u r a l i n e s (department s t o r e )

ideal t real frequency ideal t real frequency

occurred o n l y i n

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

1

1

4

3

1

3

3

5

-

4.75

5.00

7

1

This s e t i s comparable w i t h (A2).

I t should

4.50 8

be noted t h a t p r i c e s ending w i t h

- 2 5 have been avoided and a r e e s s e n t i a l l y l e s s f r e q u e n t than those ending w i t h .75 w h i l e i n o t h e r cases t h e .25 values a r e p r e f e r r e d : 1) A r t i c l e s o f f e r e d w i t h d i f f e r e n t t a s t s have been counted o n l y once.

277

THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

l e a t h e r blousons (department s t o r e )

Example 3: ideal

200

225

250

275

300

325

350

375

400

425

real

195

-

250

-

295

325

350

-

395

425

Another candy has a c o n s i d e r a b l y h i g h e r degree o f prominence, comparable w i t h (83): Example 4:

c h o c o l a t e eggs (department s t o r e )

ideal t real

40

(45)

50

1 0 1

frequency

(55)

4

60

(65)

2 1 4

1

70 7

75 1

80

1

. .. 100 ... 120

3

1

1

T h a t t h e r e a r e a l s o p r i c e systems where t h e prominence 2 i s p r e f e r r e d t o

2.5

i s i l l u s t r a t e d by

Example 5:

d r e s s m a t e r i a l (reduced p r i c e s o f seasonal s a l e s )

ideal t real frequency

2

3

28

24

(dept. s t o r e )

8

10

12

15

18

20

25

7 9 1 17 76

19

18

11

6

4

1

4

5

6

The r e l a t i v e prominence o f these p r i c e s i s comparable w i t h ( B ) b u t i t i s exceptionally high a t

2 and

3.

T h i s may be a consequence o f t h e q u i c k c a l -

c u l a t i o n i n t h e s a l e s s i t u a t i o n o r r e s u l t f r o m t h e p r i n c i p l e t o a v o i d broken numbers. Another example o f a p r i c e system a c c o r d i n g t o (B) i s Example 6:

shoes ( d i s c o u n t e r )

ideal

10.00

12.50

15.00

20.00

25.00

30.00

35.00

40.00

50.00

60.00

real

9.99

12.98

15.98

19.98

25.98 25.00 24.98

29.98

35.95 35.00

39.95

49.95 49.50

59.95 59.50

70.00 69.95

75.00 75.95

80.00 79.95

90.00 89.50

100.00

ideal real

-

110.00 109.00

The l a s t example g i v e s a p r i c e system which i s somewhat more e x a c t t h a n ( B ) . I t p e r m i t s i n s i g h t i n t o some a d d i t i o n a l c o n d i t i o n s under w h i c h numbers w i t h a l o w e r prominence a r e s e l e c t e d :

278

SOCIAL INTERACTION

Example 7: a l l a r t i c l e s of a l e a f l e t with do-it-yourself materials and equipment (discounter) 1) ideal real

1.25

2.00

2.50

3.00

3.50

4.00

5.00

5.50

6.00

(6.25)

1.29

1.99

2.49

2.99

3.49

3.99

4.99

5.49

5.99

6.29

frequ.

1

1

1

3

1

1

5

1

3

1

ideal real

6.50

7.00

7.50

8.00

9.00

12.00

12.50(?)

15.00

6.49

6.99

7.49

7.99

8.99

10.00 9.99

11.98

12.98

14.98

frequ.

1

3

3

2

7

5

1

3

ideal

17.00

17.50(?)

19.00

20.00

22.00

25.00

27.00

30.00

35.00

real

16.98

17.98

18.98

19.98

21.98

24.98

26.98

29.98

34.95

1

3

3

65.00 65.00

70.00 69.95

75.00 75.00

80.00 79.95

1

5

1

2

frequ.

1

2

ideal real

40.00 39.95

45.00 44.95

frequ.

3

4

2 50.00 49.95

7 55.00 55.00

4

1

2

60.00 59.95 2, 5

4

2

ideal

85.00

90.00

100.00

110.00

120.00

130.00

140.00

150.00

160.00

real

85.00

89.50

99.50

-

119.00

129.00

139.00

149.00

1

1

4

-

2

1

3

4

-

f requ , ideal

170.00

180.00

190.00

200.00

real

169.00

179.00

189.00

198.00

1

1

f reau

.

3

2

... 240.00 .. . 260.00 239.00

259.00

1

1

I t should be remarked t h a t the prices with a comparably high degree f xactness are s e t f o r a r t i c l e s t o which similar or comparable a r t i c l e s existed with 15 a r t i c l e s from the food section which were in a separate board have been omitted. 1 a r t i c l e had the price 59.50.

279

THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

p r i c e s w i t h a h i g h e r prominence: c u r t a i n r a i l s ( w i t h d i f f e r e n t l e n g t h s ) had the prices (180 cm), points

18.98

(120 cm),

where t h e

20.00

and

19.98

(140 cm),

140 cm and t h e 25.00.

21.98

(160 cm)

and

24.98

180 cm v e r s i o n s g i v e t h e r e f e r e n c e

The p r i c e s o f s p o t l i g h t s were

two on a beam

I t can be e a s i l y r e a l i z e d t h a t t h e p r i c e o f t h e s i n g l e p i n e v e r s i o n s e t s t h e

reference p o i n t

20.00.

The s i n g l e oak v e r s i o n i s c a l c u l a t e d one s t e p (10 I =

2.00) higher. The p r i c e s o f t h e double and t h r e e f o l d v e r s i o n s are s e t cheaper than t h e corresponding

m u l t i p l e s o f t h e s i n g l e versions, however, t h e se-

l e c t e d degree o f prominence d i d n o t p e r m i t a r e d u c t i o n o f t h e double p i n e version. The p r i c e o f t h e p i n e r o n d e l l v e r s i o n i s again a number w i t h a h i g h prominence (10)

(75),

h i g h e r than

whereas t h e oak r o n d e l l g o t a p r i c e which i s one s t e p 75.

These two examples e x p l a i n most o f t h e p r i c e s w i t h a comparably l o w prominence, namely 18.98, 21.98 ( 2 x ) , 55.00 and 85.00. A l s o t h e p r i c e o f 6.29 (per square meter o f r o c k wool) r e s u l t s from a p r i c e o f 5.00 p e r runn i n g meter f o r t h e normal w i d t h o f

3.

.80

meters.

The prominence selection r u le

The observation t h a t goods w i t h t h e same sales p r i c e b u t d i f f e r e n t margins d i f f e r i n t h e exactness o f c a l c u l a t i o n

suggests another approach t o t h e pre-

d i c t i o n o f t h e prominence o f p r i c e o r value systems. The main c r i t e r i o n on t h i s approach i s t h e range o f reasonable a l t e r n a t i v e s o f t h e d e c i s i o n problem. prominence s e l e c t i o n r u l e : F o r a given range

[ x,

y

I

o f reasonable

a l t e r n a t i v e s t h e prominence i s s e l e c t e d i n such a way t h a t t h e range includes a t l e a s t t h r e e numbers w i t h h i s prominence. Although t h i s r u l e seems q u i t e tough i t p o i n t s b e t t e r p r e d i c t i o n s than t h e

280

SOCIAL INTERACTION

-

10 % r u l e o f t h e p r e c e d i n g paragraph. To e x p l a i n t h e b e h a v i o r a l ideas behin d t h i s r u l e we c o n s i d e r an example o f p r i c e s e t t i n g :

A s e l l e r r e f l e c t s upon t h e adequate s a l e s p r i c e o f an a r t i c l e . From h i s experienc e he has some i d e a o f p o s s i b l e p r i c e s on t h e market. He can d e s c r i b e t h e corres ponding range o f p r i c e s r o u g h l y by an upper and a lower bound. Genera lly , t hes e bounds have a comparably h i g h prominence. Now o u r assumption i s t h a t t h e s e l l e r s e l e c t s t h e prominence o f t h e s c a l e i n such a way t h a t bes i d e s t h e extremes a t l e a s t one a l t e r n a t i v e between them becomes prominent. F i g u r e 1 g i v e s range and prominence f o r a s e l e c t i o n o f a r t i c l e s f rom t h e food market. The c o r r e s p o n d i n g d a t a were c o l l e c t e d i n 68 shops a t B i e l e f e l d i n May 1982. We s e l e c t e d a l l branded a r t i c l e s o f t h i s survey which had a s e l l i n g p r i c e above

2.00

and f o r which t h e prominence s t r u c t u r e was n o t

perturb ed by a p r i c e recommendation. The range o f reasonable a l t e r n a t i v e s i s est imat e d by t h e range o f observed p r i c e s o m i t t i n g t h e 10 % t a i l s o f b o t h s ides . I n f i g u r e 1 t h e c l o s e d p o i n t s r e f e r t o branded a r t i c l e s ( s p e c i a l brands o f c o f f e e , honey, brandy, champagne, i n s t a n t f o o d and pre-cooked meal ) w h i l e t h e c i r c l e s belong t o o t h e r goods ( a p p l e s ( " g o l d e n d e l i c i o u s " ) , tomatoes, cheap l i q u o r and s l i c e d bread o f a c e r t a i n q u a l i t y ) . The r e s u l t s show t h a t t h e range has a s i z e between

2 A and

5 A

where A

i s t h e prominence o f t h e s e t o f a c t u a l l y s e l e c t e d p r i c e s . T h i s support s t h e prominence s e l e c t i o n r u l e p e r m i t t i n g a maximal e x t e n s i o n o f t h e range o f reasonable a l t e r n a t i v e s between 2 A and 5 A , i f t h e boundaries o f t h e range a r e assumed t o have a t l e a s t prominence A . To i l l u s t r a t e t h i s : i f t h e range o f reasonable a l t e r n a t i v e s i s {x, 2 5 x 5 4) t h e n A = 1 and t h e range has an ext e ns ion o f 2 A ,if t h e r e a s o n a b l e a l t e r n a t i v e s a r e I x , 1. 5 5 x 2 3.51 then A = .5 and t h e range has an e x t e n s i o n o f 5 A , It i s i n t e r e s t i n g t h a t t h e nonbounded a r t i c l e s have a l a r g e r r e l a t i v e range ( > 5 A , t n e d i f f e r e n c e t o t h e branded a r t i c l e s i s s i g n i f i c a n t on t h e 0.02

level).

T h i s can be e x p l a i n e d b y t h e f a c t t h a t t hese a r t i c l e s do n o t have a

comparable homogenity o f c a l c u l a t i o n . The o n l y e x c e p t i o n (cheap c o f f e e ) which i s unbranded b u t has a r e l a t i v e range o f 4.3 A seems t o have a q u i t e homogeneous c a l c u l a t i o n .

THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

281

cac

*.oc

3.00

2 .oo

1.00

4,

Euatmtion of pqj0,f.f differences by r e a m of t h e p m i n e n c e

Results from game t h e o r e t i c a l experiments i n d i c a t e t h a t the prominence s t r u c t ure of a decision problem does n o t o n l y serve t o s e l e c t possible numerical a l t e r n a t i v e s f o r a decision b u t i s a l s o used t o evaluate d i f f e r e n c e s between possible outcomes. The f o l l o w i n g r e s u l t s which s h a l l be reported i n s h o r t seem t o admit t h i s i n t e r p r e t a t i o n ,

SOCIAL INTERACTION

282

We r e f e r t o a s e r i e s o f n-person c h a r a c t e r i s t i c f u n c t i o n games w i t h f r e e comm u n i c a t i o n and t r a n s f e r a b l e p a y o f f s . Each o f t h e s e n-person games was p l a y e d by

n2 p l a y e r s . Each p l a y e r p l a y e d t h e game

n

times i n

n

subsequent

rounds m e e t i n g no o t h e r p l a y e r t w i c e . The games can be d e s c r i b e d b y t h e s e t o f p l a y e r s

N = I 1, 2,

...,

n }

(here

n = 4 o r 5 ) and t h e p a y o f f s o f t h e c o a l i t i o n s S 5 N which can be d i s t r i b u t e d among t h e p l a y e r s of t h e c o a l i t i o n . Here t h e s e p a y o f f s were o f t h e f o l l o w i n g type:

A

v

(s)

=

where A

2

if l € S

N

B

if S =

o

otherwise

B 5 A/n

and S

L

111

D d

111

a r e p o s i t i v e r e a l numbers.

1 has a r e l a t i v e l y s t r o n g p o s i t i o n

I n t h e s e games p l a y e r

s i n c e he needs

o n l y one o t h e r p l a y e r as a p a r t n e r w h i l e t h e o t h e r p l a y e r s can c o a l e s c e witho u t player

1 o n l y i f a l l o f them s t i c k t o g e t h e r . However, t h i s symmetric

weakness o f p l a y e r s

2, 3,

...,

...,n

causes them t o c o n s i d e r t h e c o a l i t i o n

{ 2, 3, n } as t h e i r f i r s t c o a l i t i o n a l t e r n a t i v e w h i c h i s s t r o n g l y supp o r t e d b y s o c i a l arguments (compare ALBERS [1978]). P l a y e r 1 can t r y t o c o r r u p t one o f t h e o t h e r t o f o r m a c o a l i t i o n w i t h him.

W i t h i n t h e " s o c i a l " c o a l i t i o n I 2, 3, ..., n I t h e p a y o f f was d i s t r i b u t e d e q u a l l y . I n many cases t h i s c o a l i t i o n has been extended t o t h e grand coa l i t i o n I 1, 2, ..., n I a g a i n a c c o r d i n g t o t h e equal share p r i n c i p l e . There a r e two arguments f o r such an e x t e n s i o n : one i s n o t t o e x c l u d e p l a y e r 1 f o r s o c i a l reasons, and t h e o t h e r i s t h a t t h e c o a l i t i o n i s s a f e r i f no p l a y e r i s o u t s i d e ( t h e grand c o a l i t i o n

N

has n o t been broken i n any o f o u r expe-

riments).

A c e n t r a l problem o f t h e game was whether a s o c i a l c o a l i t i o n s h o u l d be formed o r one o f t h e two-person c o a l i t i o n s . The i n c e n t i v e s f o r t h i s d e c i s i o n can be c h a r a c t e r i z e d ex p o s t by t h e mean p a y o f f s

xl,

xi

o f player

1 and

h i s p a r t n e r i n t h e t w o - p l a y e r c o a l i t i o n and t h e outcomes

...,

al, ai ( i = 2, 3, n ) i n t h e s o c i a l c o a l i t i o n . Here we s e t al = ai = A/n s i n c e i n n e a r l y a l l cases where a s o c i a l c o a l i t i o n became

THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

283

N was formed. (Only i n one sequence o f games I 2 , 3, ..., n 1 t o t h e grand c o a l i t i o n . F o r t h i s

s t a b l e t h e grand c o a l i t i o n t h e players p r e f e r r e d sequence we d e f i n e d

ai

by B/n

-

1.)

I t seems reasonable t o assume t h a t t h e i n c e n t i v e o f a p l a y e r j t o l e a v e t h e s o c i a l c o a l i t i o n and f o r m a two person c o a l i t i o n i s p r o p o r t i o n a l t o

-

t h e corresponding payoff d i f f e r e n c e x a E m p i r i c a l r e s u l t s show t h a t j j* t h i s i n c e n t i v e ( w h i c h can be measured i n d i r e c t l y be t h e p r o p o r t i o n o f nons o c i a l r e s u l t s i n t h e f i n a l outcomes o f t h e games) remains more o r l e s s un-

A

changed i f t h e c o n s t a n t s

and

d e f i n i n g t h e value s t r u c t u r e o f t h e

B

10 A

game a r e s i m u l t a n e o u s l y r e p l a c e d by

and

10 B.

On t h e o t h e r hand,

t h e y c o u l d be e s s e n t i a l l y i n f l u e n c e d by t h e degree o f i n v o l v e m e n t o f t h e p l a y e r s i n t h e game (measured by t h e r e l a t i v e prominence o f t h e s e t o f n u m e r i c a l proposal 5 ) . Both r e s u l t s can be e x p l a i n e d by t h e assumption t h a t t h e e v a l u a t i o n o f t h e p a y o f f d i f f e r e n c e s i s i n f l u e n c e d by t h e prominence i n such a way t h a t t h e incentives t o leave t h e social c o a l i t i o n a r e

d. = J where

A

'j - a j A

i s t h e prominence o f t h e a n a l y s i s (measured by t h e prominence o f

t h e s e t o f proposals).

O f course,

A

s h o u l d be measured i n d i v i d u a l 1 . y b u t

we had n o t enough o b s e r v a t i o n s f o r such an a n a l y s i s . We, t h e r e f o r e , assumed t h a t a l l p l a y e r s a n a l y s e t h e game w i t h more o r l e s s t h e same prominence and we e s t i m a t e d A f o r each t y p e o f t h e game by t h e prominence o f t h e s e t o f a l l proposals o f a l l players. By t h i s f o r m u l a t h e i n c e n t i v e i s g i v e n by t h e number o f prominence s t e p s between x

and a f o r i n s t a n c e i f x = 500, a . = 300 and A = 50, j j ' j J t h e n t h e i n c e n t i v e i s 4, s i n c e t h e r e a r e 4 prominence s t e p s ( 350, 400,

450,

500 ) f r o m

300

to

500.

I f i n t h e same s i t u a t i o n w i t h t h e same expected outcomes a n o t h e r p l a y e r

( o r group o f p l a y e r s ) o n l y p l a y s w i t h a prominence A = 100, t h e n t h e i n c e n t i v e t o l e a v e t h e s o c i a l c o a l i t i o n i s o n l y h a l f as high, s i n c e f o r t h i s

284

SOCIAL INTERACTION

prominence t h e r e a r e o n l y two s t e p s ( 400

,

500 )

from

300 t o

500.

I t can be seen f rom t a b l e 3 t h a t i n two cases t h e same game has been p l a y -

ed w i t h d i f f e r e n t prominences. I n b o t h cases t h e corresponding m o t i v a t i o n a l s i t u a t i o n s became c o m p l e t e l y d i f f e r e n t and v e r y d i f f e r e n t degrees o f cooper a t i o n resulted. o f a p l a y e r j t o decide f o r t h e twoWe assume t h a t t h e p r o b a b i l i t y p j person c o a l i t i o n i s p r o p o r t i o n a l t o h i s i n c e n t i v e d and our dat a show j t h a t t h e p r o p o r t i o n a l i t y f a c t o r has t h e s i z e o f about .1 ( l e a s t square a n a l y s i s g i v e s an o p t i m a l v a l u e o f

pj =

.1

*

d . = .1 J

.094). So we o b t a i n x *

j

- a

j

A

(since p i s a p r o b a b i l i t y , i t must be r e p l a c e d by j a h i g h e r value.)

1 i f the formula gives

Assuming t h a t t h e d e c i s i o n s o f t h e p l a y e r s between a two-person c o a l i t i o n and a s o c i a l c o a l i t i o n a r e made i n d e p e n d e n t l y , we g e t t h e f o l l o w i n g f o r m u l a f o r t h e p r o b a b i l i t y t h a t a n o n s o c i a l c o a l i t i o n i s formed.

n P = P 1 ( 1 -

i=2

( l - p i ) )

n ( l - pi) g i v e s t h e p r o b a b i l i t y t h a t a l l weak p l a y e r s decide t o f o r m a s o c i a l c o a l i t i o n , 1 - n ( 1 - pi) i s the probability that a t least one weak p l a y e r decides f o r a n o n s o c i a l c o a l i t i o n and t h e f i n a l e x p r e s s i o n s e l e c t s t h os e cases where p l a y e r 1 a l s o d e c i d es t o p l a y i n a n o n s o c i a l way. where

Figure 2 gives the r e s u l t s o f

8

such s e r i e s o f experiments. (They i n -

volve d 155 s u b j e c t s who p l a y e d 155 games.) A has been measured as t h e prominence of t h e r e s p e c t i v e s e t s of proposals, x j as t h e mean outcome i n t h e two-person c o a l i t i o n o f p l a y e r 1 w i t h a n o ther p l a y e r , a i s the j equal s hare i n t h e s o c i a l c o a l i t i o n . Using t h e se values our t h e o r y g i v e s

285

THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

.IO

.lo

.ID

.YO

.SO

.kb

.+o

.PO

90

+

1.00

p r e d i c t e d n o n c o o p e r a t i v i t y ( b y prominence)

F i g u r e 2:

P r e d i c t e d ( b y prominence) and observed n o n c o o p e r a t i v i t y i n games o f Apex t y p e

o f n o n c o o p e r a t i v e r e s u l t s . On t h e o t h e r

a prediction o f the proportion p hand,

p can be measured by t h e percentage o f n o n s o c i a l r e s u l t s i f outcomes

which do n o t i n v o l v e t h e s o c i a l c o a l i t i o n o r t h e two-person c o a l i t i o n ( t h e s e were l e s s t h a n

10 %

o f the r e s u l t s ) a r e omitted.

The e r r o r o f t h e percentage o f n o n s o c i a l r e s u l t s i s e s t i m a t e d by t h e asA:___

LL-L

sumprion r n a c

, r 1.3

__

OT

LO__

m e

q r

13

-

clr

~3

. - -7 1 .* - _ _ _ _ _ _ _ resuics OT a sequence

._ _._.__ .-.,>

OT

games c o u i a

have a s o c i a l i n s t e a d o f a n o n s o c i a l r e s u l t o r v i c e versa. The e r r o r o f t h e prominence i s e s t i m a t e d by t h e p r o b a b i l i t y a c c o r d i n g t o t h e g e o m e t r i c mean o f t h e observed prominence and t h e n e x t h i g h e r ( r e s p e c t i v e l y l o w e r ) promi-

SOCIAL INTERACTION

206

~~

noncoop. r e s u l t s n

v(N)

v (234)

5

100 1000

100

5

5

1000

20

5 5 5

1000 1000 1000

1000

50 50

predicted

prominence

1000 1000

100

10 5

4

100 1)

4 4

160

75 165

160

165

10

4 4

320 1000

315

20 50

1000

I observed

.50

I

.87 .31 .32

.73 .43 .25

.44

.47

Table 3: P r e d i c t e d ( v i a t e m p t a t i o n ) and observed degree o f n o n c o o p e r a t i v i t y i n games o f Apex type.

nence. The e r r o r o f t h e mean outcome

x

j

c o u l d be n e g l e c t e d because i t was

comparably s m a l l . The d a t a g i v e

a correlation o f

which i s s i g n i f i c a n t on t h e

.83

o f observed and p r e d i c t e d outcomes

.01 l e v e l .

Maybe i t cannot be f i n a l l y d e c i d e d whether t h e prominence r e a l l y i n f l u e n c e s the evaluation o f the payoff differences x - a o r i f both, t h e e v a l u j j a t i o n o f t h e d i f f e r e n c e s and t h e prominence,depend on an u n d e r l y i n g beh a v i o r a l p a t t e r n . However, t h e observed r e s u l t s i n d i c a t e t h a t t h e prominence influences the evaluation since the e r r o r o f of

A

p

caused by t h e i n e x a c t n e s s

seems t o be e s s e n t i a l l y l o w e r t h a n a d m i t t e d by t h e " n a t u r a l " e r r o r

bounds g i v e n by t h e geometric means t o t h e n e x t h i g h e r o r l o w e r prominences.

I n t h i s game we s e t v ( 1 ) = 25

THE PROMINENCE STRUCTURE OF THE DECIMAL SYSTEM

References

Albers, W.: 1983. Bloc Forming Tendencies as C h a r a c t e r i s t i c s o f t h e Barg a i n i n g Behavior i n D i f f e r e n t Versions o f Apex Games, i n H. Sauermann (ed.): C o a l i t i o n Forming Behavior, Tiibingen, 172-203. Albers,W. and Albers G.: 1983. Prominence and A s p i r a t i o n Adjustment i n Location Games, i n R.Tietz (ed.): A s p i r a t i o n Levels i n Bargaining and Economics Decision Making, B e r l i n , Heidel berg, New York, 243-258. Albers, W. and Albers, G.: 1983a. S e l e c t i o n o f optimal numbers o f t h e decimal system i n i n d i v i d u a l d e c i s i o n making, Working Paper o f t h e I n s t i t u t e o f Mathematical Economics, U n i v e r s i t y o f B i e l e f e l d . 1969. Some Psychological Aspects o f P r i c e , i n T a y l o r - W i l l s Emory, F.: (eds.), P r i c i n g Strategy, Princeton. Ginzberg, E.:

1936.

Customary Prices, American Economic Review 25, 296.

1971. Comments on "Psycholophysics Gabor, A., Grange, J.W.J., Sowter, A.P.: o f Prices, Journal Marketing Research 8 (May), 251-252. CroRmann, H.J., Tietz,R.: 1983. Market Behavior on A s p i r a t i o n Levels, i n R. T i e t z : A s p i r a t i o n Levels i n Bargaining and Economic Decision Making, Lecture Notes i n Economics and Mathematical Systems, Springer-Verlag, B e r l in-Heidel berg-New York-Tokyo, 170-185. 1970. Psychophysics o f P r i c e s , Journal o f MarKamen, J.M., Toman, R.J.: k e t i n g Research 7 (February), 27-31. Kamen, J.M., Toman, R.J.: 1971. "Psychophysics o f P r i c e s " :A R e a f f i r m a t i o n , Journal o f Marketing Research 8 (May), 252-257. Monroe, K.B., Venkatesau,M. : 1969. The concept o f p r i c e l i m i t s and psychop h y s i c a l measurement: A Laboratory Experiment, i n Proceedings F a l l Conference , American Marketing A s s o c i a t i o n , Chicago 1969. Oesterwal besloh, K.H. , Rembde, M. ( i n p r e p a r a t i o n ) : Prominenzstrukturen und Preissetzungsverhalten i m Einzelhandel, diploma t h e s i s , I n s t i t u t e o f Mathematical Economics, U n i v e r s i t y o f B i e l e f e l d 1968. How Housewives Form P r i c e Impressions, Journal o f Oxenfeldt, A.R.: A d v e r t i s i n g Research 8 (September) , 9-17. Sauermann, H., Selten, R.: 1962. Anspruchsanpassungstheorie.der Unternehmung, Z e i t s c h r i f t f u r d i e gesamte Staatswissenschaft, 118, 577-597. S c h e l l i n g , T.C.:

1960.

The S t r a t e g y o f C o n f l i c t , Cambridge.

Schmitz, G.: 1964. Z w i s c h e n b e t r i e b l i c h e r V e r g l e i c h d e r Einzelhandelspreise s o r t e n g l e i c h e r Konsumwaren , KO1n , Opl aden. Simon, H.A.: 1955. A behavioral model o f r a t i o n a l choice. Q u a r t e r l y Journal o f Economics 69, 99-118. T i e t z , R. : 1978: Entscheidungsprinzipien der b i l a t e r a l e n Anspruchsanpassung, i n : E. Helmstadter (ed.): Neuere Entwicklungen i n den Wirtschaftswissenschaften, S c h r i f t e n des Vereins f i r S o c i a l p o l i t i k , Neue Folge, Bd. 98, B e r l i n , 431-455.

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DEVELOPMENT AND EPISTEMOLOGY

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

291

CONCEPTUAL AND THEORETICAL ISSUES IN DEVELOPMENTAL RESEARCH ON THE ACQUISITION OF THE PROBABILITY CONCEPT1)

ROLAND W. SCHOLZ

MANFRED WALLER

Universitat Bielefeld lnstitut fur Didaktik der Mathematik F. R. Germany

Universitat Bielefeld Abteilung fur Experimentelle und Angewandte Psychologie F. R. Germany

P s y c h o l o g i c a l t h e o r i e s a r e b e i n g developed about t h e c o g n i t i o n s o r a t t i t u d e s e t c . towards c e r t a i n o b j e c t s , o r about b e h a v i o r i n c e r t a i n s i t u a t i o n s . When d e v e l o p i n g a p s y c h o l o g i c a l t h e o r y one s h o u l d c l a r i f y " f o r what", i.e.,

about

which o b j e c t o r f o r which s i t u a t i o n i t i s intended. T h i s i s a l s o t r u e f o r a t h e o r y about t h e a c q u i s i t i o n o f t h e p r o b a b i l i t y concept. However, one has t o r e a l i z e t h a t t h i s has n o t been done

- w i t h the exception o f

PIAGET and

INHELDER (1951/1975). As t h e p r o b a b i l i t y concept can be i n t e r p r e t a t e d i n e s s e n t i a l l y d i f f e r e n t ways we b e g i n w i t h an o u t l i n e o f t h e ( m a i n - ) v a r i a n t s o f t h e p r o b a b i l i t y concept i n s e c t i o n 1. B e f o r e t u r n i n g t o an a n a l y s i s o f t h e developmental t h e o r e t i c a l approaches i n t o t h e a c q u i s i t i o n o f t h e probab i l i t y concept, we p i c k o u t which t y p e s o f p r o b a b i l i t y a r e i n v e s t i g a t e d i n section 2 . B e f o r e t h e t h e o r e t i c a l r e s e a r c h approaches on t h e o n t o g e n e s i s o f t h e proba b i l i t y concept can be analysed, we have t o s t a r t w i t h a s l i g h t l y more det a i l e d s t u d y o f t h e p r o b a b i l i t y concept and how i t i s i n t e r p r e t a t e d w i t h i n developmental p s y c h o l o g i c a l r e s e a r c h . T h i s seems t o us t o be necessary, as a comprehension o f a concept s h o u l d precede an u n d e r s t a n d i n g o f t h e o n t o g e n e t i c development o f t h i s concept.

1) We w i s h t o t h a n k P r o f . I a n Hacking f o r h i s h e l p f u l comments and f o r h i s and Jonathan Harrow's a s s i s t a n c e i n p r o d u c i n g a r e a d a b l e E n g l i s h t e x t .

DEVELOPMENT AND EPISTEMOLOGY

292

1. Variants of probabiZity The main d i v i d i n g 1 i n e i n i n t e r p r e t i n g p r o b a b i l i t y i s s t i l l t h a t between t h e objectivist and t h e subjectivist i n t e r p r e t a t i o n s o f t h i s concept. These can be i d e n t i f i e d s i n c e t h e emergence of probabizity. " P r o b a b i l i t y has two aspects. It i s connected w i t h t h e degree o f b e l i e f w a r r a n t e d by evidence, and i t i s connected w i t h t h e tendency d i s p l a y e d by some random devices, t o produce stab1 e f r e q u e n c i e s .

I'

(HACKING,1975

, p.1)

B o t h s i d e s of p r o b a b i l i t y a r e , i n o u r o p i n i o n , i n d i s p e n s i b l e and must be i n t e g r a t e d i n t o an u n d e r s t a n d i n g o f p r o b a b i l i t y t h a t encompasses b o t h aspects. Nevertheless, these d i f f e r e n t I n t e r p r e t a t i o n s a r e f r e q u e n t l y c o n s i d e r e d t o be complementary by t h e schools o f t h o u g h t d i s s e m i n a t i n g them.

A p r o t o t y p e o f t h e objectivist c o n c e p t i o n i s t h e s o - c a l l e d f r e q u e n t i s t approach o f R i c h a r d von MISES (1928). For t h i s p o s i t i o n , i t i s fundamental t o comprehend p r o b a b i l i t y as t h e l i m i t o f a r e l a t i v e f r e q u e n c y e s t a b l i s h e d by an experiment which can be r e p e a t e d a t w i l l . Beyond t h a t , i t i s d e c i s i v e f o r t h i s i n t e r p r e t a t i o n t h a t t h e s e f r e q u e n c i e s a r e o b s e r v a b l e and independ e n t o f each o t h e r , and t h a t t h e mechanism o f chance i s e x p l i c i t . From a p u r i s t i c a l l y o b j e c t i v i s t i c p o i n t o f view, c o m b i n a t o r i c a l a s p e c t s a r e i n e s s e n t i a l . They m e r e l y can p l a y a p a r t i n model c o n i d e r a t i o n s . Urns, d i c e , and c o i n s as i d e a l i z e d symmetrical o b j e c t s a r e t h e models f o r c e r t a i n random s i t u a t i o n s f u l f i l l i n g t h e o b j e c t i v i s t c r i t e r i a o f p r o b a b i l i t y . Hence such c a l c u l a t i o n s as t h o s e p l a y i n g a c e n t r a l r o l e i n LAPLACE's c l a s s i c a l concept o f p r o b a b i l i t y a r e o n l y a d m i s s i b l e if t h e e l e m e n t a r y p r o b a b i l i t i e s prove t o be e q u a l l y p r o b a b l e i n t h e f r e q u e t i s t sense. For t h e subjectivist, t h e f o l l o w i n g statement h o l d s t r u e : " S u b j e c t i v e proba b i l i t y i s o n e ' s degree o f b e l i e f i n an outcome based on an e v a l u a t i o n making t h e b e s t use o f a l l t h e i n f o r m a t i o n a v a i l a b l e t o him and h i s own skill."

(DE FINETTI,1974,

p. 16) The b a s i c i d e a e s t a b l i s h i n g t h e s u b j e c t i -

v i s t t h e o r y o f p r o b a b i l i t y i s t h e b e t t i n g o r odds paradigm which r e l a t e s

293

ACQUISITION OF THE PROBABILITY CONCEPT

t h e p r o b a b i l i t y concept t o t h e concept o f u t i l i t y . However, one has t o r e g a r d t h e t h e o r y o f p e r s o n a l i s t o r s u b j e c t i v i s t p r o b a b i l i t y as d e a l i n g w i t h ideaZized persons. These persons a r e c o n s i s t e n t and f r e e o f l o g i c a l c o n t r a d i c t i o n s i n t h e i r judgments. I t i s a d m i s s i b l e , however, t o c o n s t r u e d i f f e r e n t i d e a l i z e d persons who d i f f e r i n t h e i r o p i n i o n s a b o u t one and t h e same s i t u a t i o n . C o n t r a r i l y , w i t h i n CARNAP's l o g i c a l t h e o r y o f p r o b a b i l i t y f o r a g i v e n s i t u a t i o n o n l y one r a t i o n a l person can be c o n s t r u c t e d r e l e v a n t t o a g i v e n body o f i n f o r m a t i o n . The s u b j e c t i v i s t conception, however, a l s o p r o v i d e s a d e f i n i t e i n t e r p r e t a t i o n o f c l d s s i c a l p r o b a b i l i t y experiments l i k e u r n s o r t o s s i n g c o i n s ,

i f t h e i n f o r m a t i o n i s e x h a u s t i v e and t h e s i t u a t i o n i s d e f i n i t e l y i n t e r p r e t e d by means o f symmetry. Under t h i s c o n d i t i o n , t h e s u b j e c t i v i s t t h e o r y d e a l s w i t h one person. " I n t h o s e s i t u a t i o n s where t h e s u b j e c t i v i s t b e l i e v e s t h e r e i s symmetry he would a c t as i f he a c c e p t e d c l a s s i c a l proba b i l i t y , f o r example

leading t o t h e Laplacian

...'I

(FINE,1973, p. 2 3 4 f )

We should l i k e t o p o i n t o u t h e r e t h e difference b e t m e n mathematicaZ

probabilities and t h e probabiZities which psychoZogists attempt to measure by d i f f e r e n t methods and means i n reaZ i n d i v i d u a l s , i n experiments o r under f i e l d c o n d i t i o n s . Mathematical t h e o r y must be c o n s i d e r e d as a l o g i c o f d e c i s i o n s , w h i l e t h e p s y c h o l o g i c a l access t o ' s u b j e c t i v e "

probabilities

must be considered as an a t t e m p t t o d e t e r m i n e an i n d i v i d u a l ' s c o g n i t i o n s and judgmental processes i n which mathematical concepts, e s p e c i a l l y mathem a t i c a l p r o b a b i l i t i e s , may be i n c l u d e d . I t i s by no means expected t h a t a r e a l i n d i v i d u a l i s f r e e o f c o n t r a d i c t i o n s and c o n s i s t e n t i n h i s / h e r judgments, even I f he/she uses c e r t a i n elements o f a mathematical t h e o r y . I t must be n o t e d t h a t t h e " s u b j e c t i v e proba b i l i t i e s " s t u d i e d i n psychology may be made t h e o b j e c t o f a c o n t r a d i c t i o n f r e e , c o n s i s t e n t mathematical t h e o r y .

As p s y c h o l o g i s t s we b e l i e v e t h a t a human b e i n g as an i n d i v i d u a l i s a c t i v e l y r e c o n s t r u c t i n g h i s environment w i t h t h e h e l p o f h i s i n f o r m a t i o n and

knowledge, t h u s assuming t h a t "normal people" b u i l d a model o f t h e i r environment i n t h e same way as s c i e n t i s t s .

294

DE VELOPMEN T A ND €PIS TEMO L OG Y

Within t h e s u b j e c t ' s model, s u b j e c t i v e p r o b a b i l i t y may a l s o be assigned t o events which i n t h e o b j e c t i v i s t sense a r e n o t p r o b a b i l i s t i c . "The a d j e c t i v e ' s u b j e c t i v e ' r e f e r s " , here, " t o t h e s u b j e c t and n o t t o t h e a r b i t r a r y assessment which i s independent o f t h e o b j e c t i v e measure. Subj e c t i v e p r o b a b i l i t i e s i n psychology

... a r e

t o be understood as t h e

o b j e c t i v e measure o f s u b j e c t i v e assessments." (HURZ,1980,

p. 190, t r a n s -

l a t i o n by t h e a u t h o r s ) ' These s u b j e c t i v e assessments a c c o r d i n g t o HURZ (1980) a r e t h e r e s u l t o f a ' W i d e r s p i e g e l u n g ' ( p r o c e s s o f r e f l e x i o n ) . " W i t h i n t h i s process, v e r i f i e d t h e o r i e s , e m p i r i c a l f a c t s , models o f v a r i o u s kinds, and hypotheses p l a y a p a r t . " (HtlRZ,1980,

p. 190)

2. The probability concept within developmentai! research A f t e r these fundamental c o n s i d e r a t i o n s we w i l l now t u r n t o t h e q u e s t i o n how these i n t e r p r e t a t i o n s o f t h e p r o b a b i l i t y concept as d i s t i n g u i s h e d above a r e d e a l t w i t h i n t h e i n v e s t i g a t i o n s i n t o t h e a c q u i s i t i o n o f t h e p r o b a b i l i t y concept. We should l i k e t o s t a r t t h i s s e c t i o n w i t h t h e f o 1owi n g thesis: I n r e s e a r c h i n t o t h e a c q u i s i t i o n o f t h e p r o b a b i l i t y concept, o n l y t h e o b j e c t i v i s t concept o f p r o b a b i l i t y has ( w i t h one e x c e p t i o n ) been s t u d ed. Now f o r t h e j u s t i f i c a t i o n o f t h i s t h e s i s . T r a d i t i o n a l l y , one can d i s t i n g u i s h between two paradigms: r e s e a r c h i n t o t h e understanding of classical chance

experiments, and r e s e a r c h i n t o probabiZity learning. Developmental r e s e a r c h i n t o t h e a c q u i s i t i o n o f t h e p r o b a b i l i t y concept was i n i t i a t e d by PIAGET and INHELDER (1951, see a l s o PIAGET,1950).

I n their

experiments t h e y p r e d o m i n a n t l y used random s i t u a t i o n s t y p i c a l f o r t h e obj e c t i v i s t p r o b a b i l i t y concept. Most f r e q u e n t l y one f i n d s u r n o r c o n t a i n e r experiments and s p i n n e r t a s k s . The s i t u a t i o n s a r e p r o t o t y p e s o f t a s k s where i n t h e p r o b a b i l i t y o f events i s determined e i t h e r s p a t i o - t e m p o r a l l y , logico-arithmetically.

or

I n t h e case o f t h e c o n t a i n e r experiment, t h e r e i s

t h e c l a s s i c a l d i s t i n c t i o n between t h e s o - c a l l e d one c o n t a i n e r and t h e two c o n t a i n e r t a s k . One d i s t i n g u i s h e s between s i n g l e o r mu1 t i p l e d r a w i n g w i t h

ACQUISITION OF THE PROBABILITY CONCEPT

295

and w i t h o u t replacement. A c c o r d i n g l y , experiments o f t h e s p a t i o - t e m p o r a l t y p e were c a r r i e d o u t w i t h t h e "one s p i n n e r " t a s k on t h e one hand, and w i t h t h e "two s p i n n e r " t a s k on t h e o t h e r . As a whole, t h e r e p e r t o i r e o f chance s i t u a t i o n s used i n developmental experiments i s r a t h e r l i m i t e d . H a r d l y any e x p e r i m e n t a l t a s k o t h e r t h a n t h e c l a s s i c a l one has been designed. I t i s p a r t i c u l a r l y bad t h a t t h e r e a r e no a t t e m p t s t o use s i t u a t i o n s w i t h e x p l i c i t chance mechanisms w i t h o u t symmetry. L e t us now t u r n t o t h e second paradigm, t o p r o b a b i l i t y l e a r n i n g . A p a r t i c u l a r l y r e v e a l i n g d e s c r i p t i o n o f p r o b a b i l i t y l e a r n i n g which shows t h e amazingly "broad" scope o f t h i s branch o f r e s e a r c h i s g i v e n by MESSICK and SOLLEY (1957, submissions by t h e a u t h o r s ) . "The s y s t e m a t i c

change i n o u t p u t p r o b a b i l i t y as a f u n c t i o n o f h a v i n g more and more exp e r i e n c e w i t h an i n p u t i s c a l l e d p r o b a b i l i t y l e a r n i n g . S t u d i e s i n t h i s a r e a have used e i t h e r a d u l t c o l l e g e s t u d e n t s as r a t s

... o r

goldfish

...I'

... o r

subhuman species such

What t h i s d e f i n i t i o n shows i s t h a t t h e

c o n t e x t - f r e e t a s k i n t e r p r e t a t i o n was dominant f o r a l o n g t i m e . The s u b j e c t s were c o n f r o n t e d w i t h d e v i c e s ( b l a c k boxes) and were asked t o p r e d i c t t h e f l a s h i n g on o f lamps e t c . w i t h o u t b e i n g i n f o r m e d about t h e i n s i d e o f t h e apparatus and t h u s about t h e n a t u r e o f t h e t a s k . The i m p l i c a t i o n s o f t h i s t a s k i n t e r p r e t a t i o n were p o i n t e d o u t r a t h e r e a r l y by FLOOD (1954). Even a t t h e b e g i n n i n g o f r e s e a r c h i n t o p r o b a b i l i t y l e a r n i n g i t was n o t e d i n f a c t t h a t s u b j e c t s , i n t a s k s w i t h two a l t e r n a t i v e s t i m u l i , do n o t a l ways choose t h e more f r e q u e n t a l t e r n a t i v e . I n s t e a d o f t h i s , t h e y a d j u s t t h e i r response f r e q u e n c i e s t o t h e f r e q u e n c i e s o f t h e s t i m u l i . T h i s response b e h a v i o r i s denoted as p r o b a b i l i t y matching. Comparing s u b j e c t performance by t h e o p t i m a l p r e d i c t i o n s t r a t e g y i n s t a t i o n a r y random sequences i t was e v a l u a t e d as b e i n g sub-optimal o r f a l l a c i o u s . W i t h r e g a r d t o t h i s e v a l u a t i o n , we should l i k e t o quote two p o i n t s r e v e a l i n g t h e main problems associated w i t h research i n t o p r o b a b i l i t y learning.

1) Tasks a r e c o g n i t i v e demands and t h u s co-determined by an i n d i v i d u a l I s a c t i v e r e c o n s t r u c t i o n . One and t h e same s i t u a t i o n s may be p e r c e i v e d v e r y

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d i f f e r e n t l y . As GOODNOW (1955) was a b l e t o show, t h e d e v i a t i o n o f an i n d i v i d u a l ' s performance f r o m t h e o p t i m a l response s t r a t e g y may o f t e n be e x p l a i n e d by assuming t h a t s u b j e c t s do n o t r e c o g n i z e random sequences as such, b u t i n s t e a d t r y t o d i s c o v e r and p r e d i c t r e g u l a r i t i e s and cont i n g e n c i e s i n t h e sequence by means o f a p r o b l e m - s o l v i n g s t r a t e g y ( c f . WEIR, 1964). 2 . I n d i v i d u a l s t a c k l i n g p r o b a b i l i t y l e a r n i n g t a s k s and h y p o t h e s i z i n g t h a t t h e s e a r e "genuine random sequences", g e n e r a l l y a r e n o t so wrong i n t h e i r assumption. T h i s i s because i n t h e e x p e r i m e n t a l d e s i g n o f probab i l i t y l e a r n i n g t a s k s n o t a l l sequences a r e u s u a l l y p e r m i t t e d , b u t o n l y those which "make a chance i m p r e s s i o n " . Under c e r t a i n c o n d i t i o n s , t h i s may have t h e consequence t h a t

i t i s some-

t i m e s b e t t e r n o t t o s e l e c t themore f r e q u e n t a l t e r n a t i v e i n 100 p e r c e n t o f t h e cases. Now l e t us r e t u r n t o o u r t h e s i s . The sequences o f s t i m u l i i n p r o b a b i l i t y l e a r n i n g t a s k s meet i m p o r t a n t c r i t e r i a o f random sequences i n t h e o b j e c t i v i s t sense. T h i s i s e s p e c i a l l y t r u e w i t h r e g a r d t o p r o b a b i l i t y l e a r n i n g t a s k s i n developmental r e s e a r c h . To i n f e r from t h i s t h a t t h e s u b j e c t s p e r c e i v e t h e t a s k s i n t h e same o b j e c t i v i s t way i s i n a d m i s s i b l e . As we have a l r e a d y p o i n t e d o u t , t h e s u b j e c t s i n t h e e x p e r i m e n t a l s i t u a t i o n s were o f t e n n o t i n f o r m e d about t h e n a t u r e o f t h e t a s k and about t h e way e v e n t s were b e i n g generated. Hence,,determinstic

reconstructions o f the

t a s k s w e r e ' n o t unreasonable. T h i s p o s i t i o n i s t a k e n up f o r i n s t a n c e by

W E I R (1964). We mentioned above t h a t t h e r e i s o n l y one work i n t h i s f i e l d t h a t r e f e r s t o t h e s u b j e c t i v i s t concept o f p r o b a b i l i t y . As r e p o r t e d by Lee ROSS (1980), one o f h i s s t u d e n t s (JOSEPHSON. 1977) presented c h i l d r e n ' s d e s c r i p t i o n s o f persons a c t i n g i n t h e "Boy meets Dog" s t o r y . F i r s t , c h i l d r e n r e c e i v e d a d e s c r i p t i o n o f a person o f t h e f o l l o w i n g k i n d : " T h i s i s a p i c t u r e o f Johnny. (E shows S a photograph o f a c h i l d . ) When

ACQUISITION OF THE PROBABILITY CONCEPT

Johnny goes t o t h e p a r k w i t h h i s mother, he d o e s n ' t go on t h e h i g h s l i p p e r y s l i d e , because h e ' s scared. When h e ' s a t school, he d o e s n ' t p l a y w i t h k i d s who a r e b i g g e r t h a n he i s , because sometimes t h e y push him around. Johnny i s a l s o scared o f t h e d a r k . " (JOSEPHSON,1977, quoted a c c o r d i n g t o ROSS, 1981, p. 30) A f t e r that, c h i l d r e n got a d e s c r i p t i o n o f a s i t u a t i o n : "Now, h e r e ' s a p i c t u r e o f a dog. (E shows S a photograph o f a l a r g e German Shepherd.) Suppose Johnny was p l a y i n g by h i m s e l f i n f r o n t o f h i s house one day, and t h i s dog i n t h e p i c t u r e came r u n n i n g up t o him and barked and growled p r e t t y l o u d l y . " (JOSEPHSON,1977,

q u o t i n g f r o m ROSS, 1981, p. 30)

There were f o u r v e r s i o n s o f t h e s t o r y . A f t e r each v e r s i o n , t h e c h i l d r e n were asked t o g i v e a judgment about t h e expected r e a c t i o n o f t h e a c t o r , and " t h e n t o i n d i c a t e how s u r e t h e y were about t h e i r answer." (ROSS,1981, p.21) I t seems obvious t h a t t h i s t y p e o f t a s k i s something q u i t e d i f f e r e n t f r o m

c o n t a i n e r experiments and p r o b a b i l i t y l e a r n i n g t a s k s . S t o r i e s l i k e "Boy meets Dog" have n o t h i n g more t o do w i t h t h e p r o b a b i l i t y concept, i n t h e o b j e c t i v i s t sense because no chance mechanism a t a l l can be determined. 3. Developmental theories and research approaches

3.1 The cognitive-developmental approach of PIAGET and INHELDER

The most comprehensive r e s e a r c h approach d e a l i n g w i t h t h e development o f t h e p r o b a b i l i t y concept i s w i t h o u t doubt t o be found w i t h i n t h e c o g n i t i v e developmental t h e o r y o f Jean PIAGET (1950). More t h a n t h i r t y y e a r s ago, PIAGET and INHELDER (1951) s y s t e m a t i c a l l y analysed t h e p r o b a b i l i t y concept i n c h i l d r e n and f o r m u l a t e d a t h e o r y t o e x p l a i n i t ' s development. I t i s w e l l known t h a t , a c c o r d i n g t o PIAGET, c o g n i t i v e development c o n s i s t s

o f a s e q u e n t i a l t r a n s f o r m a t i o n process d u r i n g which e l e m e n t a r y s e n s o r i motor schemata a r e r e s t r u c t u r e d i n t o p r o g r e s s i v e l y more complex c o g n i t i v e s t r u c t u r e s , i n i t i a l l y i n t o p r e - o p e r a t i v e and t h e n i n t o o p e r a t i v e ones.

297

298

DEVELOPMENT AND EPISTEMOLOGY

W i t h i n PIAGET's t h e o r y o p e r a t i v e c o g n i t i v e s t r u c t u r e s c o n s t i t u t e t h e most i m p o r t a n t developmental p r e r e q u i s i t e f o r t h e emergence o f logical-mathemat i c a l concepts. T h i s i s because t h e growing c h i l d i s o n l y by means o f l o g i c a l o p e r a t i o n s a b l e t o d i s c o v e r and t o r e c o n s t r u c t t h e q u a n t i t a t i v e l o g i c a l r e l a t i o n s i n h e r e n t i n problems. T h i s a l s o a p p l i e s t o t h e c h i l d ' s c o n s t r u c t i o n o f t h e p r o b a b i l i t y concept. According t o PIAGET and INHELDER t h e v e r y b a s i s f o r t h e a c q u i s i t i o n o f t h i s concept i s t h e a b i l i t y t o d i s t i n g u i s h between chance and n e c e s s i t y

.

The p r e - o p e r a t i v e c h i l d i s u n a b l e t o make t h i s d i s t i n c t i o n . A t t h i s s t a g e o f development t h e c h i l d s t i l l l a c k s t h e a b i l i t y t o c o n s t r u c t l o g i c a l r e l a t i o n s such as cause and e f f e c t o r s i m i l a r d e d u c t i v e r e l a t i o n s , t h a t a r e necessary f o r an u n d e r s t a n d i n g o f why events o c c u r . I n o t h e r words, t h e p r e - o p e r a t i v e c h i l d has n o t y e t developed

t h e c o g n i t i v e frame on t h e

b a s i s o f which he/she can g r a s p t h e d i f f e r e n c e between necessary ( p r e d i c t a b l e ) and random ( n o n - p r e d i c t a b l e ) e v e n t s . Random e v e n t s , i n h i s / h e r u n d e r s t a n d i n g a r e s t i l l s u b j e c t t o t h e same d e t e r m i n i s t i c o r d e r , which a l s o c o n t r o l s necessary events. I n o r d e r t o be a b l e t o understand t h e n e c e s s i t y v e r s u s t h e randomness o f e v e n t s , t h e c h i l d must have o r g a n i z e d c o g n i t i v e s t r u c t u r e s i n t o c o n c r e t e o p e r a t i o n s o f t h i n k i n g . These enable t h e c h i l d t o comprehend t h e c o n t i n g e n c i e s o f d e t e r m i n i s t i c e v e n t s by means o f b u i l d i n g l o g i c a l r e l a t i o n s , and t h u s t o r e c o g n i z e t h e l o g i c a l n e c e s s i t y o f t h e i r occurence. On t h e o t h e r hand, t h e c o n t i n g e n c i e s o f s t o c h a s t i c a l events cannot be r e presented i n c o n c r e t e - o p e r a t i v e s t r u c t u r e s because o f t h e i r i r r e v e r s i b i 1 i t y . Hence, t h e c h i l d cannot i n f e r t h e i r occurrence, and b y r e c o g n i z i n g t h e d i s t i n c t p r e d i c t a b i l i t y o f necessary v e r s u s random events he/she becomes a b l e t o d i f f e r e n t i a t e between them. I n a f u r t h e r developmental s t e p beyond t h i s conceptual d i s t i n c t i o n , t h e c h i l d a c q u i r e s t h e a b i l i t y t o make chance events o p e r a t i v e l y c a l c u l a b l e . The c o n c r e t e - o p e r a t i v e c h i l d may a l r e a d y be a b l e t o make some q u a n t i t a t i v e e s t i m a t e s o f p r o b a b i l i t y , b u t o n l y , when t h e number o f events i s l i m i t e d , and when, s i m u l t a n e o u s l y , t h e p r o p o r t i o n between f a v o u r a b l e and non-favoura b l e cases can be c o n s t r u c t e d by a d d i t i v e l y 1 i n k i n g s i m p l e p a r t - w h o l e - r e l a t i o n s h i p s . I n o r d e r t o overcome t h e s e l i m i t a t i o n s t h a t a r e s e t upon con-

ACQUISITION OF THE PROBABILITY CONCEPT

299

C r e t e o p e r a t i o n s i n p r o b a b i l i t y c a l c u l u s , t h e c h i l d has t o be a b l e t o c o n s t r u c t and r e p r e s e n t t h e t o t a l i t y o f a l l events and t o t a k e m u l t i p l e r e l a t i o n s i n t o account when he/she i s c a l c u l a t i n g p r o p o r t i o n s . The p r e r e q u i s i t e s f o r d o i n g t h i s a r e a c q u i r e d i n t h e f o r m a l - o p e r a t i v e stage, d u r i n g which t h e c o n r e t e o p e r a t i o n s a r e c o - o r d i n a t e d and t r a n s f o r m e d i n t o a c o m p l e t e l y c l o s e d and r e v e r s i b l e system o f f o r m a l - l o g i c a l o p e r a t i o n s . 3.2

FISCHBEIN's learning developmental approach

A c c o r d i n g t o FISCHBEIN (1975, 1976), PIAGET and INHELDER's s t a g e model o f t h e development o f t h e p r o b a b i l i t y concept has two s e r i o u s d e f i c i e n c i e s : F i r s t , t h e f i n d i n g s on t h e growth o f p r o b a b i l i t y l e a r n i n g ( c f . GOULET & GOODWIN,

1970), p a r t i c u l a r l y t h o s e made i n t h e s i x t i e s , cannot be i n t e -

g r a t e d i n t o t h i s model. Secondly, i n FISCHBEIN's o p i n i o n , PIAGET and INHELDER u n d e r e s t i m a t e t h e importance o f t h e l e a r n i n g processes l e a d i n g t o t h e f o r m a t i o n o f t h e p r o b a b i l i t y concept. As an a l t e r n a t i v e , FISCHBEIN f o r m u l a t e s a l e a r n i n g - d e v e l o p m e n t a l approach t o t h e a c q u i s i t i o n o f t h e p r o b a b i l i t y concept. T h i s approach i s based on s t u d i e s on t h e growth o f p r o b a b i l i t y l e a r n i n g t h a t r e v e a l t h a t c h i l d r e n a l r e a d y show p r o b a b i l i t y matching a t preschool age. T a k i n g t h e s e f i n d i n g s i n t o account, FISCHBEIN p o s t u l a t e s t h a t c h i l d r e n , even i n t h e p r e - o p e r a t i v e stage, have a p r e conceptual u n d e r s t a n d i n g o f b o t h r e l a t i v e f r e q u e n c i e s and p r o b a b i l i t i e s a t t h e i r d i s p o s a l . T h i s pre-conceptual u n d e r s t a n d i n g i s founded on primary

i n t u i t i o n s . I n FISCHBEIN's o p i n i o n p r i m a r y i n t u i t i o n s a r e a t y p e o f i m p l i c i t knowledge t h a t grows spontaneously o u t o f a c h i l d ' s everyday experience, f o r example from t h e p e r c e p t i o n o f s t o c h a s t i c a l e v e n t s i n everyday l i f e s i t u a t i o n s . As t h e ' c h i l d grows o l d e r p r i m a r y i n t u i t i o n s t h a t , a c c o r d i n g t o FISCHBEIN, a r e developed w i t h r e g a r d t o t h e concepts o f r e l a t i v e f r e quency, chance and probability a r e s u c c e s s i v e l y t r a n s f o r m e d i n t o an operat i v e concept o f p r o b a b i l i t y . T h i s t r a n s f o r m a t i o n process i s n o t t h e spontaneous and q u a s i - a u t o m a t i c r e s u l t o f a s e l f - r e g u l a t i n g process o f growth, b u t r a t h e r i s mediated by i n s t r u c t i o n a l i n t e r v e n t i o n a t school ( f o r i n s t a n c e by e x p l a i n i n g t o t h e c h i l d t h e b a s i c i d e a s o f p r o b a b i l i t y t h e o r y , by t e a c h i n g s t r a t e g i e s and r u l e s o f p r o b a b i l i t y c a l c u l u s , e t c . ) . W i t h o u t such p u r p o s e f u l i n t e r v e n t i o n , an o p e r a t i v e concept o f p r o b a b i l i t y cannot be developed, even i n a d u l t s .

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3.3 Information processing approaches

W i t h i n t h e frame o f i n f o r m a t i o n p r o c e s s i n g t h e o r i e s , t h r e e r e c e n t approaches can be i d e n t i f i e d t h a t a r e independent developments based on heterogeneous t h e o r i e s of c o g n i t i v e development. They p r o v i d e a much more r a d i c a l a l t e r n a t i v e t o t h e c o g n i t i v e - d e v e l o p m e n t a l s t a g e model t h a n FISCHBEIN's l e a r n i n g developmental approach. These approaches a r e a l l b u i l t on a p r o p o s a l t h a t i s c o n t r a r y t o PIAGET and INHELDER. According t o t h i s p r o p o s a l t h e a c q u i s i t i o n o f p r o b a b i l i t y competencies i s n o t a f u n c t i o n o f t h e development o f c o g n i t i v e s t r u c t u r e s r e s p . t h e i r o r g a n i z a t i o n i n t o o p e r a t i v e systems, b u t r a t h e r depends, on b o t h t h e growth o f i n f o r m a t i o n p r o c e s s i n g space, and on t h e changing o f i n f o r m a t i o n p r o c e s s i n g and problem s o l v i n g s t r a t e g i e s t h a t occurs d u r i n g t h e course o f development. W i t h i n t h e frame o f t h e i n f o r m a t i o n p r o c e s s i n g approaches, a comprehensive and u n i t a r y t h e o r y o f t h e a c q u i s i t i o n o f t h e p r o b a b i l i t y concept has n o t been f o r m u l a t e d u n t i l now. There are, however, some s e p a r a t e s t u d i e s t h a t f o c u s on t h e developmental a n a l y s i s o f s t r a t e g i e s and t h e a b i l i t i e s i n s o l v i n g p r o b a b i l i t y problems. SCARDAMALIA (1977) attempted t o show h a t t h e a b i l i t y t o cope w i t h combinat o r i a l problems i s a f u n c t i o n o f t h e i n t e r a c t i o n between a s u b j e c t ' s a v a i l a b l e i n f o r m a t i o n p r o c e s s i n g space and t h e i n f o r m a t i o n p r o c e s s i n g demands o f a r e s p e c t i v e problem t a s k . S u b j e c t s w i t h d i f f e r e n t i n f o r m a t i o n p r o c e s s i n g spaces were asked t o work on c o m b i n a t o r i a l problems

which,

a c c o r d i n g t o t h e r e s u l t s o f a t a s k a n a l y s i s , were designed i n such a way t h a t , holding t h e i r l o g i c a l s t r u c t u r e i n v a r i a n t , t h e i r information proc e s s i n g demands v a r i e d . R e s u l t s showed t h a t even c o n c r e t e - o p e r a t i v e c h i l d r e n a r e r e a d y t o s o l v e c o m b i n a t o r i a l problems p r o v i d e d t h a t t h e t a s k demands do n o t exceed t h e i r i n f o r m a t i o n p r o c e s s i n g space. C o n t r a r y t o PIAGET and INHELDER's s u p p o s i t i o n , SCARDAMALIA's f i n d i n g s i n d i c a t e t h a t f o r m a l - o p e r a t i v e t h i n k i n g i s n o t a p r e r e q u i s i t e f o r b e i n g a b l e t o cope w i t h c o m b i n a t o r i a l problems. BRAINERD (1981) proposed

t h a t t h e development o f p r o b a b f l i t y judgments

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i n container tasks i s adequately

e x p l a i n e d as b e i n g a f u n c t i o n o f t h e

growth o f w o r k i n g memory. Taking i n t o account t h e a v a i l a b l e evidence on t h e m n e s t i c r e p r e s e n t a t i o n o f f r e q u e n c y d a t a r e p o r t e d by BOGARTZ (1969)

,

BRAINERD a t f i r s t designed a general model o f t h e i n f o r m a t i o n p r o c e s s i n g

t a k i n g p l a c e i n w o r k i n g memory d u r i n g t h e s o l v i n g o f c o n t a i n e r t a s k s . T h i s model c o n t a i n s a t o t a l o f f o u r s t o r a g e and t h r e e p r o c e s s i n g o p e r a t i o n s . I n a second s t e p , he m o d i f i e d t h i s model w i t h r e g a r d t o s e v e r a l v a r i a t i o n s o f t h e container task ( f o r instance w i t h regard t o tasks with-versusw i t h o u t replacement), i n such a way t h a t s p e c i f i c e f f e c t s o f each o f t h e f o u r s t o r a g e and t h e t h r e e p r o c e s s i n g o p e r a t i o n s on s u b j e c t ' s responses c o u l d be p r e d i c t e d and t e s t e d by a n a l y s i n g t h e f r e q u e n c y o f h i t s and f a i l u r e s observed.

S t a r t i n g f r o m these p r e d i c t i o n s , BRAINERD conducted a s e r i e s o f

t w e l v e experiments w i t h s i x v a r i a t i o n s o f t h e c o n t a i n e r t a k s . He used p r e school and p r i m a r y school c h i l d r e n . C h i l d r e n ' s responses i n d i c a t e d t h a t a l l age groups d i d w e l l i n s t o r i n g f r e q u e n c y d a t a . However, younger c h i l d r e n f a i l e d t o r e t r i e v e t h i s c o r r e c t l y s t o r e d d a t a . I n o t h e r words, t h e wrong p r e d i c t i o n s observed were p r i m a r i l y due t o r e t r i e v a l e r r o r s . As BRAINERD proposes, t h e s e r e t r i e v a l e r r o r s a r e n o t dependent on a l a c k o f s p e c i f i c c o g n i t i v e s t r u c t u r e s , b u t r a t h e r on l i m i t a t i o n s o f memory space. T h e r e f o r e , i n h i s o p i n i o n t h e decrease o f f a i l u r e s i n o l d e r c h i l d r e n i s s u f f i c i e n t l y e x p l a i n e d as a f u n c t i o n o f t h e i r growing memory space. F i n a l l y , p r o c e e d i n g f r o m h i s ruZe-assessment-approach,

SIEGLER (1981)

attempted t o prove t h a t i n f o r m a t i o n p r o c e s s i n g r u l e s f o r s o l v i n g p r o b a b i l i t y t a s k s such as c a r d deck experiments change i n t h e c o u r s e o f a c h i l d ' s development. SIEGLER p o s t u l a t e s t h a t c h i l d r e n a p p l y f o u r r u l e s w i t h an i n c r e a s i n g degree o f c o m p l e x i t y , b o t h when w o r k i n g on p r o b a b i l i t y t a s k s i n t h e c a r d deck experiment and when s o l v i n g

n o n - p r o b a b i l i s t i c problems.

These r u l e s a r e d e f i n e d i n t h e f o l l o w i n g way. Rule I: Only t h e f r e q u e n c i e s o f t h e f a v o u r a b l e elements a r e ovserved. A c c o r d i n g l y , t h e deck c o n t a i n i n g t h e l a r g e r number o f f a v o u r a b l e elements i s chosen. Rule 11: Rule I i s adhered t o e x c e p t when t h e f r e q u e n c y o f t h e p r e f e r r e d elements i s t h e same f o r b o t h decks. I n t h i s case, t h e f r e q u e n c y o f t h e n o n - f a v o u r a b l e elements i s a l s o t a k e n i n t o account.

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Rule 111: The d i f f e r e n c e between f a v o u r a b l e and n o n - f a v o u r a b l e elements i s determined f o r each deck, and t h e deck w i t h t h e l a r g e r a l g e b r a i c number i s chosen. Rule IV:

The p r o p o r t i o n s o f f a v o u r a b l e versus n o n - f a v o u r a b l e elements i n b o t h decks a r e c a l c u l a t e d .

According t o SIEGLER t h e growing c h i l d a c q u i r e s t h e s e f o u r r u l e s s t e p b y s t e p . As t h e c o g n i t i v e development i s p r o c e e d i n g t h e y a r e i n t e g r a t e d i n t o i n c r e a s i n g l y complex systems o f r u l e s analogous t o b i n a r y d e c i s i o n - t r e e models. Which r u l e i s a p p l i e d i n s o l v i n g a p r o b a b i l i t y problem w i l l depend b o t h on t h e c o m p l e x i t y o f a s u b j e c t ' s system o f r u l e s , and on t h e s t r u c t u r e o f t h e t a s k t o be s o l v e d (e.g.,

on t h e r e l a t i v e f r e q u e n c i e s o f f a v o u r a b l e

versus n o n - f a v o u r a b l e elements). I n o r d e r t o t e s t t h e v a l i d i t y o f h i s assumptions, SIEGLER designed a t o t a l o f s i x deck t a s k s . Favourable v e r s u s n o n - f a v o u r a b l e elements were combined i n such a manner t h a t t h e percentage o f h i t s f o r any o f t h e f o u r r u l e s c o u l d be p r e d i c t e d w i t h i n a l l t a s k s w i t h r e s p e c t t o each o f t h e f o u r r u l e s . These p r e d i c t i o n s were t h e n compared w i t h t h e t a s k responses o f 3- t o 20-year-old

subjects.

S u b j e c t ' s responses agreed v e r y w e l l w i t h t h e p r e d i c t i o n s made f o r Rules and IV.

I

However, t h e y o n l y p a r t l y f i t t e d t h e response p a t t e r n s t h a t were

expected f o r Rules 111 and 11. Furthermore, i t was q u i t e o b v i o u s t h a t p r e school c h i l d r e n , w i t h one e x c e p t i o n , were a p p l y i n g o n l y Rule I, whereas even t h e 8-year-olds,

and more pronouncedly t h e o l d e r s u b j e c t s , made t h e i r

p r o b a b i l i t y judgments p r e d o m i n a n t l y a c c o r d i n g t o Rule I V and p a r t l y t o Rule 111, t h u s q u a s i s k i p p i n g Rule 11. 4.

Discussion

I n a s t u d y o f t h e t h e o r i e s and r e s e a r c h approaches o u t l i n e d above one can f i n d o n l y s l i g h t disagreement as t o t h e age p e r i o d w i t h i n which c h i l d r e n s h o u l d be r e a d y t o work s u c c e s s f u l l y w i t h p r o b a b i l i t y t a s k s . However, t h e r e i s much d i v e r g e n c e o f o p i n i o n as t o how c h i l d r e n cope w i t h such t a s k s .

ACQUISITION OF THE PROBABILITY CONCEPT

303

The main i s s u e has been what w e i g h t s h o u l d be assigned t o conceptual knowl e d g e versus s t r a t e g i c a b i l i t i e s w i t h r e g a r d t o t h e s o l v i n g o f p r o b a b i l i t y problems. PIAGET and INHELDER d e f i n i t e l y presume t h a t o n l y by a c q u i r i n g t h e p r o b a b i l i t y concept i n t h e course o f t h e development o f o p e r a t i v e - c o g n i t i v e s t r u c t u r e s , w i l l t h e s u b j e c t be enabled t o s o l v e t h e p r o b a b i l i t y t a s k s i n an adequate way, i . e . by means o f l o g i c a l and c o m b i n a t o r i a l t h i n k i n g . The v a l i d i t y o f t h i s assumption, however, seems t o be q u e s t i o n a b l e . A t f i r s t , i t i s d i s p u t e d by t h e above mentioned f i n d i n g s o f SCARDAMALIA (1977) which

c l e a r l y i n d i c a t e t h a t c o r r e c t s o l u t i o n s o f c o m b i n a t o r i a l problems do n o t n e c e s s i t a t e formal o p e r a t i v e t h i n k i n g . Secondly, ROSS and HOEMANN ( 1 9 7 5 ) were a b l e t o show t h a t f o r m a l o p e r a t i v e a d o l e s c e n t s o n l y r a r e l y a p p l y c o m b i n a t o r i a l s t r a t e g i e s when w o r k i n g on p r o b a b i l i t y t a s k s i n a c o n t a i n e r r e l a t e d experiment, b u t s t i l l o f t e n use s t r a t e g i e s which c h a r a c t e r i z e the t h i n k i n g o f t h e concrete-operative c h i l d . A s i m i l a r problem-solving b e h a v i o r may be shown by a d u l t s i n p r o p o r t i o n a l r e a s o n i n g w i t h i n e v e r y day c o n t e x t s ( c f . CAPON & KUHN, 1979). These and o t h e r f i n d i n g s s t r o n g l y suggest t h a t even when competencies a r e a v a i l a b l e f o r f o r m a l l y c o r r e c t s o l u t i o n s , these competencies may n o t be r e a l i z e d immediately, due t o m i s s i n g s o l u t i o n s t r a t e g i e s . Hence, t h e y v e r y s t r o n g l y emphasize t h e r o l e o f s t r a t e g i c a b i l i t i e s i n s o l v i n g p r o b a b i l i s t i c problems, and t h e r e f o r r t h e y p o i n t t o t h e n e g l e c t o f j u s t t h e s e a b i l i t i e s by PIAGET and INHELDER. Additionally,

a l o t o f evidence seems t o i n d i c a t e t h a t t h e development

o f t h e p r o b a b i l i t y concept may be enhanced by t h e a s k i n g o f c h i l d r e n t o work on p r o b a b i l i s t i c problems and a l s o by t h e s t r a t e g i c a l knowledge which t h e y a c q u i r e d u r i n g problem s o l v i n g . Taken i n combination, t h e r e s e a r c h f i n d i n g s s u b m i t t e d u n t i l now s t r o n g l y s u p p o r t FISCHBEIN's approach a c c o r d i n g t o which t h e a c q u i s i t i o n o f an operat i v e p r o b a b i l i t y concept i s promoted and f o s t e r e d by t e a c h i n g c h i l d r e n r u l e s and s t r a t e g i e s o f p r o b a b i l i t y c a l c u l u s . Does t h i s a l s o mean, as BRAINERD and SIEGLER propose, t h a t c h i l d r e n may o n l y cope s u c c e s s f u l l y w i t h p r o b a b i l i t y t a s k s by means o f a c q u i r e d s t r a t e g i e s , t h a t i s w i t h o u t knowledge? We do n o t t h i n k so. F i r s t o f a l l , one has t o answer t h e q u e s t i o n as t o how a s u b j e c t w o r k i n g on a c o n c r e t e t a s k may accomplish t h e s e l e c t i o n o f an o p t i m a l r u l e o r s t r a t e g y . According t o S I E G -

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LER, t h e s t r u c t u r e o f a t a s k p l a y s an i m p o r t a n t p a r t i n t h i s r e g a r d . B u t does t h i s n o t a l s o mean t h a t t h e s u b j e c t must a n a l y t i c a l l y p e n e t r a t e a t a s k and generate hypotheses about i t ' s s t r u c t u r e , f o r i n s t a n c e hypotheses about the s i g n i f i c a n c e o f r e l a t i v e differences o f frequencies? I f t h i s i s true, i n o r d e r t o s o l v e a p r o b a b i l i s t i c problem t h e s u b j e c t must have u s a b l e conc e p t u a l knowledge. SIEGLER's f i n d i n g t h a t c h i l d r e n s k i p Rule 11, i n d e e d c o n t r a d i c t s t h e assumption t h a t t h e s u b j e c t b l i n d l y proceeds a l o n g t h e d e c i s i o n t r e e o f h i s own r u l e - s y s t e m w i t h o u t any conceptual knowledge. Rather, i t s u p p o r t s t h e s u p p o s i t i o n t h a t n o t o n l y t h e s e l e c t i o n , b u t a l s o t h e a c q u i s i t i o n o f t h e r u l e s i s mediated by t h e a c q u i s i t i o n o f c o n c e p t u a l know1edge. The r e s u l t s o f BRAINERD's s t u d y p o i n t i n t h e same d i r e c t i o n . BRAINERD q u i t e e m p h a t i c a l l y propose; t h a t t h e a c t u a l i z a t i o n o f a p p r o p i a t e r e t r i e v a l s t r a t e g i e s i n p r o b a b i l i t y judgments depends e x c l u s i v e l y on t h e a g e - s p e c i f i c g r o w t h o f memory space. I n o u r o p i n i o n , however, t h i s c o n c l u s i o n i s t o h a s t y . One has t o t a k e i n t o account t h e f a c t t h a t b o t h t h e a g e - s p e c i f i c e x t e n s i o n o f memory space (CASE, 1978) and t h e a c t u a l i z a t i o n o f a p p r o p r i a t e r e t r i e v a l s t r a t e g i e s (KOBASIGAWA,

1977) a r e c l o s e l y l i n k e d t o t h e degree o f

o r g a n i z a t i o n and t h e encoding f o r m a t o f s t o r e d i n f o r m a t i o n . B o t h s h o u l d be c o n s i d e r e d t o be a f u n c t i o n o f t h e a c q u i s i t i o n o f semantic knowledge. Therefore, one cannot e x c l u d e t h e p o s s i b i l i t y t h a t a g e - s p e c i f i c d i f f e r e n c e s i n p r o c e s s i n g f r e q u e n c y d a t a a r e a l s o due, c o n t r a r y t o BRAINERD's supposit i o n , t o t h e corresponding differences i n the r e c o g n i t i o n o f s p e c i f i c t a s k c h a r a c t e r i s t i c s as b e i n g r e l e v a n t f o r t h e o r g a n i z a t i o n o f f r e q u e n c y d a t a . For example, t h e conceptual knowledge about t h e d i f f e r e n c e between d r a w i n g w i t h , and w i t h o u t replacement s h o u l d d e f i n i t e l y govern t h e s e l e c t i o n o f s t o r i n g and r e t r i e v a l s t r a t e g i e s . I f o u r r e - i n t e r p r e t a t i o n o f BRAINERD's and SIEGLER's f i n d i n g s i s c o r r e c t ,

i t must be concluded t h a t b o t h conceptual knowledge and s t r a t e g i c a b i l i t i e s

are p r i n c i p l e requirements f o r coping successfully w i t h p r o b a b i l i s t i c problems. Moreover, as s t a t e d above, i t

has t o be t a k e n f o r g r a n t e d t h a t

b o t h t h e knowledge and t h e s t r a t e g i c components m u t u a l l y i n f l u e n c e and s t i m u l a t e one a n o t h e r d u r i n g t h e course o f development.

ACQUISITION OF THE PROBABILITY CONCEPT

305

Regarding t h e l a t t e r suggestion, t h e r e i s as y e t o n l y one p i e c e o f evidence, but

i t i s v e r y s i g n f i c a n t . I n a c a r e f u l l y conducted i n v e s t i g a t i o n NOEL-

TING (1980) was a b l e t o show t h a t t h e development o f t h e concept o f p r o p o r t i o n may be regarded as a h i e r a r c h i c a l process i n which i n c r e a s i n g l y complex r u l e s a r e e s t a b l i s h e d by which r e l a t i o n s a r e c o n s t r u c t e d between t h o s e elements t h a t c o n s t i t u t e p r o p o r t i o n s . The degree o f c o m p l e x i t y o f t h e r u l e s which a r e a t a c h i l d ' s d i s p o s a l a t a s p e c i f i c p o i n t o f h i s / h e r development a r e supposed t o depend on t h e p a r t i c u l a r s t a t u s o f h i s / h e r cogn i t i v e s t r u c t u r e s r e s p . conceptual knowledge, whereas, on t h e o t h e r hand, t h e a p p l i c a t i o n o f t h e r u l e s s t i m u l a t e s and promotes t h e development o f conceptual knowledge. We t h i n k t h a t NOELTING's s t u d y marks o u t t h e b a s i c l i n e s o f a r e s e a r c h s t r a t e g y which may h e l p us t o o b t a i n n o t o n l y more p r e c i s e i n f o r m a t i o n about t h e presumed i n t e r a c t i o n s between t h e a c q u i s i t i o n o f conceptual knowledge and s t r a t e g i c a b i l i t i e s , b u t a l s o h e l p t o answer t h e q u e s t i o n o f a t which p o i n t i n t h e course o f development t h e c h i l d w i l l be a b l e t o s o l v e which p r o b a b i l i t y problems i n which way. Such i n f o r m a t i o n would n o t o n l y be o f general i n t e r e s t f o r developmental psychology, b u t c o u l d a1 so be a p p l i e d t o mathematics e d u c a t i o n . Beyond t h e d i s p u t e c o n c e r n i n g t h e impact o f conceptual knowledge and/or s t r a t e g i c a b i l i t i e s t h e r e i s a f u r t h e r more fundamental t h e o r e t i c a l c o n t r o v e r s y . I t concerns t h e q u e s t i o n as t o whether t h e development o f p r o b a b i l i t y should be conceived o f as a c o n t i n u o u s o r d i s c o n t i n u o u s process. W h i l e

PIAGET and INHELDER, and l a t e r HOEMANN and ROSS (1982)

and o t h e r s have

c o n s i d e r e d t h i s process t o be a d i s c o n t i n u o u s one, FISCHBEIN proposes t h a t t h e p r o b a b i l i t y concept develops c o n t i n u o u s l y o u t o f t h e primary

intuitions t h a t a r e a c q u i r e d i n e a r l y c h i l d h o o d . These p o i n t s o f v i e w seem t o be i r r e c o n c i l a b l e . I f , however, t h e c o n c e p t o f p r o b a b i l i t y w i t h i n developmental psychology i s conceived t o be o f two k i n d s , j u s t l i k e t h e m a t h e m a t i c a l - p h i l o s o p h i c a l concept, t h e s e o p p o s i t e p o s i t i o n s may be made compatible. On t h e one s i d e i n t u i t i v e components o f P r o b a b i l i t y can be i d e n t i f i e d . On

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t h e o t h e r s i d e a concept o f p r o b a b i l i t y e x i s t s which i s founded on e x p l i c i t c a l c u l a t i o n s performed by f o r m a l o p e r a t i o n s . W i t h i n t h e stage approach o f PIAGET and INHELDER, o p e r a t i v e c o g n i t i v e s t r u c t u r e s a r e c o n s i d e r e d t o be a necessary developmental p r e r e q u i s i t e o f t h e p r o b a b i l i t y concept. T h i s conc e p t , as has been noted, i s m a n i f e s t e d i n c l a s s i c a l random s i t u a t i o n s l i k e u r n s and c o i n s . T h i s s t r o n g l y i n d i c a t e s t h a t PIAGET and INHELDER have an o b j e c t i v i s t i c , c a l c u l a b l e concept o f p r o b a b i l i t y i n mind. FISCHBEIN's p o s i t i o n on t h i s i s s u e i s l e s s c l e a r . There can be no d o u b t t h a t , l i k e PIAGET and INHELDER,, he c o n s i d e r s t h e o p e r a t i v e concept o f p r o b a b i l i t y t a u g h t by i n s t r u c t i o n a l i n t e r v e n t i o n i n school t o be an o b j e c t i v i s t i c one. However, how t h e primary probability

i n t u i t i o n s p o s t u l a t e d by him a r e t o be i n t e r -

p r e t e d i n t h i s r e s p e c t , remains an open q u e s t i o n . FISCHBEIN does n o t e x p l a i n h i s s t a n d p o i n t i n t h i s r e g a r d . He s i m p l y s t a t e s : " P r i m a r y i n t u i t i o n s a r e c o g n i t i v e a c q u i s i t i o n s which a r e d e r i v e d from t h e e x p e r i e n c e o f t h e i n d i v i dual

, without

p. 117)

t h e need f o r any s y s t e m a t i c i n s t r u c t i o n . ' '

(FISHBEIN, 1975,

S i m i l a r i l y Hans FREUDENTHAL (1975), a famous mathematics e d u c a t o r ,

r e g a r d s such i n t u i t i o n s t o be necessary p r e r e q u i s i t e s f o r t h e c o n s t i t u t i o n o f such mathematical o b j e c t s as t h e p r o b a b i l i t y concept. U n f o r t u n a t e l y , n e i t h e r FISCHBEIN n o r FREUDENTHAL d e s c r i b e t h e process f o r t h e emergence o f i n t u i t i v e probabiZity judgments, n o r do t h e y e x p l i c i t l y d e f i n e t h e i r structure. I f one c o n s i d e r s t h e f i n d i n g s w i t h r e g a r d t o c h i l d r e n ' s p r o b a b i l i t y l e a r n i n g f r o m which FISCHBEIN i n f e r s t h a t t h e r e a r e p r o b a b i l i t y i n t u i t i o n s i n e a r l y c h i l d h o o d , i t can be shown t h a t t h e s e i n t u i t i o n s presumably a r e based on s i m p l e judgmental h e u r i s i t i c s . As has been shown by WEIR (1964), younger c h i l d r e n a p p l y a s o - c a l l e d . s t r a t e g y o f m a x i m i z i n g r e i n f o r c e m e n t i n p r o b a b i l i t y l e a r n i n g t a s k s . I n o t h e r words, t h e y p r e f e r t h e response i t e m which i s most f r e q u e n t l y a s s o c i a t e d w i t h a c o n t i n g e n t o r non-cont i n g e n t r e i n f o r c e r . According t o WEIR, t h i s s t r a t e g y i s based on t h e s i m p l e h y p o t h e s i s t h a t t h e r e i n f o r c e r i s t h e s t i m u l u s which o c c u r s most f r e q u e n t l y . I n our o p i n i o n

t h i s response s t r a t e g y can be c o n s i d e r e d as a p r o t o -

f o r m o f a n a i v e judgmental h e u r i s t i c i n t h e sense o f KAHNEMAN and TVERSKY (1982). I n a s i m i l a r way t h e p r o b a b i l i t y judgments i n t h e s p i n n e r e x p e r i ments, a r e based, even w i t h o l d e r c h i l d r e n , on a r e a e s t i m a t e s ( c f . HOEMANN

307

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& ROSS, 1971 and HOEPIANY! & ROSS,l982), which may be governed by

simple

judgmental h e u r i s t i c s . Taking t h i s c o n c l u s i o n t o be t r u e , t h e primary probability i n t u i t i o n s o f younger c h i l d r e n a r e t o be conceived o f as most l i k e l y b e i n g a k i n d o f degree o f b e l i e f and hence a preconeeptuaZ s u b j e c t i v e p r o b a b i l i t y concept, and n o t t h e r e s u l t o f a conscious r e c o n s t r u c t i o n o f r e p e a t e d events. Consequently, t h e development o f t h e p r o b a b i l i t y concept may be d e s c r i b e d as a process o f d i f f e r e n t i a t i o n o f p r o g r e s s i v e l y more e l a b o r a t e d subjec-

t i v e probability judgments, which a r e based on i n c r e a s i n g l y more complex h e u r i s t i c s . A t t h e e n d p o i n t o f t h i s d i f f e r e n t i a t i o n process t h e heuristic-

based s u b j e c t i v e p r o b a b i l i t y judgments may be r e p l a c e d by l o g i c a l - m a t h e m a t i c a l ones, b u t o n l y i n s i t u a t i o n s o r t a s k s which e x h i b i t t h e above mentioned c r i t e r i a o f t h e o b j e c t i v i s t i c p r o b a b i l i t y concept. I t seems a t l e a s t p l a u s i b l e t h a t t h i s process s t a r t s w i t h i n t u i t i v e p r o b a b i l i t y estimates

based on s i m p l e m o t i v a t i o n a l , p e r c e p t u a l , and s i m i l a r j u d g -

mental h e u r i s t i c s . Taking t h i s developmental d i f f e r e n t i a t i o n as g r a n t e d , FISHBEIN'S i n t e r p r e t a t i o n t h a t t h e a c q u i s i t i o n o f t h e p r o b a b i l i t y conc e p t i s a c o n t i n u o u s process seems t o be j u s t i f i e d . A t t h e same t i m e , t h e o p p o s i t e p o s i t i o n t a k e n by PIAGET and INHELDER and o t h e r s i s o b v i o u s l y due t o a one-sided o b j e c t i v i s t i c i n t e r p r e t a t i o n o f t h e p r o b a b i l i t y

concept. L e t us c l o s e t h i s c o n t r i b u t i o n w i t h two q u e s t i o n s t h a t r e l a t e i t more c l o s e l y t o t h e c e n t r a l i s s u e o f t h i s symposium. The f i r s t q u e s t i o n concerns t h e i s s u e : f i y and when a ZogicaZ-mathematical concept of probability develops out of i n t u i t i v e judgmental heuristics? The q u e s t i o n as t o when w i l l n o t be answered by f o c u s s i n g upon a s p e c i f i c age p e r i o d , b u t r a t h e r by making e x p l i c i t two i m p o r t a n t developmental c o n d i t i o n s which, i n o u r o p i n i o n , have t o precede t h e emergence o f a l o g i c a l mathematical p r o b a b i l i t y concept. The f i r s t c o n d i t i o n i s t h e conceptual u n d e r s t a n d i n g o f chance and v a r i a b i l i t y as w e l l as t h e a b i l i t y t o d e t e r mine and c o n s t r u c t t h e event-space.

The second and more fundamental con-

d i t i o n i s t h a t o f r e f l e x i v i t y o f thinking. It i s only the a b i l i t y t o

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r e f l e c t on o u r own t h i n k i n g and t o r e c o g n i z e t h e i n s u f f i c i e n c y o f o u r intuitive

judgments t h a t enables us t o generate a l t e r n a t i v e and b e t t e r

s t r a t e g i e s f o r handling

problems. These p r o p o s i t i o n s may y i e l d an

answer t o t h e why. They may a l s o have i m p o r t a n t consequences f o r t h e t e a c h i n g o f p r o b a b i l i t y i n s c h o o l s . We t h i n k t h a t s h a k i n g s t u d e n t s n a i v e judgmental h e u r i s t i c s and making them aware o f t h e i r f a l l a c i e s by means o f s u i t a b l e problem t a s k s w i t h i n t h e school t e a c h i n g - p r o c e s s may be an a p p r o p r i a t e

-

and perhaps even an i n d i s p e n s i b l e

-

i n t e r v e n t i o n , s t r a t e g y by which t h e a c q u i s i t i o n o f an o p e r a t i v e p r o b a b i l i t y concept can be promoted. The second q u e s t i o n r e f e r s t o t h e r e l a t i o n s h i p between i n t u i t i v e j u d g mental h e u r i s t i c s and t h e m a t h e m a t i c a l - l o g i c a l p r o b a b i l i t y concept, and whether t h e r e a r e i m p l i c a t i o n s f o r an u n d e r s t a n d i n g o f d e c i s i o n s under u n c e r t a i n t y . The development o f t h e l o g i c a l - m a t h e m a t i c a l concept o f probability

-

a t l e a s t i n western c u l t u r e

-

i s fostered predominantly, i f n o t

e x c l u s i v e l y , by means o f , and w i t h r e s p e c t t o , a narrow c l a s s o f s i t u a t i o n s , namely s i t u a t i o n s which a r e p a r a d i g m a t i c f o r t h e o b j e c t i v i s t conc e p t o f p r o b a b i l i t y . R e s t r i c t i n g t h e a p p l i c a t i o n o f t h e p r o b a b i l i t y conc e p t t o such s i t u a t i o n s , does n o t , however, t a k e i n t o account a l l t h e p o s s i b l e r e c o g n i t i o n s t r a t e g i e s o f t h e s u b j e c t . T h i s means f o r i n s t a n c e , t h a t even i n s i t u a t i o n s o f d i f f e r e n t t y p e s o f u n c e r t a i n t y ( c f . KAHNEMAN & TVERSKY, 1982 und HUMPHREYS & BERKELEY, i n t h i s volume), t h e i n d i v i d u a l i s a b l e t o d e f i n e elementary e v e n t s , c o n s t r u c t an e v e n t space, i n which t o a s s i g n p r o b a b i l i t i e s t o these events, and i n t h i s way b u i l d an image o r model o f t h e environment. W i t h i n t h i s model, f o r m a l o p e r a t i o n s o f t h e l o g i cal-mathematical concept o f p r o b a b i l i t y may be a p p l i e d , t h u s r e n d e r i n g p o s s i b l e a s u b j e c t i v e s t o c h a s t i c a l r e c o n s t r u c t i o n o f d e c i s i o n s under uncertainty.

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Josephson, J.: 1977. The c h i l d ' s use o f s i t u a t i o n a l and personal i n f o r m a t i o n i n p r e d i c t i n g t h e b e h a v i o r o f a n o t h e r . Unpublished d o c t o r a l d i s s e r t a t i o n , Stanford U n i v e r s i t y Kahneman, D- & Tversky, A.: 1982a. Judgment under u n c e r t a i n t y . I n : D. Kahneman/ p. S l O V i C A - T v e r s k y (Eds. ), Judgment under u n c e r t a i n t y : h e u r i s t i c s and b i a s e s . Cambridge: Cambridge U n i v e r s i t y Press, 3 20

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Kahneman, 0. fi Tversky, A,: 1982b. V a r i a n t s o f u n c e r t a i n t y . I n : 0. Kahneman/ P. S l o v i c & A. Tversky (Eds.), Judgment under u n c e r t a i n t y : h e u r i s t i c s and b i a s e s . Cambridge: Cambridge U n i v e r s i t y Press, 509-520 Kobasigawa, A.: 1977. R e t r i e v a l s t r a t e g i e s i n t h e development o f memory. I n : R.V. K a i l & J.W. Hagen (Eds.), P e r s p e c t i v e s on t h e development o f memory. H i l l s d a l e , N.J.: Lawrence Erlbaum, 177-201 Messick, S.J. & S o l l e y , C.M.: 1957. P r o b a b i l i t y l e a r n j n g i n c h i l d r e n : some e x p l o r a t o r y s t u d i e s . J o u r n a l o f Genetic Psychology, Vol. 90, 23-32 Mises von, R . : 1928. W a h r s c h e i n l i c h k e i t , S t a t i s t i k und Wahrheit. Wien: Springer N o e l t i n g , G . : 1980a. The development o f p r o p o r t i o n a l r e a s o n i n g and t h e r a t i o concept, p a r t I D i f f e r e n t i a t i o n o f stages. Educational Studies i n Mathematics 11, 217-253

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N o e l t i n g , G.: 1980b. The development o f p r o p o r t i o n a l reasoninc, and t h e r a t i o concept, p a r t I1 P r o b l e m - s t r u c t u r e and t h e mechanism o f a d a p t i v e r e s t r u c t u r i n g . E d u c a t i o n a l S t u d i e s i n Mathematics 11, 331-363

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P i a g e t , J.: 1950. Une experiment sup l a p s y c h o l o g i e du hasard chez l ' e n f a n t : La t i r a g e au s o r t des couples. Acta Psychologica, Vol , 7, 323-336 P i a g e t , J. & I n h e l d e r , B.: 1951. La genese de l ' i d e e de hasard chez 1 ' e n f a n t . P a r i s : Presses U n i v e r s i t a i r e s de France (1975. T r a n s l a t e d as: The o r i g i n o f t h e i d e e o f chance i n c h i l d r e n . New York: N o r t o n ) Ross, L.: 1981. The " i n t u i t i v e s c i e n t i s t " f o r m u l a t i o n and i t s development a l i m p l i c a t i o n s . I n : J.H. F l a v e l l & L. Ross (Eds.), S o c i a l c o g n i t i v e development. F r o n t i e r s and p o s s i b l e f u t u r e s . Cambridge: Cambridge U n i v e r s i t y Press, 1-42

ROSS, B.M. & Hoemann, H.W.: 1975. The a t t a i n m e n t o f f o r m a l o p e r a t i o n s : A comparison o f p r o b a b i l i t y concepts i n d e a f and h e a r i n g a d o l e s c e n t s . I n : Genetic Psychology Monography 91 , 61-119 Scardamalia, M.:

1977. I n f o r m a t i o n p r o c e s s i n g c a p a c i t y and t h e problem o f

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h o r i z o n t a l d'ecalage: A d e m o n s t r a t i o n u s i n g c o m b i n a t o r i a l r e a s o n i n g t a s k . I n ' C h i l d Development 48, 28-37 S i e g l e r , R . S . : 1981. Developmental sequences w i t h i n and between c o n c e p t . I n : Monograbhsof t h e S o c i e t y f o r Research i n C h i l d Development 46, 2 W e i r , M.W.: 1964. Developmental changes i n p r o b l e m - s o l v i n g s t r a t e g i e s . P s y c h o l o g i c a l Review 71, 473-490

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GEORGES GLAESER Universite' Louis Pasteur I R EM de Strasbourg France

T h i s i s t h e t i t l e o f a d o c t o r a l t h e s i s s u b m i t t e d i n S t r a s b o u r g i n June 1982 by t h e Mexican r e s e a r c h e r Jesus A l a r c o n

nicknamed " P a p i n i " .

Our a t t e n t i o n has been c a l l e d t o t h i s s t u d y f o r t w o reasons: F i r s t . t h e t o p i c t r e a t e d i n t h i s s t u d y concerns t h e t e a c h e r and t h e educator. T h i s d o c t o r a l t h e s i s i n v e s t i g a t e s t h e ( c o r r e c t o r f a l s e ) i d e a s young p u p i l s may have a c q u i r e d w i t h r e g a r d t o c e r t a i n phenomena o f chance w e l l b e f o r e t h e y began t o study p r o b a b i l i t i e s i n t h e classroom. When one s t a r t s t o t e a c h them t h i s s u b j e c t . t h e y a r e c e r t a i n l y n o t w i t h o u t p r e v i o u s knowledge: from i n f a n c y , everybody i s f a m i l i a r i z e d w i t h s i t u a t i o n s o f b e t t i n g , o f drawing l o t s

o r w i t h d e c i s i o n s under

u n c e r t a i n t y . The o n l y b a s i s f o r e f f i c i e n t t e a c h i n g i s t h e teacher's awareness o f h i s p u p i l s ' p r e c o n c e i v e d i d e a s which he w i l l seek t o d i s c o u r a g e o r promote. Developing t h e concept o f p r o b a b i l i t y i s much more c r i t i c a l t h a n imagined by t h o s e who a l r e a d y have a s s i m i l a t e d i t s b a s i c n o t i o n s . Humanity has had a g r e a t d i f f i c u l t y i n m a s t e r i n g t h e phenomena o f chance and i n g e t t i n g t o t h e b o t t o m o f t h e magical p r e j u d i c e s which were a t t a c h e d t o them. The h i s t o r y of p r o b a b i l i t y c a l c u l u s has seen d i f f i c u l t b e g i n n i n g s h e r a l d e d by

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t h e names o f CAKDANO. PASCAL. FERMAT. d'ALEMBERT. etc.

.. For

LAPLACE. KOLMOGOROV

a l o n g t i m e . t h i s s c i e n t i f i c d i s c i p l i n e was t h e p r e f e r e n t i a l

domain o f paradoxes and erroneous c o n c l u s i o n s . And i t i s reasonable t o t h i n k t h a t many o f t h e d i f f i c u l t i e s which f o r m e r l y f r u s t r a t e d t h e g r e a t e s t s c i e n t i s t s must b a f f l e our p u p i l s s t i l l today. Second. ALARCON has c a r r i e d o u t h i s s t u d y making use o f an exemplary e x p e r i m e n t a l method. He r e f r a i n s from f o r m u l a t i n g c o n c l u s i o n s which a r e n o t based on e s t a b l i s h e d f a c t s . He a n t i c i p a t e s r e a s o n a b l e p o t e n t i a l o b j e c t i o n s and t a k e s e x p e r i m e n t a l p r e c a u t i o n s i n advance which enable him t o r e f u t e these. I n my o p i n i o n . t h i s d o c t o r a l t h e s i s c o n s t i t u t e s t h e p r e s e n t model o f what enables us t o s t a t e today t h a t mathematical d i d a c t i c s has become a science.

I s h o u l d l i k e t o add t h i s i n a l l modesty. While t h i s t h e s i s has been p r e p a r e d under t h e s u p e r v i s i o n o f t h e S t r a s b o u r g mathematics educators. ALARCON has c a r r i e d i t t h r o u g h " a g a i n s t " o u r team: t h e c a n d i d a t e always p r o v e d t o be more e x a c t i n g and more s u b t l e t h a n we were. Our r o l e was c o n f i n e d t o h e c k l i n g him. E v e n t u a l l y , however. i t was m o s t l y " P a p i n i " who i n f l u e n c e d us! A conflict of logics

The elementary t e a c h i n g o f p r o b a b i l i t y g e n e r a l l y begins w i t h d e s c r i b i n g s i t u a t i o n s o f drawing w i t h replacement which a r e r e a l i z e d w i t h u r n s c o n t a i n i n g p w h i t e b a l l s and q b l a c k b a l l s . It i s t h e n p o s t u l a t e d more o r l e s s e x p l i c i t l y t h a t a l l answers t o q u e s t i o n s i n t h a t c o n t e x t depend o n l y on t h e p r o p o r t i o n p/(p+q) and n o t on t h e t o t a l number p+q o f b a l l s . ALARCON. however. has e s t a b l i s h e d t h a t t h i s p o s t u l a t e i s f a r from b e i n g e v i d e n t t o t h e b e g i n n e r ! More p r e c i s e l y . a f t e r t h i s b e l i e f i s e s t a b l i s h e d . i t s t i l l remains u n s t a b l e : a s l i g h t m o d i f i c a t i o n i n t h e f o r m u l a t i o n o f

PERCEPTION OF PROBABILITY SITUATIONS

q u e s t i o n s w i l l s u f f i c e t o cancel i t o u t (as i t w i l l be seen w i t h r e g a r d t o one c o n d i t i o n o f t h e experiment shown i n f i g . 2). One r e s u l t , i n p a r t i c u l a r . i s s u r p r i s i n g . t h e p e r s i s t e n c e o f answering s t r a t e g i e s which a r e n o t based e x c l u s i v e l y on t a k i n g i n t o account t h e r e l a t i o n s h i p p /(p+q) i s e s p e c i a l l y e v i d e n t f o r p u p i l s c o n s i d e r e d more advanced t h a t t h e average o f t h e i r peers. I n f a c t . b e g i n n i n g t o l e a r n p r o b a b i l i t i e s uncovers a c o n f l i c t o f l o g i c s . An example: i n t h e c o u r s e o f a game o f heads and t a i l s . one may b e t heads. B u t i f chance s h o u l d make heads come o u t , nobody w i l l c o n s i d e r t h i s t o be an i n t e l l e c t u a l outrage. But i f , as opposed t o t h a t , t h e keys o f a hand-held c a l c u l a t o r a r e punched 12+7 and t h e r e s u l t i n d i c a t e d i s 15. everybody w i l l be aware t h a t something has gone wrong.

From t h e C a r t e s i a n p o i n t o f view. i t i s p e r f e c t l y c o n c e i v a b l e t h a t i n t h e c o u r s e o f a b r i d g e game. a hand o f 13 spades c o u l d b e d e a l t . From t h e p r o b a b i l i s t - s p o i n t o f view, however. t h i s s m e l l s o f m i r a c l e o r t r i c k e r y . ALARCON shows t h a t t h e r e i s an o p p o s i t i o n between c o m p a t i b i l i s t i c and p r o b a b i l i s t i c a t t i t u d e s . The p u p i l s who have adopted t h e former w i l l r e f u s e t o choose between " p o s s i b l e " a l t e r n a t i v e s even i f one i s c l e a r l y more p r o b a b l e t h a n t h e o t h e r . Those who have adopted t h e p r o b a b i l i s t i c a t t i t u d e s w i l l be i n t e r e s t e d i n r e l a t i o n s and c o n s i d e r an event whose p r o b a b i l i t y i s very low "impossible".

( F o r i n s t a n c e t h o s e who answer

" r a t h e r bag 2" o r " c e r t a i n l y bag 2 " t o t h e q u e s t i o n i n f i g 4.)

The expem'ment A l a r c o n - s study began w i t h a p r e t e s t i n which he observed about f o r t y c h i l d r e n o f d i f f e r e n t ages i n i n d i v i d u a l i n t e r v i e w s . He s u b m i t t e d a b o u t t h i r t y q u e s t i o n s t o them. a d a p t i n g these, i f necessary. between one i n t e r v i e w and t h e next.

315

316

DE VELOPMEN TAND EPISTEMOLOGY

A f t e r t h i s stage. he designed a q u e s t i o n n a i r e which was s u b m i t t e d t o 6 c l a s s e s o f f i f t h - f o r m e r s (12 t o 13 y e a r s ) and t o 6 c l a s s e s o f forth-formers

(13 t o 14 y e a r s ) s e l e c t e d a t random i n t h e French Department

o f Bas-Rhin ( i . e .

t o about 300 p u p i l s ) .

The q u e s t i o n n a i r e was p r e s e n t e d i n t h r e e v a r i a n t s ("moda it 6 s " ) which d i f f e r e d e i t h e r i n t h e i r formulation o r i n the question order. The f o l l o w i n g i s an example o f a t y p i c a l quest on t h a t can be answered on t h e basis o f t h e c m p a t i b i l i s t i c a t t i t u d e .

1. A l a i n has t a k e n one o f t h e t w o bags shown below. and w i t h o u t e v e r l o o k i n g i n t o t h e bag

...

A l a i n has drawn a b a l l : t h e b a l l t a k e n o u t was WHITE; t h e n he has p u t t h e b a l l back i n t o t h e bag, m i x i n g w e l l . A l a i n has drawn a second t i m e : he has t a k e n o u t a WHITE b a l l has p u t t h e b a l

t h e n he

back i n t o t h e bag. m i x i n g we1

A l a n has drawn a t h i r d t i m e : he has t a k e n o u t a BLACK b a l l ; t h e n he has p u t t h e b a l

back i n t o t h e bag. m i x i n g we1

A l a i n has drawn a f o r t h t i m e : he has taken o u t a WHITE b a l has p u t t h e b a l

t h e n he

back i n t o t h e bag. m i x i n g w e l l .

A l a i n has drawn a f i f t h t i m e : he has t a k e n o u t a BLACK b a l . t h e n he has p u t t h e b a l l back i n t o t h e bag. m i x i n g w e l l . A l a i n has drawn a s i x t h t i m e : he has t a k e n o u t a BLACK b a l l . . t h e n he has p u t t h e b a l l back, m i x i n g w e l l .

PERCEPTION OF PROBABILITY SITUATIONS

317

Which bag d i d A l a i n t a k e ? Place a c r o s s i n t h e box you p r e f e r .

fig. 1

bag 1

bag 2

U ’ O

0

0

0

certainly

rather

no reason f o r

rather

certainly

p r e f e r r i n g one

bag 2

bag 2

bag 1

bag 1

o f t h e bags B u t now c o n s i d e r two q u e s t i o n s c o n c e r n i n g t h e two bags below:

fig. 2

bag

1

bag 2

Suppose t h a t a draw ( w i t h p u t t i n g back) o u t o f one o f t h e s e bags be g i v e n . and y o u a r e asked t o i n d i c a t e f r o m which bag i t i s . F o r an average p r o b a b i l i s t . t h e r e i s n o need t o t a k e note o f t h e draw. t h e bags a r e equivalent

and so t h e r e i s n o reason t o p r e f e r one o f t h e bags r a t h e r

than t h e other. The experiment. however. y i e l d s t h e f o l l o w i n g r e s u l t : i f t h e draw mentioned i s reduced t o a s i n g l e w h i t e b a l l , 98 p e r c e n t o f t h e ( o l d e r ) p u p i l s of t h e f o u r t h f o r m and 76 p e r c e n t o f t h o s e o f t h e f i f t h f o r m w i l l g i v e t h e c o r r e c t answer. But t h o s e percentages go down t o 65 and 52.

’

r e s p e c t i v e l y , i f t h e draw i n d i c a t e d i s 4 w h i t e b a l l s and 2 b l a c k b a l l s ! It seems t h a t t h e p u p i l s b e l i e v e t h a t i t i s e a s i e r t o draw 4 w h i t e b a l l s from

DEVELOPMENT AND EPISTEMO LOG Y

318

bag one t h a n f r o m bag two. T h i s i s t h e phenomenon o f i n s t a b i l i t y mentioned above. But t h i s i s o n l y a l o c a l r e s u l t o f t h i s t h e s i s . I n f a c t . t h e q u e s t i o n n a i r e harasses t h e respondent p u p i l s w i t h i t e m s which seem t o be analogous. b u t r e a l l y c a l l f o r d i f f e r e n t s t r a t e g i e s . I f t h e draw c o n t a i n s more w h i t e b a l l s t h a n b l a c k ones. t h e bag c o n t a i n i n g more w h i t e t h a n b l a c k b a l l s may be p r e f e r r e d w i t h o u t t a k i n g p r o p o r t i o n s i n t o account.0ther i n f o r m a t i o n i s o b t a i n e d on t h e b a s i s o f items i n which t h e c o n t e n t o f one o f t h e bags i s unknown. F o r example: E d i t h has t a k e n one o f t h e bags shown below, t h e n she has drawn a b a l l w i t h o u t l o o k i n g i n t o t h e bag; t h e b a l l t a k e n o u t was WHITE. Then she has p u t t h e b a l l back i n t o t h e same bag.

6

fig. 3

bag 1

bag 2 FFVFV

lot13

FFVVV

16+14

FFVVF

7+0

Now t h e r e a r e f i v e statements: which a r e t r u e . and which a r e f a l s e ? P u t a c r o s s i n t h e r e s p e c t i v e box.

I t i s i m p o s s i b l e t h a t E d i t h t o o k bag 1.

true

0 0

I t i s n o t s u r e t h a t E d i t h t o o k bag 1.

true

0

I t i s more p r o b a b l e t h a t E d i t h took bag 1. I t i s j u s t as p r o b a b l e t h a t E d i t h t o o k

true0

bag 1 as bag 2

true

It i s sure t h a t E d i t h t o o k bag 1.

true

0

false

0 0 0 0

false

0

false false

false

319

PERCEPTION OF PROBABILITY SITUATIONS

Or. a g a i n : The r e s u l t o f t h e draws was: 6 BLACK b a l l s . Which bag d i d Gerard t a k e ? Put a c r o s s i n t h e box you p r e f e r .

fig. 4

bag 1

0

0

certainly

rather

bag 1

bag 1

no reason t o p r e f e r any

0

0

rather

certainly

bag 2

bag 2

o f t h e bags fourth form

?

11 -

fifth form

?

4

18 -

18

19

19

7

5

The method chosen by A l a r c o n a v o i d s drawing c o n c l u s i o n s f r o m one p a r t i c u l a r answer t o an i s o l a t e d q u e s t i o n . On t h e c o n t r a r y . i t shows e v i d e n c e f o r coherent b e h a v i o u r w i t h r e g a r d t o t h e q u e s t i o n n a i r e as a whole. And t h e f o r m u l a t i o n o f t h e q u e s t i o n s p e r m i t s t o r e a c h v e r y a c c u r a t e c o n c l u s i ons

.

Probability and fractionaZ caZcuZus L e t us now see how A l a r c o n r e f u t e s a view v e r y widespread among t e a c h e r s and r e s e a r c h e r s i n t h e f i e l d o f education.

320

OE VELOPMENT AND EPISTEMOLOGY

It would seem p l a u s i b l e t h a t l e a r n i n g p r o b a b i l i t y t h e o r y r e q u i r e s

m a s t e r i n g f r a c t i o n a l c a l c u l u s , and i t i s common b e l i e f t h a t t h i s i s t h e o n l y indispensible prerequisite. I n order t o discuss t h i s point. Alarcon p r e s e n t e d , a t t h e b e g i n n i n g o f each o f t h e t h r e e " m o d a l i t 6 s " . i t e m s w h i c h were i n s p i r e d by t h e famous orange j u i c e t e s t developed by G e r a l d Noelting. C a t h e r i n e w i l l make up orange j u i c e f o r a p a r t y . Below each j u g . t h e r e i s an i n d i c a t i o n o f t h e q u a n t i t i e s o f orange j u i c e and w a t e r C a t h e r i n e w i l l m i x i n o r d e r t o p r e p a r e t h e d r i n k :

!zl

= a

U

= a g l a s s o f water.

?lass of orange j u i c e ;

Which d r i n k w 11 have a s t r o n g e r orange t a s t e ? P u t a c r o s s

n the

c o r r e c t box.

Fig. 5

( a box f o r "both d r i n k s t a s t e t h e same" i s also provided)

Fig. 6

PERCEPTION OF PROBABILlTY SITUATIONS

I n o r d e r t o p e r m i t comparisons. t h e experimenter has chosen t h e same numbers p and q o f g l a s s e s o f c o n c e n t r a t e d orange j u i c e and water as b l a c k and w h i t e b a l l s i n t h e bags. It would seem t h a t N o e l t i n g - s experiment was a f u l l success. w h i l e many s i g n i f i c a n t f a i l u r e s a r e observed i n t h e random b a l l -drawing t e s t . The experiment f u r n i s h e s a c o n v i n c i n g e x p e r i m e n t a l r e f u t a t i o n o f t h e c l a i m t h a t mastering f r a c t i o n a l c a l c u l u s i s s u f f i c i e n t t o understand t h e c a l c u l u s o f elementary p r o b a b i l i t i e s . Even b e t t e r , t h e f o l l o w i n g t e s t i s one o f t h o s e which c o n t a i n a s t r o n g and unexpected p r o p o r t i o n o f f a i l u r e s , w h i l e t h e items r e f e r r i n g t o bags o f t h e same composition a r e f a i r l y s u c c e s s f u l on t h e whole.

0

0

0

same t a s t e i n t h e two j u g s

fourth

2

0

49

fifth

19

0

27

321

DEVELOPMENT AND EPISTEMOLOGY

322

8

Fig. 8

bag 1

0

0

certainly

rather

bag 1

bag 1

bag 2

0

0

no reason t o

rather

certainly

bag 2

bag 2

p r e f e r one o f t h e bags

fourth

10

29

16

fifth

10

18

9

DeveZopment w i t h age As a general r u l e , an improvement o f performance can be observed when p a s s i n g from p u p i l s o f t h e f i f t h f o r m t o t h o s e o f t h e f o u r t h . But A l a r c o n i s t o o e x a c t i n g t o draw c o n c l u s i o n s i f t h e improvements a r e n o t s i g n i f i c a n t ( i n t h e s t a t i s t i c a l sense). Besides

i t would have been

necessary, i n t h i s case. t o use r e a l age groups r a t h e r t h a n r e f e r r i n g e x c l u s i v e l y t o t h e o p p o s i t i o n between f i f t h form ( p u p i l s aged 12 t o 13) and f o u r t h form ( p u p i l s aged 13 t o 14). F o r t h e cases where t h i s p r o g r e s s was p r o v e d e x p e r i m e n t a l l y , t h e r e a d e r i s referred t o the thesis i n detail. T h i s r e s u l t , i n any case, c o n f l i c t s w i t h many accepted i d e a s o f o f f i c i a l pedagogy: a c e r t a i n m a t u r i n g can be e s t a b l i s h e d a t p r e c i s e p o i n t s between t h e ages o f 12 and 14 w i t h r e g a r d t o n o t i o n s which a r e n o t t h e o b j e c t o f any t e a c h i n g i n school.

PERCEPTION OF PROBABILITY SITUATIONS

Anyway. t h i s summary o f A l a r c o n - s d o c t o r a l t h e s i s i s much t o o b r i e f t o g i v e more t h a n a glimpse a t t h e r i c h f i n d i n g s o f t h i s e x p e r i m e n t a l study

A l l t h o s e who s h o u l d l i k e t o g e t a more p r o f o u n d view o f t h e o p p o r t u n i t i e s o f f e r e d by t h e experimental method i n e d u c a t i o n a l r e s e a r c h a r e a d v i s e d t o r e a d Jesus ALARCON-s t h e s i s c a r e f u l ly.

323

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor)

0 Elsevier Science Publishers B.V. (North-Holland), 1983

325

THE PSYCHOLOGY OF INFERENCE AND THE TEACHING OF PROBABILITY AND STATISTICS: TWO SIDES OF THE SAME COIN?

J. MICHAEL SHAUGHNESSY San Diego State University Department of Mathematical Sciences USA

For many y e a r s teachers o f p r o b a b i l i t y and s t a t i s t i c s have been s t r u g g l i n g t o improve t h e p l i g h t o f t h e n a i v e s t a t i s t i c i a n , t h e s t u d e n t i n an i n t r o d u c t o r y course i n p r o b a b i l i t y and s t a t i s t i c s .

Such s t u d e n t s may have a

f i r s t encounter w i t h s t a t i s t i c s e i t h e r a t t h e secondary l e v e l o r i n c o l l e g e , and a r e p r o b a b l y p o o r l y prepared m a t h e m a t i c a l l y f o r t h e i r plunge i n t o t h i s new subject.

They may b e l i e v e t h a t a l l outcomes f r o m an experiment a r e e q u a l l y

l i k e l y , f o r example t h a t t h e chance o f g e t t i n g a sum o f 4 on a r o l l o f two d i c e i s t h e same as t h a t o f g e t t i n g a sum of 7.

They have l i t t l e e x p e r i e n c e

w i t h t h e r a p i d growth and decay o f p a r t i a l products, and so t h e y make v e r y l o w e s t i m a t e s f o r t h e number o f d i f f e r e n t p o s s i b l e b a t t i n g o r d e r s f o r a n i n e person b a s e b a l l team, and g i v e v e r y h i g h e s t i m a t e s f o r t h e number o f p e o p l e needed t o have an even chance o f matching b i r t h d a y s . Indeed, t h e n a i v e s t u d e n t o f p r o b a b i l i t y and s t a t i s t i c s possesses a m y r i a d o f mathematical misconcept i o n s which p r e s e n t a f o r m i d a b l e c h a l l e n g e t o t h e t e a c h e r o f s t a t i s t i c s . W i t h i n t h e l a s t t e n y e a r s a c o n s i d e r a b l e amount o f i n v e s t i g a t i o n by cogn i t i v e and s o c i a l p s y c h o l o g i s t s i n t h e area o f judgment and d e c i s i o n making has produced evidence which suggests t h a t many m i s c o n c e p t i o n s o f p r o b a b i l i t y and s t a t i s t i c s have p s y c h o l o g i c a l r o o t s .

Mere exposure t o t h e t h e o r e t i c a l

laws o f p r o b a b i l i t y and s t a t i s t i c s may n o t be s u f f i c i e n t t o overcome probab i l i s t i c misconceptions (TVERSKY and

KAHNEMAN, 1971).

consider the naive teacher o f s t a t i s t i c s .

Thus we a r e l e d t o

Students show up i n an i n t r o d u c t o r y

s t a t i s t i c s course equipped w i t h a f o r m i d a b l e a r r a y o f mathematical misconcep-

326

DEVELOPMENT AND EPISTEMO L OG Y

t i o n s and d e f i c i e n c i e s .

On t o p o f t h a t , we f i n d out, t h e s e same s t u d e n t s use

i n t u i t i v e h e u r i s t i c s such as a v a i l a b i l i t y and r e p r e s e n t a t i v e n e s s when e s t i m a t i n g t h e l i k e l i h o o d o f events. They i g n o r e base r a t e s when making p r e d i c t i o n s . They a r e slow t o change o p i n i o n s i n t h e f a c e o f new d a t a and t h e y a r e poor a t d e t e c t i n g c o v a r i a t i o n .

They may n o t even " b e l i e v e " i n p r o b a b i l i t y .

I n short, they are a s t a t i s t i c a l disaster.

I f t h e students' misconceptions

a r e n ' t enough t o d e t e r our n a i v e t e a c h e r s o f s t a t i s t i c s , t h e y can be c o m f o r t e d by t h e f a c t t h a t they too a r e s u b j e c t t o t h e same p s y c h o l o g i c a l b i a s e s and misconceptions as t h e i r students. The s t u d e n t i s caught i n t h e m i d d l e as t h e j u d g e and d e c i s i o n maker o f t h e f u t u r e , t h e f u t u r e user ( o r non-user) o f s t a t i s t i c s .

How can r e s e a r c h i n

t h e l e a r n i n g o f s t a t i s t i c s and p r o b a b i l i t y h e l p t h e s t u d e n t as n a i v e judge? How can r e s e a r c h i n judgment and d e c i s i o n making h e l p t h e s t u d e n t as n a i v e statistician? of statistics? education.

How can r e s e a r c h from b o t h s i d e s o f t h e c o i n h e l p t h e t e a c h e r These q u e s t i o n s a r e posed from t h e p e r s p e c t i v e o f mathematics

I m p l i c i t i n t h e above q u e s t i o n s a r e t h e b e l i e f s t h a t 1)

it i s

p o s s i b l e t o improve p e o p l e ' s u n d e r s t a n d i n g o f p r o b a b i l i t y and s t a t i s t i c s ;

2)

i t i s p o s s i b l e t o improve p e o p l e ' s judgment and decision-making processes;

3 ) i t e m s 1) and 2) a r e i n t r i c a t e l y connected w i t h each o t h e r , and so t h e y s h o u l d be connected i n t h e c u r r i c u l u m . I n t h i s paper we w i l l e x p l o r e t h e i n t e r r e l a t i o n s h i p o f judgment and d e c i s i o n making w i t h p r o b a b i l i t y and s t a t i s t i c s i n t h r e e p a r t s .

P a r t one w i l l

examine r e s e a r c h on o b s t a c l e s t o t h e use o f s t a t i s t i c a l p r i n c i p l e s when making i n f e r e n c e s from t h e p e r s p e c t i v e s o f psychology and mathematics education. P a r t two w i l l r e v i e w some s u g g e s t i o n s from p s y c h o l o g i s t s and mathematics educ a t o r s f o r i n c r e a s i n g p e o p l e ' s knowledge and r e l i a n c e upon s t a t i s t i c s when making i n f e r e n c e s .

F i n a l l y , p a r t t h r e e c o n s i s t s o f some recommendations f o r

f u r t h e r research.

I.

Decision Making Under Uncerta
t a s k o f s u r v e y i n g t e n y e a r s o f l i t e r a t u r e on judgment and d e c i s i o n making. Several e x c e l l e n t reviews and syntheses o f t h i s l i t e r a t u r e have r e c e n t l y been pub1 i s h e d (SLOVIC, FISCHHOFF, and LICHTENSTEIN 1977; NISBETT and ROSS, 1980;

327

INFERENCE AND TEACHING PROBABILITY

KAHNEMAN, SLOVIC, and TVERSKY, 1982).

However t h e r e a r e aspects o f t h i s

r e s e a r c h t h a t may be o f p a r t i c u l a r i n t e r e s t t o r e s e a r c h e r s i n mathematics education.

We s h a l l t o u c h upon s e v e r a l o f these aspects b r i e f l y . The use of heuristics. C o n s i d e r a b l e evidence has been gathered t o show t h a t people t e n d t o r e l y upon s i m p l i f y i n g h e u r i s t i c s when making e s t i m a t e s f o r t h e l i k e l i h o o d o f events. 1974).

(KAHNEMAN and TVERSKY,

1972, 1973a, 1973b,

People's use o f such h e u r i s t i c s as r e p r e s e n t a t i v e n e s s , anchoring,

and a v a i l a b i l i t y t o make i n f e r e n c e s has been r e p l i c a t e d and extended many t i m e s s i n c e t h e s e i n i t i a l s t u d i e s were published. 1982, SHAUGHNESSY,

1977).

(BAR-HILLEL,

1982; TAYLOR,

The use o f t h e s e h e u r i s t i c s has been found n o t o n l y

among s t a t i s t i c a l l y - n a i v e s u b j e c t s b u t a l s o among supposed s t a t i s t i c a l e x p e r t s (TVERSKY and KAHNEMAN,

1971).

A l t h o u g h these s e a t - o f - t h e - p a n t s

heuristics

o f t e n guide one adequately i n d a i l y a f f a i r s , t h e y can be m i s l e a d i n g , p a r t i c u l a r l y when i n f e r e n c e s a r e t o be made from s t a t i s t i c a l data.

Thus, people a r e

i n s e n s i t i v e t o t h e e f f e c t s o f sample s i z e on p r e d i c t i v e v a l i d i t y , unaware t h a t d a t a t e n d t o r e g r e s s t o t h e mean and may be o v e r c o n f i d e n t o f p r e d i c t i o n based upon i n v a l i d i n p u t d a t a i f t h e y employ r e p r e s e n t a t i v e n e s s .

They may t e n d t o

r e l y on v i v i d anecdotol data r a t h e r t h a n c a r e f u l l y c o l l e c t e d s t a t i s t i c s i f t h e y employ a v a i l a b i l i t y .

They may " l o c k i n " t o an i n i t i a l e s t i m a t e and be

slow t o r e v i s e when faced w i t h new d a t a i f t h e y employ an anchor.

The e x i s -

t e n c e and p r e v a l e n c e o f these h e u r i s t i cs poses added chal 1enges t o t h e t e a c h e r o f p r o b a b i l i t y and s t a t i s t i c s .

The disregard f o r base-rate data.

I n many s i t u a t i o n s when p e o p l e must

combine several p i e c e s o f i n f o r m a t i o n t o make an e s t i m a t e f o r t h e l i k e l i h o o d o f an event, t h e y t e n d t o i g n o r e b a s e - r a t e data.

This behavior i s p a r t i c u l a r -

l y e v i d e n t i n many s i t u a t i o n s where an a p p l i c a t i o n o f Bayes l a w would g i v e a

good n o r m a t i v e p r e d i c t i o n .

Research on problems such as t h e t a x i c a b problem

(TVERSKY and KAHNEMAN, 1980) has c r e a t e d a f l u r r y o f t h e o r e t i c a l a c t i v i t y i n an attempt t o e x p l a i n why people a t t e n d t o i n d i v i d u a t i n g d a t a and i g n o r e b a s e - r a t e data.

Here i s one v e r s i o n o f t h e t a x i c a b problem which has been

used by many researchers.

A cab was i n v o l v e d i n a h i t - a n d - r u n a c c i d e n t a t n i g h t . Two cab compani e s , t h e Green and t h e Blue, o p e r a t e i n t h e c i t y . You a r e g i v e n t h e f o l l o w i n g data : (i)

85% o f t h e cabs i n t h e c i t y a r e Green and 15% a r e Blue.

( i i ) A w i t n e s s i d e n t i f i e d t h e cab as a Blue cab.

The c o u r t t e s t e d h i s

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DEVELOPMENT AND EPISTEMOLOGY

When a b i l i t y t o i d e n t i f y cabs under t h e a p p r o p r i a t e v i s i b i l i t y c o n d i t i o n s . presented w i t h a sample o f cabs ( h a l f o f which were B l u e and h a l f o f which were Green) t h e w i t n e s s made c o r r e c t i d e n t i f i c a t i o n s i n 80% o f t h e cases and e r r e d i n 20% o f t h e cases. What i s t h e p r o b a b i l i t y t h a t t h e cab i n v o l v e d i n t h e a c c i d e n t was B l u e r a t h e r than Green? R e s u l t s show t h a t people a t t e n d t o t h e w i t n e s s r e l i a b i l i t y i n f o r m a t i o n and i g n o r e t h e b a s e - r a t e data. a r e , thus, t o o high.

Estimates f o r t h e l i k e l i h o o d o f t h e b l u e cab

AJZEN (1977) suggests t h a t when base r a t e s have causal

i m p l i c a t i o n s f o r outcomes, p e o p l e ' s p r e d i c t i o n s 1,622 be i n f l u e n c e d by them.

NISBETT and BOGIDA (1975) and NISBETT, BORGIDA, CRANDALL, and REED (1976) s u b s t a n t i a t e t h e base-rate phenomenon when people a r e g i v e n consensus i n f o r m a t i o n and t a r g e t s p e c i f i c i n f o r m a t i o n .

They suggest t h a t people i g n o r e t h e

consensus ( b a s e - r a t e ) i n f o r m a t i o n because o f i t s a b s t r a c t n a t u r e and a t t e n d r a t h e r t o t h e c o n c r e t e a c t o r o r t a r g e t r e l a t e d i n f o r m a t i o n which i s much more vivid.

They r a i s e concern t h a t people make e r r o r s i n p r e d i c t i n g a whole e v e n t

from i t s b i a s e d p a r t s , o r c o n v e r s e l y from i n f e r i n g p a r t s o r i n s t a n c e s f r o m a whole.

They a r e a l s o concerned t h a t i g n o r a n c e o f b a s e - r a t e d a t a c o u l d have

s e r i o u s i m p l i c a t i o n s f o r t h e t e a c h i n g o f s o c i a l psychology, and r e f e r t o t h e "Oh, t h a t c o u l d never happen t o me" phenomenon i n t h e i r students.

(We suggest

t h a t i t a l s o has s e r i o u s i m p l i c a t i o n s f o r t h e t e a c h i n g o f p r o b a b i l i t y and statistics).

BAR-HILLEL (1980) found t h a t b a s e - r a t e i n f o r m a t i o n w i l l be com-

b i n e d w i t h o t h e r i n f o r m a t i o n i f t h e two k i n d s o f i n f o r m a t i o n a r e p e r c e i v e d as equally relevant t o the task.

A l t h o u g h t h e v e r y n a t u r e o f t h e t a s k i t s e l f may

have some b e a r i n g on how base r a t e s a r e used (AJZEN 1977, BAR-HILLEL 1980). LYON and SLOVIC, (1976) found t h e b a s e - r a t e phenomenon t o be h i g h l y r o b u s t across a wide v a r i e t y o f tasks.

FISCHHOFF, SLOVIC and LICHTENSTEIN (1979)

found t h a t changes i n b a s e - r a t e i n f o r m a t i o n d i d r e s u l t i n s u b j e c t s s h i f t i n g t h e i r p r e d i c t i o n s i n t h e d i r e c t i o n t h a t n o r m a t i v e t h e o r y would suggest, however, i n s u f f i c i e n t l y .

Corresponding changes i n sample s i z e d i d n o t produce

such s h i f t s . For t h e mathematics educator and t e a c h e r o f s t a t i s t i c s , t h e r e i s much t o be concerned about w i t h t h e i g n o r a n c e o f base r a t e s and i n s e n s i t i v i t y t o sample size.

Somehow b a s e - r a t e i n f o r m a t i o n g e t s l o s t i n t h e s h u f f l e , and

people may o r may n o t u t i l i z e i t depending on v i v i d n e s s , s a l i e n c y , c a u s a l i t y , o r perhaps knowledge about p r o b a b i l i t y and s t a t i s t i c s i t s e l f , and t h e r e l e vance o f t h a t knowledge t o t h e s i t u a t i o n .

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INFERENCE AND TEACHING PROBABILITY

Unwarranted confidence i n precition.

When c o n f r o n t e d w i t h new d a t a o r

i n f o r m a t i o n t h a t c o n f l i c t s w i t h b e l i e f s , people a r e slow t o change

nferences

NISBETT and ROSS (1980) reviewed and summarized t h e

iterature

o r decisions.

on t h e o r y maintenance and change (see Chapter 8), and concluded t h a t p e o p l e t e n d t o adhere t o t h e o r i e s t h a t a r e f a r from n o r m a t i v e l y j u s t i f i a b l e . evidence seems t o c a r r y more w e i g h t t h a n subsequent evidence.

Initial

When p r e s e n t e d

w i t h c o n f l i c t i n g evidence, s u b j e c t s t e n d t o p o l a r i z e i n f e r e n c e s and base them on a priori b e l i e f systems ( w i t n e s s t h e e f f e c t s o f r e p o r t s o f t h e Vietnam War on t h e American p e o p l e ) . People commit p r e m a t u r e l y t o a p o s i t i o n , and r e v i s e s l o w l y from a p o s i t i o n .

A p p a r e n t l y people t e n a c i o u s l y m a i n t a i n t h e i r

own microparadigms (KUHN 1962) which a r e as r e s i s t a n t t o change i n i n d i v i d u a l s as t h e s c i e n t i f i c paradigms a r e t o change i n s c i e n t i f i c t h e o r y .

So, t h e

t e a c h e r o f p r o b a b i l i t y and s t a t i s t i c s n o t o n l y has p s y c h o l o g i c a l b i a s e s and mathematical misconceptions t o contend w i t h , b u t a l s o p e r s o n a l b e l i e f systems among students.

There are a variety of individua2 inference strategies.

Recent r e s e a r c h

by WELL, POLLATSEK, and KONOLD (1982) suggests t h a t t h e t y p e s o f misconcept i o n s t h a t occur among n a i v e s u b j e c t s when c o n f r o n t e d w i t h Bayesian problems, such as t h e t a x i c a b problem, a r e much more c o m p l i c a t e d and v a r i e d t h a n s i m p l e n e g l e c t o f b a s e - r a t e d a t a o r use o f some causal schemas.

Much o f t h e r e s e a r c h

on misconceptions o f probabi 1 it y has r e p o r t e d o n l y average responses--means o r medians--and has n o t asked t h e s t u d e n t s why t h e y responded i n a p a r t i c u l a r way.

WELL c a r r i e d o u t f o l l o w - u p c l i n i c a l i n t e r v i e w s w i t h s t u d e n t s who had

been given Bayesian problems, l i k e t h e t a x i c a b problem, i n o r d e r t o d i s c l o s e t h e i r mental processes.

He found l a r g e i n d i v i d u a l d i f f e r e n c e s i n t h e t y p e s o f

s t r a t e g i e s employed and i n t h e k i n d o f m i s c o n c e p t i o n s t h a t were apparent among t h e students.

I n p a r t i c u l a r , he found l i t t l e evidence t o s u p p o r t t h e causal

schemas c o n t e n t i o n o f TVERSKY and KAHNEMAN (1980).

Rather, some s t u d e n t s

used n o n - p r o b a b i l i s t i c s t r a t e g i e s t o e s t i m a t e t h e l i k e l i h o o d o f events, s t r a t e g i e s which may be based on a d e t e r m i n i s t i c model o r even on p e r s o n a l b e l i e f systems. answer.

WELL says, "It i s d i f f i c u l t t o i n f e r a s t r a t e g y from a numerical We have observed s u b j e c t s w i t h markedly d i f f e r e n t s t r a t e g i e s ( i n t h e

i n t e r v i e w s ) a r r i v e a t t h e same numerical answer, and s u b j e c t s w i t h t h e same s t r a t e g y a r r i v e a t d i f f e r e n t answers." An e s t i m a t e f o r t h e l i k e l i h o o d o f an event may n o t be based upon a p r o b a b i l i s t i c strategy a t a l l .

For example, suppose someone decides t h a t t h e

DEVELOPMENT AND EPISTEMO LOG Y

330

w i t n e s s is r i g h t i n t h e cab problem.

They w i l l t h e n conclude t h a t t h e b l u e

cab was i n v o l v e d i n t h e a c c i d e n t even though t h e base r a t e i s green-cab dominated.

Chance i s n o t i n v o l v e d i n such d e c i s i o n s .

I n t h e i n t e r v i e w s WELL and

h i s c o l l e a g u e s o b t a i n e d e s t i m a t e s as h i g h as 90-95% t h a t i t was a b l u e cab. When asked why, t h e s e s u b j e c t s s a i d , "It h i n k t h e w i t n e s s was r i g h t , " o r "I t h i n k i t ms a b l u e cab."

I n such cases, t h e d e c i s i o n may be based on a be-

l i e f or d e t e r m i n i s t i c decision. The percentage e s t i m a t e g i v e n by t h e s u b j e c t i s more l i k e a c o n f i d e n c e r a t i n g t h a n a p r o b a b i l i t y e s t i m a t e . B a s e - r a t e i n f o r m a t i o n may be n e g l e c t e d because i t v i o l a t e s p r e c o n c e p t i o n s o r b e l i e f s o f t h e subject. This t y p e o f m i s c o n c e p t i o n may pose t h e most d i f f i c u l t c h a l l e n g e o f a l l t o t h o s e who t e a c h p r o b a b i l i t y and s t a t i s t i c s .

What do y o u do i f y o u r s t u -

d e n t s j u s t p l a i n don't beZieve in probabiZity o r chance?

WELL e t . a l . (1982)

r e p o r t s a p r o t o c o l e x c e r p t i n which a c o l l e g e s t u d e n t i s asked t o p i c k which s i d e he t h i n k s w i l l t u r n up most o f t e n i f an i r r e g u l a r l y - s h a p e d d i e ( a c t u a l l y a bone) i s r o l l e d .

D u r i n g t h e i n t e r c h a n g e t h e s t u d e n t e v e n t u a l l y s e t t l e s on

s i d e C as t h e most l i k e l y t o come u p - - f o r no apparent p r o b a b i l i s t i c reason. ( I n f a c t , t h e r e i s no evidence o f an u n d e r s t a n d i n g o f chance by t h e s t u d e n t d u r i n g t h e e n t i r e episode.)

The s t u d e n t i s t h e n shown d a t a from 1000 r o l l s o f

t h e d i e which c l e a r l y suggest s i d e which would be t h e most l i k e l y . C."

D i s most l i k e l y t o occur, and t h e n asked

"You found D most l i k e l y .

...I

s t i l l think

This s t u d e n t has e v i d e n t l y formed a b e l i e f t h a t C is most l i k e l y , and

experimental data a r e n o t g o i n g t o a l t e r h i s b e l i e f . kELL's r e s e a r c h suggests c a u t i o n i n a t t r i b u t i n g i n f e r e n c e s t r a t e g i e s such as r e p r e s e n t a t i v e n e s s o r a v a i l a b i l i t y , o r n e g l e c t o f b a s e - r a t e d a t a ( f o r whate v e r reason) t o i n d i v i d u a l s based upon summary numerical r e s u l t s .

There may

indeed be many different strategies employed by t h e i n d i v i d u a l s which a r e l o s t i n summary t y p e s t a t i s t i c s .

C l i n i c a l interviews o f subjects' reasoning pro-

cesses p r o v i d e an a l t e r n a t i v e methodology t o i n v e s t i g a t e s t r a t e g i e s .

Several

a u t h o r s have suggested t h a t accuracy o f s u b j e c t s ' i n t r o s p e c t i v e r e p o r t s i s i n s e r i o u s q u e s t i o n when t h e y make judgments. NISBETT and WILSON, 1977).

(NISBETT and BELLOWS, 1977;

S u b j e c t s appear t o be unaware t h a t t h e i r judgments

were a f f e c t e d by c e r t a i n s t i m u l i , and t h u s a r e perhaps u n r e l i a b l e i n t h e i r i n t r o s p e c t i v e report.

However, much use has been made o f " t h i n k i n g a l o u d "

t e c h n i q u e s i n t h e f i e l d o f problem s o l v i n g i n r e c e n t y e a r s (NEWELL and SIMON, 1972; SCHOENFELD, 1979).

WELL and h i s c o l l e a g u e s posed t h e t a x i c a b problem

INFERENCE AND TEACHING PROBABILITY

331

t o t h e i r s u b j e c t s as a problem t o be solved, r a t h e r t h a n as a judgment t o be made.

I n t h e c o n t e x t o f problem s o l v i n g , i n t r o s p e c t i o n may indeed r e l i a b l y

d e t e c t t h e s t r a t e g i e s employed by s u b j e c t s . R e f l e c t i o n s from Mathematical Educati on The p s y c h o l o g i c a l l i t e r a t u r e o f judgment and d e c i s i o n making has p r i m a r i l y used a d u l t s u b j e c t s , o f t e n t h e "freshman psych m a j o r " p o p u l a t i o n , f o r i t s

investigation. I n c o n t r a s t , much r e s e a r c h i n mathematics e d u c a t i o n has d e a l t w i t h young c h i l d r e n ' s i n t u i t i o n s o f p r o b a b i l i t y . Perhaps t h i s i s i n p a r t due t o t h e r e l a t i v e l y r e c e n t t r a n s l a t i o n i n t o E n g l i s h o f PIAGET's (1975) work on t h e o r i g i n o f chance i n c h i l d r e n . FISCHBEIN (1982, 1976, 1975; FISCHBEIN, PAMPU, and MANZAT, 1970a, 1970b) has i n v e s t i g a t e d young c h i l d r e n ' s c o n c e p t i o n o f chance and a l s o t h e e f f e c t s o f t e a c h i n g on t h e i r c o m b i n a t i o n a l and p r o b a b i l i s t i c s a b i l i t y .

He concluded

(1976) t h a t c h i l d r e n and adolescents do n o t d i s t i n g u i s h t h e necessary f r o m t h e p o s s i b l e i n many s i t u a t i o n s , and t h u s a r e i l l - e q u i p p e d t o reason p r o b a b i l i s tically.

According t o FISCHBEIN, t h i s c o u l d be an a r t i f a c t o f o u r i n s t r u c t i o n

i n science e d u c a t i o n which has emphasized d e t e r m i n i s t i c models a t t h e expense o f t h e study o f u n c e r t a i n t y .

Indeed, almost a l l o f o u r mathematics i n s t r u c -

t i o n i s based upon d e t e r m i n i s t i c models. i n s t a n c e , o r Newtonian Physics.

Consider E u c l i d e a n Geometry, f o r

Once t h e laws o r p o s t u l a t e s a r e s t a t e d ,

e v e r y t h i n g e l s e i s determined, and t h e r e i s no room f o r "chance."

It may n o t

be so s u r p r i s i n g t h a t people have so many m i s c o n c e p t i o n s o f chance and probab i l i t y , i f one l o o k s a t t h e i r e d u c a t i o n a l background. Several s t u d i e s have r e p o r t e d on misconceptions o f p r o b a b i l i t y i n o l d e r subjects.

I n o r d e r t o survey t h e i n t u i t i o n s o f chance and p r o b a b i l i t y con-

c e p t s of a d o l e s c e n t s i n England, GREEN (1982) t e s t e d f o u r thousand 11- t o 1 6 - y e a r - o l d secondary s t u d e n t s from t h e m i d l a n d c o u n t i e s .

A f t e r frequent

p i l o t i n g , he developed a p r o b a b i l i t y concepts t e x t t o examine s t u d e n t s ' n o t i o n o f chance, randomness, p r o b a b i l i t y as r a t i o , expected value, and a l s o t h e i r v e r b a l understanding and use o f such phrases as " e q u a l l y l i k e l y , " and " i n f l u e n c e d by chance."

"unlikely,"

Here a r e some o f t h e c o n c l u s i o n s l i s t e d by GREEN

which may be o f i n t e r e s t t o b o t h p s y c h o l o g i s t s and mathematics e d u c a t o r s : a)

Items r e q u i r i n g r a t i o concept a r e v e r y p o o r l y done.

b)

Various s t r a t e g i e s a r e employed by p u p i l s when answering odds p r o b lems, and t h e r e i s l i t t l e c o n s i s t e n c y i n c h o i c e o f s t r a t e g y e i t h e r

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DE VEL OPMENTAND EPISTEMOLOGY

by i n d i v i d u a l s o r by c l a s s e s . c)

Items i n v o l v i n g an a p p r e c i a t i o n o f randomness, t h e s t a b i l i t y o f f r e quencies, and i n f e r e n c e s were p a r t i c u l a r l y p o o r l y done, w i t h ZittZe

improvement with age. d)

P u p i l s ' v e r b a l a b i l i t y i s o f t e n inadequate f o r a c c u r a t e l y d e s c r i b i n g probabilistic situations.

e)

Very o f t e n , c e r t a i n t y and h i g h p r o b a b i l i t y a r e equated, as a r e impos-

f)

s i b i l i t y and low p r o b a b i l i t y . Most p u p i l s do n o t a t t a i n t h e l e v e l o f formal o p e r a t i o n s i n p r o b a b i l i s t i c t h i n k i n g b e f o r e t h e y l e a v e secondary school.

GREEN'S r e s u l t s w i t h secondary s t u d e n t s s u b s t a n t i a t e some o f t h e f i n d i n g s o f

t h e psychological l i t e r a t u r e . o f randomness o r i n f e r e n c e .

I n p a r t i c u l a r , t h e s t u d e n t s had l i t t l e n o t i o n Also, t h e i s s u e o f d e t e r m i n i s m i s once a g a i n

r a i s e d as G R E E N ' S s u b j e c t s f e l t t h e " v e r y l i k e l y " was necessary. Evidence f o r misconceptions o f randomness among o l d e r s u b j e c t s has been documented by FALK (1981a).

S u b j e c t s were shown sequences o f 21 c a r d s o f t w o

c o l o r s , say 10 green and 11 y e l l o w , and were asked t o r a t e t h e sequences f o r randomness.

S u b j e c t s committed b o t h t y p e I and t y p e I 1 e r r o r s , s e e i n g p a t -

t e r n s where none e x i s t e d and i n f e r r i n g randomness when a non-chance sequence was t o be judged.

FALK concludes t h a t q u i t e o f t e n people r e j e c t t h e chance

h y p o t h e s i s when making judgments and search f o r a l t e r n a t i v e e x p l a n a t i o n s .

In

a n o t h e r r e p o r t FALK (1981b) p o i n t s o u t t h a t " m i r a c l e s " and "coincidences," such as chance meetings between f r i e n d s , a r e much more l i k e l y t o o c c u r t h a n we think.

People t e n d t o l o o k a t t h e i n t e r s e c t i o n of a l l t h e m i n u t e d e t a i l s t h a t

surround a coincidence, and t h u s j u d g e i t t o have a v e r y l o w p r o b a b i l i t y o f occurring.

I f events s u r r o u n d i n g a s o - c a l l e d c o i n c i d e n c e a r e viewed disiunc-

tiveZy i n s t e a d o f conjunctively, i t i s p o s s i b l e t o demonstrate m a t h e m a t i c a l l y t h a t t h e event i s much more l i k e l y t o occur t h a n we imagine. The s t u d i e s o f FISCHBEIN, GREEN, and FALK suggest t h a t , a t l e a s t i n p a r t , p e o p l e ' s m i s c o n c e p t i o n s o f p r o b a b i l i t y may be due t o a l a c k o f e x p e r i e n c e i n p r o b a b i l i s t i c t h i n k i n g , a l a c k o f exposure t o p r o b a b i l i s t i c models i n o u r schools, and an overemphasis upon d e t e r m i n i s t i c r e a s o n i n g i n society.

our s c h o o l s and

KAHNEMAN and TVERSKY (1971) have v o i c e d concern t h a t even p e o p l e

t r a i n e d i n p r o b a b i l i t y and s t a t i s t i c s s t i l l a r e s u b j e c t t o t h e same b i a s e s and m i s c o n c e p t i o n s as n a i v e s u b j e c t s , and so perhaps mere exposure t o t h e laws and p r i n c i p l e s o f p r o b a b i l i t y and s t a t i s t i c s w i l l n o t change p e o p l e ' s manner o f

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making i n f e r e n c e s and judgments,

However, l e t us c o n s i d e r h a , a s t u d e n t

u s u a l l y encounters p r o b a b i l i t y and s t a t i s t i c s i n , say, a c o l l e g e course. P r o b a b i l i t y may be i n t r o d u c e d t h r o u g h s e t s , sample space, events, and formal c o u n t i n g p r i n c i p l e s . The model o f t e a c h i n g i s most l i k e l y a formal l e c t u r e where t h e s t u d e n t watches P r o f e s s o r X " s o l v e problems."

I n addition t o the

f o r m a l l e c t u r e s and t h e formal mathematized model o f p r o b a b i l i t y , t h e s t u d e n t has t o cope w i t h a brand new formal language--combination, p e r m u t a t i o n , e v e n t , sample space, and so on. F i r s t encounters w i t h s t a t i s t i c s a r e p r o b a b l y no l e s s formal. The s t u d e n t i s p r e s e n t e d w i t h some " d a t a " and t o l d t h a t t h e r e a r e some i m p o r t a n t c a l c u l a t i o n s t o be done w i t h t h e s e numbers--mean, v a r i a t i o n , s t a n d a r d d e v i a t i o n - - a n d t h a t he had b e t t e r know them.

Why s h o u l d

s t u d e n t s be good a t making i n f e r e n c e s i f t h e importance o f u s i n g d a t a t o make i n f e r e n c e s i s never mentioned i n t h e i r course work o r i n t h e i r experience?

To

borrow a phrase from N i s b e t t , what may be needed h e r e i s a " v i v i d , c o n c r e t e " p r e s e n t a t i o n o f data t o focus t h e a t t e n t i o n o f s t u d e n t s on t h e importance and usefulness o f s t a t i s t i c s . I n an a t t e m p t t o overcome n a i v e c o l l e g e s t u d e n t s ' r e l i a n c e upon t h e heuri s t i c s of a v a i l a b i l i t y and r e p r e s e n t a t i v e n e s s , SHAUGHNESSY (1977) developed an experimental course i n p r o b a b i l i t y and s t a t i s t i c s ,

The course c o n s i s t e d o f a

s e r i e s of a c t i v i t i e s which emphasized experiments, d a t a c o l l e c t i o n , s i m u l a t i o n o f experiments, making i n f e r e n c e s from e x p e r i m e n t a l data, model b u i l d i n g , and comparing t h e o r e t i c a l r e s u l t s w i t h experimental r e s u l t s .

The course was con-

d u c t e d w i t h s t a t i s t i c a l l y - n a i v e c o l l e g e freshmen ( m o s t l y ) and met f o r t e n weeks, f i v e t i m e s a week f o r an hour. elsewhere (SHAUGHNESSY 1977, 1982).

The a c t i v i t i e s have been d e s c r i b e d Students worked i n small groups o f t h r e e

o r four, made p r e l i m i n a r y guesses f o r t h e l i k e l i h o o d o f events, c a r r i e d o u t an experiment and t h e n t a b u l a t e d , organized, and analyzed t h e i r d a t a t o make t h e i n f e r e n c e s requested i n t h e a c t i v i t i e s .

There was no t r a d i t i o n a l l e c t u r i n g

i n t h e experimental s e c t i o n s o f t h e course, as t h e i n s t r u c t o r s were p r i m a r i l y f a c i l i t a t o r s f o r t h e a c t i v i t y i n t h e small groups.

Several s e c t i o n s o f t h e

experimental course were compared t o s e c t i o n s o f a more t r a d i t i o n a l l e c t u r e course on p r o b a b i l i t y and s t a t i s t i c s , t o t e s t f o r s t u d e n t r e l i a n c e upon t h e h e u r i s t i c s of a v a i l a b i l i t y and r e p r e s e n t a t i v e n e s s , b o t h b e f o r e and a f t e r exposure t o p r o b a b i l i t y .

The r e s u l t s o f t h e s u b j e c t s ' performance on some o f

t h e e a r l y t a s k s of KAHNEMAN and TVERSKY (1972, 1973) i n d i c a t e t h a t s t u d e n t s i n t h e small group a c t i v i t y - b a s e d course r e l i e d l e s s on r e p r e s e n t a t i v e n e s s and a v a i l a b i l i t y t h a n s t u d e n t s i n t h e usual l e c t u r e approach.

DEVELOPMENT AND EPISTEMO LOG Y

334

11.

Some Suggestions for I m p r o v i n g R e l i a n c e upon S t a t i s t i c s when Making Inferences

Suggestions From Psycho1 ogy There have been s e v e r a l v i e w p o i n t s expressed by p s y c h o l o g i s t s on t h e poss i b i l i t y o f i m p r o v i n g judgment and d e c i s i o n making by e d u c a t i o n i n s t a t i s t i c s . We have a l r e a d y mentioned t h e s k e p t i c a l r e s u l t s o f KAHNEMAN and TVERSKY (1971) who found b i a s e s and misconceptions among t h e i r s t a t i s t i c a l l y w e l l - t r a i n e d c o l l e a g u e s . SLOVIC, FISCHHOFF, and LICHTENSTEIN (1976, 1977, 1980) suggest t h a t humans may be p r i m a r i l y t r i a l - a n d - e r r o r

l e a r n e r s when i t comes t o making

d e c i s i o n s and t h u s s u f f e r t h e consequences.

We may l a c k adequate p r o b a b i l i s -

t i c c o g n i t i v e processes t o deal w i t h d e c i s i o n making, a g r e a t danger i n t h i s n u c l e a r age.

They suggest t h a t b o t h t h e e x p e r t s and t h e p u b l i c need t o be

educated i n t h e s u b t l e t i e s o f s t a t i s t i c a l t h i n k i n g s t a r t i n g as e a r l y as t h e l o w e r grades i n school.

They do n o t , however, h o l d o u t much hope f o r t h e

e f f e c t s o f e d u c a t i o n on p e o p l e ' s judgmental a b i l i t y .

People t e n d t o be o v e r -

c o n f i d e n t o f t h e i r a b i l i t y and possess b e l i e f s t h a t a r e e x t r a o r d i n a r i l y r e s i s t a n t t o change.

Furthermore, e d u c a t i o n may j u s t make people aware o f "more

t h i n g s t h a t can go wrong." NISBETT and ROSS (1980) h o l d o u t hope t h a t p e o p l e ' s i n f e r e n c e e r r o r s a r e more f r e q u e n t l y c o g n i t i v e t h a n emotional ( a l s o r e p o r t e d i n LA BECQUE, 1980). They c l a i m t h a t t h e emotional and p s y c h o - a n a l y t i c aspects o f d e c i s i o n making have perhaps r e c e i v e d over-emphasi s. o f probabi 1 it y and s t a t i s t i cs.

They, t o o , recommend t h e e a r l y t e a c h i n g

Secondary and c o l 1ege courses i n p r o b a b i 1 it y

should p r o v i d e a b e t t e r grounding i n i n f e r e n t i a l r u l e s t h a t emphasize t h e c o n t r a s t between formal and i n f o r m a l i n t u i t i v e s t r a t e g i e s .

Concrete examples

A t t h e end o f t h e i r book (1980) NISBETT and ROSS o u t l i n e what t h e y r e f e r t o as t h e " f o r t u n e c o o k i e

o f i n t u i t i v e e r r o r s should be g i v e n i n such courses.

d i d a c t i c " which should be employed when making d e c i s i o n s .

Roughly, t h e f o r -

t u n e cookie d i d a c t i c suggests t h a t we: 1)

2)

Look a t judgmental q u e s t i o n s e m p i r i c a Z Z y - - o b t a i n and a n a l y z e data. Pay c l o s e a t t e n t i o n t o how samples a r e obtained--which h a t d i d t h e y come from?

3)

Check f o r c o v a r i a t i o n - - w h a t

4)

Beware o f t h e fundamental a t t r i b u t i o n e r r o r when making judgments o r

do t h e o t h e r t h r e e c e l l s l o o k l i k e ?

i n f e r e n c e s - - b e h a v i o r may be due t o s i t u a t i o n s r a t h e r t h a n p e r s o n a l disposition.

INFERENCE AND TEACHING PROBABILITY

335

5) Since e x c e p t i o n s t o o u r personal t h e o r i e s a r e t o o e a s i l y e x p l a i n e d away, use new d a t a , e s p e c i a l l y i f i t p e r s i s t s and i s r e p l i c a b l e . The f o r t u n e c o o k i e d i d a c t i c c o n t a i n s recommendations t h a t a r e r e m i n i s c e n t o f t h o s e found i n D a r r e l H u f f ' s book How t o Lie with S t a t i s t i c s (1954). Although NISBETT and ROSS suggest c o g n i t i v e approaches t o i m p r o v i n g d e c i s i o n making, NISBETT, BORGIDA, CRANDALL, and REED (1976) remind us t h a t p e o p l e a r e n o t moved by d r y s t a t i s t i c a l data.

Effective, concrete, emotionally

i n t e r e s t i n g ways of p r e s e n t i n g d a t a need t o be used t o persuade p e o p l e t o pay a t t e n t i o n . They p r o v i d e several anecdotes a t t h e end o f t h e i r a r t i c l e t o demonstrate t h e e f f e c t o f v i v i d c o n c r e t e i n f o r m a t i o n on d e c i s i o n making. I n c l u d e d are r e f e r e n c e s t o t h e huge i n c r e a s e i n v i s i t s t o cancer d e t e c t i n g agencies s i n c e former f i r s t l a d y B e t t y F o r d ' s o p e r a t i o n , and t h e e f f e c t ' o f l i v e t e l e v i s i o n coverage o f t h e Vietnam war on t h e American p u b l i c . b o t h h o t and c o l d c o g n i t i o n s - - e m o t i o n a l

Perhaps

and c o g n i t i v e processes--are e s s e n t i a l

t o t h e improvement o f judgment and d e c i s i o n making.

I n o r d e r t o g e t a judge

t o use s t a t i s t i c a l data, one f i r s t has t o g e t him t o attend t o it.

(One

cannot h e l p b u t r e c a l l t h e anecdote o f t h e farmer who helped h i s n e i g h b o r t o g e t h i s donkey moving.

The farmer watched f o r a w h i l e as no amount o f c o a x i n g

and y e l l i n g by h i s n e i g h b o r c o u l d g e t t h e donkey going. p i c k e d up a board and h i t t h e donkey over t h e head. attention first,"

he s a i d .

F i n a l l y , t h e farmer

"You have t o g e t h i s

I t may be q u i t e t h e same w i t h p e o p l e and t h e i r use

o f p r o b a b i l i t y when making i n f e r e n c e s . ) Suggestions From Mathematics Education During t h e p a s t s e v e r a l y e a r s , t h e r e has been a r e k i n d l i n g o f concern about t h e t e a c h i n g o f p r o b a b i l i t y and s t a t i s t i c s a t a l l school l e v e l s , element a r y , secondary, and i n t r o d u c t o r y c o l l e g e .

T h i s concern f o r s t a t i s t i c a l

l i t e r a c y has up u n t i l now n o t been much i n f l u e n c e d by r e s e a r c h r e s u l t s f r o m psychology.

Rattier, t h e r e has been a growing concern i n t h e mathematical

e d u c a t i o n community f o r i m p r o v i n g t h e s t a t i s t i c a l e d u c a t i o n o f o u r s t u d e n t s who must f a c e an i n c r e a s i n g l y t e c h n o l o g i c a l j o b market, and become f u t u r e consumers who a r e c o n f r o n t e d w i t h masses o f " s t a t i s t i c a l " d a t a i n newspaper, r a d i o , and t e l e v i s i o n .

U n f o r t u n a t e l y , t h e t e a c h i n g o f p r o b a b i l i t y and s t a t i s -

t i c s i n t h e schools has taken a back s e a t t o o t h e r t o p i c s i n t h e c u r r i c u l u m f o r q u i t e some time. T h i s i s q u i t e apparent i n t h e U n i t e d States, where s t u dents t e n d t o e i t h e r drop out o f mathematics a f t e r a y e a r o f a l g e b r a and a y e a r o f geometry, o r t o f o l l o w t h e " c o l l e g e prep" t r a c k and t a k e a second y e a r of a l g e b r a and a y e a r o f c a l c u l u s and a n a l y t i c geometry.

This curriculum,

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DEVELOPMENT AND EPISTEMOLOG Y

which has been entrenched f o r many y e a r s i n s c h o o l s i n t h e s t a t e s , l e a v e s v e r y l i t t l e space and t i m e f o r t o p i c s from p r o b a b i l i t y and s t a t i s t i c s .

Some r e c e n t

developments have attempted t o r e f o c u s t h e a t t e n t i o n o f mathematics e d u c a t o r s The 1981 Yearbook o f t h e

upon t h e importance o f p r o b a b i l i t y and s t a t i s t i c s .

N a t i o n a l Council o f Teachers o f Mathematics was devoted t o t h e t e a c h i n g o f P r o b a b i l i t y and S t a t i s t i c s (SHULTE, 1981).

T h i s book c o n t a i n s a number o f

i n t e r e s t i n g approaches t o t h e t e a c h i n g o f s t a t i s t i c s which a r e a p p r o p r i a t e f o r Emphasis i s p l a c e d upon experiments and s i m u l a t i o n s . grades 5-12. A s e r i e s of e x p l o r a t o r y a c t i v i t i e s c a l l e d Statistics and Information

Organization (Mathematics Resource P r o j e c t , 1977) was developed i n t h e U n i t e d A t t h e same t i m e a p r o j e c t S t a t e s f o r m i d d l e school s t u d e n t s (grades 5-8). i n Great B r i t a i n has been d e v e l o p i n g t o p i c a l modules f o r t e a c h i n g s t a t i s t i c s t o secondary s t u d e n t s (HOLMES and TURNER, 1981). Indeed, i t i s n o t f o r want o f e x c e l l e n t m a t e r i a l s t h a t p r o b a b i l i t y and s t a t i s t i c s i s s l i g h t e d i n t h e schools.

The problem i s t o d i s s e m i n a t e t h e s e m a t e r i a l s and t o c o n v i n c e more

s c h o o l s t h a t t e a c h i n g p r o b a b i l i t y and s t a t i s t i c s i s i m p o r t a n t .

As a f i r s t

s t e p toward t h i s goal, t h e F i r s t I n t e r n a t i o n a l Conference on t h e Teaching o f S t a t i s t i c s ( I C O T S I ) was h e l d i n S h e f f i e l d i n t h e summer o f 1982.

Sessions a t

t h i s conference i n c l u d e d papers on i n t u i t i v e n o t i o n s o f chance among c h i l d r e n and ado1 escents. Several a u t h o r s have made s p e c i f i c recommendations f o r t h e t e a c h i n g o f p r o b a b i l i t y and s t a t i s t i c s i n o r d e r t o improve s t a t i s t i c a l l i t e r a c y and conf r o n t misconceptions o f p r o b a b i l i t y t h a t a r e o f a p s y c h o l o g i c a l n a t u r e .

After

i n v e s t i g a t i n g t h e p r o b a b l i s t i c s concepts o f 11- t o 16-year-olds GREEN (1982) gave a number o f recommendations f o r t h e t e a c h i n g o f p r o b a b i l i t y and s t a t i s t i c s i n t h e schools.

Among h i s suggestions were:

1) There i s a g r e a t need f o r practicaZ activity a t an e a r l y age, emphas i z i n g guided d i s c o v e r y .

2)

Emphasis should be p l a c e d upon mathematical m o d e l l i n g and i n t e r p r e t a -

t i o n i n school mathematics. 3 ) R a t i o i s a p r e r e q u i s i t e o f many complex p r o b a b i l i s t i c s i t u a t i o n s , and t h u s more emphasis should be p l a c e d upon s t u d e n t s ' u n d e r s t a n d i n g o f f r a c t i o n s and r a t i o . 4 ) The concept o f randomness i s s o r e l y n e g l e c t e d i n o u r d e t e r m i n i s t i c a l l y o r i e n t e d e d u c a t i o n a l system. T h i s i s unreasonable g i v e n t h a t we a r e surrounded by so many random occurrences. 5)

Students have no e x p e r i e n c e with drawing i n f e r e n c e s from data, and

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t h i s r e q u i r e s immediate a t t e n t i o n i n t h e c u r r i c u l u m .

STEINBRING and VON HARTEN (1982) suggest t h a t s t a t i s t i c s be i n t r o d u c e d They a l s o

v i a t h e concept of r e l a t i v e frequency, s t a r t i n g w i t h a p p l i c a t i o n s .

promote m u l t i p l e approaches t o t h e t e a c h i n g o f s t o c h a s t i c s , b o t h o b j e c t i v e ( f r e q u e n t i a l ) and s u b j e c t i v e (Bayesian). According t o them, s t o c h a s t i c s a t t h e school l e v e l should emphasize b o t h t h e t r a n s f o r m a t i o n o f d a t a i n t o i n s i g h t s and t h e t h e o r y o f d e c i s i o n s .

FALK (1980) suggests t h a t we c o n t i n u a l l y

c a r r y o u t s u b j e c t i v e s e n s i t i v i t y a n a l y s i s w i t h o u r s t u d e n t s i n t h e classroom, changing t h e v a l u e o f parameters i n o r d e r t o h e l p s t u d e n t s improve t h e i r i n t u i t i o n s as t o how d a t a can a f f e c t i n f e r e n c e s . SHAUGHNESSY (1981, 1982) has s e t f o r t h a number o f s u g g e s t i o n s f o r t e a c h -

i n g i n t r o d u c t o r y p r o b a b i l i t y and s t a t i s t i c s t o secondary o r c o l l e g e s t u d e n t s . These i n c l u d e : 1. Teach from guesses, t o experiments, t o simuzation, t o theory. I n t r o d u c t o r y p r o b a b i l i t y and s t a t i s t i c s should be a c t i v i t y - b a s e d and experimental.

We should proceed from guesses t o experiments t o s i m u l a t i o n s o f

experiments, and o n l y a f t e r c o n s i d e r a b l e i n t u i t i o n has been developed, t o a formal mathematical p r e s e n t a t i o n . which misconceptions a r i s e .

We can c o n f r o n t s t u d e n t s w i t h problems i n

R e s u l t s from a p r e v i o u s s t u d y (SHAUGHNESSY, 1977)

suggest t h a t s t u d e n t s have a b e t t e r chance o f overcoming m i s c o n c e p t i o n s o f p r o b a b i l i t y t h a t a r e due t o i n t u i t i v e h e u r i s t i c s such as r e p r e s e n t a t i v e n e s s o r a v a i l a b i l i t y i f they are introduced t o p r o b a b i l i t y i n experimental a c t i v i t y based mode r a t h e r t h a n i n a formal l e c t u r e mode.

Students can work i n s m a l l

groups, make guesses f o r t h e l i k e l i h o o d o f events, p e r f o r m an experiment, o r g a n i z e and analyze t h e data, and make e s t i m a t e s f o r t h e l i k e l i h o o d o f events based on t h e i r data.

More advanced s t u d e n t s can perhaps a t t e m p t t o b u i l d a

t h e o r e t i c a l p r o b a b i l i t y model f o r t h e experiment and t h e n compare t h e o r e t i c a l p r e d i c t i o n s t o experimental outcomes.

The reasons f o r any d i s c r e p a n c i e s

between experimental r e s u l t s and t h e o r e t i c a l p r e d i c t i o n s , such as sources of b i a s i n t h e d a t a - g a t h e r i n g process, can be discussed.

An e x p e r i m e n t a l

approach t o p r o b a b i l i t y may, i n f a c t , be t h e o n l y way t o break t h e " b e l i e f b a r r i e r " f o r those s t u d e n t s who do n o t understand chance a t a l l , and who r e l y upon causal schemas o r o t h e r d e t e r m i n i s t i c e x p l a n a t i o n s o f p r o b a b i l i s t i c phenomena.

Emphasize the confounding roZe t h a t beZief systems can pZay i n ju&ment and decision making. A t t h e same time, t e a c h t h e importance and u s e f u l 2.

ness of s t a t i s t i c s f o r h e l p i n g t o make o b j e c t i v e judgments and d e c i s i o n s .

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Suppose t h a t you a r e t a k i n g a s t a t i s t i c s course which has weekly q u i z z e s w i t h 10 q u e s t i o n s each. Your average a l l y e a r has been 7 c o r r e c t out o f 10. For t h e n e x t q u i z , you s t u d y w i t h a f r i e n d and get 9 o u t o f 10. Did s t u d y i n g w i t h a f r i e n d h e l p ? One p o s s i b l e response t o t h i s q u e s t i o n i s : b e t t e r than 7?"

" O f course i t helped, i s n ' t 9

Another p o s s i b l e response i s :

"Probably n o t , 9 c o u l d have

j u s t happened t h a t t i m e s i n c e y o u r average o f 7 i s c l o s e t o 9."

I f you ask

t h i s q u e s t i o n o f s t u d e n t s i n an i n t r o d u c t o r y course i n s t a t i s t i c s , you w i l l g e t b o t h t h e answers above, i n a d d i t i o n t o a spectrum o f " i t depends" answers i n between,

*These responses r a r e l y i n d i c a t e t h e need f o r any s o r t o f c a l c u l a -

I n o r d e r t o r e s o l v e t h e argument i n an objective way, t h e c l a s s needs

tion.

some data.

This i s an e x c e l l e n t o p p o r t u n i t y t o c a r r y out a s i m u l a t i o n , say

w i t h random numbers, and see j u s t how l i k e l y i t i s t o o b t a i n t h e e v e n t 9 o u t o f 10 c o r r e c t i f you have p r o b a b i l i t y o f success P(s) = . 7 on any one item. ( I t i s n o t necessary--or a d v i s a b l e i n t h i s a u t h o r ' s o p i n i o n - - t o do a c a l c u l a t i o n u s i n g t h e b i n o m i a l d i s t r i b u t i o n t h e f i r s t t i m e a s t u d e n t sees t h i s t y p e o f problem. ) Suppose t h a t a consumers' group f e e l s t h a t t h e gas t a n k s on a c e r t a i n model o f automobile a r e l i k e l y t o explode on impact from t h e r e a r . T h i s group has been formed because o f t h e u n f o r t u n a t e d e a t h o f a f r i e n d i n j u s t such an a c c i d e n t . The consumers' group has formed a belief about a c a r model. They may even have found o t h e r i n s t a n c e s o f such an a c c i d e n t . Should t h e c a r be r e c a l l e d ? We must a t t e m p t t o make an o b j e c t i v e d e c i s i o n , i f p o s s i b l e .

Clearly there

i s some need f o r more data, and a c l a s s i n i n t r o d u c t o r y s t a t i s t i c s c o u l d p e r haps be l e d t o suggest some reasonable experiment t h a t can be c a r r i e d o u t t o determine t h e s a f e t y o f t h e gas tanks.

I n a problem such as t h i s , t h e impor-

t a n c e o f sampling i n making d e c i s i o n s f r o m s t a t i s t i c s can a l s o be b r o u g h t out. (Would we do our experiments on c a r s a l l from one p l a n t ?

Would we smash a l l

t h e cars?) I f we focus on t h e u s e f u l n e s s and importance o f s t a t i s t i c s f o r making obj e c t i v e d e c i s i o n s , we may be a b l e t o temper t h e e f f e c t s o f b e l i e f systems o r d e t e r m i n i s t i c approaches t o p r o b a b i l i t y i n o u r students. 3.

Sensitize students t o misuses of s t a t i s t i c s .

I f we demonstrate t h e importance o f s t a t i s t i c s i n making o b j e c t i v e d e c i s i o n s we a r e a l s o o b l i g e d t o show o u r s t u d e n t s how s t a t i s t i c s can be used t o m i s l e a d o r t o " i n f l u e n c e " o u r b e l i e f systems.

Books such as Ha, t o Lie with

S t a t i s t i c s (HUFF, 1954) should be r e q u i r e d r e a d i n g i n a l l i n t r o d u c t o r y p r o -

INFERENCE AND TEACHING PROBABILITY

339

b a b i l i t y and s t a t i s t i c s courses. Such books can s e n s i t i z e o u r s t u d e n t s t o misuses o f s t a t i s t i c s , d e l i b e r a t e o r o t h e r w i s e , and encourage them t o a n a l y z e p r o b a b i l i s t i c statements c r i t i c a l l y . Students can be asked t o keep a l o g o f misuses o f s t a t i s t i c s t h a t t h e y A r t i c l e s can be taken from newspapers o r p e r i o d i c a l s and analyzed

discover.

f o r c o r r e c t o r i n c o r r e c t use o f s t a t i s t i c s .

Students who have k e p t such a l o g

have f e r r e t e d out m i s l e a d i n g graphs, i n f l a t e d percentages, b i a s e d samples, i n s u f f i c i e n t sample s i z e , and verbose s t a t i s t i c a l q u a n t i t a t i v e d e s c r i p t i o n s t h a t had l i t t l e o r no b a s i s i n f a c t .

One s t u d e n t d i s c o v e r e d an a r t i c l e on

t h e f r o n t page o f t h e l a r g e s t - s e l l i n g newspaper i n a m a j o r U.S.

c i t y that

mentioned t h a t 95 p e r c e n t o f t h e p e o p l e who were i n t e r v i e w e d were a g a i n s t c r o s s - d i s t r i c t b u s i n g t o achieve r a c i a l i n t e g r a t i o n i n t h e schools.

A con-

t i n u a t i o n o f t h e a r t i c l e (on page 14) mentioned t h a t 70 p e r c e n t o f t h e p e o p l e i n t e r v i e w e d d i d n o t have c h i l d r e n i n t h e schools. Even " l i t e r a t e " newspaper c o l u m n i s t s a r e prone t o h u r t l e s t a t i s t i c s about i n defense o f t h e i r p o s i t i o n , w i t h t o t a l d i s r e g a r d f o r t h e d a t a base.

I n an

a r t i c l e c e l e b r a t i n g a new h i g h school a l g e b r a t e x t b o o k f o r i t s s u c c e s s f u l b a t t l e a g a i n s t a l g e b r a i c i l l i t e r a c y , W i l l i a m F. BUCKLEY (1982) t o s s e s p e r c e n t ages about w i t h no r e f e r e n c e t o t h e o r i g i n a l data, and perhaps l i t t l e b a s i s i n

He c l a i m s f i r s t - y e a r a l g e b r a s t u d e n t s u s i n g t h i s new t e x t o u t s c o r e d

fact.

second-year a1 gebra s t u d e n t s "by an a s t o n i s h i n g 200%."

We should encourage

o u r students t o be v e r y c r i t i c a l o f such loose f l o a t i n g percentages.

Teaching

t h e misuses o f s t a t i s t i c s i s a v e r y concrete, v i v i d (NISBETT e t . al.,

1976)

approach which emphasizes t h e v a l u e o f s t a t i s t i c s . 4. Give students chaZZenge probZems which may require them t o gather d a t e t h e i r own data, and then make a decision.

OP

sim-

A f t e r s e v e r a l weeks i n an i n t r o d u c t o r y p r o b a b i l i t y and s t a t i s t i c s course, c h a l l e n g e problems can be g i v e n which c o n f r o n t m i s c o n c e p t i o n s o f s t a t i s t i c s , demonstrate p r a c t i c a l usefulness, and encourage s i m u l a t i o n o f experiments. Here i s a sample o f such problems g i v e n by t h e a u t h o r i n t h e p a s t . Sources f o r t h e problems i n c l u d e MOSTELLER (1962), MOSTELLER e t . a l . (1973), and RADE (1977), as w e l l as some problems from t h e l i t e r a t u r e o f judgment and d e c i s i o n making, and some c r e a t e d by t h e author.

StimuZation by SimuZation The q u e s t i o n s i n Stirnulation by SimuZatim a r e somewhat open-ended i n nature.

I n many cases i t i s necessary t o f i r s t d e f i n e c l e a r l y what t h e

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problem i s , and t h e n t o adopt a mode o f s i m u l a t i o n which w i l l e n a b l e you t o "model" an experiment which may be t o o cumbersome t o a c t u a l l y c a r r y o u t . Before d o i n g any o f these problems, aZW4s.vrite down P u p best guess

first, and o n l y t h e n c a r r y out a s i m u l a t i o n o f t h e problem. A simulation o f probabi 1 it y experiment i n v o l ves : a ) modeling an experiment by u s i n g some apparatus w i t h known p r o b a b i l i t i e s b)

( i .e., c o i n s , d i c e , spinners, random numbers) "performing" t h e experiment many t i m e s w i t h y o u r apparatus, t h u s p o o l i n g

c)

r e s u l t s o f s e v e r a l small groups may h e l p t o g e t a " l a r g e enough" sample. g a t h e r i n g , o r g a n i z i n g , and a n a l y z i n g t h e d a t a (some s t a t i s t i c a l s k i l l s )

d)

c a l c u l a t i n g experimental p r o b a b i l i t i e s , o r o t h e r e x p e r i m e n t a l outcomes (i.e.,

e)

f r e q u e n c i e s ) from t h e d a t a

making i n f e r e n c e s o r drawing c o n c l u s i o n s from t h e e x p e r i m e n t a l outcomes, t h a t i s , l o o k i n g back.

Frobterns for StimuZation by Simulation 1.

A c e r e a l company has p u t p l a s t i c models o f 5 S t a r Wars c h a r a c t e r s i n boxes o f SPACY-0's. The company p u t s one c h a r a c t e r i n each box. How many would you expect t o have t o buy i n o r d e r t o g e t a l l f i v e c h a r a c t e r s ? ( W r i t e down a guess f i r s t . )

2.

F i v e f r i e n d s a r e s i t t i n g around a t a b l e a t a p a r t y d i s c u s s i n g t h e m e r i t s of a s t r o l o g y . What i s t h e p r o b a b i l i t y t h a t a t l e a s t two o f them have t h e same a s t r o l o g i c a l s i g n ? (Assume t h e s i g n s a r e equal i n d u r a t i o n . )

3.

Some d a t a were o b t a i n e d from t h e r e c o r d s o f t h e Bureau o f Meteorology i n P e r t h , A u s t r a l i a . A sunny day was d e f i n e d as a day on which t h e o f f i c i a l l y recorded number o f hours o f sunshine was 70% o r more o f t h e maximum p o s s i b l e hours of sunshine f o r t h a t day o f t h e year. The months A p r i l t o September i n 1973 seemed t o be t y p i c a l months i n terms o f a v e r age number o f hours of sunshine. A c l a s s i f i c a t i o n o f each day i n t h i s p e r i o d as "sunny" o r " d u l l " ( S o r D) and r e c o r d i n g o f t h e e v e n t s l e d t o the following: D followed D followed S followed S followed

by by by by

D: S: D: S:

79 33 33 37

occasions occasions occasions occasions

Thus t h e p r o p o r t i o n o f d u l l days f o l l o w e d by d u l l days was 79/112 = .70, and t h e p r o p o r t i o n of sunny days f o l l o w e d by sunny days was 37/70 = .53.

4.

a)

What i s t h e "average" l e n g t h o f runs o f c o n s e c u t i v e d u l l days?

b)

What i s t h e "average" l e n g t h o f runs o f c o n s e c u t i v e sunny days?

Two p o i n t s (numbers) between

0

and

1 a r e chosen a t random.

The

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INFERENCE AND TEACHING PROBABILITY

p o i n t s d i v i d e t h e i n t e r v a l from 0 t o 1 i n t o t h r e e l i n e segments. What i s t h e p r o b a b i l i t y t h a t t h e t h r e e segments w i l l form a t r i a n g l e ?

5.

About one-half t h e babies born a r e boys. Would t h e r e be more days d u r i n g t h e calendar y e a r when a t l e a s t 60% o f t h e babies b o r n were boys: i )

i n a small h o s p i t a l

ii) i n a l a r g e h o s p i t a l iii) 6.

makes no d i f f e r e n c e

Suppose you a r e p l a y i n g a game i n which you have a 50% chance o f winning. Bet a d o l l a r . Each t i m e you l o s e , double y o u r bet. Each t i m e you win, go back t o b e t t i n g a d o l l a r . I s t h i s a good b e t t i n g system?

7.

Fast A1 suggests t o Slow Sam t h a t t h e y p l a y t h e f o l l o w i n g game. They w i l l I f Sam throws one and A1 t h r o w s two, each t h r o w e i t h e r one o r two f i n g e r s . Sam wins $30. I f Sam throws two and A1 throws one, Sam wins $10. I f t h e y t h r o w t h e same number o f f i n g e r s , A1 w i n s $20. a)

If each p l a y e r p l a y s a 50-50 s t r a t e g y between t h e i r two choices, who wins i n t h e l o n g run?

b)

What i s t h e b e s t s t r a t e g y f o r Slow Sam, i.e., choices?

how should he d i v i d e h i s

8.

From where he stands, one s t e p toward a c l i f f would send t h e drunken man o v e r t h e c l i f f . He t a k e s random steps, e i t h e r towards t h e c l f f o r away f r o m it. A t any step, t h e p r o b a b i l i t y t h a t he t a k e s a s t e p toward t h e c l i f f i s 1/3, and t h e p r o b a b i l i t y t h a t he s t e p s away f r o m i t i s 2/3. What i s h i s chance o f escaping a f a l l ?

9.

16 teams a r e e n t e r e d i n a double e l i m i n a t i o n s o f t b a l l tournament. t h e expected number o f games a team w i l l p l a y :

10.

i )

if t h e team has a 50% chance o f w i n n i n g each game?

ii)

i f t h e team has a 70% chance o f w i n n i n g each game?

What i s

Several y e a r s ago a c e r t a i n beer company c l a i m e d t h a t t h e y " t r i e d t h e i r famous High L i f e t a s t e t e s t on many a beer e x p e r t , and no one had y e t c o r r e c t l y i d e n t i f i e d which beer came from a can, which from a b o t t l e , and which from a tap." How l i k e l y i s i t t h a t t h e beer company was t e l l i n g t h e t r u t h ? O r , perhaps another way t o p u t i t , how many beer e x p e r t s do you suppose t h e company r e a l l y tested?

11.

Suppose a man faces n o r t h and t o s s e s a c o i n t o d e c i d e whether t o t a k e one s t e p n o r t h o r one s t e p south. The man does t h i s f o r t w e n t y - f i v e t r i e s .

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DEVELOPMENT AND EPISTEMOLOGY

i ) On t h e average, how f a r i s he from t h e s t a r t i n g p o i n t ? ii) iii) 12.

13.

5.

On how many steps i s he on t h e n o r t h s i d e o f h i s s t a r t i n g p o i n t ? On t h e s o u t h s i d e ? How o f t e n does he r e t u r n t o t h e s t a r t i n g p o i n t ?

Three towers a r e b u i l t o f 8, 6, and 4 b l o c k s . dom and a b l o c k removed.

A t o w e r i s chosen a t r a n -

a)

What i s t h e expected number o f b l o c k s t h a t w i l l be removed b e f o r e one o f t h e towers has been " l e v e l e d " (i.e., no b l o c k s l e f t ) ?

b)

What i s t h e p r o b a b i l i t y t h e 4-block tower w i l l be d e s t r o y e d ?

c)

Can you t h i n k o f a " r e a l w o r l d " s i t u a t i o n f o r which t h i s tower game i s a model?

(The b i g - b a d - t r i a n g l e , o r t h r e e ' s a crowd.) Three d u e l e r s , t h e Good, t h e Bad, and t h e Ugly, a r e t o f i g h t a t h r e e - c o r n e r e d p i s t o l d u e l . A l l know t h a t t h e U g l y ' s chance o f h i t t i n g h i s t a r g e t i s .30, t h e B a d ' s i s .50, and t h e Good ( o f c o u r s e ! ) never misses. They a r e t o f i r e a t t h e i r c h o i c e o f t a r g e t i n succession i n t h e o r d e r Ugly, Good, Bad, c y c l i c a l l y u n t i l o n l y one man i s l e f t u n h i t . A h i t man l o s e s f u r t h e r t u r n s , and i s no l o n g e r shot a t . Ugly has t h e f i r s t shot. What s h o u l d he do t o have t h e b e s t chance o f s u r v i v a l ? (Running away d o e s n ' t c o u n t ! )

Expose Students t o S t a t i s t i c a l Paradoxes We have a1 ready mentioned t h e importance o f s e n s i t i z i n g s t u d e n t s t o t h e

misuses o f s t a t i s t i c s and emphasizing t h e r o l e o f s t a t i s t i c s i n making j u d g ments and d e c i s i o n s .

There a r e t i m e s when t h e way we choose t o a n a l y z e d a t a ,

o r t h e u n d e r l y i n g assumptions we make i n our s t a t i s t i c a l model, w i l l i n f l u e n c e judgments o r d e c i s i o n s .

It i s v e r y i m p o r t a n t t h a t we expose o u r s t u d e n t s t o

i n s t a n c e s o f these s o - c a l l e d "paradoxes" so t h a t t h e y remain h i g h l y c r i t i c a l o f i n t e r p r e t a t i o n o f d a t a when r e a d i n g r e s u l t s .

I f t h e r e i s more t h a n one

p o s s i b l e i n t e r p r e t a t i o n f o r t h e data, we should encourage our s t u d e n t s t o examine b o t h i n t e r p r e t a t i o n s .

Several r e c e n t a r t i c l e s p r e s e n t i n s t a n c e s o f

s o - c a l l e d paradoxes t h a t a r e a c c e s s i b l e t o secondary school s t u d e n t s i n an i n t r o d u c t o r y p r o b a b i l i t y and s t a t i s t i c s c l a s s (REINHARDT 1981; WAGNER 1981). How, f o r example, i s i t p o s s i b l e f o r b a s e b a l l p l a y e r A t o have a b e t t e r b a t t i ng average t h a n p l a y e r

B both a g a i n s t r i g h t - h a n d e d p i t c h i n g and a g a i n s t

l e f t - h a n d e d p i t c h i n g , and y e t p l a y e r B g e t s t o s t a r t games because she has a b e t t e r overaZ1 average than p l a y e r A?

T h i s example, t a k e n from REINHARDT's

a r t i c l e , p o i n t s o u t t h e importance o f a weighted average, and t h e r e s u l t i n g

343

INFERENCE AND TEACHING PROBABILITY

p o s s i b l e d i 1 emma.

FALK and BAR-HILLEL (1980) mentioned how weighted average

l e a d s t o t h e paradox t h a t t h e average IQ i n a l l 50 U.S.

s t a t e s can be r a i s e d

m e r e l y by r e d i s t r i b u t i n g t h e p o p u l a t i o n . Another paradox p r e s e n t e d i n b o t h t h e REINHARDT and WAGNER a r t i c l e s i n v o l v e s t h e c o n v i c t i o n o f a c o u p l e i n a c o u r t case based upon s i x c h a r a c t e r i s t i c s t h e w i t n e s s e s t e s t i f i e d were p r e s e n t i n t h e c o u p l e a t t h e scene o f t h e crime. The p r o s e c u t i o n computed t h e odds as 1 i n 12 m i l l i o n t h a t t h e s e c h a r a c t e r i s t i c s would s i m u l t a n e o u s l y occur, and o b t a i n e d a c o n v i c t i o n based p r i m a r i l y on t h i s calculation.

The v e r d i c t was subsequently o v e r t u r n e d when a mathemati-

c i a n e x p l a i n e d t h a t t h e s i x c h a r a c t e r i s t i c s mentioned by t h e w i t n e s s e s were f a r from being c l e a r independent events, and t h a t i n f a c t one c o u l d p r o b a b l y f i n d q u i t e a few couples near t h e scene o f t h e c r i m e w i t h t h e same s i x charact e r i s t i c s (Time, 1968).

The roZe of asswnptions i n o u r i n t e r p r e t i v e models i s

c r u c i a l , and we should impress t h i s upon o u r students.

Indeed, c o i n c i d e n c e s

may n o t be as c o i n c i d e n t a l as t h e y appear t o us (FALK, 1981b).

We shouZd expose both teachers and students o f probability and s t a t i s t i c s t o the research r e s u l t s i n the Ziterature of judgment and deciiion making. 6.

It i s t h e a u t h o r ' s e x p e r i e n c e t h a t v e r y few s t a t i s t i c i a n s o r t e a c h e r s o f

p r o b a b i l i t y and s t a t i s t i c s a r e even aware o f t h e l i t e r a t u r e o f judgment and It i s

d e c i s i o n making, o r o f t h e i m p l i c a t i o n s of t h i s r e s e a r c h f o r t e a c h i n g .

n o t enough t o l o o k a t t h e s t u d e n t from o n l y one s i d e o f t h e coin, and decide, "They a r e j u s t s t a t i s t i c a l l y i g n o r a n t . things."

We have o n l y t o t e a c h them t h e s e

Our s t u d e n t s a l s o have i n t u i t i v e h e u r i s t i c s and schemas f o r d e a l i n g

w i t h i n f e r e n c e s , and many o f t h e s e a r e p s y c h o l o g i c a l l y based.

NISBETT and

ROSS (1980) r e p o r t t h a t a s e t o f s t a t i s t i c a l n o t e s i s b e i n g developed a t t h e

U n i v e r s i t y o f Michigan which emphasizes judgmental h e u r i s t i c s and t h e c o n f l i c t between i n t u i t i v e and formal s t a t i s t i c s .

This approach s h o u l d become much

more widespread (DOUMOUCHEL and KRANTZ, 1979). 111.

Some Suggestions for Further Research Researchers i n judgment and d e c i s i o n making a r e concerned about under-

s t a n d i n g and i m p r o v i n g t h e process o f human i n f e r e n c e under u n c e r t a i n t y . On t h e o t h e r hand, r e s e a r c h e r s and t e a c h e r s i n s t a t i s t i c a l e d u c a t i o n a r e concerned about t h e l e v e l o f s t a t i s t i c a l i l l i t e r a c y t h a t i s so p r e v a l a n t among our school c h i l d r e n , c o l l e g e s t u d e n t s , and t h e general p u b l i c .

This

34.4

DEVELOPMENTAND EPISTEMOLOGY

suggests t h a t c o o p e r a t i ve r e s e a r c h e f f o r t s between t h e two d i s c i p l ines may h e l p both sides o f the issue.

Here a r e some q u e s t i o n s t h a t w a r r a n t f u r t h e r

investigation. 1.

How much o f t h e problem o f r e l i a n c e upon h e u r i s t i c s and b i a s e s i s due t o

ignorance o f t h e s u b j e c t o f s t a t i s t i c s and i n q x p e r i e n c e w i t h randomness, and how much i s due t o p s y c h o l o g i c a l l y - b a s e d schemas o r s c r i p t s (ABELSON 1976) t h a t would be beyond change by reasonable e d u c a t i o n a l e f f o r t s ? SCHOLZ (1981), f o r example, has conducted an a n a l y s i s o f n i n e t e e n d i f f e r e n t s t o c h a s t i c problems from b o t h a p s y c h o l o g i c a l and a t e a c h i n g p o i n t o f view.

Such s t u d i e s a r e

e s s e n t i a l i n o r d e r t o separate p s y c h o l o g i c a l i n t u i t i o n s f r o m l a c k o f background knowledge as c o n t r i b u t o r s t o poor i n f e r e n c e s among humans.

Those who

r e s e a r c h from t h e p o i n t o f view o f s t a t i s t i c a l e d u c a t i o n should pay a t t e n t i o n t o t h e p s y c h o l o g i c a l r e s u l t s , and i n c o r p o r a t e judgment and decision-making t a s k s i n t o t h e i r r e s e a r c h designs.

On t h e o t h e r hand, p s y c h o l o g i s t s c o u l d

g a t h e r base-rate d a t a on t h e l e v e l o f s t a t i s t i c a l competence o f t h e i r subj e c t s , and c o n t r o l f o r t h a t v a r i a b l e i n t h e i r i n v e s t i g a t i o n s .

S p e c i f i c ques-

t i o n s which should be i n v e s t i g a t e d c o o p e r a t i v e l y i n c l u d e : a ) Can i n s t r u c t i o n which i s aimed a t i n t e g r a t i o n o f b a s e - r a t e i n f o r m a t i o n overcome p e o p l e ' s tendency t o i g n o r e b a s e - r a t e data? For example, would t h e i n t r o d u c t i o n o f s t a t i s t i c s v i a random v a r i a b l e s s i m u l a t e d by s p i n n e r s - - a s developed by BEN2 (1982)--improve p e o p l e ' s a b i l i t y t o combine i n f o r m a t i o n i n judgmental t a s k s ?

Would t h e i n t r o d u c t i o n o f Bayes Theorem f i r s t t h r o u g h sim-

u l a t i o n experiments and t h e n t h r o u g h 2x2 c o n t i n g e n c y t a b l e s w i t h r e l a t i v e f r e q u e n c i e s h e l p people t o focus on t h e restricted sample space which p l a y s

so v i t a l a r o l e i n c o n d i t i o n a l p r o b a b i l i t y problems?

That i s , suppose we had

o u r s t u d e n t s t r a n s l a t e percentages o r p r o b a b i l i t i e s i n t o r e l a t i v e f r e q u e n c i e s i n problems such as t h e cab problem.

We c o u l d focus on making i n f e r e n c e s f r o m

tables l i k e

Blue

120

30 Sampling o f 1000 Cabs

Cab Green

170

680

345

INFERENCE AND TEACHING PROBABILITY

r a t h e r t h a n h i t t i n g s t u d e n t s w i t h a f o r m a l i s t i c l a w l i k e Bayes Theorem. Would t h e y r e c o g n i z e such t a s k s i n judgmental s i t u a t i o n s ? b ) Can i n s t r u c t i o n be t a i l o r e d i n such a way t h a t t h e e f f e c t s o f h e u r i s t i c s such as avai 1a b i 1 it y , r e p r e s e n t a t i veness and a n c h o r i n g can be d i m i n i shed i n t h o s e i n s t a n c e s where t h e y m i s l e a d t h e e s t i m a t o r ? Can t h e study (SHAUGHNESSY 1977), which demonstrated some success i n overcoming r e l i a n c e upon such heuri s t i c s by implementing an a c t i v i t y - b a s e d course i n p r o b a b i l i t y and s t a t i s t i c s , Can i n s t r u c t i o n such as t h i s m a i n t a i n i t s e f f e c t s o v e r time,

be r e p l i c a t e d ?

or a r e such r e s u l t s j u s t an "end o f course" phenomenon? 2.

Causal schemas (KAHNEMAN and TVERSKY 1980) and d e t e r m i n i s t i c reason-

i n g (FISCHBEIN 1976; WELL e t . a l . 1982: GREEN 1982) have f r e q u e n t l y been c i t e d as reasons f o r t h e i n a b i l i t y o f humans t o make adequate i n f e r e n c e s under uncertainty.

Is re1 ia n t e upon causal schemas or d e t e r m i n i s t i c r e a s o n i n g a

western c u l t u r a l phenomenon, or can s i m i l a r r e s u l t s be o b t a i n e d i n c r o s s -

Is t h e causal-schema phenomenon an a r t i f a c t o f t h e heavy

c u l t u r a l studies?

emphasis upon d e t e r m i n i s t i c models i n o u r western c u l t u r e ?

I f so, can i n -

s t r u c t i o n which emphasizes model b u i l d i n g and which e l u c i d a t e s t h e d i f f e r e n c e s between s t a t i s t i c a l and d e t e r m i n i s t i c models make a d i f f e r e n c e i n p e o p l e ' s inferential ability? 3.

Most o f t h e r e s e a r c h on h e u r i s t i c s and b i a s e s has been done w i t h i n d i -

vidual subjects. individuals?

Are groups s u b j e c t t o t h e same h e u r i s t i c s and b i a s e s as

I f a tempering e f f e c t on p r o b a b i l i s t i c m i s c o n c e p t i o n s occurs i n

group processes, s h o u l d we perhaps t e a c h p r o b a b i l i t y and s t a t i s t i c s and j u d g ment and d e c i s i o n making i n small groups? A program c a l l e d Guided Des'ign (WALES 1976) teaches d e c i s i o n making i n small groups i n i n t r o d u c t o r y e n g i n e e r i n g courses a t West V i r g i n i a U n i v e r s i t y .

4.

Are p r o f e s s i o n a l s t a t i s t i c i a n s s u b j e c t t o t h e s e same h e u r i s t i c s and

biases, or do t h e y approach judgment and decision-making t a s k s f r o m a normat i ve p o i n t o f view?

KAHNEMAN and TVERSKY (1971 ) r e p o r t e d m i s c o n c e p t i ons among

psychologists t r a i n e d i n s t a t i s t i c s . a tool.

For most p s y c h o l o g i s t s , s t a t i s t i c s i s

For p r o f e s s i o n a l s t a t i s t i c i a n s however, s t a t i s t i c s i s t h e i r l i f e ' s

blood, a work o f a r t .

It would be w o r t h w h i l e i n v e s t i g a t i n g t h e i r responses

on some o f t h e judgment and decision-making t a s k s r e p o r t e d i n t h e l i t e r a t u r e .

5.

The r o l e of t h e t a s k i t s e l f and i t s i n f l u e n c e upon t h e judgmental

process w a r r a n t s f u r t h e r i n v e s t i g a t i o n .

Changes i n t h e causal a t t r i b u t i o n

v a l u e o f v a r i a b l e s , or changes i n t h e emotiona, impact o f data, seem t o a f f e c t

DEVELOPMENTAND EPISTEMOLOGY

346

Can

t h o s e f a c t o r s t o which people a t t e n d when making judgments o r d e c i s i o n s . changes i n t h e t a s k o p t i m i z e , o r a t l e a s t improve, i n f e r e n c e ?

I f so, what a r e

t h e i m p l i c a t i o n s o f t h i s f o r t h e way we p r e s e n t s t o c h a s t i c problems t o o u r s t u d e n t s o r t o p o t e n t i a l judges? 6.

C l i n i c a l i n t e r v i e w s s h o u l d be c a r r i e d o u t w i t h s u b j e c t s t o a t t e m p t t o

g a t h e r v a l i d a t i n g i n f o r m a t i o n f o r t h e t h e o r i e s we o f f e r as reasons f o r t h e i r i n f e r e n t i a l behavior.

WELL e t . a l .

(1982) has warned about t h e dangers o f

o v e r r e l i a n c e on summary nummeri c a l d a t a f o r t h e o r y devel opment. I n summary, much i n t e r d i s c i p l i n a r y i n v e s t i g a t i o n needs t o t a k e p l a c e t o assess t h e s t a t e o f human s t a t i s t i c a l i l l i t e r a c y . poor d e c i s i o n makers, o r can we improve o u r l o t ?

Are we f o r e v e r doomed t o be The (perhaps b i a s e d ) p o i n t

o f view o f a t e a c h e r i s t h a t many, i f n o t a l l , o f t h e m i s c o n c e p t i o n s o f p r o b a b i l i t y can be addressed c o g n i t i v e l y .

(This teacher i s not y e t w i l l i n g t o

admit t h a t we a r e f o r e v e r doomed t o be i n e p t a t i n f e r e n c e t a s k s . ) I t would be i n t e r e s t i n g t o have a p s y c h o l o g i s t and a mathematics e d u c a t o r

team-teach an i n t r o d u c t o r y course i n p r o b a b i l i t y and s t a t i s t i c s t o b r i n g b o t h s i d e s o f t h e c o i n t o bear upon o u r s t u d e n t s .

The general p u b l i c i s n o t y e t

s u f f i c i e n t l y aware o f t h e importance o f p r o b a b i l i t y and s t a t i s t i c s , o r o f t h e prevalence o f judgmental t a s k s i n d e c i s i o n making,. o r o f t h e i r i n t e r r e l a t i o n ship.

Teachers need t o become aware o f t h e i m p l i c a t i o n s o f judgment and

decision-making r e s e a r c h f o r t h e t e a c h i n g o f p r o b a b i l i t y and s t a t i s t i c s .

We

cannot w a i t f o r some v i v i d , c o n c r e t e event (NISBETT e t . a l . 1976) t o o c c u r which w i l l convince everyone o f t h e importance o f p r o b a b i l i t y and s t a t i s t i c s when making d e c i s i o n s .

Nor can we c o n t i n u e t o t o l e r a t e t h e t r i a l - a n d - e r r o r

process o f human d e c i s i o n making i n r i s k y s i t u a t i o n s (SLOVIC e t . a l . 1976). We must b e g i n t o spread t h e word o u r s e l v e s t h r o u g h c o o p e r a t i v e e f f o r t s .

References Abelson, R. P.: 1976. S c r i p t p r o c e s s i n g i n a t t i t u d e f o r m a t i o n and d e c i s i o n making. I n J. S. C a r r o l l and J. W. Payne (Eds.), C o g n i t i v e and S o c i a l Beh a v i o r . H i l l s d a l e : Lawrence Erlbaum. Ajzen, I c e k : 1977. I n t u i t i v e t h e o r i e s o f events and t h e e f f e c t s o f b a s e - r a t e i n f o r m a t i o n on p r e d i c t i o n . J o u r n a l o f P e r s o n a l i t y and S o c i a l Psychology, 35, 5, 303-314. B a r - H i l l e l , M.: 1980. The b a s e - r a t e f a l l a c y i n p r o b a b i l i t y judgments. Psychologica, 44, 211-233.

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B a r - H i l l e l , M.: 1982. S t u d i e s o f r e p r e s e n t a t i v e n e s s . I n D. Kahneman, P. S l o v i c , and A. Tversky (Eds.) Judgment and U n c e r t a i n t y : H e u r i s t i c s and Biases. Cambridge: Cambridge Uni v e r s i t y Press. B a r - H i l l e l , M. and F a l k , R.: 1982. Some t e a s e r s c o n c e r n i n g c o n d i t i o n a l probab i l i t i e s , C o g n i t i o n , 11, 109-122. Benz, H. J.: 1982. S t o c h a s t i c s - T e a c h i n g based on common sense. Osnabrucker S c h r i f t e n z u r Mathemati k, U n i v e r s i t x t Osnabruck. Buckley, W. F.:

1982. Column i n t h e New York D a i l y News, J u l y 27.

Doumouchel, W. H. and Krantz, D. H.: 1979. S t a t i s t i c s notes. Unpublished m a n u s c r i p t , U n i v e r s i t y o f M i c h i gan. Falk, R.: 1980. P r o b a b i l i s t i c r e a s o n i n g as an e x t e n s i o n o f common sense t h i n k i n g , Presented a t t h e I n t e r n a t i o n a l Conference on Mathematical Educat i o n , I C M E I V , Berkeley, CA. F a l k , R.: 1981a. P e r c e p t i o n o f Randomness. From t h e proceedings o f t h e f i f t h i n t e r n a t i o n a l conference f o r t h e psycho1 ogy o f mathematics e d u c a t i o n , Gren o b l e , France, 222-229. F a l k , R.:

1981b. On Coincidences.

The S k e p t i c a l I n q u i r e r , 6, 2, 18-31.

F a l k , R. and B a r - H i l l e l , M.: 1980. Magic p o s s i b i l i t i e s o f t h e weighted a v e r age. Mathematics Magazine, 106-107. F i s c h b e i n , E.: 1975. The i n t u i t i v e sources o f p r o b a b i l i s t i c t h i n k i n g i n c h i l dren. Dordrecht-Boston: 0. Reidel P u b l i s h i n g Company. F i s c h b e i n , E. P r o b a b i l i s t i c t h i n k i n g i n c h i l d r e n and adolescents. I n : I D M ( H r s g ) : Forschung zum Prozess des Mathematiklernen, Reihe: M a t e r i a l i e n und S t u d i e n , Band 2, 23-42. F i s c h b e i n , E.: 1982. Can t h e t e a c h i n g o f p r o b a b i l i t y improve p r o b a b i l i s t i c i n t u i t i o n s ? From t h e proceedings o f t h e F i r s t I n t e r n a t i o n a l Conference on t h e Teaching o f S t a t i s t i c s , ICOTS I , S h e f f i e l d , England. F i s c h b e i n , E., Pampu, J., and Manzat, J.: 1970a. Companion o f r a t i o s and t h e chance concept i n c h i l d r e n . C h i l d Development, 41, 377-389. F i s c h b e i n , E., Pampu, J., and Manzat, J.: 1970b. E f f e c t s o f age and i n s t r u c t i o n on combinatory a b i l i t y i n c h i l d r e n . B r i t i s h J o u r n a l o f E d u c a t i o n a l Psychology, 40, 261-270. F i s c h h o f f , B., S l o v i c , P., and L i c h t e n s t e i n , S.: 1979. S u b j e c t i v e s e n s i t i v i t y a n a l y s i s . O r g a n i z a t i o n a l Behavior and Human Performance, 23, 339-359. Green, D. R. P r o b a b i l i t y concepts i n 11-16 y e a r o l d p u p i l s . Report f r o m t h e Center f o r Advancement o f Mathematical Education i n Technology, Loughborough Uni v e r s i t y o f Techno1 ogy. Holmes, P. and Turner, D.:

1981. Teaching S t a t i s t i c s t o 11-16 y e a r o l d s . I n

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A. S h u l t e (Ed.), Teaching S t a t i s t i c s and P r o b a b i l i t y . The Yearbook o f t h e N a t i o n a l C o u n c i l o f Teachers o f Mathematics. H u f f , D.:

1954. How t o L i e w i t h S t a t i s t i c s .

London: W.

W.

Norton.

Kahneman, D., and Tversky, A.: 1972. S u b j e c t i v e p r o b a b i l i t y : a judgment o f r e p r e s e n t a t i v e n e s s . C o g n i t i v e Psychology, 3, 430-454. Kahneman, D., and Tversky, A.: 1973a. A v a i l a b i l i t y : a h e u r i s t i c f o r j u d g i n g frequency and p r o b a b i l i t y . C o g n i t i v e Psychology, 5, 207-232. Kahneman, D., and Tversky, A.: 1974. Judgment under u n c e r t a i n t y : h e u r i s t i c s and biases. Science, 185, 1124-1131. Kahneman, D., and Tversky, A. : 1973. On t h e psychology o f p r e d i c t i o n . Psychol o g i c a l Review, 80, 237-251. Kahneman, D., S l o v i c , P., H e u r i s t i c s and Biases.

and Tversky, A.: 1982. Judgment under U n c e r t a i n t y : New York: Cambridge U n i v e r s i t y Press.

Kuhn, T. S.: 1962. The S t r u c t u r e o f S c i e n t i f i c R e v o l u t i o n s . Chicago: U n i v e r s i t y o f Chi cago Press. La Brecque, M.: 1980. On r e a c h i n g sounder judgments: s t r a t e g i e s and snares. Psychology Today, June, 33-42. Lyon, D., and S l o v i c , P.: 1976. Dominance o f accuracy i n f o r m a t i o n and n e g l e c t o f base-rates i n p r o b a b i l i t y e s t i m a t i o n . Acta Psychologica, 40, 287-298. Mathematics Resource P r o j e c t . S t a t i s t i c s and I n f o r m a t i o n O r g a n i z a t i o n . A l t o : C r e a t i v e P u b l i c a t i o n s , 1977.

Palo

M o s t e l l e r , F.: 1962. F i f t y C h a l l e n g i n g Problems i n P r o b a b i l i t y w i t h S o l u t i o n s . Reading: Addison-Wesley. M o s t e l l e r , F., K r u s k a l , W. H., L i n k , R. F., P e i t e r s , R. S., 1973. S t a t i s t i c s by Example. Reading: Addison-Wesley. Newell, A. and Simon, H.: Hal 1.

and R i s i n g , G. R.:

1972. Human Problem S o l v i n g . New J e r s e y : P r e n t i c e

N i s b e t t , R. E. and Borgida, E.: 1975. A t t r i b u t i o n and t h e psychology o f p r e d i c t i o n . Journal o f P e r s o n a l i t y and S o c i a l Psychology, 32, 932-943. N i s b e t t , R. E., Borgida, E., C r a n d a l l , R., and Reed, H.: 1976. P o p u l a r i n d u c t i o n : I n f o r m a t i o n i s n o t n e c e s s a r i l y i n f o r m a t i v e . I n J. S. C a r r o l l and J. Payne (Eds.), C o g n i t i o n and S o c i a l Behavior. H i l l s d a l e : Lawrence Erlbaum. N i s b e t t , R. E. and Bellows, N.: 1977. Verbal r e p o r t s about causal i n f l u e n c e s on s o c i a l judgments: P r i v a t e access versus p u b l i c t h e o r i e s . J o u r n a l o f Pers o n a l i t y and S o c i a l Psychology, 35, 613-624. N i s b e t t , R. E. and Wilson, T. D.: 1977. T e l l i n g more t h a n we can know: Verbal r e p o r t s on mental processes. P s y c h o l o g i c a l Reviews, 84, 231-259.

349

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N i s b e t t , R. E. and Ross, L.: 1980. Human I n f e r e n c e : S t r a t e g i e s and S h o r t comings o f Soci a1 Judgment. Engl ewood C1 if f s : P r e n t i ce Hal 1

.

Ride, L.: 1977. Take a Chance w i t h Your C a l c u l a t o r : P r o b a b i l i t y Problems f o r Programmable C a l c u l a t o r s . F o r e s t Grove, Oregon: D i l i t h i u m Press. R e i n h a r d t , H. E.: 1981. Some S t a t i s t i c a l Paradoxes. I n A. S h u l t e (Ed.), Teachi n g S t a t i s t i c s and P r o b a b i l i t y , t h e 1981 N a t i o n a l C o u n c i l o f Teachers o f Mathematics Yearbook, 100-109. Ross, M. and S i c o l y , F.: 1982. E g o c e n t r i c b i a s e s i n a v a i l a b i l i t y and a t t r i b u t i o n . I n Kahneman, D., S l o v i c , P., and Tversky, A. (Eds.), Judgment Under U n c e r t a i n t y : H e u r i s t i c s and Biases. New York: Cambridge U n i v e r s i t y Press. Schoenfeld, A.: 1979. E x p l i c i t h e u r i s t i c s t r a i n i n g as a v a r i a b l e i n problem s o l v i n g performance, Journal o f Research i n Mathematics Education, 10, 3. Scholz, R.: 1981. S t o c h a s t i s c h e Problemaufgaben: Analysen aus D i d a k t i s c h e r und P s y c h o l o g i s c h e r P e r s p e k t i v e . I n s t i t u t fur D i d a k t i k d e r Mathematik d e r U n i v e r s i t a t B i e l e f e l d , M a t e r i a l i e n und Studien, Band 23. Shaughnessy, J. M.: 1977. Misconceptions o f p r o b a b i l i t y : an experiment w i t h a small-group, a c t i v i t y - b a s e d model b u i l d i n g approach t o i n t r o d u c t o r y probab i l i t y . E d u c a t i o n a l S t u d i e s i n Mathematics, 8, 295-316. Shaughnessy, J. M.: 1981. Misconceptions o f p r o b a b i l i t y : f r o m s y s t e m a t i c e r r o r s t o s y s t e m a t i c experiments and d e c i s i o n s . I n A. S h u l t e (Ed.), Teaching S t a t i s t i c s and P r o b a b i l i t y , t h e 1981 N a t i o n a l C o u n c i l o f Teachers o f Mathem a t i c s Yearbook, 90-100. Shaughnessy, J. M.: 1982. Misconceptions o f p r o b a b i l i t y , s y s t e m a t i c and o t h e r w i s e : Teaching p r o b a b i l i t y and s t a t i s t i c s so as t o overcome some misconcept i o n s . Presented a t t h e F i r s t I n t e r n a t i o n a l Conference on t h e Teaching o f S t a t i s t i c s , S h e f f i e l d , England. Shulte, A.: 1981. Teaching S t a t i s t i c s and P r o b a b i l i t y . The 1981 yearbook o f t h e N a t i o n a l Council o f Teachers o f Mathematics. S l o v i c , P., F i s c h h o f f , B., and L i c h t e n s t e i n , S.: 1976. C o g n i t i v e processes and s o c i e t a l r i s k t a k i n g . I n J. S. C a r r o l l and J. W. Payne (Eds.), C o g n i t i o n and S o c i a l Behavior. H i l l s d a l e : Lawrence Erlbaum. S l o v i c , P., F i s c h h o f f , B., and L i c h t e n s t e i n , S.: t h e o r y . Annual Review o f Psychology, 28, 1-39. S l o v i c , P., F i s c h o f f , B., and L i c h t e n s t e i n , S . : Psychology Today, June, 14, 44-48.

1977. B e h a v i o r a l d e c i s i o n

1980. R i s k y assumptions.

S l o v i c , P.: 1982. Toward u n d e r s t a n d i n g and i m p r o v i n g d e c i s i o n s . I n W. Howell (Ed. ) , Human Performance and P r o d u c t i v i ty. H i 11sdal e : Lawrence E r l baum. T a y l o r , S. E.: 1982. The a v a i l a b i l i t y b i a s i n s o c i a l p e r c e p t i o n and i n t e r a c t i o n . I n D. Kahneman, P. S l o v i c , and A. Tversky (Eds.), Judgment Under U n c e r t a i n t y : H e u r i s t i c s and Biases, New York: Cambridge U n i v e r s i t y Press.

350

" T r i a l by Mathematics":

1968. Time, 26, A p r i l , p. 41.

Tversky, A., and Kahneman, D.: 1971. B e l i e f i n t h e Law o f Small Numbers. P s y c h o l o g i c a l B u l l e t i n , 76, 105-110. Tversky, A., and Kahneman, D. Causal Schemata i n Judgment under U n c e r t a i n t y . I n M. F i s c h b e i n (Ed.), Progress i n S o c i a l Psychology. Wagner, C. H.: 1981. Paradoxes i n Sampling. I n Teaching S t a t i s t i c s and P r o b a b i l i t y , t h e 1981 N a t i o n a l Council o f Teachers o f Mathematics Yearbook, Ed. by A. S h u l t e , 162-167. Well, A. D., P o l l a t s e k , A,, and Konold, C. P r o b a b i l i t y E s t i m a t i o n and t h e Use and Neglect o f Base-Rate I n f o r m a t i o n . D r a f t copy o f a paper p r e s e n t e d a t t h e r e s e a r c h p r e s e s s i o n o f t h e 1982 N a t i o n a l Council o f Teachers o f Mathematics meeting i n Toronto. Under r e v i s i o n a t t h e U n i v e r s i t y o f Massa c h u s e t t s , Amherst.

DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science PublishersE.V. (North-Holland), 1983

351

(MIS-) INTERPRETATION OF STOCHASTIC MODELS

GEORG SCHRAGE Universitat Dortmund lnstitut fur Didaktik der Mathematik F. R. Germany

I am n o t an expert i n the theory o f d e c i s i o n making, r a t h e r I am concerned w i t h mathematics education and teacher t r a i n i n g . Within the frame o f t h i s task one o f my special i n t e r e s t s i s teaching s t a t i s t i c s . Statistics t o use a d e f i n i t i o n o f W. A. W a l l i s and H. W. Roberts

-

-

is

a body o f methods f o r making wise decisions i n the face o f uncertainty. And t h a t i s the subject o f our conference.

In german schools, teaching s t a t i s t i c s i s p a r t o f mathematics education. To be precise, t h e r e are n o t special courses i n s t a t i s t i c s b u t s t a t i s t i c a l a p p l i c a t i o n s are involved i n a stochastic curriculum t h a t i s p a r t o f the mathematics curriculum from primary school up t o c o l l e g e l e v e l . That may sound very good and c e r t a i n l y t h e r e has been an important development during the past decade. But r e a l i t y of s t a t i s t i c s education i n german schools i s s t i l l f a r from being s a t i s f a c t o r y and I doubt t h a t i t b r i n g s about an improved decision behavior o f students. Some r e l a t e d t e s t r e s u l t s are l a t e r presented.

I n my judgement the i n t e g r a t i o n o f s t o c h a s t i c elements has succeeded best i n primary schools, where c h i l d r e n gather experiences w i t h random behav i o r by performing and observation o f r e a l experiments. The aim a t t h i s primary l e v e l should be t o e s t a b l i s h fundamental concepts and i n t u i t i v e f e e l i n g f o r the r e g u l a r i t i e s o f random outcome.

3 52

DEVELOPMENTAND EPISTEMOLOGY

A t the l e v e l o f age ten t o sixteen

-

the german Sekundarstufe I

-

these

experiences should be ordered and made e x p l i c i t . Q u a n t i t a t i v e aspects a r e emphasized. P r o b a b i l i t i e s o f events t h a t are connected w i t h simple Laplace-experiments are calculated. S t a t i s t i c s i n these classes should be mainly d e s c r i p t i v e s t a t i s t i c s . There are very s t i m u l a t i n g appropriate m a t e r i a l s f o r classroom i n v e s t i g a t i o n s . Especially Prof. Winter from the U n i v e r s i t y i n Aachen made a l o t o f e x c i t i n g proposals how s t a t i s t i c s can be i n t e g r a t e d i n mathematics education a t t h i s l e v e l . But r e a l i t y as I see i t - i s t h a t most p u p i l s a t t h i s age are never confronted a t

-

school w i t h any s t a t i s t i c a l problem.

The main reasons f o r t h i s l a c k a r e

teaching t r a d i t i o n s , i n f l e x i b i l i t y o f t h e german school systems and i n s u f f i c i e n t prepared teachers. A t the upper secondary l e v e l

-

the german Sekundarstufe I 1

-

students

have t o choose t h e i r courses. Nowadays many schools o f f e r s t o c h a s t i c courses and I estimate t h a t between 10 and 20 percent o f p u p i l s p a r t i cipate. But these courses a r e o f t e n u n s a t i s f a c t o r y because s t a t i s t i c a l inference i s reduced t o mathematics. Students get knowledge o f the most common d i s t r i b u t i o n functions, they l e a r n t o estimate parameters, t o make t e s t s depending on a f i x e d l e v e l o f s i g n i f i c a n c e and perhaps t o c a l c u l a t e i n t e r v a l s o f confidence. Examples are most o f such a k i n d t h a t t h e r e i s no doubt about the corresponding mathematical model, so t h a t standardized methods o f inference can be applied. I n most german t e x t books the character o f s t a t i s t i c s as a methodology f o r making decisions, t a k i n g i n t o consideration the consequences o f possible e r r o r s , i s comp l e t e l y hidden o r n o t e x i s t e n t . So s t a t i s t i c s becomes a branch o f mathematics l i k e calculus, geometry o r algebra. But mathematics i s o n l y one face o f s t a t i s t i c s . Usually t h e r e a l d i f f i c u l t i e s o f a s t a t i s t i c a l inves t i g a t i o n have t o be solved before the process o f decision making can be reduced t o a mathematical problem. Therefore I t h i n k t h a t s t a t i s t i c s education must i n c l u d e the nonmathematical aspects o f s t a t i s t i c a l p r o j e c t s : The precise formulation o f the problem, the process o f modeling, t h e d i s cussion o f reasonable a1 t e r n a t i v e decisions and t h e i r p o s s i b l e consequences, questions r e f e r i n g t o r i s k and personal r e s p o n s i b i l i t y , the process o f sampling and the i n t e r p r e t a t i o n o r even the a p p l i c a t i o n o f the r e s u l t s

/MIS)INTERPRETATION OF STOCHASTIC MODELS received by s t a t i s t i c a l inference. This can be done best by c a r r y i n g o u t some r e a l p r o j e c t s . A t the symposium on stochastics i n education, t h a t took place 1980 i n Klagenfurt, I presented a course, emphasizing decisiont h e o r e t i c a l aspects and developing fundamental ideas o f s t a t i s t i c a l i n ference by means o f some r e a l i s t i c p r o j e c t s t h a t are s u i t a b l e t o be rea-

lized a t school. It i s a matter o f f a c t , and I w i l l prove i t by some examples, t h a t even people w i t h some mathematical o r s t a t i s t i c a l background may use formal methods i n misleading ways. To overcome t h i s s i t u a t i o n , educators must get more knowledge on d e f e c t i v e concepts and s t r a t e g i e s . What we would need i s an elaborated phenomenology o f frequent e r r o r s and misleading s t r a t e g i e s . There are already many systematical i n v e s t i g a t i o n s on stochas t i c conceptions and misconceptions. E s p e c i a l l y I r e f e r t o the work o f M r . Shaughnessy and h i s s t i m u l a t i n g a r t i c l e "Misconceptions o f P r o b a b i l i -

t y : From Systematic Errors t o Systematic Experiments and Decisions" i n t h e 1981 Yearbook on Teaching S t a t i s t i c s and P r o b a b i l i t y o f t h e NCTM. But we do n o t know enough o f the reasons f o r misconceptions and we do n o t know enough o f t h e influence o f education on misconceptions. A t the S h e f f i e l d conference on teaching s t a t i s t i c s i n August o f t h i s year, Prof. Fischbein reported on some t e s t s and pointed o u t t h a t stochastics educat i o n can even be responsible f o r misconceptions and l o s s o f good i n t u i t i o n . A t the same conference M r . Green o f the Loughborough U n i v e r s i t y o f Technology reported on a p r o j e c t t o i n v e s t i g a t e s t o c h a s t i c conceptions o f about 3000 c h i l d r e n from 11 t o 16. I w i l l present you two o f the t e s t

iterns. 1. Which o f the f o l l o w i n g r e s u l t s i s more l i k e l y ? a) Getting 7 o r more boys o u t o f t h e f i r s t 10 babies born i n a new h o s p i t a l . b) Getting 70 o r more boys o u t o f t h e f i r s t 100 babies born i n a new h o s p i t a l . (A) They are e q u a l l y l i k e l y (B) 7 o r more o u t o f 10 i s more l i k e l y (C) 70 o r more o u t o f 100 i s more l i k e l y (0)

No one can say.

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DEVELOPMENT AND EPISTEMOLOG Y

I t h i n k t h i s i s a very good question t o t e s t students understanding o f l o n g run s t a b i l i t y o f r e l a t i v e frequencies. The r e s u l t : (A) (€3) (C) (D)

-

-

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25 % 8 %

5 % 61 %

I asked the same question t o 153 o f my teacher students a t the U n i v e r s i t y o f Dortmund, o m i t t i n g the p o s s i b i l i t y (D).

The r e s u l t : (A) (6) (C) no answer

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60 %

14 % 15 % 10 %

The r e s u l t i s s t i l l worse than i t seems, because a g r e a t p a r t o f t h e s t u dents who answered c o r r e c t l y gave wrong arguments f o r t h e i r answer. Nobody r e f e r e d t o long r u n s t a b i l i t y o f frequencies. From 17 students, who had p a r t i c i p a t e d a t a stochastic course a t school, o n l y one gave the c o r r e c t answer. Many students c a l c u l a t e d t h e p r o b a b i l i t y f o r both events t o be 7

m.

By the way, M r . Shaughnessy i n the above mentioned a r t i c l e r e p o r t s on the same question and s i m i l a r r e s u l t s .

Some weeks ago I had a d e s i l l u s i o n i n g experience. A t the beginning o f my l e c t u r e s on stochastics I t a l k e d i n d e t a i l on the empirical background o f p r o b a b i l i t y theory. I spoke on long run s t a b i l i t y o f frequencies, I showed the graph o f a r e a l experiment's outcome as below and I p u t a t t h e black board the empirical law o f l a r g e numbers. Then I asked my students again t h e above question. The r e s u l t was approximately the same as i n the t e s t . My t e s t also contained items t o prove s t u dents understanding o f independence. The r e s u l t was n o t b e t t e r then the r e s u l t o f t h e j u s t mentioned item.

(MIS-) INTERPRETATION OF STOCHASTIC MODELS

E

2 . A bag has i n i t some white and some black b a l l s .

A

boy p i c k s o u t a b a l l ,

notes i t s c o l o u r and puts i t back. He then shakes the bag. He does t h i s f o u r times. He picks a black b a l l every time. Then he p i c k s o u t one more ball. What a colour do you t h i n k he i s l i k e l y t o get?

(A) Black i s more l i k e l y again (B) Black and white are e q u a l l y l i k e l y ( C ) White i s more l i k e l y t h i s time.

The answer should be ( A ) .

I t h i n k t h i s i s a bad question. As a consequence o f t h e absence o f any a p r i o r i p r o b a b i l i t y one can say a b s o l u t e l y nothing on the p r o b a b i l i t y t o take a white o r a black b a l l o u t o f t h e bag. I f I say t h a t i t i s more l i k e l y t o get a black one, then t h a t i s a pure s u b j e c t i v e judgement. The c l a s s i f i c a t i o n as " t r u e " o r " f a l s e " i s n o t appropriate f o r t h i s s i t u a t i o n .

I f I had t o make a decision, perhaps I would decide f o r "black". But mak i n g a decision and making a statement on the p r o b a b i l i t y o f an event are two completely d i f f e r e n t p a i r s o f shoes. B e l i e v i n g t h a t a hypothesis t h a t has been r e j e c t e d by a s t a t i s t i c a l t e s t i s u n l i k e l y t o be t r u e i s a very common m i s i n t e r p r e t a t i o n . This i s an e r r o r s i m i l a r t o base r a t e f a l l a c i e s . I n an a r t i c l e on teaching p r o b a b i l i t y theory and s t a t i s t i c s I found t h e following interpretation o f a test:

355

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DEVELOPMENT A N 0 EPISTEMO LOG Y

I f Ho i s t r u e then the event E i s nearly sure

The outcome o f an experiment was

T

So Ho i s n e a r l y impossible

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I n 1975 I observed a murder t r i a l t h a t took place i n Wuppertal a town about 100 m i l e s away from B i e l e f e l d where t h i s way o f t h i n k i n g n e a r l y had produced an e r r o r o f j u s t i c e . An unknown man had murdered a woman,

-

and there was a defendant who was commited f o r t r i a l since two years. The law c o u r t asked some experts on t h e i r opinion about some accusing c i r cumstances. One expert a member o f the Bundeskriminalamt ( t h e german had found a t the defendant a t i n y proof a m i l l i o n t h part o f a FBI)

-

-

-

gram o f s t u f f o f the same k i n d as the clothes o f the k i l l e d woman. The expert now argued i n the f o l l o w i n g way: The p r o b a b i l i t y t o f i n d proofs o f such s t u f f a t a person who was n o t i n contact w i t h the k i l l e d woman i s 0,06 percent. ( I d o n ' t know how he got t h i s number.) So the defendant had contact w i t h the k i l l e d woman and must be considered t o be the murderer w i t h p r o b a b i l i t y 0,9994. I n the same way argued a doctor o f medicine who c a l l e d himself a b i o s t a t i s t i c a l expert. He had analysed a t i n y p r o o f o f blood, t h a t was used up by the analysis. He c a l c u l a t e d a p r o b a b i l i t y o f 0,979 t h a t t h e defendant was g u i l t y . A l l the world i n c l u d i n g the t r i b u n a l believed t h a t t h e defendant's g u i l t was mathematically proved. I n t h i s s i t u a t i o n I placed myself a t the disposal o f the a t t o r n e y f o r t h e defence. I n my comment I contradicted t h e argumentation o f the experts by the f o l l o w i n g example: Some time ago a bank was robbed by a red-haired man About 1 percent o f the german population i s red-haired. So, i f I was n o t the robber, t h e p r o b a b i l i t y f o r a red-haired robber would have been 1 percent. Consequently t h e experts would c a l c u l a t e a p r o b a b i l i t y o f 99 percent t h a t I was the robber. This i s an example how an erroneous k i n d o f argument can be unmasked by o n l y changing the context. By t h e way, t h e defendant was l a t e r acquitted, since he was proven t o be innocent. I believe t h a t t h e most frequent types o f m i s i n t e r p r e t a t i o n o f s t o c h a s t i c models are generated by imprecise common use o f language. v. Neumann and Morgenstern i n t h e i r book on game theory described t h i s problem: "There a r e notions which are q u i t e fundamental f o r our discussion but t h e use o f which i n everyday language i s h i g h l y ambiguous. The words which describe

357

(MIS-) INTERPRETATION OF STOCHASTIC MODELS

them are used sometimes i n one sense, sometimes i n another and occasionly worst o f a l l as i f they were synonyms."

-

-

We speak o f t h e p r o b a b i l i t y o f an event. But an event f o r i t s e l f does n o t obey any a t t r i b u t e t h a t can be c a l l e d i t s p r o b a b i l i t y . The p r o b a b i l i t y o f an event i s p a r t o f a model t h a t describes e i t h e r frequencies depending on c e r t a i n experimental conditions, o r the s u b j e c t i v e degree o f b e l i e f t h a t may be r e f l e c t e d by the b e t t i n g behavior o f a person. So i f we make a model o f a s t o c h a s t i c s i t u a t i o n , the i n t e r p r e t a t i o n o f corresponding s t a t e ments must be very c l e a r . I w i l l g i v e you an example where the r e l a t i o n s h i p between the problem t h a t should be investigated and the stochastic model has been t o t a l l y confused. I n 1972 a j o u r n a l f o r mathematics teachers published an a r t i c l e w i t h the t i t l e " M i t welcher Wahrscheinlichkeit g i l t der Fermat-Satz?" (What i s t t h e p r o b a b i l i t y t h a t Fermat's conjecture i s t r u e ? ) Fermat's conjecture i s one of t h e challenging unsolved problems o f number theory. I t s t a t e s t h a t t h e equation xn + yn = zn has no s o l u t i o n i n natural numbers i f 1123. The aut h o r o f the mentioned a r t i c l e c a l c u l a t e d the p r o b a b i l i t y o f the t r u t h o f the conjecture as f o l l o w s : b

b

1

2

1

...

k

He considered an i n t e r v a l o f n a t u r a l numbers

INk

= (1,2,..

. ,k3.

L e t MkI

be

t h e subset o f numbers t h a t can be represented i n t h e form xn t yn f o r cert a i n n and MZk the subset o f numbers t h a t can be represented i n t h e form n . z , i.e., Mlk

M

n n {a E: INk I a = x + y , x,y ~ { ~a G t=i k 1 a = zn , Z E N .

=

E

IN}

Fermat's conjecture asserts t h a t M2 k "Mlk

and

=

0

f o r any k i f n23.

The author considered the i n t e r v a l INk t o be an u r n and made a stochastic model f o r the experiment t o take f i r s t a subset A k clNk such t h a t

=# Ak

-

=*Mlk-assigning equal p r o b a b i l i t y t o each possible outcome and then a second subset B kc INk such t h a t # B k = * MZk. He c a l c u l a t e d the p r o b a b i l i t y Pk = P(A kf l Bk = 0). This value was i n t e r p r e t e d t o be the pro-

b a b i l i t y t h a t no counterexample t o Fermat's conjecture can be found i n t h e u r n INk.

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DEVELOPMENT AND EPISTEMOLOGY

And t h e r e s u l t l i m Pk = 1 was a c c o r d i n g l y i n t e r p r e t e d as t h e p r o b a b i l i t y k-tm o f t h e t r u t h o f Fermat's conjecture. But t h e r e i s no j u s t i f i c a t i o n f o r t h i s i n t e r p r e t a t i o n . The model r e p r e sents a h y p o t h e t i c a l s t o c h a s t i c experiment and t h e r e s u l t o f t h e c a l c u l a t i o n may be i n t e r p r e t e d as a s t a t e m e n t on f r e q u e n c i e s r e l a t e d t o t h i s experiment. The t r u t h o f Fermat's c o n j e c t u r e however i s c o m p l e t e l y i n d e pendent o f any random experiment and i t s p o s s i b l e outcome. We cannot c a l c u l a t e any o b j e c t i v e p r o b a b i l i t y f o r Fe r m a t ' s c o n j e c t u r e . Moreover t h e above i n t e r p r e t a t i o n l e a d s t o t h e absurd consequence t h a t t h e p r o b a b i l it y o f a counterexample i n t h e urnlNk i s d e c r e a s i n g when k i s i n c r e a s i n g .

Neve rt heles s

-

and t h i s i s t h e i m p o r t a n t t h i n g

-

t h e r e was no p r o t e s t o f

t h e re aders . Q u i t e on t h e c o n t r a r y , a n e d i t o r i a l committee i n c l u d i n g some u n i v e r s i t y mathematicians awarded a p r i z e t o t h i s a r t i c l e .

I w i l l come t o an end w i t h an example t h a t demonstrates some e x p e r t s l a c k o f commonsense. German s t u d e n t s t o g e t admission f o r s t u d y i n g medicine must pass t h ro ugh a t e s t . The t e s t s a r e developed w i t h g r e a t expense b y a c e n t r a l i n s t i t u t e . I n 1979 t h i s i n s t i t u t e made p u b l i c some t y p i c a l items, among them t h e f o l l o w i n g : ' ) The t a b l e below shows t h e f r e q u e n c i e s ( i n p e r c e n t ) o f f i v e symptoms w i t h r e s p e c t t o t h e ma la d i e s 1 t o 5. malady symptoms Insomnia Headache Trembl ing Falls Skin rash

I ) VUL S p i e g d ,

Nh.

14/I979

(MIS) INTERPRETATION OF STOCHASTIC MODELS

M r . Meyer comes t o the doctor and complains o f insomnia, f a l l s and tremb1 ing. His malady w i l l probably be

1

2

3

4

5

o o n o o The question was considered o f moderate d i f f i c u l t y . The " c o r r e c t " answer should be 4. It should be c l e a r t h a t there i s n o t enough information t o answer t h e

question. We know nothing on t h e base r a t e o f the d i f f e r e n t maladies and nothing on the dependence o f symptoms. Moreover, under the l i t t l e r e a l i s t i c assumptions t h a t a l l maladies have t h e same base r a t e and t h a t a l l symptoms are mutually indepent, both 1 and 4 have the same p r o b a b i l i t y 0,404. How d i d t h e experts o f the i n s t i t u t e g e t t h e i r answer? They used the f o l l o w i n g algorithm: For each o f the maladies add t h e p r o b a b i l i t i e s o f the p a t i e n t ' s symptoms. The i l l n e s s w i t h maximal corresponding sum w i l l be t h e most probable one. Why? I do n o t know. This a l g o r i t h m has absolutel y nothing t o do w i t h the problem. For example, i t ignores the f a c t t h a t

the absence o f c e r t a i n symptoms gives information on t h e k i n d o f malady as w e l l as the presence o f o t h e r symptoms. Since i n t e l l i g e n t students w i l l be misled by such poor t e s t items, I wrote a l e t t e r t o the people who prepared t h e t e s t . I e x h i b i t e d t h e f o l l o w i n g example t o demonstrate the wrongness o f t h e i r t h i n k i n g : L e t us assume t h a t 99 % o f people s u f f e r i n g from cancer o f lung sometimes g e t a f i t o f coughing while the same symptom i s observed a t 50 % o f people who caught a cold. ma1ady symptom coughing Now suppose M r . Meyer gets a f i t o f cough w h i l e he consults t o a doctor who learned decision making according t o the j u s t mentioned algorithm.

359

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DEVELOPMENT AND EPISTEMO LOG Y

My example d i d n o t convince the experts o f the t e s t i n s t i t u t e . I got an answer i n c l u d i n g d e t a i l e d mathematical explanations t h a t I d i d n o t under-

stand. But t h e r e was a f u r t h e r remarkable argument: "The t e s t i t e m was examined by more than t h i r t y u n i v e r s i t y teachers and p r a c t i t i o n e e r s i n c l u d i n g experts o f biomathematics and s t a t i s t i c s . None o f these experts c r i t i s i z e d the question o r the k i n d o f solution. Experiences w i t h t h e t e s t proved t h a t the question i s c o r r e c t l y understood and answered by most students. "

I t e s t e d t h i s statement w i t h my students, and r e a l l y most o f them used the same algorithm as the t e s t engineers.

I t h i n k i t should be a c e n t r a l aim o f s t a t i s t i c s education t h a t students recognize such serious abuse o f stochastic instruments. But what can we do t o reach t h i s aim? One p o i n t should be t o discuss i n the classroom examples o f misleading s t r a t e g i e s t o warn students never t o l o s e the meaning o f mathematical operations.

To overcome misconceptions, Mr. Shaughnessy a t the S h e f f i e l d conference recommended the o r i e n t a t i o n o f stochastics teaching a t some guiding p r i n c i p l e s t h a t I w i l l repeat:

1. I n t r o d u c t o r y courses should be a c t i v i t y based and experimental, 2. Emphasize judgement and d e c i s i o n making, 3. Sensitize students t o the misuse o f s t a t i s t i c s ,

4. Give very s t i m u l a t i n g problems. I agree t o t h i s program. I t should be r e a l i z e d by s t a t i s t i c s teachers i n t h e i r d a i l y work i n the classroom and so i t i s the task o f my colleagues and me t o prepare teachers t o be capable t o accomplish such a program.

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References Schrage, G . : 1980. Schwierigkei ten m i t stochastischer Modellbildung. Zwei Beispiele aus der Praxis. I n : JDM 1, 86-101

-

Schrage, G.: 1981. Entscheiden und begrunden L e i t l i n i e n f u r den S t a t i s t i k u n t e r r i c h t . I n : W. D o r f l e r & R. Fischer (eds.), Stochastik i m Schulunt e r r i c h t . S c h r i f t e n r e i h e D i d a k t i k der Mathematik, U n i v e r s i t g t f u r B i l dungswissenschaften i n Klagenfurt, Band 3, Wien Shaughnessy, J.M. : 1981. Misconceptions o f p r o b a b i l i t y : From systematic e r r o r s t o systematic experiments and decisions. I n : A. Shulte (ed.), Teaching S t a t i s t i c s and P r o b a b i l i t y . The 1981 Nationdl Council o f Teachers o f Mathematics Yearbook, 90-100

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Winter, H.: 1981. Zur Beschreibenden S t a t i s t i k i n der Sekundarstufe I Rechtfertigungsgrunde und Mogl i c h k e i t e n z u r I n t e g r a t i o n der Stochasti k i n den Mathematikunterricht. I n : W. D o r f l e r & R. Fischer (eds.), Stochastik i m Schulunterricht. S c h r i f t e n r e i h e D i d a k t i k der Mathematik, U n i v e r s i t a t f u r Bildungswissenschaften i n Klagenfurt, Band 3, Wien

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

363

- RANDOMNESS AND STOCHASTIC INDEPENDENCE ON THE RELATIONSHIP BETWEEN INTUITIVE MOTION MATHEMAT ICA L DEFINIT ION

GERD v. HARTEN

HEINZ STEINBRING

Universitat Bielefeld lnstitut fur Didaktik der Mathematik F. R. Germany

1.

Introduction

I n stochastics education, one o f the most important questions i s how students handle the concepts o f chance and o f randomness. I t r e f e r s t o two aspects: f i r s t , t o the preconcepts and i n t u i t i o n s o f p r o b a b i l i t y , and second, t o the mathematical c a l c u l i and techniques used t o adapt and t o develop these i n t u i tions further. The concept o f chance o r o f randomness i s c e n t r a l f o r stochastics education f o r the very reason t h a t t h i s concept, as opposed t o o t h e r mathematical theor i e s and concepts, i s the s p e c i f i c f e a t u r e o f p r o b a b i l i t y theory. Stochastics independence has emerged, during the h i s t o r i c a l development o f p r o b a b i l i t y theory, as one c e n t r a l mathematical technique o f assigning a mathematically precise meaning t o randomness. Nevertheless, i t should be mentioned t h a t t h i s does n o t solve the problem o f i n t u i t i o n and o f calcul u s o f chance. Independence i t s e l f r e f l e c t s the same problematical r e l a t i o n ship:"What i s r e a l l y involved i s a d e f i n i t i o n o f independence and a b e l i e f (borne out by experience and experiment, t o be sure) t h a t t h e d e f i n i t i o n i s applicable t o a p a r t i c u l a r s i t u a t i o n . There i s , thus, independence i n a

364

DEVELOPMENT AND EPISTEMO LOG Y

vague and i n t u i t i v e sense, and t h e r e i s 'independence' i n t h e narrow b u t w e l l - d e f i n e d sense t h a t t h e r u l e o f m u l t i p l i c a t i o n o f p r o b a b i l i t y i s app l i c a b l e . I t was t h e vague and i n t u i t i v e n o t i o n t h a t p r o v i d e d f o r a l o n g t i m e t h e main m o t i v a t i o n and d r i v i n g f o r c e b e h i n d p r o b a b i l i t y t h e o r y . ' ' (KAC, 1959, p.10) Chance and randomness i s a common fundamental assumption b o t h o f s t o c h a s t i c s e d u c a t i o n and a g r e a t p a r t o f p s y c h o l o g i c a l r e s e a r c h i n d e c i s i o n making under u n c e r t a i n t y .

I n t h e f o l l o w i n g paper, we s h a l l a n a l y z e some a s p e c t s o f t h i s t o p i c f r o m an e d u c a t i o n a l p e r s p e c t i v e , and s h a l l t r y t o f i n d o u t some s i m i l a r i t i e s o r d i f f e r e n c e s between t h e s e two approaches; i . e . we s h a l l see what t h e s p e c i f i c e d u c a t i o n a l requirements f o r i n v e s t i g a t i n g p u p i l s ' i n t u i t i o n s and p r e concepts a r e . 2.

How can rmdomness be tested?

One c e n t r a l problem i n p s y c h o l o g i c a l r e s e a r c h about s t o c h a s t i c a l t h i n k i n g i s how t o e v a l u a t e i n d i v i d u a l judgments o f randomness. U s u a l l y , v a r i o u s b i n a r y sequences a r e p r e s e n t e d t o t h e s t u d e n t who i s asked t o e s t i m a t e w h i c h o f t h e s e sequences i s random. The s t u d e n t s w i l l m o s t l y o n l y be a b l e t o g i v e i n t u i t i v e judgments as t h e y l a c k d e t a i l e d knowledge i n p r o b a b i l i t y t h e o r y . The adequacy o f t h e s e answers i s d i s c u s s e d as compared t o t h a t o f answers g i v e n b y an e x p e r t o r o b t a i n e d by f o r m a l c a l c u l u s . The l a t t e r m i g h t p r o v i d e t h e c o r r e c t answer, i . e . which sequence i s random, and w h i c h i s n o t . David Green made a s u r v e y o f p r o b a b i l i t y concepts o f 3000 p u p i l s aged 11-16 y e a r s . I n h i s work he asked t h e f o l l o w i n g q u e s t i o n s :

A t e a c h e r asked C l a r e and Susan each t o t o s s a c o i n a l a r g e number o f t i m e s and t o r e c o r d e v e r y t i m e whether t h e c o i n landed Heads o r T a i l s . F o r each 'Heads' a

1i s

r e c o r d e d and f o r each ' T a i l s ' a

0is

recorded.

RANDOMNESS AND INDEPENDENCE

365

Here are the two s e t s of results: Clare:

01011001100101011011010001110001101101010110010001 01010011100110101100101100101100100101110110011011 01010010110010101100010011010110011101110101100011

Susan:

10011101111010011100100111001000111011111101010101 11100000010001010010000010001100010100000000011001 01000001111100001101010010010011111101001100011000

Now one g i r l did i t properly, by tossing the coin. The other g i r l cheated and j u s t made i t up.

( a ) Which g i r l cheated? ( b ) How can you t e l l ?

Answer Answer

................................ ................................

The results are shown in the following table: Year

Clare

Susan

1 2 3 4 5 A1 1

38 34 30 40 35 35

53 56 59 46 49 53

'Cannot t e l l 1 2 4 3 5 3

I

Omit 7 7 7 9 9 8

Posing such or similar problems serves t o thematize the re1 a tionshi p between individual intuitions and objective, or theoretically correct answers. Research into i n t u i t i v e preconcepts of randomness directly refers t o naive ideas about randomness and chance in the sense of "undetermined", "no repet i t i v e or regular structure", "no possibility of predicting further outcomes", etc.

In the attempt t o precisely define the concept of randomness, the so-called theory of computational complexity adopts a quite similar view of chance ( C f . FINE 1973, p.118 f f ) . Roughly speaking, a f i n i t e binary sequence i s called random i f the smallest algorithm capable of specifying i t t o a computer has about the same number of information b i t s as the sequence i t s e l f

366

DEVELOPMENT AND EPISTEMOLOGY

(CHAITIN 1975, p.48).

T h i s i s j u s t what would be n a i v e l y expected, as a

sequence f o r which an a l g o r i t h m ( o r a s p e c i f i c g e n e r a t i n g l a w ) can be cons t r u c t e d w h i c h i s " s m a l l e r " t h a n t h i s sequence, condensing it, s h o u l d n o t be random i n t h e sense o f t h i s d e f i n i t i o n . I t i s c l e a r t h a t computer-genera t e d random numbers, a c c o r d i n g t o t h i s d e f i n i t i o n , a r e n o t r e a l l y random b u t r a t h e r pseudo-random. I t i s seen t h a t t h i s approach t o randomness w i t h i n t h e t h e o r y o f computa-

t i o n a l c o m p l e x i t y corresponds t o some aspects o f p r e c o n c e p t s and i n t u i t i o n s o f chance and randomness. The a t t e m p t a t f o r m a l i z i n g such

n t u i t i o n s i n a t h e o r e t i c a l frame p e r m i t s

t o draw some i n t e r e s t i n g e p i s t e m o l o g i c a l (and m a t h e m a t i c a l ) c o n c l u s i o n s . These c o n c l u s i o n s w i l l shed some l i g h t o n t o t h e d i f f i c u l t r e l a t i o n between i n t u i t i o n and t h e o r e t i c a l techniques. CHAITIN (1976) has shown t h a t t h i s def i n i t i o n o f randomness c o r r e s p o n d i n g t o i n t u i t i o n cannot be proved i n t h e f o l l o w i n g sense: There a r e sequences o f zeros and ones, o f which we c a n n o t t e l l whether t h e y a r e random, o r n o t . I n one word, t h i s i s unprovable. The conclude t h a t o u r d e f i n i t i o n o f randomness i s g e n e r a l l y unacceptable would mean emptying t h e baby o u t w i t h t h e b a t h . We s h o u l d l i k e t o propose t h e f o l l o w i n g d i s t i n c t i o n : on t h e one hand, t h e u n p r o v a b i l i t y o f randomness i m p l i e s t h a t we cannot a b s o l u t e l y o r u n i v e r s a l l y d e c i d e whether a sequence i s random o r n o t ; on t h e o t h e r hand, however, we have t o be c o n t e n t w i t h a r e l a t i v e d e c i s i o n about randomness. I n q u i t e t h e same sense as i t i s used i n s t a t i s t i c a l p r a c t i c e , a sequence i s p r a g m a t i c a l l y c a l l e d random i f i t s a t i s f i e s a l l t h e known o r r e l e v a n t s t a t i s t i c a l t e s t s . R e l a t i v e t o t h e body o f t e s t techniques and i n s t r u m e n t s , a sequence w i l l be more o r l e s s random. There i s no randomness i n an a b s o l u t e sense. T h i s d i s t i n c t i o n e x p l a i n s why a computer-generated random sequence sense

-

-

which i s n o t random i n an a b s o l u t e

i s random i n a r e l a t i v e sense, and why such sequences w i l l do i n

s t a t i s t i c a l simulation. One m i g h t draw on analogy t o t h e p r i n c i p l e o f r e l a t i v i t y s t a t e d i n t h e f o l i f a space s h i p i s d r i f t i n g a l o n g a t a u n i f o r m speed, a l l l o w i n g form:"

...

experiments performed i n t h e space s h i p and a l l phenomena i n t h e space s h i p

367

RANDOMNESS AND INDEPENDENCE

w i l l appear t h e same as i f t h e s h i p were n o t moving

..." (FEYNMAW,

1970,

p. 15-1). T h i s means, t h e t o o l s i . e . experiments f o r d e t e c t i n g m o t i o n become c r u c i a l . How c o u l d t h i s a n a l y s i s c o n t r i b u t e f r u i t f u l l y t o t h e problem o f s t u d y i n g and d e v e l o p i n g t h e concept o f randomness i n t h e u n d e r s t a n d i n g o f p u p i l s ? F i r s t o f a l l , we s h o u l d keep an eye on t h e c l o s e i n t e r r e l a t i o n between i n t u i t i v e i d e a s a b o u t randomness and t h e body o f t e s t t e c h n i q u e s a v a i l a b l e t o deal w i t h t h i s concept. I n p a r t i c u l a r t h i s means t h a t b o t h i d e a s and i n t u i t i o n s , as w e l l as mathematical t e c h n i q u e s and r u l e s , have t o be t a k e n i n t o account a t e v e r y l e v e l o f u n d e r s t a n d i n g , no m a t t e r how p r o v i s i o n a l t h e y may be. I n t u i t i o n and c a l c u l u s o f chance a r e n o t c o n t r a d i c t o r y , b u t c a l c u l u s s h o u l d be used t o f u r t h e r develop and c l a r i f y i n t u i t i o n s which, however, w i l l n o t be d e f i n i t e . For stochastics education s t u d i e s as w e l l

-

-

and perhaps i n some aspects f o r p s y c h o l o g i c a l

we have t o draw t h e f o l l o w i n g c o n c l u s i o n s : I n o r d e r t o

unfold stochastical thinking, i t i s not s u f f i c i e n t t o confront individuals w i t h s i n g l e random s i t u a t i o n s . Besides, i t i s r e q u i r e d t o be aware o f t h e means a v a i l a b l e t o d e s c r i b e and a n a l y z e t h i s s i t u a t i o n . I n school mathemati c s , we have t o c o n s t r u c t n o t s i n g l e t a s k s , b u t r a t h e r connected sequences o f t a s k s i n which t h e degree o f d i f f i c u l t y , t h e scope o f t e c h n i q u e s and mathematical concepts r e q u i r e d a r e v a r i e d and which w i l l show t h e c e n t r a l meaning o f d i f f e r e n t approaches and v a r i a b l e model 1i n g s f o r t h e i n t e r p r e t a t i o n o f r e a l random events. 3.

Independence

- intuitive

notion and mathematical caZcuZus

A t y p i c a l problem i n t h e s t o c h a s t i c s s e c t i o n o f school t e x t b o o k s reads as fol1ows:"Two d i c e a r e thrown. What i s t h e p r o b a b i l i t y t h a t t h e sum o f scores i s g r e a t e r o r equal 10 i f i t i s known t h a t one d i e shows a 4?" P u p i l s a r e expected t o f i n d t h e f o l l o w i n g so1ution:"There a r e 36 p o s s i b l e outcomes o f t h i s experiment. I f i t i s known t h a t one o f t h e d i c e shows a 4, t h e n t h e r e

368

DEVELOPMENTAND EPISTEMOLOGY

are o n l y 11 outcomes l e f t . Among these, o n l y 2 w i l l y i e l d a sum o f scores greater o r equal 10. Hence, the p r o b a b i l i t y i s 2/11."

H. Dinges reports t h a t he presented t h i s problem t o some o f h i s colleagues o f the mathematics department w i t h the r e s u l t t h a t they e i t h e r asked him t o describe the s i t u a t i o n more p r e c i s e l y , o r gave the s o l u t i o n o f 1/6. This answer i s due t o the f o l l o w i n g i n t e r p r e t a t i o n o f t h i s s i t u a t i o n : Two dice have been thrown, and an observer has glimpsed t h a t one o f them shows a 4. With which p r o b a b i l i t y may the observer expect a sum o f scores higher than lo? I t i s seen t h a t t h e textbook problem i s open t o representation by d i f f e r e n t

models. While the textbook authors appear t o have presented the s i t u a t i o n as a paradigm o f applying c o n d i t i o n a l p r o b a b i l i t y , experts confronted w i t h the problem t r i e d t o imagine a r e a l s i t u a t i o n . L e t us now consider a concept which i s a t the very core o f p r o b a b i l i t y theory and which ought t o be o f prime importance f o r the school curriculum as

well

-

the concept o f independence which i s t h e c e n t r a l instrument t o handle

randomness i n school mathematics. The d e f i n i t i o n o f stochastic independence i s usually given f o r two events as f o l l o w s : Two events A, B a r e independent i f

P(AnB) = P(A).P(B).

To use t h i s concept f o r s o l v i n g simple problems, we must have some i n t u i t i v e notion o f independence which t e l l s us whether o r n o t a chosen model i s adequate f o r a c e r t a i n s i t u a t i o n . I n school, such models are used e.g. f o r twostage experiments as the throwing o f two dice. I n more s o p h i s t i c a t e d s i t u a t i o n s , the model character o f independence becomes more obvious. L e t us take a look a t the f o l l o w i n g textbook example: " B i l l and Joe are two frequent guests i n a pub. The innkeeper knows by ex-

perience t h a t B i l l w i l l take a morning p i n t w i t h a p r o b a b i l i t y o f 30 percent, w h i l e Joe's p r o b a b i l i t y i s 40 percent. The two o f t e n q u a r r e l . Hence, the i n n keeper p r e f e r s e i t h e r B i l l o r Joe coming alone t o take a morning p i n t . What i s the p r o b a b i l i t y f o r t h i s event?"

369

RANDOMNESS AND INDEPENDENCE

P ( B n J ) = 0.12

p ( B ~ 5 )= 0 . 1 8

0.3

The s o l u t i o n o f t h i s example i s based on t h e h y p ot hesis o f independence. I n dependence i s used as a t o o l t o c o n s t r u c t a p r o b a b i l i t y space f o r t h e compound event; w i t h o u t t h i s h y p o t h e s i s , t h e r e i s no s o l u t i o n . We can c o n s i d e r t h i s use t o be t h e c o n s t r u c t i v e a s p e c t o f independence. L e t us c o n s i d e r ano t h e r prob1em:"A c o i n i s t o s s ed and two observers X, Y a r e wa t c h ing t h e outcome E. The two o b s e r v e r s make no c o l l u s i o n . Are t h e y ' indep endent '?'' L e t H = heads, T = t a i l s , P ( E = T ) = p itatement

I

t r u e outcome

1

-

p ro ba b i 1it i e s

u probabi 1it i e s

DEVELOPMENT AND EPISTEMO LOG Y

370

Now we can e a s i l y compute t h a t the events " X t e l l s t h e t r u t h ' ' and " Y t e l l s the t r u t h " are n o t independent ( o n l y i n t h e exceptional cases o f p = O o r 1).

outcome o f the coinexperiment

X

"outcome" probabi 1it y

says

o f the

Y says

P ( X says the t r u t h ) = ( 1 - p ) t 1 t ilp 1 P(Y says the t r u t h ) =

q

1 -zp

= 1=

"who1 e" experiment

p1

$

P(X says the t r u t h and Y says the t r u t h ) = P(X says the t r u t h )

. P(Y says the t r u t h )

$) =

1

1

1 1

-zp(l-$) = ~ ( ~ 2 - v ~ )

This r e s u l t i s evident as the p r o b a b i l i t y o f these events i s l i n k e d t o the experiment's outcome. The events are o n l y c o n d i t i o n a l l y independent, w i t h r e spect t o A, t h e outcome o f the experiment. Thus we are l e d t o t h e f o l l o w i n g

RANDOMNESS AND INDEPENDENCE

371

well-known n o t i o n : Two e v e n t s A , B a r e c o n d i t i o n a l l y independent g i v e n an e v e n t C i f f P(AnB1C) = P(A1C)

*

(P(B1C)

I n t h i s example, we have used t h e f o r m a l mathematical concept o f independence

t o d i s c o v e r t h a t some events a r e n o t independent, a l t h o u g h t h e o b s e r v e r s have n o t been i n c o l l u s i o n , o r m i g h t even have been i n d i f f e r e n t rooms. (Condorcet, Laplace, Poisson, Cournot, Boole have drawn some c o n c l u s i o n s w i t h r e g a r d t o judgments founded on t h e assumption t h a t i f two judges a r e n o t i n c o l l u s i o n t h e events o f t h e i r t e l l i n g t h e t r u t h w i l l be s t o c h a s t i c a l l y i n d e p e n d e n t . ) T h i s example shows a n o t h e r a s p e c t o f independence, which we w i l l c a l l t h e exp l o r a t o r y a s p e c t o f independence. T h i s r e f l e c t s t h e f a c t t h a t independence i s n o t o n l y used t o d e s c r i b e b u t t o d i s c o v e r independence. L e t us come t o a l a s t example b e f o r e we m i g h t draw some e d u c a t i o n a l p r o p o s i t i o n s : " I n an u r n , t h e r e a r e 50 b a l l s : 49 b l a c k and 1 w h i t e . Two persons a l t e r n a t e l y draw w i t h o u t replacement. He who draws t h e w h i t e b a l l f i r s t has won. Does t h e f i r s t t o b e g i n o r t h e second have t h e b e t t e r chance t o w i n ? " To s o l v e t h i s problem one can assume t h a t b o t h p l a y e r s c o n t i n u e t o draw unt i l t h e u r n i s empty. Now a l l p o s s i b l e sequences o f b l a c k and w h i t e b a l l s have t h e same p r o b a b i l i t y . T h e r e f o r e g e t t i n g a w h i t e b a l l a t an even draw o r an uneven i s e q u a l l y l i k e l y , p r o v i d e d t h e number o f draws i s even. T h i s c o n d i t i o n , we have n o t i c e d i n t h i s s p e c i a l case, i s g e n e r a l l y known as t h e concept o f e x c h a n g e a b i l i t y d e f i n e d u s u a l l y i n t h e f o l l o w i n g way: A f a m i l y {Xi)

o f random v a r i a b l e s i s c a l l e d exchangeable, i f f o r e v e r y f i n i t e c o l -

l e c t i o n and e v e r y p e r m u t a t i o n u :

where

F denotes t h e d i s t r i b u t i o n f u n c t i o n . Thus f r o m such a s i m p l e problem,

p u p i l s m i g h t g a i n f i r s t i n s i g h t s i n a fundamental concept as e x c h a n g e a b i l i t y which p l a y s a c e n t r a l r o l e i n s u b j e c t i v e p r o b a b i l i t y t h e o r y .

372

4.

DEVELOPMENT AND EPISTEMOLOGY

Educational proposition

A l l t h e examples we have g i v e n s e r v e d t o i l l u s t r a t e t h e f o l l o w i n g p o i n t : t o a p p l y t h e mathematical concepts o f independence, we must possess an i n t u i t i v e n o t i o n ; where t h e l a t t e r may be inadequate, mathematical concepts as independence f o r example w i l l s e r v e as a t o o l f o r i n v e s t i g a t i o n . We a r e c a l l e d t o c o n s t r u c t a more o r l e s s adequate model o f a r e a l s i t u a t i o n . F o r t h i s purppse, we want t o r e c a l l t h e d i f f e r e n c e between a t h e o r y and i t s a p p l i c a t i o n . I n t h e 1 8 t h c e n t u r y , such a d i f f e r e n c e d i d n o t e x i s t . P r o b a l i t y was d i r e c t l y l i n k e d t o reason; p r o b a b i l i t y t h e o r y was i n t e n d e d t o de-

s c r i b e how a reasonable b e i n g would a c t . I f c a l c u l u s p r e d i c t e d an unreasonable r e s u l t , c a l c u l u s was t o blame. D u r i n g t h e subsequent development o f p r o b a b i l i t y t h e o r y , i t was i m p o s s i b l e t o m a i n t a i n t h i s k i n d o f d i r e c t l i n k between a t h e o r y and i t s a p p l i c a t i o n , between concept and c o n t e n t , between t h e t h e o r e t i c a l t e r m and t h e o b j e c t . I n t h e end, Kolmogorov's system of axioms was t o d e s c r i b e p r o b a b i l i t y o n l y i m p l i c i t l y as a f u n c t i o n f u l f i l l i n g c e r t a i n r e l a t i o n s . From an e d u c a t i o n a l p o i n t o f view a t l e a s t , t h i s a x i o m a t i c c h a r a c t e r i z a t i o n i s n o t v e r y s a t i s f a c t o r y , as i t i s i m p o s s i b l e i n school t o t e a c h b u t t h e use o f a concept w i t h o u t e x p l a i n i n g i t s c o n t e n t . As t h e above examples have shown, some i n s i g h t i n t o t h e meaning o f say, i n dependence i s r e q u i r e d b e f o r e t h i s concept can be used. On t h e o t h e r hand, t h e t h e o r e t i c a l t e r m o f s t o c h a s t i c a l independence cannot be d e s c r i b e d b y j u s t t e l l i n g something about i t s meaning.

A f i r s t e d u c a t i o n a l c o n c l u s i o n suggested m i g h t be t h e f o l l o w i n g : P u p i l s must l e a r n t o c o n s t r u c t models; t h i s can be achieved b y c o n f r o n t i n g them w i t h open problems e t c . As we have seen, t h e problems t a k e n f r o m t e x t b o o k s a r e u s u a l l y understood t o have a r a t h e r c l e a r - c u t s o l u t i o n . To t h e p u p i l , i t i s o b v i o u s t h a t t o s o l v e t h i s problem he/she must a p p l y , f o r example, t h e f o r mula o f independence, c o n d i t i o n a l p r o b a b i l i t y , e t c . There a r e b u t two opt i o n s : e i t h e r t h e p u p i l knows t h e s e concepts and i s a b l e t o s o l v e t h e prob-

373

RANDOMNESS AND INDEPENDENCE

lem, o r he/she w i l l f a i l . Now t h i s c o u l d be e a s i l y overcome: j u s t p r o v i d e t h e p u p i l s w i t h open problems, where d i f f e r e n t models a r e p o s s i b l e . We do n o t c o n s i d e r t h i s s o l u t i o n v e r y sound, however. I t i s n o t w i t h o u t reason t h a t t h e c o n t e x t o f problems p r e s c r i b e d i n t h e school c u r r i c u l u m i s narrow. F i r s t , p u p i l s a r e n o t r e s e a r c h e r s ( a f a c t sometimes o v e r l o o k e d by t h o s e p r o p a g a t i n g p r o b l e m - s o l v i n g l e s s o n s ) , and f i n d i n g new theorems i s n o t t h e o b j e c t i v e o f i n s t r u c t i o n . Second, whether o r n o t a problem i s solved, o r a t l e a s t a s o l u t i o n adequate i n some sense i s a t t a i n e d , w i l l depend o f t h e t o o l s used ( b y t h i s , we n o t o n l y mean t o o l s i n t h e m a t e r i a l sense, b u t a l s o s c i e n t i f i c c o n c e p t s ) . Hence, knowledge cannot be c o n f i n e d t o p r o c e d u r a l s k i 11s. I n o u r view, a p o s s i b l e way o u t o f t h i s a p p a r e n t dilemma i s t o t r e a t i n school n o t o n l y s i n g u l a r problems w i t h i n a s t r i c t l y f i x e d c o n t e x t , b u t t o c o n s i d e r , w i t h r e s p e c t t o a c e r t a i n f f e a t u r e o f a t h e o r e t i c a l concept o r r u l e , systems o f problems which p e r m i t t o e s t a b l i s h a more open c o n t e x t and which r e f l e c t , and use as a d r i v i n g f o r c e , t h e i n h e r e n t d i f f e r e n c e between t h e o r e t i c a l terms and t h e i r a p p l i c a t i o n s .

References Athen, H. & G r i e s e l , H. ( e d s . ) : Mathematik h e u t e (Bd.10). Hannover u.a.

1976

C h a i t i n , G.J.: Randomness and mathematical p r o o f . S c i e n t i f i c American, 1975, 232, 47-52. Dinges, H.: S c h w i e r i g k e i t e n m i t d e r Bayesschen Regel, i n : Math.Phys.Sem., Bd. X X I , 1978, 113-156. Feynman, R.P. Fine, T.L.:

u.a.:

L e c t u r e s on Physics, V o l . I . Reading Mass.) u.a.

T h e o r i e s o f P r o b a b i l i t y . New York

1970.

1973.

Green, D.: P r o b a b i l i t y Concepts i n 11-16 Year o l d P u p i l s . Manuscri p t , Loughborough, May 1982. Kac, M.: S t a t i s t i c a l Independence i n P r o b a b i l i t y , A n a l y s i s and Number Theory. London u.a. 1964. Keynes,

J.M.: Ober W a h r s c h e i n l i c h k e i t . L e i p z i g 1926.

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (No.rth-Holland), 1983

375

DECISION-THEORETIC PARADOXES

GEORG PFLUG

Universitat Giessen Fachbereich Mathematik F. R. Germany

The h i s t o r y o f i n d u c t i v e s t a t i s t i c s up t o t h e m a t h e m a t i c a l f o u n d a t i o n o f d e c i s i o n t h e o r y b y Abraham WALD (around t h e m i d d l e o f t h e 20th c e n t u r y ) was c h a r a c t e r i z e d by t h e fecL t h a t i n t u i t i v e p r i n c i p l e s were c o n s i d e r e d t o be o f equal importance as mathematical d e d u c t i o n . Such " i n t u i t i v e p r i n c i p l e s " were f o r i n s t a n c e : t h e o p t i m a l i t y of t h e maximum l i k e l i h o o d (ML-') e s t i m a t o r , t h e a r i t h m e t i c mean as b e s t e s t i m a t o r f o r l o c a t i o n , t h e c o n s i s t e n c y o f a l l u s u a l estimators. B u t soon s e v e r a l examples were found showing t h a t some i n t u i t i v e concepts o f o p t i m a l i t y o f s t a t i s t i c a l procedures needed t o be r e v i s e d . Such examples

some o f them a r e p r e s e n t e d i n t h i s paper

-

-

a r e a t t h e same t i m e counterexam-

p l e s and paradoxes. The q u a l i f i c a t i o n o f an example t o be a s t o n i s h i n g , c o n t r a i n t u i t i v e o r paradox depends on t h e g e n e r a l s t a t e o f knowledge and i s t h e r e f o r e r e l a t i v e . I t can happen t h a t n o example amazes t o d a y an e x p e r t i n e s t i m a t i o n t h e o r y even ifhe n o t i c e s t h i s example f o r t h e f i r s t t i m e . T h i s r e l a t i v e v a l i d i t y however i s common t o a l l paradoxes, a f a c t w h i c h i s e.g.

described

by SCHOLZ (1981) f o r t h e w e l l known paradox o f ZENON. N e v e r t h e l e s s some of t h e p r e s e n t e d examples shake t h e t h e o r y o f s t a t i s t i c a l d e c i s i o n s t o i t s v e r y f o u n d a t i o n s . T h i s e f f e c t i s a l s o common t o a l l paradoxes (Compare t h e e f f e c t o f RUSSEL'S paradox t o t h e development o f mathematical l o g i c ) . Charles S t e i n d e s c r i b e s i n h i s paper on i n a d m i s s i b i l i t y how s c e p t i c a l he was about t h e c o r r e c t n e s s o f h i s own r e s u l t s

-

h e c o u l d n o t b e l i e v e such

a c o n t r a - i n t u i t i v e s i t u a t i o n . And L i n d l e y ' s Paradox i n d i c a t e s t h a t t h e Bayesian s t a n d p o i n t can l e a d t o c o n t r a d i c t i o n s w i t h c l a s s i c a l s t a t i s t i c s .

DE VELOPMEN T A N D EPISTEMOLOGY

376

Why does i t happen t h a t contra-intuitive situations occur? An partial answer t o that question i s given by the theory of restricted rationality. This theory explains the phenomena by the limited capacitive and/or operative abili t i e s of the human thinking. Although such an explanation i s sufficient f o r the understanding of simple f a l l a c i e s f o r children i t i s n o t able to explain f a l lacies of experts. Those f a l l a c i e s can better be explained by the argument t h a t the paradox examples come mostly from very extreme case. Our intuition which was trained with rather normal cases will be mislead by very unusual examples. I t i s not surprising t h a t Edgeworth's counterexample (Example 1) makes use of a unusual density and t h a t Stein's Paradox (Example 2 ) is valid only for n L 3 . So we conjecture that i f a s t a t i s t i c a l situation i s considered in an intuitive manner, simple and beautiful examples are kept in mind a t the same time, a f a c t which can lead t o the conclusion that there are more analogies t h a n actually are. Before going into the presentation of the examples some basic f a c t s concerning WALD's s t a t i s t i c a l decision theory are discussed. Let f ( x , e ) be a family of probability densities which depend on an unknown parameter 8. A function g ( e ) , mapping the parameter onto a point on the real axis i s t o be estimated by an estimator T ( x ) which depends only on the observations. The "costs" f o r the discrepancy between the estimated value T ( x ) and the true value g ( e ) are measured by the loss function

L(T(x)

9

s(e))

the mean loss, i . e . the expectation of the loss function f ( x , e ) i s called r i s k function

with respect to

R ( T y 8 ) := Ee(L(T( * ) ,g( 0 ) 1

The estimator

T1

i s called better t h a n

T2

if

for all

e

An estimator i s called admissible, i f there i s no better estimator. Example 1.An iron trading company has bought shock-absorber tubes a t a very low price from a factory which was i n bankruptcy. The tubes were without any internal cylinder. UnfortunateZy the diameter of these tubes proved t o be very different. (Perhaps that was the reason why the producing company went i n t o a

377

DECISION- THEORE TIC PARADOXES

bankrmptcy?) The d i s t r i b u t i o n of the d i m e t e r s is unimodal and s y m e t r i c . A sample of s i z e n i s drawn t o determine the mean diameter according t o which the appropriate cylinders should be produced. I t i s possible that the very f i r s t observation i s a b e t t e r estimate than the arithmetic mean of repeated measurements. Explanation. This i s Edgeworth's Paradox. The f a c t t h a t one s i n g l e observation may be b e t t e r than the a r i t h m e t i c mean o f repeated observations stands i n e v i dent c o n t r a d i c t i o n t o t h e i n t u i t i v e p r i n c i p l e o f t h e o p t i m a l i t y o f t h e a r i t h metic mean which was generally accepted i n the lgth century. The o r i g i n a l formulation o f t h i s example by EDGEWORTH makes use o f the density family f(x,e)

=

-3

1 (~+lx-e\)~

and a r i s k f u n c t i o n chosen as follows: Since both p a r t s f i t together o n l y i f some tolerance l i m i t c i s not exceeded i t makes sense t o s e t R(T,e) = P{)T-e S E ) By some simple c a l c u l a t i o n s one r e a l l y f i n d s t h a t f o r the density defined above

and f o r

E

T1 = X1

i.e.

T1

< 0.1

i s a b e t t e r estimator than

T2 =

x1+x2 2

has smaller r i s k . Analogously, f o r any given

n, i t i s p o s s i b l e t o

construct examples t h a t

T1 = X1

i s a b e t t e r estimator than

T2 =

x 1+x 2+...+xn n

The explanation f o r t h i s phenomenon i s simple. Edgeworth's density has a sharp peak a t t h e center o f symmetry. Therefore i t i s more probable t h a t l i e s i n the considered confidence i n t e r v a l than

x1+x2 , t h e density 2

XI

o f which

has n o t a s i m i l a r peak. However, t h i s f a c t i s h a r d l y seen i n t u i t i v e l y . Sometimes i t i s argued t h a t the example shows t h a t i n some s i t u a t i o n s the a r i t h m e t i c mean o f independent observations makes no sense a t a l l . This argu-. ment i s i n t h e author's opinion n o t convincing. A d i f f e r e n t example which goes back t o the l o g i c i a n Venn i s b e t t e r s u i t e d f o r t h i s purpose:

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DE VELOPMENTAND EPISTEMOLOGY

mo sailors from a ship are sent t o s t e a l i n t o a f o r t and take t h e calibre f-

of a cannon. We may suppose that t h e f o r t i s going t o be taken and used

against the enemy, and that it i s therefore of importance t o provide shot of the r i g h t calibre. One sailor says t h e calibre was eight inches and t h e other reported it was nine. C l e a r l y the decision t o produce shots w i t h diameter 8 112 inches makes no sense a t a l l , since t h i s c o n t r a d i c t s the observations o f both.

A good choice f o r t h e loss f u n c t i o n i n t h i s example i s

L =

("

if

1

otherwise

For such a loss f u n c t i o n t h e a r i t h m e t i c mean i s u s u a l l y n o t admissible. I n t h i s respect the s i t u a t i o n i s s i m i l a r t o Edgeworth's paradox.

Exwnple 2. A wine-producing cooperative has received from a wine-grower a delivery of 1000 kilograms of each of t h e following v a r i e t y of grapes: V e l t l i n e r , Traminer, Rotgipfler and Zweigelt. The sugar content of t h e grapes i s determined by picking up a bunch of grapes from each variety. The rates of sugar content are the following: 1,s; 3,O; 2,5; 1,O. The vector as a r e s u l t of t h i s i s inadmissible. An admissible estwiator brings t o the rates 1,33; 2,67; 2,38; 0,89. Explanation. This phenomenon i s c a l l e d S t e i n ' s Paradox and i t i s based on t h e f o l l o w i n g assumptions. A sample o f variables

X1,...Xn

n

independent normally d i s t r i b u t e d random

w i t h unknown means

el, ...,en

and known variance

observed. The loss f u n c t i o n o f t h e estimator T=(T1,.

..Tn)

for

cr2

is

g=(el,. .. ,On)

i s quadratic, i . e .

For

i s a t l e a s t admisXn) f a i l s t o be admissible. There i s

n = l the estimator T=X1 i s optimal. For n=2 T=(X1,X2)

s i b l e . I f n 2 3 the estimator T=(X1, a u n i f o r m l y b e t t e r estimator, namely

...,

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DECISION- THEORETIC PARADOXES

( James

- Stein estimator). This improved estimator is formed by multiplying

the naive estimation vector (Xl,...,Xn) with a constant smaller than one and thus by reducing the norm of the naive estimator. But this is most paradox: Since the origin is completely arbitrary, it can be replaced by any other point provided that this point i s chosen in advance. By introducing some arbitrariness, the estimator i s improved! For an expert in estimation theory the phenomenon can be explained by the fact, that all admissible estimates are Bayes-estimates and that the JamesStein estimator i s a Bayes-estimator for the normal distribution with mean zero as prior. This is however rather little intuitive. Therefore the opinion, that a similar phenomenon can only happen for dimension n z 3 and the quadratic loss function prevailed for a long time. This is one more argument for the above cited assertion that our intuition is trained by simple one- or twodimensional examples and that we intend to transfer their simple features to more general situations. BERGER has however, given a full variety of cases (cf. BROWN 1980) in which non-admissibility holds even if n i 2 . From the today's knowledge the admissibility of the intuitive estimator (XI,. . .Xn) seems to be rather the exception than the rule. ExwnpZe 3. The Zocation parameter (mean value) 0 of a bimodaZ s y m e t r i c d i s t r i bution i s t o be estimated. I t i s known t h a t the two modes of the density are a t distance 2 one from the other. A sample (X,,X2) of s i z e tuo i s observed. The efficiency of tuo d i f f e r e n t estimators i s t o be compared. We take T=- X I +x2 1

2

and

T2= min(Xl,X2)

+

1

I t may happen t h a t

( i . e . T 2 Zies with higher probabiZity closer t o t h e true vatue) and a t the same time 2 2 E ( ( T , - 0 ) I E ( ( T ~ - e) 1. fi.e.

TI has smaZler r i s k ) . Which estimator i s preferable?

Explanation. In order to simplify the explanation it is assumed that the random

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DEVELOPMENTAND EPISTEMOLOGY

variables Xi;

i = 1,2

xi Then

P{Tf

}=

are discrete with d i s t r i b u t i o n with probability

"2

with probability

'/2

=

'14

but

2

E ( T f ) = ' 1 2 < 1 = E(T2 ) .

Analogous examples w i t h random v a r i a b l e s possessing a density can be constructed i n a s i m i l a r manner. I n p a r t i c u l a r the p r o b a b i l i t y t h a t the estimator w i t h lower r i s k deviates more from t h e t r u e value can be a r b i t r a r i l y c l o s e t o one. The reason f o r t h i s paradoxon i s simply t h e f a c t t h a t t h e marginal d i s t r i b u t i o n s o f a two dimensional random v a r i a b l e s do n o t a t a l l determine the common d i s t r i b u t i o n . A f u r t h e r example i n t h i s d i r e c t i o n i s h e l p f u l :

It may happen that P{L1 > L z ) is larbitrariZyl close to I aZthough L I is stochasticaZZy smaZZer than L2 (i.e. P{ L1 6 a}> P { L 2 6 a3 f o r aZI a). BLYTH (1972) gives t h e f o l l o w i n g example: The common density f u n c t i o n f(x,y)

of

L1

and

L2 w i t h i n the u n i t square i s given by 0 sy-x 6 E x-y > 1- E otherwise

Then

P I L1>L23 = 1-E

but

L1

i s s t o c h a s t i c a l l y smaller than L2. BLYTH

describes a l s o an i n t e r e s t i n g a p p l i c a t i o n : By observing two runners, who a l ways run together, o n l y one separate from the other, n o t h i n g can be s a i d about the presumable winner o f the contest. It may happen t h a t runner 1 has a lower mean running time although runner 2 i s w i t h very high p r o b a b i l i t y t h e winner i n d i r e c t comparison. Transfering t h i s example i n t o t h e framework o f WALD's s t a t i s t i c a l decision theory we can say t h a t r i s k minimal estimators could be i n f e r i o r i n pairwise comparison. Thus some care i s needed f o r t h e formulation o f optimality criteria. S i m i l a r paradoxa come from the n o n - t r a n s i t i v ' i t y o f t h e r e l a t i o n P{ L1< L2) > l / 2 . I t may happen t h a t t h e r e l a t i o n s

P I L2 < L3 3>1/2 and P{ L3 < L1)> l / 2

P{ L1< L2)>1/2,

h o l d a t t h e same time. Such examples

are w e l l known i n t h e context o f voting-paradoxa.

I n knockout tournaments

DECISION- THEORETIC PARADOXES

38 I

such paradoxical situations can arise as well (cf. ISRAEL (1981)). The choice of the best among three alternative estimators T1, T2 and T3 seems to be ambigous, if the corresponding loss functions L1, L2 and L3 exhibit such an intransitive behavior. Evidently, risk minimality is better than a criterion based on pairwise comparisons since the latter can lead to inconsistencies. Example 4 . A window was smashed when a burglaq has been committed i n a jewe l l r y . A piece of broken glass was found i n a suspect. The breaking index of the shop's window-pane was estimated t o 1,518458. The breaking index of the glass splinter found i n the suspect m s investigated and calculated t o the rate of 1,518472 with an standard deviation of 0,000004 due t o the measurement error A classical s t a t i s t i c a l t e s t rejects the hypothesis of equality. Since the breaking indexes of window-panes vary between 1,51 and 1,53 the hypotheses of equality can be rejected from a bayesian standpoint, and thus the culprit could be caught. Which conclusion i s the correct one?

.

Explanation. This phenomenon is called LINDLEY's Paradoxon. It can be described as follows: It may occur that a hypothesis is rejected with high significance by a classical test and is doubtless accepted by a bayesian standpoint. If, for instance, X is normally distributed with unknown mean 8 and known variance u then the hypothesis e =eo is rejected if IX-eol > 4u (say). If however a normal distribution with variance T is taken as prior fore and T > > u , than X may lie in the acceptance region. The paradox conclusion is: The more vaguethe prior information is, the more diffuse the prior distribution is to be chosen and the more likely is the acception of the null hypothesis. Or to put it into the language of the example: The less there is known about the distribution of the breaking indices of glass, the more likely is the guilty of the suspect. The different positions (classical versus bayesian standpoint) can be characterized as follows: The attorney uses the argument: The breaking indices of the two glasses are such different one from the other, compared to the error of measurement, that it is impossible that the suspect had committed the crime. The reply of the public prosecutor is: There are so different types of glass windows in our town that it is rather unlikely that the suspect had a piece of

DEVELOPMENT AND EPISTEMO LOG Y

302

g l a s s i n h i s pocket, which i s so s i m i l a r t o t h a t i n t h e j e w e l l e r y . How t o s o l v e t h i s dilemma? SHAFER (1982) u t i l i z e s t h e t h e o r y o f be1 i e f - f u n c t i o n s

, which

was i n v e n t e d

by h i m s e l f , t o r e s o l v e t h e problem. H i s conclusion i s t h a t t h e p r i o r informat i o n on t h e d i s t r i b u t i o n i s l e s s r e l e v a n t than t h e measurement. LINDLEY comments on t h a t by saying t h a t SHAFER was l e d by a s u b t i l e form o f base-rate f a l l a c y (TVERSKY & KAHNEMANN 1974). From LINDLEY's v i e w p o i n t t h e bayesian standp o i n t i s s u p e r i o r t o t h e non-bayesian i f t h e p r i o r i n f o r m a t i o n i s e m p i r i c a l l y based. There i s no general way o u t o f t h e dilemma. To t h e c o n t r a r y : The q u e s t i o n o f t h e use o f p r i o r p r o b a b i l i t i e s i n s t a t i s t i c a l e s t i m a t i o n separates t h e s c i e n t i s t s i n two d i s j o i n t groups. The d e c i s i o n , t o which group one adheres, e l i m i n a t e s t h e paradoxon on t h e i n d i v i d u a l l e v e l . To t h e a u t h o r ' s o p i n i o n however, t h i s way o u t o f t h e c o n t r a d i c t i o n i s n o t q u i t e s a t i s f a c t o r y . References

B l y t h , C.R.: 1972. Some p r o b a b i l i t y paradoxes i n choice f r o m among random a l t e r n a t i v e s . I n : J . Amer. S t a t i s t i . Assoc. 67, 366-373 Brown, L.: 1980. Example o f B e r g e r ' s phenomenon i n t h e e s t i m a t i o n o f independent normal means. I n : Ann. S t a t i s t . 8, 572-585 Edgeworth, Y.:

1983. The law o f e r r o r . I n : P h i l o s o p h i c a l Magazine 16, 300-309

I s r a e l , R.B.: 1981. Stronger p l a y e r s need n o t t o w i n more knockout tournaments. I n : J. Amer. S t a t i s t . Assoc. 76, 950-951 James, W. & S t e i n , Ch.: 1961. E s t i m a t i o n w i t h q u a d r a t i c l o s s . Proceedings o f the 4 t h Berkeley Symposium, Vol. 1, 361-379 Lindley, D.V.:

1977. A problem i n f o r e n s i c science. I n : Biometrika 64, 207-213

Pflug, G.Ch. : 1980. Paradoxien d e r Wahrscheinlichkeitsrechnung. I n : Stochas t i k i m S c h u l u n t e r r i c h t . Holder, P i c h l e r , Tempsky, Wien

-

Scholz, R.W. : 1981. Stochastische Problemaufgaben Analysen aus d i d a k t i s c h e r und psychologischer S i c h t . I D M M a t e r i a l i e n und Studien Bd. 23, B i e l e f e l d Shafer, G.:

1982. L i n d l e y ' s paradox. I n : J . Amer. S t a t i s t . Assoc. 77, 325-334

Stein, C.: 1956. I n a d m i s s i b i l i t y o f t h e usual e s t i m a t o r f o r t h e mean o f a m u l t i v a r i a t e normal d i s t r i b u t i o n . Proceedings o f t h e 3 r d Berkeley Symposium, Vol. 1, 197-206

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S t i g l e r , S.:

383

1980. An Edgeworth curiosum. I n : Ann. S t a t i s t . 8, 931-934

Tversky, A. & Kahneman, D. : 1974. Judgement under u n c e r t a i n t y : H e u r i s t i c s and biases. In: U t i l i t y , P r o b a b i l i t y , and Human Decision Making, 141-162 Wald, A.:

1950. S t a t i s t i c a l d e c i s i o n f u n c t i o n s . New York: John Wiley

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COMMENTS

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz [editor)

0 Elsevier Science Publishers B.V. (North-Holland), 1983

387

THE POSSIBLE AND THE PROFITABLE Some remarks on the symposium's impact on decision research

BARUCH FISCHHOFF Decision Research A Branch of Perceptronics USA

One common response t o f e e l i n g s o f unease about t h e c u r r e n t s t a t e i n a f i e l d research i s t o search f o r a best method f o r t h e f u t u r e . I n t h i s meeting, we have heard a v a r i e t y o f persuasive and i n c o n s i s t e n t arguments f o r how the study o f i n d i v i d u a l decision making should proceed. P r o j e c t i n g from my own response, these appeals have f a l l e n on open minds, b u t unreceptive hearts.

The ensuring discussions have shown g r e a t w i l l i n g n e s s t o

take other people's r e s u l t s seriously, b u t l i t t l e readiness t o s h i f t one's own research program. I suspect t h a t such changes o f course are as much a f u n c t i o n o f c o n s t i t u t i o n as o f conviction. Some o f us l i k e t o ask people a l o t o f questions face t o face; others l i k e t o keep subjects a t arms' length. Some o f us l i k e t o proceed by l i t t l e steps; others l i k e t o p l o t grand r e search strategies. Some p r e f e r t h e r i s k o f c r e a t i n g t h e o r i e s t h a t cannot be f a l s i f i e d ; others p r e f e r the r i s k o f producing empirical r e s u l t s o f unknown general ity. I f we a r e t o have anarchy, r a t h e r than j u s t chaos, then we need t o know how best t o l e a r n from one another and t o e x p l o i t one anothers' l o o s e l y cou-

p l e d experience.

One d i s t i n c t i o n t h a t helps me t o understand t h e relevance

o f t h e diverse work described here t o my own work i s between top-down and bottom-up approaches. The top-downers l i k e t o l a y o u t a complete frame o f reference i n c l u d i n g terms, methods, theories, assumptions, and domain o f i n -

388

COMMENTS

quiry, before collecting any data ( i f there i s any time l e f t t o do so or any experiment that will meet a l l these c r i t e r i a ) . The bottom-uppers l i k e t o discover solid, replicable e f f e c t s that can be accounted f o r by modest local theories, before worrying about formal definitions f o r the semi-intuitive concepts that they use ( i f there i s time l e f t to do so and a clear correspondence can be made between t h e i r experiment and any theory). One possible analogy i s w i t h the differences between those who t r y to understand a forest by aerial photography and those who t r y t o do so by exploring clearings on the ground. Like the botanists who have used these contrasting procedures, we t e n d t o find t h a t f o r e s t s with a f u l l canopy often do not look so healthy on the ground. There may be l i t t l e regeneration, parasitic diseases in incipient stages, o r soil compactment that are hidden by the foliage of a standing forest. Conversely, the problems seen in one locality may not be found widely, may n o t threaten the f o r e s t as a whole, and may even be necessary f o r the s t a b i l i t y of the e n t i r e ecosystem. For those of us working on the ground, the c r i t i c a l question i s where the t r e e s t h a t we know so well f i t into the overall f o r e s t . I t i s always easy t o speculate about the boundary conditions f o r any observed phenomenon. Classic ad hoc claims a r e that the phenomenon would disappear i f the question were posed differently, i f subjects were better motivated, or i f subj e c t s could be induced t o adopt the experiment's view on how the problem should be structured. A bottom-up strategy f o r addressing such claims i s t o review available studies t o see whether any of these additional factors consistently makes a difference (which would give i t the s t a t u s of an explanatory rather t h a n a nuisance variable). A more top-down procedure i s t o compare notes with those who study d i f ferent s t r a t a on the f o r e s t o r aspects of i t s functioning. A much stronger claim can be made f o r the necessity o f reconciling one's own results w i t h other robust r e s u l t s than for taking seriously mere speculations about what e l s e might be happening. In t h i s regard, the conference has been extremely productive. From the point of view of research into individual decision making behavior, we have seen a variety of useful a l t e r n a t i v e cuts a t our

DECISION RESEARCH

common problem.

389

One i s t h e study o f t h e ways and frequency w i t h which group

s i t u a t i o n s can i n t e r f e r e w i t h i n d i v i d u a l r a t i o n a l i t y . Another i s t h e d i f f i c u l t i e s t h a t a r i s e when the demands o f a decision-making f a i l t o match t h e decision maker's a b i l i t i e s or organizational s e t t i n g . A t h i r d i s some of v a r i e t y of a l t e r n a t i v e ways i n which subjects might reasonably (mis)construe our s t i m u l i . A f o u r t h i s t h e k i n d o f educational and developmental experiences t h a t may have shaped the c o g n i t i v e s k i l l s t h a t subjects b r i n g t o our tasks. A f i f t h i s t h e k i n d o f judgmental d i f f i c u l t i e s t h a t have been found even w i t h people who should be experts and w i t h problems t h a t should demand t h e best performance of which they a r e capable. Our task now i s t o go back home and q u i e t l y d i g e s t t h i s m a t e r i a l , i n order t o see which of these p o s s i b i l i t i e s a r e a c t u a l l y p r o b a b i l i t i e s .

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

39 1

THE IMPORTANCE OF THE FICTIONAL FOR THEORIES OF KNOWLEDGE Some remarks on the symposium's impact on cognitive psychology

ALF C. ZIMMER Universitat Munster Psychologisches lnstitut F. R. Germany

The q u e s t of Tversky, Kahneman, and o t h e r s f o r p e r s i s t e n t b i a s e s i n human judgment ( f o r overviews see KAHNEMAN, SLOVIC, and TVERSKY, 1982, o r NISBETT and ROSS, 1980) i s r e m i n i s c e n t o f HELMHOLTZ' (1896) approach t o t h e i n v e s t i g a t i o n o f human p e r c e p t i o n : i n b o t h cases d i s c r e p a n c i e s between t h e " r e a l " on t h e one hand and t h e " p e r c e i v e d " o r "judged" on t h e o t h e r hand have been used as s t a r t i n g p o i n t s f o r t h e i n v e s t i g a t i o n o f t h e d i f f e r e n c e s between human i n f o r m a t i o n p r o c e s s i n g and t h e way t h e w o r l d i s d e s c r i b e d i n terms o f p h y s i c s o r l o g i c . HELMHOLTZ' a n a l y s i s o f o p t i c a l i l l u s i o n s i n f l u e n c e d him t o postulate the theoretical construct

I'

unconcious i n f e r e n c e s

'I

which i n

t u r n l e d t o t h e i n t e r p r e t a t i o n of p e r c e p t i o n as t h e r e s u l t o f processes a n a l o gous t o computation (PYLYSHYN, 1980). The approach t o human d e c i s i o n making t a k e n b y TVERSKY and o t h e r s f o l l o w s an analogous r e s e a r c h program b y i n v e s t i g a t i n g under which circumstances human (unaided) judgment f a i l s e i t h e r t o process a l l t h e necessary aspects o f i n f o r m a t i o n a v a i l a b l e o r t o r e p r e s e n t them v e r i d i c a l l y . GIBSON (e.g.

1979) has c h a l l e n g e d

t h e HELMHOLTZian paradigm by p o i n t i n g o u t

t h a t f r o m an e v o l u t i o n a r y p o i n t o f view i t i s n o t t h e mismatches between t h e g i v e n and t h e p e r c e i v e d t h a t a r e o f i n t e r e s t b u t t h e matches. T h a t i s , t h e v e r i d i c a l usage o f t h e a f f o r d a n c e s p r o v i d e d t o us by o u r environment a r e o f i n t e r e s t . I have suggested (ZIMMER, n o t e 1 ) t h a t t h e s e seemingly opposing

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COMMENTS

views can be regarded as d i f f e r e n t a s p e c t s o f t h e same process, i n one case f r o m a bottom-up p e r s p e c t i v e and i n t h e o t h e r case f r o m a top-down p e r s p e c t i v e . Both aspects seem t o be necessary f o r a complete u n d e r s t a n d i n g o f complex p e r c e p t u a l o r judmental t a s k s . Complementary t o t h e approach t a k e n by TVERSKY and

o'tihers, I propose t o t a k e t h e c o n d i t i o n s f o r v e r i d i c a l judgments as a

framework i n which b i a s e s can be t r a c e d back t o t h e c o n d i t i o n s under which such judgments a r e e c o l o g i c a l l y v a l i d . ( see LOPES, 1982 ) I n o r d e r t o develop a t h e o r y o f knowledge i n t h i s framework one has t o acc o u n t f o r an a s p e c t o f human knowledge t h a t i s u s u a l l y d i s r e g a r d e d : t h e cap a c i t y t o g e n e r a t e v i r t u a l e x p e r i e n c e s by u s i n g i d e a l i z e d models o f t h e r e a l i t y , which a r e more o r l e s s f i c t i o n a l . T h i s f i c t i o n a l a s p e c t o f knowledge can be used i n o r d e r t o s k e t c h f u t u r e outcomes o f d e c i s i o n s o r t o d e r i v e o t h e r consequences. The matching o f such v i r t u a l e x p e r i e n c e s d e r i v e d f r o m f i c t i o n a l knowledge i n t u r n seems t o u n d e r l y what has been termed t h e expectanc y e f f e c t i n p e r c e p t i o n . T h a t i s , what we see i s s t r o n g l y i n f l u e n c e d by exp e c t a t i o n s we have a b o u t t h e r e c y l a r i t i e s o f o u r environment, which a r e a t l e a s t p a r t i a l l y f i c t i o n s o f o u r mind. R e l y i n g on t h e s e e x p e c t a n c i e s a l l o w s f o r a v e r y f a s t p r o c e s s i n g o f i n f o r m a t i o n . The c o s t f o r p r o c e s s i n g i n f o r m a t i o n

so f a s t i s t h a t t h e r e sometimes a r e d e l e t i r i o u s s i d e e f f e c t s , w h i c h become e s p e c i a l l y apparent i n t h e t r i c k s m a g i c i a n s p l a y ( GREGORY,

1970, MULHOL-

LAND, 1927 ), where t h e audience i s t r i c k e d i n t o e x p e c t i n g an a c t i o n , which does n o t happen, b u t which works as a camouflage f o r what i s r e a l l y happening. V i r t u a l experiences due t o t h e f i c t i o n a l aspects o f knowledge may i n f l u e n c e t h e s e l e c t i o n o f c o n j e c t u r e s and p r e d i c t i o n s . V i r t u a l e x p e r i e n c e s may a l s o p l a y an i m p o r t a n t r o l e i n h i n d s i g h t e x p l a n a t i o n s . These ideas f o l l o w i n some r e s p e c t V A I H I N G E R (1911) who i n h i s " P h i l o s o p h y o f "as i f " developed a t h e o r y o f knowledge founded on " t h e o r e t i c a l , p r a c t i c a l , and r e l i g i o u s f i c t i o n s o f mankind". He showed how knowledge can be expanded by p r e t e n d i n g "as i f " . F o r i n s t a n c e " L e t ' s p r e t e n d as i f t h e r e were two t r u l y p a r a l l e l l i n e s " o r " L e t ' s p r e t e n d a l l t h e mass o f a body were i n i t s c e n t e r of g r a v i t y . " As we know these "as i f " p r e t e n s i o n s have been c r u c i a l f o r t h e development o f geometry o r p h y s i c s , were i d e a l i z a t i o n s abound. What i s m i s s i n g i n VAIHINGER's t h e o r y i s a c o n n e c t i o n between f i c t i o n a l knowledge and knowledge due t o experiences. T h e o r i e s on t h e s u b j e c t i v e assessment o f one's

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own knowledge (How good a r e my reasons f o r assuming t h a t a g i v e n i d e a l i z a t i o n o r a f i c t i o n a l s c e n a r i o t u r n s o u t t o be an a p t t o o l f o r d e s c r i b i n g o r p r e d i c t i n g events?) m i g h t be a b l e t o b r i d g e t h i s gap. Such s u b j e c t i v e assessments

of t h e f i c t i o n a l a s p e c t s o f knowledge can be done i n terms o f p o s s i b i l i t y (ZADEH, 1970), p l a u s i b i l i t y ( RESCHER, 1973), o r s u b j e c t i v e p r o b a b i l i t y . T h e r e f o r e t h e y s h o u l d be accounted f o r by models f o r human i n f o r m a t i o n p r o c e s s i n g and t h e o r i e s o f a c t i o n . I have argued (ZIMMER 1 ) t h a t i t i s p l a u s i b l e t o assume t h a t t h i s "knowledge a b o u t knowledge" i s r e p r e s e n t e d i n t e r n a l l y by means of a p r o p o s i t i o n a l l a n g u a g e - l i ke code..

The i n t e r p r e t a t i o n o f v e r b a l e x p r e s s i o n s f o r t h i s knowledge p l a y s t h e r e f o r e an i m p o r t a n t r o l e i n t h e u n d e r s t a n d i n g o f such elements o f c o n v e r s a t i o n w h i c h cannot be reduced t o d e m o n s t r a t i v e procedures and c o n s e c u t i v e o b s e r v a t i o n s . Up t o t h i s p o i n t o n l y t h e i n d i v i d u a l knowledge has been t a k e n i n t o account b u t much o f t h e knowledge used i n m a s t e r i n g t h e environment i s n o t due t o t h e i n d i v i d u a l s ' o w n e x p e r i e n c e s b u t i s due t o v i c a r i o u s e x p e r i e n c e s made by o t h e r people and shared a f t e r w a r d s . How t h e n e c e s s i t y t o share knowledge imposes c o n s t r a i n t s upon i n f o r m a t i o n p r o c e s s i n g has been i n v e s t i g a t e d i n s h a r e a b i l i t y t h e o r y (FREYD, i n p r e s s ) . One i m p o r t a n t consequence o f t h i s t h e o r y f o r t h e i n v e s t i g a t i o n o f t h e e f f e c t i v e n e s s o r biasedness o f human judgment i s t h a t i t f o r c e s us t o d i s t i n g u i s h between t h e e f f e c t s due t o a l a c k o f knowledge ( c o n t e n t and procedures) and t h o s e due t o t h e f a c t t h a t c e r t a i n r e a l and v i r t u a l e x p e r i e n c e s do n o t f o l l o w s h a r e a b i l i t y c o n s t r a i n t s . The k i n d o f r e p a i r s i n c o n v e r s a t i o n used i n o r d e r t o overcome t h e misunders t a n d i n g s caused by t h e s e e f f e c t s m i g h t p r o v i d e an e m p i r i c a l approach t o t h i s problem. F o r i n s t a n c e ifone assumes t h a t a m i s u n d e r s t a n d i n g i s due t o a l a c k o f knowledge t h e n an a p p r o p r i a t e r e p a i r would be t o g i v e a d e f i n i t i o n , on t h e o t h e r hand i f t h e m i s u n d e r s t a n d i n g i s assumed t o be due t o d i s r e g a r d i n g s h a r e a b i l i t y c o n s t r a i n t s an a p p r o p r i a t e r e p a i r c o u l d c o n s i s t i n suggest-

'see ZIMMER, A. Verbal vs. n u m e r i c a l p r o c e s s i n g o f s u b j e c t i v e p r o b a b i l i t i e s , i n t h i s volume.

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ing: "Imagine a s i t u a t i o n where the f o l l o w i n g i s t r u e :

... .2 I1

Reference Notes 1. What makes the eye i n t e l l i g e n t ? manuscript, Stanford U n i v e r s i t y , Department o f Psychology, 1981 References Freyd, J.J.: S h a r e a b i l i t y : the s o c i a l psychology o f epistemology. I n : Cogn i t i v e Science, i n press Gibson, J.J.: 1979. The ecological approach t o v i s u a l perception. Boston: Houghton and M i f f l i n Gregory, R.L.:

1970. The i n t e l l i g e n t eye. New York: MacGraw-Hill

Helmholtz, H. von: 1896. Handbuch der physiologischen Optick (2nd ed.). Hamburg und Leipzig: Voss Kahneman, D./Slovic, P. & Tversky, A. (eds.): 1982. Judgment under uncert a i n t y : H e u r i s t i c s and biases. Cambridge: Cambridge U n i v e r s i t y Press Lopes, L.L.: 1982. Doing the impossible: A note on i n d u c t i o n and the experience o f randomness. I n : Journal o f Experimental Psychology / Learning, Memory, and Cognition 8, 626-636 Mulholland, J.: 1927. Quicker than the eye. Indianapolis: Bobbs-Merril Nisbett, R. & Ross, L.: 1980. Human Inference: Strategies and Shortcomings o f Social Judgments. Englewood C l i f f s : Prentice H a l l Pylyshyn, Z.: 1980. Computation and cognition: Issues i n the foundation o f c o g n i t i v e science. I n : The B r a i n and Behavioral Science 9, 111-169

These r a t h e r t h e o r e t i c a l considerations about t h e connection between knowledge and s u b j e c t i v e p r o b a b i l i t i e s permit the deduction o f educational consequences concerning the use o f f a l l a c i e s i n human judgment as a means of teaching s t a t i s t i c s o r decision theory. Among others one b e n e f j c i a r y e f f e c t might be the demonstration t h a t immediate impressions tend t o be misleading because o f the expectancy e f f e c t . But inducing students t o produce judgmental f a l l a c i e s might have unwanted negative e f f e c t s too, because students might experience t h i s behavior as a k i n d o f non-cooperative discourse. This i n t u r n might lead t o the consequence t h a t i t i s n o t the s e n s i t i v i t y f o r matches vs. mismatches o f one's own v i r t u a l experiences w i t h real s i t u a t i o n s t h a t i s heightened. I t might be t h a t instead o f t h i s merely the v i c a r i o u s experience i s made how somebody can be t r i c k e d i n t o inconsistency.

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Rescher,

N.: 1973.

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The Coherence Theory o f Truth. Oxford: Clarendon Press

Vaihinger, H. : 1911. Die Philosophie des Als-Ob. B e r l i n : Reuther und Reichard Zadeh, L.: 1978. Fuzzy s e t s as a basis f o r a theory o f p o s s i b i l i t y . In: Fuzzy Sets and Systems 1, 3-28

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0 Elsevier Science Publishers B.V. (North-Holland), 1983

WHY ARE DEVELOPMENTAL PSYCHOLOGISTS BIASED WITH REGARD TO THE BIASEDNESS OF THE HUMAN SUBJECT? Some remarks on the symposium's impact on developmental psychology MANFRED WALLER Universitat Biele feld A bteilung fur Experimentelle und A ngewandte Psychologie F. R. Germany

That human judgements a r e biased, as impressively substantiated by many o f t h e c o n t r i b u t i o n s t o t h i s conference, i s a t r u i s m f o r the developmental psycholo g i s t . He i s almost d a i l y confronted w i t h i t i n manifold forms when he i n v e s t i gates the growth o f knowledge and judgement s t r u c t u r e s i n t h e growing c h i l d , f o r example, i n the form o f i n f a n t i l e egocentrism (PIAGET, 1955). That t h i s a l s o applies i n s i m i l a r ways f o r the judgements made by the a d u l t , f u l l y developed, subject ( f o r a review c.f. KAHNEMAN, SLOVIC & TVERSKY, 1982) may, on the other hand, be a somewhat s u r p r i s i n g r e s u l t o f t h i s conference f o r the developmental psychologist. I t presents him w i t h a challenge, f o r i t seems t o c o n t r a d i c t one of h i s i m p l i c i t basic assumptions, namely the assumption that, i n the course o f h i s development, the growing c h i l d surmounts step by step t h e obvious d e f i c i e n c i e s and biases o f h i s judgement a b i l i t y , and a t the completion o f h i s development becomes an almost p e r f e c t being, a l o g i c a l l y and r a t i o n a l l y ( i .e. unbiased) t h i n k i n g quasi-scientist

.

as PIAGET (1970) postu-

l a t e s i n h i s c o g n i t i v e developmental theory This s i m i l a r i l y applies f o r t h e development o f o t h e r functions; f o r example, speech development according t o CHOMSKY (1965) and moral development according t o KOHLBERG (1976). Regarding the evident c o n t r a d i c t i o n between t h i s view and t h e main f i n d i n g s o f t h e conference, the developmental psychologist has t o consider SHAKLEE's (1979) c r i t i c a l question, whether t h e p o s t u l a t e of a quasi-unl i m i t e d developmental progression can be maintained.

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I n order t o answer t h i s question adequately, i t i s f i r s t necessary t o p o i n t o u t t h a t t h e r e i s a basic d i f f e r e n c e between t h e biased judgement o f a c h i l d , such as the non-recognition o f the invariance o f a q u a n t i t y o f l i q u i d t h a t i s poured from one container i n t o another, and the f a l s e judgement o f an a d u l t based on judgement d i s t o r t i o n s o r judgemental h e u r i s t i c s ( c . f . KAHNEMAN & TVERSKY, 1982). A c h i l d ' s judgement b i a s i s generally a question o f competence. This means t h a t before a d e f i n i t e stage o f development i s reached, a c h i l d i s b a s i c a l l y n o t able t o solve an invariance problem because o f s p e c i f i c l i m i t a t i o n s o f h i s c o g n i t i v e system ( f o r example, lack o f knowledge, l i m i t a t i o n s o f working memory and o f a t t e n t i o n a l resources , etc. ). A d d i t i o n a l l y , because o f t h e n o n - r e f l e x i v i t y o f h i s t h i n k i n g , he i s not capable o f becoming aware o f t h e one-sidedness and b i a s o f h i s f a l s e judgement. However, the biased judgement o f the a d u l t subject has t o be assessed d i f f e r e n t l y . I t i s generally n o t due t o a s u b j e c t ' s l a c k o f competence, b u t i s a question o f performance conditions. I n o t h e r words, t h e a d u l t i s b a s i c a l l y able t o c o r r e c t h i s e r r o r given s u i t a b l e task conditions, e s p e c i a l l y because he has the a b i l i t y t o recognize t h e incorrectness o f h i s judgement and c r i t i c a l l y r e f l e c t upon it. Because o f t h i s , i t seems t o be problematic, and f o r a developmental psycholo g i s t u n j u s t i f i a b l e , t o a t t r i b u t e c h i l d r e n ' s judgement b i a s t o t h e same judgement h e u r i s t i c s as those o f adults, as ROSS (1981) f o r example, attempts. ROSS r e f e r s t o f i n d i n g s reported by JOSEPHSON (1977, quoted by ROSS, 1981) who was able t o show t h a t i n t h e p r e d i c t i o n o f s i t u a t i o n - s p e c i f i c behaviour o f o t h e r persons, 5 year o l d c h i l d r e n place greater emphasis on s i t u a t i v e f a c t o r s than on d i s p o s i t i o n a l ones, j u s t as 20 year o l d s do, and t h a t they t h e r e f o r e make b e t t e r behaviour p r e d i c t i o n s than a group o f c h i l d r e n o f a mean in-between age who p u t more emphasis on d i s p o s i t i o n a l factors. According t o ROSS these r e s u l t s i n d i c a t e t h a t the judgements o f the 5 year o l d s are (as y e t ) l e s s d i s t o r t e d by t h e "fundamental a t t r i b u t i o n a l e r r o r " than those formed by o l d e r c h i l d r e n . Such an i n t e r p r e t a t i o n i s inadmissable. I t doesn't take i n t o account t h a t pre-school age c h i l d r e n e x c l u s i v e l y take n o t i c e o f external, i.e.

situ-

a t i v e and/or physical o r accessory features, and t h a t they are s t i l l g e n e r a l l y b l i n d t o d i s p o s i t i o n a l features such as t r a i t s and t h e l i k e . This i s c l e a r l y shown by research i n t o t h e development o f person perception ( f o r a review, c.f.

LIVESLEY & BROMLEY, 1973, and SECORD & PEEVERS, 1974).

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I t would t h e r e f o r e be wrong t o draw t h e conclusion from the biased judgements o f a d u l t subjects t h a t they have made no progress i n t h e i r c o g n i t i v e development when compared w i t h the erroneously judging c h i l d . Developmental

psychological research has repeatedly and unanimously shown what these developmental advances are. They c o n s i s t on the one hand i n t h e enlargement o f s p e c i f i c competencies and s k i l l s (e.g. i n the enlargement o f the e f f e c t i v e capacity o f working memory, i n t h e a c q u i s i t i o n o f knowledge and c o n t r o l s t r a t egies, etc.),

and on the other hand, i n the growing a b i l i t y t o c o n t r o l

and

c r i t i c a l l y r e f l e c t upon h i s own t h i n k i n g ,as a r e s u l t o f h i s meta-cognitive development (c.f. BROWN & DeLOACHE, 1978). Certainly, however, such developmental advances obviously do n o t r u l e o u t the judgement e r r o r s made by adults. This i s very c l e a r l y documented by t h e r e s u l t s o f t h i s conference, b u t i s a l s o w e l l known t o t h e developmental psychologist. I n p a r t i c u l a r , c r o s s - c u l t u r a l studies ( i n summary c . f . COLE & SCRIBNER, 1974) reveal t h a t i t i s o n l y under optimal development and s o c i a l i z a t i o n conditions t h a t the adolescent subject a t t a i n s the s t a t e o f development t h a t CHOMSKY (1965),. KOHLBERG(1976), PIAGET (1970) and others, consider t o be t h e f i n a l stage o f t h e developmental progression. I n a d d i t i o n t o t h i s , i n t e r i n d i v i d u a l d i f f e r e n c e s must a l s o be taken i n t o account. However, i n t h e i r research p r a c t i c e , developmental psychologists o f t e n pursue a strategy t h a t aims a t f u l l y e x p l o r i n g the c a p a b i l i t i e s and competencies o f the growing subject by i n v e s t i g a t i n g t h e a b i l i t i e s o f c h i l d r e n o r young people under optimal i f n o t idea2 performance conditions ( c . f . SHAKLEE, 1979). I n developmental psychological research, t h e subjects being i n v e s t i g a t e d a r e gene r a l l y prepared f o r t h e i r task w i t h great care, and a r e therefore, as a r u l e , h i g h l y motivated. The problems they have t o solve can o f t e n be worked on w i t h o u t time l i m i t s and also, due t o the r e s u l t s o f more o r l e s s d e t a i l e d task analyses, these problems are mostly presented i n such a form t h a t a l l i n f o r mations r e l e v a n t f o r t h e i r s o l u t i o n are a v a i l a b l e t o t h e subject. Such performance conditions, t h a t so favour a s u b j e c t ' s responses, a r e seldom given f o r the s o l u t i o n o f everyday problem s i t u a t i o n s , a t least, n o t as a r u l e . A d d i t i o n a l l y , t h e s c i e n t i f i c i n t e r e s t o f developmental psychologists i s t o o o f t e n , a s f o r example i n the case o f PIAGET, p r i n c i p a l l y aimed a t general o r modal development trends, w i t h the consequence t h a t i n t e r i n d i v i d u a l d i f f e r -

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ences a r e l o s t f r o m view. Both t h e s p e c i f i c s c i e n t i f i c i n t e r e s t o f t h e developmental p s y c h o l o g i s t , and t h e p a r t i c u l a r f e a t u r e s o f h i s r e s e a r c h s t r a t e g y , may r e s u l t i n t h e g r o w i n g s u b j e c t becoming an idea2 s u b j e c t w i t h i n t h e realms o f developmental p s y c h o l ogy: a s u b j e c t who due t o t h e n e g l e c t o f i n t e r i n d i v i d u a l d i f f e r e n c e s , and by

t h e i m p l i c i t assumption o f o p t i m a l developmental c o n d i t i o n s , t e n d s t o become a quasi-scientist, w i t h u n l i m i t e d and unbiased judgement c a p a b i l i t y . Thereby t h e ecologica,l

v a l i d i t y o f t h e r e s u l t s o f developmental psychology r e s e a r c h

may be g r e a t l y reduced. T h i s conference can draw t h e a t t e n t i o n o f developmental p s y c h o l o g i s t s t o e x a c t l y t h i s problem, and h o p e f u l l y s t i m u l a t e them t o i n v e s t i g a t e t h e developi n g c a p a b i l i t i e s o f t h e growing s u b j e c t under c o n d i t i o n s t h a t p l a c e more emp h a s i s on e c o l o g i c a l f a c t o r s .

References Brown, A.L. & DeLoache, J.S.: 1978. S k i l l s , plans, and s e l f - r e g u l a t i o n . I n : R.S. S i e g l e r (Ed.), C h i l d r e n ' s t h i n k i n g : What develops? H i l l s d a l e , N.J.: Lawrence Erlbaum, 3 35

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Chomsky, N . : 1965. k p e c t s o f t h e t h e o r y o f s y n t a x . Cambridge, Mass.: HIT Press Cole,

M./ Scribner,S. : 1974. C u l t u r e and t h o u g h t : A p s y c h o l o g i c a l i n t r o d u c t i o n . London: W i l e y

Josephson, J.: 1977. The c h i l d ' s use o f s i t u a t i o n a l and personal i n f o r m a t i o n i n p r e d i c t i n g t h e b e h a v i o r o f a n o t h e r . Unpublished d o c t o r a l d i s s e r t a t i o n . Stanford: Stanford U n i v e r s i t y Kahneman, 0. , S l o v i c , P. & Tversky, A. (Eds.) : 1982. Judgment under unc e r t a i n t y : h e u r i s t i c s and b i a s e s . Cambridge: Cambridge U n i v e r s i t y Press Kahneman, D. & Tversky, A.: 1982. Judgment under u n c e r t a i n t y . I n : D. Kahneman/ P. S l o v i c & A. Tversky (Eds.), Judgment under u n c e r t a i n t y : h e u r i s t i c s and b i a s e s . Cambridge: Cambridge U n i v e r s i t y Press, 3 20

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Kohl berg, L. : 1976. Moral . s t a g e s and m o r a l i z a t i o n . The c o g n i t i v e - d e v e l o p mental approach. I n : Th. L i c k o n a (Ed.), Moral development and behav i o r . Theory, research, and s o c i a l i s s u e s . New York: H o l t , R i n e h a r t and Winston, 31 53

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L i v e s l e y , W.J. & Bromley, D.B.: 1973. Person p e r c e p t i o n i n c h i l d h o o d and adolescence. London: W i l e y P i a g e t , J.: 1955. The language and t h o u g h t o f t h e c h i l d . Cleveland: Mer i d i a n Books P i a g e t , J.: 1970. P i a g e t l s t h e o r y . I n : P.H. Mussen (Ed.) C a r m i c h a e l ' s ma732 nual o f c h i l d psychology. V o l . 1. New York: Wiley, 703

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Ross, L.: 1981. The " i n t u i t i v e s c i e n t i s t " f o r m u l a t i o n and i t s development a l i m p l i c a t i o n s . Jn: J.H. F l a v e l l & L. Ross (Eds.), S o c i a l c o g n i t i v e development. F r o n t i e r s and p o s s i b l e f u t u r e s . Cambridge: Cambridge U n i v e r s i t y Press, 1 42

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Secord, P.F. & Peevers, B.H.: 1974. The development and a t t r i b u t i o n o f p e r son concepts. I n : T. Mischel (Ed.), Understanding o t h e r persons. Oxford: B l a c k w e l l , 117 142

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Shaklee, H.: 1979. Bounded r a t i o n a l i t y and c o g n i t i v e development: Upper lim i t s on growth? I n : C o g n i t i v e Psychology 11, 327 345

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DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor)

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0 Elrevier Sciencv Publishers B.V. (North-Holland), 1983

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INDIVIDUAL DECISION MAKING AND SOCIAL PSYCHOLOGY Some remarks on the symposium's impact on social psychology

KATRIN BORCHERDING Universitat Mannheim Sonderforschungsbereich 24 F. R. Germany

The area o f s o c i a l psychology can r o u g h l y be d e f i n e d as ' i n v e s t i g a t i n g and explaining i n d i v i d u a l behavior i n s o c i a l c o n t e x t s ' . Since ' i n d i v i d u a l decis i o n making under u n c e r t a i n t y ' i s i n c r e a s i n g l y analysed i n s o c i a l s i t u a t i o n s , i t becomes a l s o a f i e l d o f s o c i a l psychology. Close i n t e r r e l a t i o n between p a r t s o f s o c i a l psychology and d e c i s i o n r e s e a r c h s h a l l be demonstrated by two t o p i c s which were discussed d u r i n g t h e symposium, namely d i s t r i b u t i o n a l d e c i s i o n s u s i n g t h e c l a s s i c a l paradigm o f n-person games ( s e e t h e c o n t r i b u t i o n s by ALBERS, by SCHULZ

& MAY and by TACK, t h i s volume) and consensus de-

c i s i o n s w i t h t h e c l a s s i c a l f i n d i n g o f c h o i c e s h i f t ( s e e CROTT & ZUBER, t h i s volume). Both t o p i c s have been t r e a t e d by many s o c i a l p s y c h o l o g i s t s as w e l l as d e c i s i o n r e s e a r c h e r s ; t h e y a r e t h e r e f o r e s u i t e d t o p o i n t o u t p a r a l l e l s as w e l l as t y p i c a l d i f f e r e n c e s i n p e r s p e c t i v e s .

Dis tributiona Z decisions D i s t r i b u t i o n a l d e c i s i o n s a r e c e n t r a l t o i n d i v i d u a l s , groups and s o c i e t i e s as a whole i n t h a t a v a i l a b l e , u s u a l l y l i m i t e d r e s o u r c e s o r 'goods' l i k e time, money, r i g h t s , e t c . have t o be a l l o c a t e d o r shared. The s u b s t i t u t a b i l i t y o f goods i s i n v e s t i g a t e d i n d e c i s i o n r e s e a r c h , e.g.

i n t h e area o f m u l t i - a t t r i -

b u t e u t i l i t y t h e o r y (KEENEY & RAIFFA, 1976), and i n s o c i a l psychology, e.g. i n t h e o r i e s o f s o c i a l exchange u s i n g economic p r i n c i p l e s t o e x p l a i n s o c i a l i n t e r a c t i o n (HOMANS, 1961).

COMMEMTS

404

I n d e c i s i o n r e s e a r c h d i s t r i b u t i o n a l d e c i s i o n s a r e s t u d i e d w i t h a f o c u s on i n d i v i d u a l aspects o f t h e s i t u a t i o n : I n game t h e o r y , where m a t r i c e s s p e c i f y t h e outcomes f o r t h e v a r i o u s p l a y e r s , i n d i v i d u a l b e h a v i o r i s analysed as a f u n c t i o n o f m a t r i x outcomes as b e i n g e.g.

r a t i o n a l , competitive o r cooperative

I n b a r g a i n i n g d e c i s i o n s , where money i s shared by t h e p l a y e r s a c c o r d i n g t o some r u l e s , i n d i v i d u a l b e h a v i o r i s analysed i n o r d e r t o l e a r n o u t p e o p l e s ' s t r a t e g i e s t o a c h i e v e t h e i r g o a l s and use t h e i r power. The s o c i a l i m p a c t o t h e s i t u a t i o n i s n o t s t r e s s e d and t h e necessary i n t e r a c t i o n s between t h e p l a y e r s a r e reduced t o a minimum and o f t e n done u s i n g t e c h n i c a l equipments w i t h o u t personal c o n t a c t . I n t h e area o f s o c i a l psychology d i s t r i b u t i o n a l d e c i s i o n s a r e a n a l y s e d accordi n g t o whether t h e y a r e e v a l u a t e d as j u s t and how p e o p l e r e a c t t o v i o l a t i o n s o f d i s t r i b u t i v e j u s t i c e . When t a l k i n g about an adequate p a r t i t i o n o f a common good, t h r e e p r i n c i p l e s o f j u s t i c e a r e discussed: p a r t i t i o n a c c o r d i n g t o cont r i b u t i o n s ( e q u i t y p r i n c i p l e ) , p a r t i t i o n a c c o r d i n g t o needs (Marxian p r i n c i p l e ) and equal p a r t i t i o n where everyone g e t s t h e same ( p a r i t y p r i n c i p l e ) . Some a u t h o r s j u s t i f y t h e e q u i t y p r i n c i p l e as u n i v e r s a l (e.g. BERSCHEID, 1978)

WALSTER, WALSTER &

o t h e r s argue t h a t d i f f e r e n t : p r i n c i p l e s a r e a p p r o p r i a t e i n

d i f f e r e n t s i t u a t i o n s (DEUTSCH, 1975; KAYSER & SCHWINGER, 1982; MIKULA, 1980;

SCHWINGER, 1980), and LEVENTHAL assumes t h a t a l l t h r e e p r i n c i p l e s e n t e r i n t o a f a i r solution,

perhaps weighted d i f f e r e n t l y (LEVENTHAL, 1976). I f s e v e r a l

p r i n c i p l e s apply e i t h e r simultanously o r a l t e r n a t i v e l y the question arises f o r t h e c o n d i t i o n s under which a s p e c i f i c p r i n c i p l e i s used o r VENTHAL model

-

-

i n t h e LE-

f o r t h e f a c t o r s t h a t i n f l u e n c e t h e r e l a t i v e w e i g h t s . The ex-

t e n t t o which a p r i n c i p l e i s e v a l u a t e d as j u s t m i g h t depend on ( a ) t h e k i n d o f t a s k f o r which t h e common 'payment' i s done ( i n t e l l e c t u a l o r n o t , easy d i v i d a b l e o r n o t , t i m e consuming o r n o t ; a r e l e v a n t taxonomy o f t a s k charact e r i s t i c s has y e t t o be developed); ( b ) t h e k i n d o f r e s o u r c e i n w h i c h t h e 'payment' occurs (money, time, l o v e , s t a t u s , e t c . ;

see FOA & FOA, 1976, f o r

a t h e o r y o f r e s o u r c e s ) ; ( c ) t h e r e l a t i o n s between t h e s u b j e c t s ( f r i e n d s , r e l a t i v e s , neighbours, s t r a n g e r s , e t c . ; see HASSEBRAUCK, 1982, f o r some r e s u l t s ) ; and ( d ) t h e s o c i a l c o n t e x t i n which a l l t h i s t a k e s p l a c e (work, l e i s u r e , etc.).

Thus, t h e same q u e s t i o n

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how do people p a r t i t i o n a common good

-

is

approached i n a formal and s t r u c t u r a l way i n d e c i s i o n r e s e a r c h and i n a more s u b s t a n t i v e way by s o c i a l p s y c h o l o g i s t s .

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Consensus decisions The f i r s t p u b l i c a t i o n s on c h o i c e s h i f t (STONER, 1961, 1968; KOGAN & WALLACH, 1964) l e d many s o c i a l p s y c h o l o g i s t s t o i n v e s t i g a t e t h e q u e s t i o n whether group judgments d i f f e r from i n d i v i d u a l judgments. They found s y s t e m a t i c d i f f e r e n c e s , which t h e y c a l l e d ' s h i f t s ' , and depending on t h e d i r e c t i o n t h e r e a r e ' c o n s e r v a t i v e ' and ' r i s k y ' s h i f t s . Since many d e c i s i o n s a r e done i n groups o r committees, e f f e c t s o f group b e h a v i o r on d e c i s i o n s a r e r e l e v a n t and knowledge about causes and consequences o f such s h i f t s i s needed. The procedure i n c h o i c e s h i f t r e s e a r c h i s ( a ) e l i c i t i n d i v i d u a l judgments, ( b ) aggregate them, ( c ) observe group judgments, ( d ) compare group judgments w i t h t h e a g g r e g a t i o n o f i n d i v i d u a l judgments, and ( e ) e x p l a i n d i f f e r e n c e s . S i n c e group judgments a r e i n t e r p r e t e d i n r e l a t i o n t o an a g g r e g a t i o n o f i n d i v i d u a l judgments, t h e adequacy o f t h e a g g r e g a t i o n a l g o r i t h m becomes c r u c i a l ; t h e r e l e v a n c e o f t h e f i n d i n g s depend on t h e e x t e n t t o which t h i s a g g r e g a t i o n can serve as a ' r a t i o n a l ' procedure. I n almost a l l cases

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w i t h o u t g i v i n g reasons and h a r d l y

e v e r q u e s t i o n e d - some a r i t h m e t i c mean was t a k e n as t h e a g g r e g a t i o n a1 g o r i t h m ( b u t see as e x c e p t i o n s BORCHERDING & KISTNER, 1982; CROTT & ZUBER, t h i s volume; JEAN, 1970; and SCHAEFER, 1978), and many t h e o r i e s a r e developed t o exp l a i n t h e r e s u l t s ( f o r an overview see SAUER, 1974; S I X , 1981). As CROTT & ZUBER have shown, t h e r e a r e a1 t e r n a t i v e a g g r e g a t i o n procedures which a r e a t l e a s t as reasonable as o t h e r s b u t wash o u t t h e e x i s t e n c e o f s h i f t s . S o c i a l p s y c h o l o g i s t s w o r k i n g i n t h e area o f c h o i c e s h i f t o r p o l a r i s a t i o n s h o u l d t h e r e f o r e c o n c e n t r a t e more on t h e a p p r o p r i a t e n e s s o f a g g r e g a t i o n procedures and i f s h i f t s s t i l l occur, t h e r e i s a need f o r t h e o r i e s which can p r e d i c t t h e d i r e c t i o n o f a s h i f t as a f u n c t i o n o f t a s k c h a r a c t e r i s t i c s f o r a v a r i e t y o f t a s k s . I t i s remarkable t h a t consensus b e h a v i o r was never analysed on an i n d i v i d u a l l e v e l . Knowledge about how easy statements can be accepted by an i n d i v i d u a l m i g h t be fundamental f o r consensus i n groups. Determinants o f t h e ' i n d i v i d u a l a c c e p t a b i l i t y o f d i v e r g e n t s t a t e m e n t s ' m i g h t be t h e importance o f t h e d e c i s i o n s i t u a t i o n f o r an i n d i v i d u a l

-

among o t h e r s

, the

-

direction

and/or magnitude o f t h e d i s c r e p a n c y from own o p i n i o n s , and t h e r e v e r s i b i l it y o f t h e p o t e n t i a l consequences. Such r e s e a r c h c o u l d be c r u c i a l t o determine whether c h o i c e s h i f t i s a phenomenon t h a t occurs o n l y i n groups o r whether i t i s already established firmly i n i n d i v i d u a l behavior.

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outlook

General conclusions a r e d i f f i c u l t and only preliminary because of the breadth of the areas. Sometimes i t appears as i f the strength and weakness of a discipline l i e s closely together. Social psychologists have investigated individual behavior i n a variety of situations and looked f o r the underlying determinants of the respective behavior. The development of substantive theories was promoted i n order to e f f i c i e n t l y and economically explain the variety of observations and r e s u l t s . B u t now and again social psychologists may be somewhat hasty t o jump'at conclusions and to invent new theories without s u f f i c i e n t l y considering the fundamentals in measurement, the specifity of the situation, a1 ternative causes, and conflicting theoretical explanations. Social psychology i s an area w i t h relevant problems t h a t need solutions; b u t social psychologists are sometimes more committed to confirm t h e i r theories than to solve the probl ems. In comparision decisions research i s more oriented towards the solution of real problems, b u t decision researchers sometimes narrowly focus on the formal structure of a situation and on measurement procedures without s u f f i c i e n t l y considering the meaningfulness of t h e i r variables, the psychological co-determinants and substantive theoretical explanations of the decision behavior. Since individuals' representation of the task i s not perfectly overlapping with the one intended by the researcher, individual behavior i s besides individual aspects always influenced by social ones as well. Evidence f o r the impact of social components on individual decisions i s given by TACK ( t h i s volume). His persuasive ' e g o i s t i c ' concept of 'individual r a t i o n a l i t y ' was violated by his subjects. Analysing the situations where most of the violations occurred led him t o conclude: "Individuals with more power n o t j u s t i f i e d by achievement might violate individual rationality i n taking social values into account besides t h e i r individual interests". Deviations from t h i s 'individual r a t i o n a l i t y ' i n order t o reduce existing interindividual differences i s a behavior in accordance to perceived distributive j u s t i c e . A higher order of i n dividual rationality could incorporate social motives l i k e the ones given by MACCRIMMON & MESSICK (1976). Research i n areas, where s c i e n t i s t s from different f i e l d s w i t h various backgrounds work on can be improved by a better communication between these re-

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searchers. This i s an o l d and somewhat t r i v i a l statement, b u t n o t so o f t e n r e a l i z e d as desirable. Symposia l i k e t h i s one a r e important and promising as issues g e t enlarged, and broader perspectives i n research w i l l improve the p o t e n t i a l impact and a p p l i c a b i l i t y o f i t s r e s u l t s .

References Borcherding, K. & Kistner, K.: 1982. Entwicklung und. Oberprufung eines Modells zur Gruppeneinigung i n Choice-Dilemma-Situationen. Z e i t s c h r i f t f u r Sozialpsychologie, 13, 323-332. Deutsch, M.: 1975. Equity, equality, and need: What determines which value w i l l be used as the basis o f d i s t r i b u t i v e j u s t i c e ? Journal o f Social Issues, 31, 137-149. Foa, E.B. & Foa, U.G.: 1976. Resource theory o f s o c i a l change. I n : J.W. Thibaut, J.T. Spence & R.C. Carson (eds.), Contemporary t o p i c s i n s o c i a l psychology. Morristown, N.Y.: General l e a r n i n g press. Hassebrauck, M. : 1982. Reaktionen a u f d i s t r i b u t i v e Ungerechtigkeit und A r t der Sozial beziehung. Eine Oberprufung des Mehr-Prinzipien-Ansatzes. Bei t r a g auf der 24. Tagung experimentell arbei tender Psychologen i n T r i e r . Homans, G.C.: 1961. Social behavior: I t s elementary forms. New York, N.Y.: Harcourt, Brace & World. Jean, R.St.: 1970. Reformulation o f the value hypothesis i n group r i s k t a k i n g . Proceedings o f the 78th Annual Convention, APA. Kayser, E. & Schwinger, T.: 1982. A t h e o r e t i c a l a n a l y s i s o f the r e l a t i o n s h i p among i n d i v i d u a l j u s t i c e concept, layman's psychology, and d i s t r i b u t i o n decisions. Journal o f the Theory o f Social Behavior, 12, 47-51. Keeney, R.L. & R a i f f a , H.: 1976. Decisions w i t h m u l t i p l e objectives: Preferences and value tradeoffs. New York, N.Y.: Wiley. Kogan, N. & Wallach, M.A.: a l i t y . New York, N.Y.:

1964. Risk taking: A study i n c o g n i t i o n and personH o l t , Rinehart & Winston.

Leventhal , G.S.: 1976. Fairness i n s o c i a l r e l a t i o n s h i p s . I n : J.W. Thibaut, J.T. Spence & R.C. Carson (eds.), Contemporary t o p i c s i n s o c i a l psychology. Morristown, N.Y.: General l e a r n i n g press. MacCrimmon, K.R. & Messick, D.M.: v i o r a l Sciences, 21, 86-100.

1976. A framework f o r s o c i a l motives. Beha-

Mikula, G.: 1980. Zur R o l l e der Gerechtigkeit i n Aufteilungsentscheidungen. I n : G. Mikula (Hrsg.), Gerechtigkeit und s o z i a l e I n t e r a k t i o n . Bern: Huber. Sauer, C.: 1974. Zur Erforschung der Gruppenextremisierung nach Diskussion. Z e i t s c h r i f t f i r Sozialpsychologie, 5, 255-273.

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Schaefer, R.E.: 1978. Eine entscheidungstheoretische Analyse des Risky S h i f t Phanomens. Zei tschri f t f d r Sozial psycho1ogie, 9, 186-205. Schwinger, T. : 1980. Gerechte Guterverteilungen: Entscheidungen zwischen drei Prinzipien. In: G. Mikula (Hrsg.), Gerechtigkeit und soziale Interaktion, Bern: Huber. S i x , U.: 1981. Sind Gruppen radikaler a l s Einzelpersonen? E i n Beitrag zum Ris i koschub-Phanomen. Darmstadt: Steinkopff.

Stoner, J.A.F.: 1961. A comparison of individual and group decisions involving risk. Unpublished master thesis. Cambridge, Mass. Stoner, J.A.F.: 1968. Risky and cautious s h i f t s i n group decisions: The i n fluence o f widely held values. Journal o f Experimental Social Psychology, 4, 442-459. Walster, E., Walster, G.W. & Berscheid, E.: 1978. Equity theory and research. Boston, Mass.: Allyn & Bacon.

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TOWARDS A THEORY OF LIMITED RATIONALITY Some remarks on the symposium's impact on economic theory

REINHARD SELTEN Universitat Bielefeld lnstitut fur Mathematische W irtschaf tsforschung F. R. Germany

The r e s u l t s on d e c i s i o n making under u n c e r t a i n t y r e p o r t e d i n t h i s c o n f e r e n c e b e l o n g t o a stream o f a c c u m u l a t i n g e v i d e n c e whose impact on economic t h e o r y may e v e n t u a l l y t u r n o u t t o be o f g r e a t s i g n i f i c a n c e . The p r e v a i l i n g p i c t u r e o f economic man needs t o be reformed i n t h e d i r e c t i o n o f l i m i t e d r a t i o n a l i t y . U n f o r t u n a t e l y e x p e r i m e n t a l r e s e a r c h d i d n o t y e t produce a s u f f i c i e n t l y powerf u l t h e o r y o f l i m i t e d r a t i o n a l i t y . I t i s t h e purpose o f t h i s s h o r t paper t o emphasize t h e n e c e s s i t y o f f u r t h e r work towards t h i s goal. The p r e v a i l i n g p i c t u r e o f r a t i o n a l man i n p r e s e n t day economic t h e o r y i s t h a t o f a Bayesian d e c i s i o n maker w i t h u n l i m i t e d c o g n i t i v e c a p a b i l i t i e s . O f course, economists a d m i t t h a t t h i s i s an i d e a l i z a t i o n which does n o t c o m p l e t e l y c o r respond t o t h e f a c t s . Defenders o f Bayesianism as a d e s c r i p t i v e t h e o r y argue, t h a t d e v i a t i o n s f r o m r e a l i t y a r e r e l a t i v e l y u n i m p o r t a n t . Moreover, t h e y maintain that

even i f i n d i v i d u a l b e h a v i o r i s f a r away f r o m Bayesian r a t i o n a l i t y

average b e h a v i o r may s t i l l be c o r r e c t l y d e s c r i b e d , s i n c e d e v i a t i o n s i n oppos i t e d i r e c t i o n s can be expected t o cancel o u t . U t i l i t y m a x i m i z a t i o n i s defended f o r t h e " r e p r e s e n t a t i v e household", even i f i n d i v i d u a l households a r e known t o make i n t r a n s i t i v e choices. I n my mind t h e r e i s no doubt t h a t such p o s i t i o n s a r e n o t t e n a b l e i n t h e f a c e o f t h e e x p e r i m e n t a l evidence. Systema t i c b i a s e s and t y p i c a l f a l l a c i e s a r e observed and n o t j u B t random d e v i a t i o n s f r o m Bayesian r a t i o n a l i t y .

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I n t h e l a s t y e a r s more and more economists have become aware o f t h e d i s c r e p ancie s between ex pe r i m e n t a l f a c t s and t h e c o n v e n t i o n a l wisdom o f economic textbook s . Phenomena l i k e t h o s e r e p o r t e d i n t h i s conf erence cannot be overl o o k e d any more. However, economic t h e o r y has been q u i t e successf ul i n s p i t e o f i t s u n r e a l i s t i c microeconomic f o u n d a t i o n s , Economists cannot a f f o r d t o r e j e c t t h e t r a d i t i o n a l view u n l e s s i t can be r e p l a c e d by a b e t t e r t h e o r y .

A g r e a t advantage o f t h e p r e v a i l i n g n e o c l a s s i c a l approach i s t h e easiness w i t h which i t generates m e a n i n g f u l t h e o r e t i c a l q uest ions. I f a new economic problem a r i s e s - say a sudden w o r l d w i d e s c a r c i t y o f f o o d - r e a d i l y a v a i l a b l e m o d e l l i n g t o o l s can be used i n o r d e r t o d e s c r i b e t h e t e c h n o l o g i c a l , i n s t i t u t i o n a l , and m o t i v a t i o n a l f e a t u r e s o f t h e s i t u a t i o n . P r o d u c t i o n f u n c t i o n s , market s t r u c t u r e s , economic p o l i n g i n s t r u m e n t s and u t i l ty f u n c t i o n s a r e combined t o f o rm a q u a n t i t a t i v e model w h i c h can be used t o e x p l o r e w e l f a r e i m p l i c a t i o n s o f a l t e r n a t i v e economic p o l i c i e s . The i n t e r e s t s o f t h o s e who have t o make d e c i s i o n s a r e d s c r i b e d b y u t i l i t y f u n c t i o n s . B ehav ior a l p r e d i c t i o n s a r e based on t h e p r i n c i p l e o f s u b j e c t i v e u t i l i t y max imiz a t io n . We l f a r e measures a r e d e f i n e d i n terms o f u t i l i t i e s . The r o l e o f t h e p r i n c i p l e o f s u b j e c t i v e l y expected u t i l i t y m a x i m i z a t i o n i s a v e r y i m p o r t a n t one. I t i s e s p e c i a l l y h a r d t o see how q u e s t i o n s concerning w e l f a r e imp1 i c a t i o n s o f p o l i c y measures can be posed w i t h o u t any recourse t o u t i l i t y theory. The ex periment a l evidence on human d e c i s i o n b e h a v i o r shows what i s wrong w i t h p r e s e n t day economic t h e o r y , b u t i t does n o t y e t show how t o improve on it. I t i s necessary t o develop a t h e o r y o f l i m i t e d r a t i o n a l i t y which can r e p l a c e t h e p r i n c i p l e o f s u b j e c t i v e l y expected u t i l i t y maximizat ion as a t o o l o f economic model b u i l d i n g . A u s e f u l t h e o r y o f l i m i t e d r a t i o n a l i t y w i l l have t o be n e a r e r t o p s y c h o l o g i c a l r e a l i t y , b u t i t may s t i l l be a v e r y i d e a l i z e d p i c t u r e o f e m p i r i c a l phenomena. It s h o u l d be s i m p l e enough t o be u n c o r p orat ed i n t o q u a n t i t a t i v e economic models amenable t o a n a l y t i c a l t r e a t m e n t . Obviously, one cannot a v o i d a compromise between r e a l i s m and s i m p l i c i t y .

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Up t o now a t t e m p t s t o m o d i f y economic t h e o r y i n t h e d i r e c t i o n o f l i m i t e d r a t i o n a l i t y have n o t been as s u c c e s s f u l as one m i g h t wish. The computer s i m u l a t i o n s t u d i e s i n s p i r e d by t h e b e h a v i o r a l t h e o r y o f t h e f i r m (CYERT & MARCH 1963) and by t h e e v o l u t i o n a r y approach (NELSON & WINTER 1974) have

produced a d m i r a b l e r e s u l t s , b u t t h e y f a i l e d t o have a m a j o r i n f l u e n c e on t h e main body o f economic t h e o r y . G e n e r a l l y , economists f e e l uneasy a b o u t c o m p l i c a t e d computer s i m u l a t i o n s ; o f t e n i t i s h a r d t o see t o which e x t e n t t h e r e s u l t s depend on s p e c i f i c parameter v a l u e s and o t h e r ad hoc f e a t u r e s o f t h e s i m u l a t i o n model. N e v e r t h e l e s s computer s i m u l a t i o n s t u d i e s may be an i m p o r t a n t source o f i n t e r e s t i n g new economic e f f e c t s w h i c h t h e n can be s t u d i e d by means o f a n a l y t i c models. F o l l o w i n g t h e seminal work o f SIMON t h e a u t h o r has p a r t i c i p a t e d i n an a t t e m p t t o develop a g e n e r a l t h e o r y o f economic d e c i s i o n making under l i m i t e d r a t i o n a l i t y (SIMON 1955, SAUERMANN & SELTEN 1962). However, t h i s approach was h i g h l y s p e c u l a t i v e and o n l y l o o s e l y based on casual e m p i r i c a l o b s e r v a t i o n s . The b a s i c i d e a o f a s p i r a t i o n l e v e l a d a p t a t i o n has p r o v e d t o be f r u i t f u l i n e x p e r i m e n t a l research, b u t t h e t h e o r y i n i t s o r i g i n a l f o r m cannot c l a i m t o be an e x p l a n a t i o n o f e m p i r i c a l f a c t s . Mere p h i l o s o p h i c a l s p e c u l a t i o n i s n o t a r e l i a b l e b a s i s f o r t h e development o f a t h e o r y o f l i m i t e d r a t i o n a l i t y . It i s necessary t o g a i n i n s i g h t on r e a l human d e c i s i o n b e h a v i o r . I n t h i s r e s p e c t e x p e r i m e n t a l r e s e a r c h i s t h e most i m p o r t a n t t o o l . A l r e a d y i n t h e l a t e f i f t i e s a s m a l l group o f economists f e l t t h a t t h e y had t o do e x p e r i m e n t a l r e s e a r c h i n o r d e r t o answer q u e s t i o n s a r i s i n g i n t h e t h e o r y o f t h e f i r m . I n Germany t h i s has l e d t o a s e r i e s o f volumes " B e i t r a g e z u r experimentel l e n W i rtschaftsforschung

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C o n t r i b u t i o n s t o e x p e r i m e n t a l eco-

nomics". The movement o f e x p e r i m e n t a l economics has come about, s i n c e i n t h e t i m e o f i t s b e g i n n i n g s p s y c h o l o g i s t s d i d n o t y e t pay much a t t e n t i o n t o beh a v i o r a l problems o f economic t h e o r y . Nowadays t h i s has changed and i t i s f a i r t o say t h a t most work i n e x p e r i m e n t a l economics i s done by p s y c h o l o g i s t s . Much o f t h e e x p e r i m e n t a l work on human d e c i s i o n making s t r o n g l y emphasizes b i a s e s and f a l l a c i e s . I n my view i t i s wrong t o c r i t i c i z e t h i s tendency as t h e e x p r e s s i o n o f an u n d u l y n e g a t i v e a t t i t u d e towards human c o g n i t i v e capab i l i t i e s . Biases and f a l l a c i e s p r o b a b l y a r e t h e key t o a b e t t e r understand-

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i n g o f the marvelous information processing m6thods which enable human beings t o f i n d p r a c t i c a l s o l u t i o n s t o problems whose complexity defies any attempt t o apply Bayesian decision theory. Undoubtedly the f u r t h e r development o f decision psychology i s n o t o n l y o f great t h e o r e t i c a l importance b u t a l s o o f considerable p r a c t i c a l s i g n i f i c a n c e . Without i n s i g h t i n t o the s t r u c t u r e o f natural d e c i s i o n bkhavior we cannot hope t o l e a r n how complex decision tasks l i k e computer administrated business games should be approached by a d e c i s i o n maker subject t o t h e usual cognit i v e c o n s t r a i n t s imposed by time pressure and task complexity. O f course, i n order t o answer such questions we need a theory o f l i m i t e d r a t i o n a l i t y which supplies q u a n t i t a t i v e answers r a t h e r than q u a l i t a t i v e ones. I t i s n o t s u f f i c i e n t t o know whether a p r i c e should be r a i s e d o r lowered a businessman has t o decide by how much t o change the p r i c e . The r e s u l t s o f the conference show t h a t experimental research on human decis i o n making i s making great progress towards a b e t t e r understanding o f l i m i t e d r a t i o n a l i t y . Everyone who works i n t h e f i e l d should be aware o f the importance o f the subject f o r economic theory. References

Cyert, R.M. & March, I.G.: 1963. A behavioral theory o f the f i r m . Englewood C l i f f s , N.J.: Prentice H a l l Nelson, R.R. & Winter, S.G.: 1974. Neoclassical vs. evolutionary t h e o r i e s o f economic growth: c r i t i q u e and prospectus. I n : The Economic Journal 84, 886-905 Simon, H.A.: 1955. A behavioral model o f r a t i o n a l choice. I n : Q u a r t e r l y Journal o f Economics 69, 99-118 Sauermann, H. (ed.): 1967, 1970, 1972, 1973, 1974, 1976, 1978. Beitrage zur experimentel 1en W i rtschaftsforschung Contributions t o Experimental Economics, Volumes I t o I X . Tubingen: I.C.B. Mohr

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Sauermann, H. & Sel ten, R. : 1962. Anspruchsanpassungstheorie der Unternehmung. I n : Z e i t s c h r i f t f u r d i e gesamte Staatswissenschaft 118, 577-597

DECISION MAKING UNDER UNCERTAINTY R.W. Scholz (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1983

TOWARDS AN EDUCATION FOR DECISION BEHAVIOR Some remarks on the symposium's impact on mathematical education

HANS-GEO RG STEl NER Universitat Bielefeld lnstitut fur Didaktik der Mathematik F. R. Germany

Before I t r y t o say something about a possible impact o f the conference on mathematical education, I would l i k e t o g i v e a b r i e f view on mathematical education as I see i t i n t e r p r e t e d and operationalized by the Instibute for

the Didactics o f Mathematics (IDM) which through O r . Scholz signs as the o r i g i n a t o r o f the conference. As you might have seen from the l a s t pages o f the book on "Stochastische Problemaufgaben" (Volume 23 o f t h e IDM-series "Material i e n und Studien", SCHOLZ 1981), the I D M i s a supraregional c e n t r a l i n s t i t u t e f o r research, development, advice, and coordination i n t h e f i e l d o f mathematical education. Using i n t e r d i s c i p l i n a r y and cooperative working structures and methods, t h e I n s t i t u t e i s p r i m a r i l y devoting i t s a c t i v i t i e s to

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research and development r e l a t e d t o the mathematics curriculum studies on the foundations o f the d i d a c t i c s o f mathematics as a d i s c i p l ine the i n v e s t i g a t i o n o f processes and i n t e r a c t i o n s i n the mathematics classroom research and development r e l a t e d t o the education and the continuing t r a i n i n g o f mathematics teachers.

I should l i k e t o add that, since i t s foundation i n 1973, the I n s t i t u t e

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has developed a n a t i o n a l and i n t e r n a t i o n a l network f o r c o o p e r a t i o n , c o o r d i n a t i o n , documentation, and i n f o r m a t i o n . An i n t e r n a t i o n a l l i b r a r y and c o l l e c t i o n o f m a t e r i a l s , media, and documents has been e s t a b l i s h e d . W i t h i n t h e s e s t r u c t u r e s t h e IDM i s h o l d i n g conferences and workshops and o f f e r i n g a v a r i e t y o f s e r v i c e s t o i n d i v i d u a l s , groups, and i n s t i t u t i o n s i n t h e a r e a s o f research, development, p r a c t i c e , and a d m i n i s t r a t i o n . Furthermore, i t i s f u n d a m e n t a l l y concerned w i t h t h e f u r t h e r a n c e o f young r e s e a r c h e r s i n t h e d i d a c t i c s o f mathematics by p e r f o r m i n g Ph.D. programs and o r g a n i z i n g r e search seminars and c o l l o q u i a . F o r t h e sake o f s i m p l i c i t y I s h a l l group my f o l l o w i n g remarks a c c o r d i n g t o t h e f o u r above mentioned main R & D-components,

s t a r t i n g w i t h the curriculum.

As a s t r u c t u r e d body o f knowledge, o r g a n i z e d w i t h r e s p e c t t o goals, cont e x t s , l e a r n i n g a c t i v i t i e s , developmental aspects, a mathematics curricuZwn r e p r e s e n t s i m p o r t a n t f e a t u r e s o f s o c i e t a l r e a l i t y and needs. D e f i n i t e l y , t h e problems r e l a t e d t o t h e theme o f t h i s conference a r e o f i n c r e a s i n g i m p o r t a n c e f o r many domains i n o u r s o c i e t y . More emphasis on t h e t e a c h i n g o f

probabiZity and statistics a t school l e v e l seems t o be an adequate consequence o f t h i s f a c t . However, i t i s q u e s t i o n a b l e whether t h e u s u a l way these topics are being presented i s

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if at all

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sufficiently contributing

t o p r o v i d i n g p u p i l s w i t h adequate knowledge, experience, and s t r a t e g i e s needed f o r a good " d e c i s i o n making under u n c e r t a i n t y " . We have h e a r d c r i t i c i s m s

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books (see SCHRAGE's p a p e r )

a t l e a s t r e l a t e d t o German s y l l a b i and t e x t -

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t h a t stochastics i s often

being taught from

a p r e d o m i n a n t l y mathematical v i e w p o i n t . I m p o r t a n t m e t h o d o l o g i c a l a s p e c t s such as a l t e r n a t i v e problem f o r m u l a t i o n s , t h e process o f modeling, competing models f o r a s i t u a t i o n given, a l t e r n a t i v e d e c i s i o n s and t h e i r consequences, r e l a t e d r i s k s and p e r s o n a l r e s p o n s i b i l i t i e s , e t c . a r e a l m o s t c o n p l e t e l y neg l ected. G. v. HARTEN and H. STEINBRING i n t h e i r paper i n d i c a t e t h a t i m p o r t a n t as-

p e c t s o f t h e genuine o r i e n t a t i o n o f p r o b a b i l i t y t h e o r y towards a p p l i c a t i o n s a r e deeply r o o t e d a l r e a d y i n t h e fundamental concepts o f t h e t h e o r y . I n p a r t i c u l a r , t h e y e x p l a i n t h i s w i t h r e s p e c t t o randomness and s t o c h a s t i c

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independence. I t i s pointed out that these facts usually are not sufficiently elaborated i n the standard approaches in school texts. The authors's suggestions are t o get away with standardized problems in s t r i c t l y fixed cont e x t s , b u t rather develop systems of problems with flexible contexts which allow t o develop a growing understanding of the interrelation between theoretical concepts and t h e i r meaning and role in various applications. Clearly, the teaching of probability and s t a t i s t i c s in the s p i r i t of "decision making under uncertainty" asks f o r a project type of a c t i v i t i e s i n school in order t o guarantee a variety of important conditions: a s u f f i c i e n t degree of similarity with real situations, the p o s s i b i l i t i e s for actually going into processes of development and modeling, the constitutive role of social factors and interactions. I want t o emphasize in particular the opportunity which we find here t o exhibit in a very specific sense the socia2 dimension of mathematics and t o give students concrete experience and methodological knowledge a b o u t t h i s dimension and i t s various aspects such as: i n dividual differences in viewing a situation and related problems, t h e i r i n fluence on basic frames and decisions in the conceptualization of the s i tuation and the formulation of problems, aspects of acceptability and agreement, personal or collective responsibility w i t h respect t o the situation given, especially t o the human beings involved and t h e i r concerns, social handling of the co-existence of competing models, e t c . So f a r as f i r s t approximations t o a methodoZogy of mathematization which includes the social components have been developed (STEINER 1976), I see very interesting material f o r further elaboration of t h i s methodology part i c u l a r l y in the paper by HUMPHREYS and BERKELEY on levels of knowledge representation in decision making b u t also in CROTT and ZUBER's paper on group s h i f t s and polarization. Concerning the e f f e c t on the learner, I find of special i n t e r e s t SHAUGHNESSY's report on a problem and process oriented project-type course p u t into comparison w i t h a traditional course in probability and s t a t i s t i c s . When dealing w i t h some of the $early) KAHNEMAN and TVERSKY tasks students in the small group activity-based course showed less reliance on representativeness and a v a i l a b i l i t y than students in the usual mathematics content oriented and teacher centered approach.

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With respect t o the overall orientation o f a contemporary fornation o f our students through the study o f mathematics and science a t school, I t h i n k i t i s very important t o n o t confine on t r a d i t i o n a l paradigms o f teachina soc a l l e d value-free knowledge and f o l l o w a naive understanding o f s o c i e t a l progress i n applying t h i s knowledge i n technology and various domains o f our modern l i f e . We should r a t h e r develop a c r i t i c a l a t t i t u d e towards these paradigms and make students acquainted w i t h more comprehensive and integrative methods of

thinking (STEINER 1982). Systems theory and studies on decision making methodology and r e l a t e d human behavior as we f i n d them i n a v a r i e t y o f cont r i b u t i o n s t o t h i s conference are important sources from which we hope t o receive substantial o r i e n t a t i o n s f o r t h i s d i f f i c u l t task. These considerations b r i n g us c l o s e r t o the second working component which i s r e l a t e d t o the foundations of didactics of mathematics as a d i s c i p l i n e . One o f the c e n t r a l concepts i n these foundations i s the concept of knowled& mathematical knowledge i n p a r t i c u l a r as a s o c i o - h i s t o r i c a l r e a l i t y . Thus epistemological h i s t o r i c a l s o c i o l o g i c a l and psychological

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studies o f mathematical knowledge i n i t s various contexts and representat i o n s p l a y an important r o l e i n the work o f the I D M (e.g., see BIEHLER 1982, JAHNKE 1978, OTTE & BROMME 1979, PFEIFFER 1981, STEINBRING 1980). Special studies have been devoted t o t h e i n t e r r e l a t i o n s between mathematics as a f i e l d o f organized development a t the u n i v e r s i t i e s o f the 19th century and mathematics as a t o p i c o f school c u r r i c u l a i n t h e same p e r i o d o f time, esp e c i a l l y from a philosophical and epistemological p o i n t o f view (JAHNKE & OTTE 1981, BEKEMEIER e t a l . 1982). Furthermore, the problem o f relating theory to practice i n teacher education and the teacher's professional work i s a problem o f various kinds o f knowledge and t h e i r i n t e r r e l a t i o n s . F i r s t there i s the problem o f coordinating content r e l a t e d knowledge t o pedagogical and psychological knowledge. Moreover, as has become evident i n recent research on teaching, the teacher needs knowZedge about knowledge (meta-knowledge) (see OTTE 1979, IDM-ARBEITSGRUPPE 1981). I n v e s t i g a t i o n s o f these problems suggest t h e usage o f various models, e s p e c i a l l y data processing models, approaches from an a r t i f i c i a l

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i n t e l l i g e n c e s t a n d p o i n t , models f r o m PIAGETian psychology, e t c . Here a r e c l o s e c o n n e c t i o n s w i t h some o f t h e c o n f e r e n c e papers, t h e paper by HUMPHREYS and BERKELEY and t h e paper by R. GRONER, M. GRONER and F. BISCHOF on "The Role of H e u r i s t i c s i n Models o f D e c i s i o n " i n p a r t i c u l a r . A l s o t h e r e a r e r e l a t i o n s t o i n v e s t i g a t i o n s on probZem soZving as a c e n t r a l theme i n mathemat i c a l education. As f o r t h e cZassroom and t h e Zearner s e v e r a l conference papers a r e d i r e c t l y concerned w i t h p u p i l s ' (and o t h e r s u b j e c t s ' ) d i f f i c u l t i e s , misconceptions, e r r o r s , and f a l l a c i e s when d e a l i n g w i t h s t a t i s t i c a l i n f e r e n c e and more gene r a l l y d e c i s i o n making under u n c e r t a i n t y (SCHRAGE, SHAUGHNESSY , GLAESER, BAR-HILLEL). GLAESER i n h i s r e p o r t on a d o c t o r t h e s i s i n v e s t i g a t i n g t h e c o r r e c t and f a l s e i d e a s young p u p i l s may have a c q u i r e d w i t h r e g a r d t o c e r t a i n phenomena o f chance w e l l b e f o r e t h e y began t o s t u d y p r o b a b i l i t y i n t h e classroom c l a i m s t h a t t h i s t y p e o f s t u d y enables us t o s t a t e today t h a t mathematics e d u c a t i o n has become a s c i e n c e . I n o t h e r papers t h e KAHNEMAN and TVERSKY f i n d i n g s , e s p e c i a l l y on base r a t e f a l l a c i e s , p l a y a c e n t r a l r o l e . SHAUGHNESSY i s p r e s e n t i n g a v a r i e t y o f t e a c h i n g s t r a t e g i e s t o improve t h e l e a r n e r s ' u n d e r s t a n d i n g and b e h a v i o r . A l s o many open r e s e a r c h problems have been i d e n t i f i e d . T h i s h o l d s i n p a r t i c u l a r w i t h r e s p e c t t o t h e r e v i e w by SCHOLZ and WALLER o f t h e e x i s t i n g r e s e a r c h on t h e p r o b a b i l i t y concept i n developmental psycho1ow. O f course, t h e cZassroom i s a domain o f socia2 interaction and one i s aware

i n t h e d i d a c t i c s o f mathematics t h a t meaning r e l a t e d t o t h e c o n t e n t o f i n s t r u c t i o n i s i n a fundamental way s o c i a l l y c o n s t i t u t e d ( s e e BAUERSFELD 1982) The t e a c h e r i s h a v i n g a c r u c i a l r o l e i n t h i s i n t e r a c t i o n w h i c h f r o m a s p e c i f i c p o i n t o f v i e w has been d e s c r i b e d as a sequence o f estabzishing and breaking didactica2 contracts between t h e t e a c h e r and i n d i v i d u a l o r groups o f p u p i l s o r a l s o among d i f f e r e n t groups o f p u p i l s , depending on t h e s o c i a l o r g a n i z a t i o n o f t h e t e a c h i n g - l e a r n i n g s i t u a t i o n (BROUSSEAU 1982). I'm s u r e t h a t s o c i a l p s y c h o l o g i c a l models as have been d i s c u s s e d d u r i n g t h i s c o n f e r ence can be used t o a n a l y s e t h e s e c o n t r a c t s i t u a t i o n s and t h e i r dynamics and t h a t i n t h i s way p r o f e s s i o n a l l y i m p o r t a n t knowledge f o r t h e teacher as a decision maker i n t h e classroom can be gained.

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COMMENTS

This f i n a l l y brings us t o the f o u r t h component o f t h e above mentioned R & D a c t i v i t i e s : the mathematics teacher. We have already p o i n t e d o u t the importance o f s p e c i f i c knowledge the teacher needs. I t i s evident t h a t many o f the conference t o p i c s stressed as being o f value f o r the d i d a c t i c s o f mathemati c s i n general should a l s o be considered w i t h respect t o t h e i r significance i n the education o f teachers. I believe t h a t the p u b l i c a t i o n o f the conference r e p o r t w i l l induce processes i n t h e community of mathematical educators which w i l l take up the many impulses we received as p a r t i c i p a n t s .

References Bauersfeld, H. (ed. ) : 1982. Analysen zum Unterrichtshandeln. IDM-Untersuchungen zum Mathematikunterricht. Koln: A u l i s Bekemeier, B. e t a l . (eds.): 1982. Wissenschaft und Bildung irn fruhen 19. Jahrhundert, Vol. I, 11. IDM-Materialien und Studien Vol. 29 and 30. Bielefeld: Universitat Bielefeld

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Eine Untersuchung aus der Biehler, R. : 1982. Explorative Datenanalyse Perspektive e i n e r deskriptiv-empirischen Wissenschaftstheorie. IDM-Mater i a l i e n und Studien Vol. 24. B i e l e f e l d : U n i v e r s i t a t B i e l e f e l d Brousseau, G. : 1982. Tendances o r i g i n a l e s des recherches en didactique des mathkvnatiques en France. Unpublished paper. IREM Bordeaux IDM-Arbeitsgruppe: 1981. Perspektiven f u r d i e Ausbildung des Mathemati klehr e r s . IDM-Untersuchungen zum M a t h e m a t i k h t e r r i c h t . Koln: Aul i s Jahnke, H.N. : 1978. Zum Verhaltnis von Wissensentwicklung und Begrundung i n der Mathemati k Beweisen a l s didaktisches Problem. IDM-Material i e n und Studien Vol. 10. B i e l e f e l d : U n i v e r s i t a t B i e l e f e l d

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Jahnke, H.N. & Otte, M. (eds.): 1981. Epistemological and Social Problems o f the Sciences i n the E a r l y Nineteenth Century. Holland: Dordrecht Otte, M. : 1979. The Education and Professional L i f e o f Mathematics Teachers. I n : B. Christiansen & H.G. S t e i n e r (eds.), New Trends i n Mathematics Teaching. Paris: UNESCO Otte, M. & Bromme, R.: 1978. Der B e g r i f f und das Problem s e i n e r Aneignung. I n : J.R. Block e t a l . (ed. ), Grundlagenkonzepte der Wissenschaftskritik a l s unterrichtsstrukturierende Momente. IPN-Arbeitsbericht Nr. 29. K i e l P f e i f f e r , H. : 1981. Zur sozialen Organisation von Wissen i m Mathematikunterr i c h t . IDM-Materialien und Studien Vol. 21. B i e l e f e l d : U n i v e r s i t a t Bielefeld

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Scholz, R.W. : 1981. Stochastische Problemaufgaben Analysen aus d i d a k t i scher und psychologischer Perspektive. IDM-Materialien und Studien

MATHEMA TlCAL EDUCA TlON

Vol. 23. Bielefeld: Universi tat Bielefeld Steinbring, H. : 1980. Zur Entwicklung des Wahrscheinlichkeitsbegriffs - Das Anwendungsproblem in der Wahrscheinl ichkei tstheorie in didaktischer Sicht. IDM-Material ien und Studien Vol. 18. Bielefeld: Universitat Bielefeld Steiner, H.G. : 19Z6. Zur Methodik des mathematisierenden Unterrichts. In: W. Ddrfler & R. Fischer, Anwendungsorientierte Mathematik i n der Sekundarstufe 11. Klagenfurt Stei ner, H. G. : 1982. Mathemati sch-naturwi ssenschaftl iche Bi ldung. IDMOccasional Paper No. 22. Bielefeld: Universitat Bielefeld

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ADDRESSES

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ADDRESSES

Gisela ALBERS, Dipl.Psych. 4800 B i e l e f e l d , FRG

, Fakultat

423

f u r Soziologie, U n i v e r s i t a t B i e l e f e l d ,

Wulf ALBERS, Prof. Dr. , I n s t i t u t f i r Mathematische Wirtschaftsforschung, U n i v e r s i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG Maya BAR-HILLEL, Dr., The Hebrew U n i v e r s i t y o f Jerusalem, Mount Scopus, Jerusalem, I s r a e l 91905, I s r a e l Dina BERKELEY, Dr., Decision Analysis Unit, The London School o f Economics and P o l i t i c a l Science, Houghton Street, London WC2A 2AE, England L!alter F. BISCHOF, Psychologisches I n s t i t u t , Laupenstr. 4, 3008 Bern, Switzerland K a t r i n BORCHERDING, Dr. L 13, 15-17, FRG

, SFB

24, U n i v e r s i t a t Mannheim, 6800 Mannheim,

Helmut-W. CROTT, Prof. Dr., Psychologisches I n s t i t u t , Albert-Ludwigs-Univ e r s i t a t , Peterhof, 7800 Freiburg i.Br. , FRG Klaus FIEDLER, Dr., Fachbereich Psychologie, U n i v e r s i t a t Giessen, 6300 Giessen, FRG Baruch FISCHHOFF, Prof. 97401, USA

, Decision

Research, 1201 Oak Street, Eugene, Oregon

Georges GLAESER, P r o f . D r . , IREM de Strasbourg, U n i v e r s i t e Louis Pasteur, 10, Rue du G'eneral Zimmer, 67084 Strasbourg, France Bernard GOITEIN, I n s t i t u t e f o r Social Research, U n i v e r s i t y o f Michigan, Ann Arbor, Michigan 48109, USA Marina GRONER, Dr., I n s t i t u t f u r Psychologie der U n i v e r s i t a t Basel , B e r n o u l l i s t r . , 4056 Basel, Switzerland Rudolf GRONER, Prof. Dr., Switzerland

Psychologisches I n s t i t u t , Laupenstr. 4, 3008 Bern,

Gerd v. HARTEN, D r . , I n s t i t u t f u r D i d a k t i k der Mathematik, U n i v e r s i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG P a t r i c k HUMPHREYS, Prof. , Decision Analysis Unit, The London School o f Economics and P o l i t i c a l Science, Houghton Street, London WC2A 2AE, England Helmut JUNGERMANN, Prof. D r . , Psychologisches I n s t i t u t , Technische Univers i t a t B e r l i n , Dovestr. 1-5, 1000 B e r l i n 10, FRG The0 MAY, D r . , Abteilung filr Experimentelle und Angewandte Psychologie, Univers i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG Georg PFLUG, Prof. Dr.,

Fachbereich Mathematik, U n i v e r s i t a t Giessen,

ADDRESSES

424

6300 Giessen, FRG Roland W. SCHOLZ, Dr. , I n s t i t u t f u r D i d a k t i k d e r Mathematik, U n i v e r s i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG Georg SCHRAGE, P r o f . D r . , I n s t i t u t f u r D i d a k t i k d e r Mathematik, U n i v e r s i t a t Dortmund, Emil-Figge-StraRe, Postfach 50 06 00, 4600 Dortmund 50, FRG u l r i c h SCHULZ, Prof. Dr. I A b t e i l u n q f u r E x p e r i m e n t e l l e und Anoewandte Psychol o g i e , U n i v e r s i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG Reinhard SELTEN, P r o f . Dr. , I n s t i t u f . f u r Mathematische Y i r t s c h a f t s f o r s c h u n g , U n i v e r s i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG Z u r SHAPIRA, School o f Business A d m i n i s t r a t i o n , The Hebrew U n i v e r s i t y o f Jerusalem, Jerusalem I s r a e l 91905, I s r a e l Michael SHAUGHNESSY, P r o f . Dr., Department o f Mathematical Sciences, C o l l e g e of Science, San Diego S t a t e U n i v e r s i t y , San Diego CB 92182-0314, USA Heinz STEINBRING, Dr., I n s t i t u t f u r D i d a k t i k d e r Mathematik, U n i v e r s i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG Hans-Georg STEINER, P r o f . D r . , I n s t i t u t f u r D i d a k t i k d e r Mathematik, U n i v e r s i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG Werner H. TACK, P r o f . D r . , Fachbereich Psychologie, U n i v e r s i t a t des Saarlandes, 6600 Saarbrucken, FRG Manfred WALLER, Prof. Dr., A b t e i l ung f u r Experimentel l e und Angewandte Psychol o g i e , U n i v e r s i t a t B i e l e f e l d , 4800 B i e l e f e l d , FRG Thomas S. WALLSTEN, P r o f . Dr., U n i v e r s i t y o f N o r t h C a r o l i n a , Chapel H i l l , Davie H a l l 013A, Chapel H i l l , N.C. 275 14, USA

A l f ZIMMER, P r o f . Dr. , Psychologisches I n s t i t u t , Abt. A l l g e m e i n e Psychologie, Westf. W i l h e l m s - U n i v e r s i t a t M i n s t e r , S c h l a u n s t r . 2, 4400 Munster, FRG Johannes A. ZUBER, Psychologisches I n s t i t u t , A1 bert-Ludwigs-Universitat, P e t e r h o f , 7800 F r e i b u r g i Br. , FRG

.

ABSTRACTS

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Wulf Albers & Gisela Albers: SYSTEM (pp. 271-287)

427

ON THE PROMINENCE STRUCTURE OF THE DECIMAL

This paper gives a f i r s t approach t o a theory o f t h e prominence o f numbers o f the decimal system. Basic components o f the theory a r e the l i m i t e d r a t i o n a l p r i n c i p l e s o f r u l e c o n s t r u c t i o n by i t e r a t e d a d d i t i o n o r s u b t r a c t i o n o f a given amount, t h e r e finement o f a scale by adding t h e means o f any two neighbours, and i t s coarsen i n g by o m i t t i n g a l l uneven elements. Applying these p r i n c i p l e s t o the powers o f 10 ( i . e . the most prominent numbers o f t h e decimal system) various scales w i t h d i f f e r e n t degrees of exactness can be constructed. Empirical observations suggest t h a t among d i f f e r e n t scales w i t h about t h e same exactness generally t h a t one i s p r e f e r r e d which can be constructed i n the easier way, where t h e c o n s t r u c t i o n p r i n c i p l e s a r e those from above and the coarsening procedure i s assumed t o have t h e same complexity as t h e t w o f o l d refinement. This reduces the r e l e v a n t scales t o (z.a*lOn, z 6 Z) w i t h n f Z and ae{1,.5,.25 o r .23. The prominence o f a number o r a s e t o f numbers i s defined as the l e n g t h o f t h e steps o f the coarsest scale c o n t a i n i n g the number o r t h e s e t . (For empirical a p p l i c a t i o n s i t i s suggested t o permit 25 percent exceptions.) Empirical r e s u l t s i n d i c a t e t h a t the prominence o f the s e t o f numbers r e s u l t i n g from a s p e c i f i c decision s i t u a t i o n i s i n many cases about one t e n t h o f t h e numbers i n question. More p r e c i s e l y the prominence ( o r exactness o f scale) o f a s p e c i f i c d e c i s i o n problem seems t o be selected according t o the r u l e t h a t i t i s maximal s u b j e c t t o the c o n d i t i o n t h a t t h e range o f reasonable a l t e r n a t i v e s contains a t l e a s t t h r e e prominent numbers. Results o f n-person bargaining experiments ( w i t h r e a l valued c h a r a c t e r i s t i c f u n c t i o n games) suggest t h a t the evaluation o f outcomes (more exactly; o f payo f f d i f f e r e n c e s ) i s done v i a the underlying prominence, such t h a t t h e distance o f two outcomes i s evaluated by the number o f prominence steps between them. By measureing i n c e n t i v e s i n t h i s way a general theory of t h e p r o b a b i l i t y t h a t a " s o c i a l " c o a l i t i o n (mainly an equal share c o a l i t i o n of a l l players) i s formed could be developed f o r games o f apex type, by which t h e empirical observation t h a t the p r o b a b i l i t i e s can be q u i t e d i f f e r e n t i f the same game i s played on scales w i t h d i f f e r e n t prominence can be explained. T h i s r e s u l t suggests t h a t t h e prominence o f the (spontaneously selected) scale can have basic importance f o r the a n a l y s i s o f i n d i v i d u a l o r group decisions on numerical values. Maya Bar-Hi1 l e l : THE BASE RATE FALLACY CONTROVERSY (pp.39-61) The a r t i c l e attempts t o sketch a conceptual and experimental h i s t o r y o f t h e base r a t e issue. The review d i s t i n g u i s h e s between a s o c i a l judgment paradigm and a textbook paradigm. Theoretical explanations o f the base r a t e phenomenon, i.e. the representativeness h e u r i s t i c s , confusing o r i n v e r t i n g condit i o n a l p r o b a b i l i t i e s , t h e s p e c i f i t y factor, the c a u s a l i t y f a c t o r , t h e v i v i d -

428

ABSTRACTS

ness f a c t o r e t c . a r e d i s c u s s e d w i t h r e s p e c t t o t h e s e paradigms. On t h e b a s i s o f a r e p o r t on v a r i o u s s t u d i e s , t h e a u t h o r h y p o t h e s i z e s t h a t t h e d i f f e r e n t paradigms e l i c i t d i f f e r e n t response tendencies, m a t c h i n g base r a t e s on t h e one hand and j u d g i n g r e p r e s e n t a t i v e n e s s on t h e o t h e r hand. She argues t h a t d i f f e r e n t base r a t e s may be j u s t i f i e d i n one and t h e same problem, depending on t h e causal s t r u c t u r e s accepted i l l u s t r a t i n g t h i s by s e v e r a l examples. The consequences o f t h i s phenomenon f o r t h e r o l e o f normativeness and i t s ( i n - ) determinacy a r e discussed. ( A b s t r a c t e d by R.W. S c h o l z ) Helmut W. C r o t t & Johannes A. Zuber: BIASES I N GROUP D E C I S I O N MAKING: ANALYSIS OF GROUP SHIFTS AND POLARIZATION I N TERMS OF SOCIAL D E C I S I O N SCHEMES (pp. 229-252) The p r e s e n t paper r e p o r t s on t h r e e experiments a b o u t group d e c i s i o n making. The experiments deal w i t h s i t u a t i o n s i n which Ss w i t h d i f f e r i n g i n d i v i d u a l p r e f e r e n c e s have t o r e a c h a common d e c i s i o n . The r e s u l t s a r e f i r s t a n a l y s e d a c c o r d i n g t o t h e c o n v e n t i o n a l methods o f s h i f t and p o l a r i z a t i o n r e s e a r c h . I n a l l t h r e e experiments t h e r e a r e s i g n i f i c a n t s h i f t - a n d / o r p o l a r i z a t i o n e f f e c t s . A f t e r t h e a p p l i c a t i o n o f an a p p r o p r i a t e S o c i a l D e c i s i o n Scheme n o s y s t e m a t i c b i a s e s a r e l e f t unaccounted f o r i n two o f t h e t h r e e e x p e r i m e n t s . The one e x c e p t i o n i s i n t e r p r e t e d as a f u n c t i o n o f i r r e g u l a r i t i e s i n p r e f e r e n c e o r d e r i n g s r a t h e r t h a n a group b i a s . The r e s u l t s demonstrate t h a t t h e d e c i s i o n schemes t h a t t a k e i n t o account p a r t i c u l a r i t i e s i n t h e o b t a i n e d i n d i v i dual p r e f e r e n c e o r d e r i n g s a r e more e f f e c t i v e i n e l i m i n a t i n g s h i f t and p o l a r i z a t i o n e f f e c t s t h a n t h o s e t h a t a r e m e r e l y based on t h e o r e t i c a l assumptions c o n c e r n i n g t h e i n d i v i d u a l p r e f e r e n c e o r d e r i ngs.

Klaus F i e d l e r :

ON THE TESTABILITY OF THE AVAILABILITY HEURISTIC (pp. 105-119)

TVERSKY and KAHNEMAN's a v a i l a b i l i t y h e u r i s t i c , a l t h o u g h o r i g i n a l l y i n t e n d e d t o account f o r o n l y f r e q u e n c y and p r o b a b i l i t y judgments, has been used t o e x p l a i n almost a l l k i n d s o f s o c i a l judgments. A c c o r d i n g l y , t h e process o f judgment f o r m a t i o n i s mediated b y t h e a v a i l a b i l i t y o f memorized i n f o r m a t i o n , t h a t i s , b y t h e ease w i t h which r e l e v a n t m a t e r i a l can be r e c a l l e d a t t h e t i m e when t h e judgment i s made. R e c a l l o p e r a t i o n s e i t h e r pure r e t r i e v a l o r rea r e regarded as t h e d e t e r m i n i n g subprocess w i t h i n t h e constructive r e c a l l process o f judgment f o r m a t i o n . I n t h i s a r t i c l e , e m p i r i c a l e v i d e n c e i s p r e s e n t e d which i s h a r d l y c o m p a t i b l e w i t h such an account. A r e a c t i o n t i m e exp e r i m e n t on t h e e g o c e n t r i c a t t r i b u t i o n phenomenon i s d e s c r i b e d s u g g e s t i n g t h a t S s l c l a i m t o c o n t r i b u t e more t o v a r i o u s s o c i a l a c t i v i t i e s t h a n t h e i r p a r t n e r i s n o t caused by Ss' tendency t o p r e d o m i n a n t l y r e c a l l examples o f t h e i r own a c t i v i t i e s . W i t h i n - j u d g e c o r r e l a t i o n s o f r e c a l l and judgment l a t e n c i e s a r e r a t h e r low. And an a n a l y s i s o f t h e f a c i l i t a t i n g e f f e c t o f p r i o r judgments on subsequent r e c a l l l a t e n c i e s ( f o r t h e same i s s u e s ) does n o t r e v e a l t h e k i n d o f p r i m i n g e f f e c t t h a t would be e x p e c t e d i f r e c a l l opera t i o n s were a l r e a d y i n v o l v e d i n t h e p r e c e d i n g judgments. N e g a t i v e e v i d e n c e f r o m some o t h e r experiments on i l l u s o r y c o r r e l a t i o n s i s a l s o mentioned. These r e s u l t s a r e d i s c u s s e d i n t h e c o n t e x t o f m e t h o d o l o g i c a l problems i n h e r e n t i n t e s t i n g s o - c a l l e d judgmental h e u r i s t i c s .

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ABSTRACTS

Baruch Fischhoff/Bernard G o i t e i n & Zur Shapira: A MODEL OF DECISION-MAKING (PD. 183-207)

SUBJECTIVE EXPECTED UTILITY:

A constant concern o f researchers who ask questions f o r a l i v i n g i s t h a t r e spondents w i l l be influenced by inadvertent aspects o f the contexts i n which t h e i r questions are set. Survey researchers, f o r example, have studied the e f f e c t o f variables such as the race o f the interviewer, the medium used (telephone, mail, i n person) and the order i n which questions are asked. When context e f f e c t s are discovered, two approaches are possible. One i s t o t r e a t them as research a r t i f a c t s t h a t need t o be n e u t r a l i z e d before questions can be meaningfully posed. The second approach i s t o t r e a t context e f f e c t s as f a c t s o f l i f e t h a t can provide useful i n s i g h t s i n t o how people organize t h e i r b e l i e f s and t r a n s l a t e them i n t o answers. Both approaches may be found i n t h e study o f peopleis preferences. Each ref l e c t s and evokes a r a t h e r d i f f e r e n t p i c t u r e o f the nature o f those p r e f e r ences. I n t h i s t a l k , the two approaches w i l l be contrasted, before t u r n i n g a t t e n t i o n t o some recent attempts t o develop t h e o r i e s o f context. These theo r i e s p r e d i c t powerful influences o f seemingly i r r e l e v a n t f a c t o r s on t h e manner i n which people express t h e i r preferences.

Georges Glaeser: THE PERCEPTION OF PROBABILITY SITUATIONS I N PUPILS AGE0 12 TO 14 (pp. 313-323) The p r o b a b i l i t y concept i s n o t simple: I t can be d i s o l v e d i n t o many components and we s h a l l describe some o f them. From the genetical p o i n t o f view, the student does n o t acquire a l l these components simultaneously, b u t r a t h e r crosses one t r e s h o l d o f the other. Hence, he/she always possesses b u t a p a r t i a l understanding. He/she w i l l be able t o answer some questions which are compatible w i t h the tresholds a l ready attained; b u t he/she w i l l systematically f a i l i n o t h e r t e s t s . Besides, t h e mere crossing o f tresholds i s n o t s u f f i c i e n t . F i x a t i o n i s i n d i s p e n s i b l e t o secure the knowledge acquired. Otherwise, the p u p i l can be asked d i v e r t i n g questions which reveal t h e i n s t a b i l i t y o f t h e knowledge acquired. This paper i s a personal view o f a study o f Jesus ALARCON (Mexico), who presented h i s doctoral t h e s i s i n Strasbourg i n Juni, 1982. This research i s n o t i n a genetical perspective, as t h i s would r e q u i r e comparing t h e r e s u l t s o f c h i l d r e n from d i f f e r e n t age groups. The study used approx. 300 12- t o 14-year-old p u p i l s who were d i s t r i b u t e d t o 3 samples o f approx. equal size, i.e., 3 questionnaires were presented w i t h s l i g h t a l t e r a t i o n s o f t h e items t o be tested. The r e s u l t s o f o n l y one o f these samples (106 p u p i l s ) w i l l be discussed here.

430

ABSTRACTS

Rudolf Groner/Marina Groner & Wa l t e r F. B i s c h o f : MODELS OF DECISION (pp. 87-108)

THE ROLE OF HEURISTICS I N

I n t h e i n i t i a l c h a p t e r a r e v i e w on t h e h i s t o r y o f t h e f i e l d o f H e u r i s t i c s i s given . I n c l a s s i c a l a n t i q u i t y , two h e u r i s t i c methods have been used, a n a l y s i s and s y nt h es is . L a t e r , p h i l o s o p h e r s such as Descartes, L e i b n i z , and 601zano i n t r o d u c e d h e u r i s t i c r u l e s and systems, f o l l o w e d by mathematicians l i k e Poincar'e, Hadamard, and Polya. The h i s t o r y o f h e u r i s t i c s i s a l s o c l o s e l y connecte d t o t h e development o f a l g o r i t h m s . I n subsequent c h apt e r s , t h e r o l e o f h e u r i s t i c s i n Psychology and A r t i f i c i a l I n t e l l i g e n c e i s b r i e f l y o u t l i n e d . Fo u r common c h a r a c t e r i s t i c s o f most approaches a r e proposed: (1) R e p r e s e n t a t i o n i n terms o f i n f o r m a t i o n processing, (2) R e s t r i c t i o n o f t h e search space, ( 3 ) L o g i c a l s u f f i c i e n c y , and (4) Computational e f f i c i e n c y . I n t h e f i e l d o f judgment and decishon research, H e r b e r t A. Simon's concepts o f "bounded r a t i o n a l i t y " and " s a t i s f i c i n g " p r o v i d e d a s t a r t i n g p o i n t f o r subsequent a p p l i c a t i o n s o f h e u r i s t i c methods. The h e u r i s t i c s o f represent at i v e n e s s and a v a i l a b i l i t y by Tversky & Kahneman a r e reviewed and discussed, f o l l o w e d by some examples o f i n f o r m a t i o n p r o c e s s i n g models i n d e c i s i o n research.

Gerd v. Ha rt en & Heinz S t e i n b r i n g : RANDOMNESS AND STOCHASTIC INDEPENDENCE ON THE RELATIONSHIP BETWEEN INTUITIVE NOTION AND MATHEMATICAL DEFINITION (pp. 363-373)

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By means o f h i s t o r i c a l i n v e s t i g a t i o n s , e p i s t e m o l o g i c a l r e f l e c t i o n s , and d i d a c t i c a l a n a l y s i s w i t h r e s p e c t t o t h e n o t i o n o f independence, we s h a l l t r y t o p r o v i d e i n s i g h t s i n t o t h e problem o f a t h e o r e t i c a l t e r m and i t s a p p l i c a t i o n s . T h i s w i l l be t h e s t a r t i n g - p o i n t f o r s t a t i n g some d i d a c t i c a l t heses about t r e a t i n g t h e n o t i o n o f independence i n t h e c u r r i c u l u m o f Sekundarstuf e I ( l o w e r secondary l e v e l ) and w i l l y i e l d examples o f t h e i r r e a l i z a t i o n . The d i f f e r e n c e between i n t u i t i v e n o t i o n and mathematical d e f i n i t i o n r e f l e c t s t h e i n s o l u b l e t e n s i o n between mathematics and r e a l i t y . T h i s should n o t be seen as a shortcoming, r a t h e r t h i s t e n s i o n has been one o f t h e p r o d u c t i v e sources f o r t h e development o f mathematics, and i t ought t o be t h e same f o r mathematics i n s t r u c t i o n .

P a t r i c k Humphreys & D i n a B e r k e l e y : PROBLEM STRUCTURING CALCULI AND LEVELS OF KNOWLEDGE REPRESENTATION I N DECISION MAKING (pp. 121-157) Much of t h e ex perime n t a l work on d e c i s i o n making has been based on t h e assumption t h a t a common u n d e r s t a n d i n g of t h e e xperiment al t a s k i s shared by experime nt e r and s u b j e c t . T h i s assumption i s c h a l l e n g e d by t h e e l a b o r a t i o n s s u b j e c t s have been f o u n d t o impose i n d e v e l o p i n g t h e i r underst andinp o f t h e d e c i s i o n problem. Seven t y p e s o f u n c e r t a i n t y a r e i d e n t i f i e d as i n need o f r e s o l u t i o n i n a c c o u n t i n g f o r d i f f e r e n c e s between t hese e l a b o r a t i o n s ; one o f them, pro c edura l u n c e r t a i n t y , i s seen as p o i n t i n g t o t h e need f o r t h e cons t r u c t i o n o f a problem s t r u c t u r i n g c a l c u l u s which w i l l f a c i l i t a t e t h e under-

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431

standing and the aiding of decision making. Previous attempts towards such a construction in terms of taxonomies of tasks by t h e i r analytic o r substant i v e features are discussed and t h e i r drawbacks outlined. A multi-level problem structure generating scheme i s proposed here and our f i r s t moves t o wards i t s definition are outlined. I t i s suggested t h a t any decision problem i s open to qualitatively different conceptualizations determined by the level of abstraction a t which the decision maker i s prepared and allowed t o operate, given the support he or she gets from his or her environment in considering the problem. An outline of the characteristics of the levels i s given; five levels of abstraction are identified and examples of typical properties of decision problems represented a t each level are provided. The paper concludes with a discussion of the implications of t h i s multi-level scheme f o r the study of individual decision making.

Helmut Jungermann: THE TWO CAMPS ON RATIONALITY (pp. 63-86) The idea of rationality i s s t i l l the core of decision theory, although the meaning of the concept has expanded considerably. In the l a s t couple of years, a vehement debate has been led by two camps on the issue of rationali t y - or, in other terms, on the issue of biases, f a u l t s , and errors in human judgment and decision making. h e camp claims that judgment and decision making under uncertainty often show systematic and serious flaws, indicating true deficiencies of the cognitive system. The other camp questions these conclusions and imputes them t o the theoretical and empirical approach taken. After a description of the two camps and t h e i r implicit or e x p l i c i t views on rationality, a resumee and outlook are given, focussing on the positive and productive aspects of the debate. The thesis and the a n t i t h e s i s seem to merge t o a synthesis which includes i n particular the recognition of the conditionality of normative models on assumptions ab o u t the environment and the cognitive representation of t h a t environment. Georg Ch. Pfl ug:

DECISION-THEORETIC PARADOXES (pp. 375-383)

All methods f o r the estimation of unknown parameters o r f o r testing of hypotheses require a twofold modelling process: One model f o r the underlying proba b i l i t y distributions and another model f o r the measurement of the efficiency of s t a t i s t i c a l procedures ( l o s s function, risk function, e t c . ) i s needed. Based on these models, which are usually found in i n t u i t i v e way, the optimal s t a t i s t i c a l procedure can be derived in most cases. A paradox situation a r i s e s i f t h i s procedure does n o t coincide with the intuitively best o r most adequate procedure f o r t h i s situation. Some examples of t h a t kind are collected in t h i s paper. The question i s : Which intuitively found model i s n o t appropriate? Or should WALD's decision theory be revised?

432

ABSTRACTS

Roland W. Scholz & Manfred Waller: CONCEPTUAL AND THEORETICAL ISSUES I N DEVELOPMENTAL RESEARCH ON THE ACQUITISTION OF THE PROBABILITY CONCEPT (pp. 291-311) This a r t i c l e provides a c r i t i c a l review o f psychological theories and research approaches on the ontogenesis of t h e p r o b a b i l i t y concept. An analysis o f t h e conceptualization and i n t e r p r e t a t i o n o f p r o b a b i l i t y w i t h i n developmental r e search reveals t h a t with only one exception an o b j e c t i v i s t i c i n t e r p r e t a t i o n o f p r o b a b i l i t y has been made. The reviewed ( t h e o r e t i c a l ) research approaches, the c o g n i t i v e developmental theory of PrAGET & INHELDER, FISCHBEIN's learning-developmental approach, and verious information processing models d i f f e r i n two main aspects. F i r s t l y on the questton whether t h e development should be considered t o be a continuous o r discontinuous process, and secondly the r o l e o f conceptual versus s t r a t e g i c knowledge f o r coping w i t h p r o b a b i l i t y problems i s disputed. The discussion tri'es t o p o i n t o u t what progress could be gained by skipping a one-sided o b j e c t i v i s t i c i n t e r p r e t a t i o n o f the p r o b a b i l i t y concept and t u r n i n g t o a conceptualization o f p r o b a b i l i t y encompassing both sides, t h e o b j e c t i v i s t i c and s u b j e c t i v i s t i c view. This i n t e g r a t i o n might a l s o lead t o a deeper understandi'ng o f t h e i n d i v i d u a l ' s conceptualization o f uncertainty. Georg Schrage:

(MIS-) INTERPRETATION OF STOCHASTIC MODELS (pp. 351-361)

" S t a t i s t i c s i s a body o f methods f o r making wise decisions i n the face of uncertainty" (W.A. Wallis and H.W. Roberts, The Nature o f S t a t i s t i c s ) . I n r e a l applications these methods are sometimes misused by incompetent "experts". Some examples o f t h i s k i n d from d i f f e r e n t f i e l d s are presented. Misleading stochastic arguments can o f t e n be detected by pure common sense because o f t h e i r consequences. Nevertheless, most people including highly c u l t u r e d people - accept even notorious nonsense i f i t i s presented i n s c i e n t i f i c disguise. This f a c t i n d i c a t e s d e f i c i e n c i e s i n stochastic t h i n k ing. I t should be a c e n t r a l aim o f s t a t i s t i c s education t o make students s e n s i t i v e t o the abuse o f i n s t r u c t i v e examples. An elaborated phenomenology o f frequent e r r o r s i n c l u d i n g misleading s t r a t e g i e s c o u l d be h e l p f u l t o teachers a t a l l educational l e v e l s .

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U l r i c h Schulz & The0 Ma : THE DEPENDANCE OF C O G N I T I O N AND BEHAVIOR I N 2x2 MATRICE GAMES (pp. 253-f70)

A model f o r the i t e r a t e d P r i s o n e r ' s Dilemma Game and the i t e r a t e d Chicken Game i s proposed. The model considers beside choice behavior a n t i c i p a t i o n o f the other p l a y e r ' s choice and s u b j e c t i v e c e r t a i n t y o f t h i s p r e d i c t i o n . Information processing i n the repeated game i s due t o p o s s i b l e changes i n a n t i c i p a t i o n and s u b j e c t i v e c e r t a i n t y . Choices depend on these variables. By experimental v a r i a t i o n o f c e r t a i n pay-off matrices i t i s demonstrated t h a t some o f the model parameters are task dependent.

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433

Michael Shaughnessy: THE PSYCHOLOGY OF INFERENCE AND THE TEACHING OF PROBABILITY AND STATISTICS: TWO S I D E S OF THE SAME COIN? (pp. 325-350)

I n t h i s paper we e x p l o r e t h e i n t e r r e l a t i o n s h i p s o f research i n judgment and d e c i s i o n making w i t h r e s e a r c h i n mathematical e d ucat ion on t h e l e a r n i n g o f p r o b a b i l i t y concepts. The p s y c h o l o g i c a l l i t e r a t u r e demonstrates t h a t people a r e s u b j e c t t o h e u r i s t i c s and b i a s e s when making i n f e r e n c e s o r p r o b a b i l i s t i c estimat e s . The l i t e r a t u r e o f mathematics e d u c a t i on i n d i c a t e s t h a t many people a r e s t a t i s t i c a l l y i l l i t e r a t e . Thus, c e n t r a l m o t i v a t i n g q u e s t i o n s f o r t h e paper a re : Can re s ea r c h i n t h e l e a r n i n g and t e a c h i n g o f p r o b a b i l i t y and s t a t i s t i c s h e l p t h e s t a t i s t i c a l l y n a i v e judge? Can r esearch i n t h e psychology o f i n f e r e n c e h e l p t h e n a i v e s t a t i s t i c i a n ? How can research f rom b o t h t hese d i s c i p l i n e s a i d t h e t e a c h e r o f p r o b a b i l i t y and s t a t i s t i c s ? The paper c o n s i s t s o f t h r e e main p a r t s . P a r t one i n v e s t i g a t e s o b s t a c l e s t o t h e use o f s t a t i s t i c s when making judgments o r i n f e r e n c e s . P a r t two d i s c u s ses some s ugges t io ns f r o m p s y c h o l o g i s t s and f r o m mathematics educators f o r i n c r e a s i n g p e o p l e ' s r e l i a n c e upon s t a t i s t i c s when making i n f e r e n c e s . Suggest i o n s c i t e d f o r t e a c h i n g s t a t i s t i c s so as t o improve p e o p l e ' s i n f e r e n c e and judgmental a b i l i t i e s i n c l u d e : 1) Teach f ro m experiments t o s i m u l a t i o n t o t h e o r y 2 ) Emphasize t h e c o n f o u n d i n g r o l e o f b e l i e f systems i n judgment and t h e objectivity o f statistics 3 ) S e n s i t i z e s t u d e n t s t o misuses o f s t a t i s t i c s i n t h e media 4 ) Give s t u d e n t s c h a l l e n g i n g problems t o model b y s i m u l a t i o n s 5 ) Expose s t u dent s t o S t a t i s t i c a l Paradoxes 6 ) Expose t e ac hers and s t u d e n t s o f p r o b a b i l i t y and s t a t i s t i c s t o r e s e a r c h r e s u l t s i n t h e l i t e r a t u r e o f judgment and d e c i s i o n making. S p e c i f i c examples, i n c l u d i n g sample problems, a r e given. I n p a r t t h re e, s ugg e s t i o n s f o r f u r t h e r r e s e a r c h a r e discussed. Among t h e questions r a i s e d and d i s c u s s e d i n d e t a i l a r e : 1) How much o f t h e r e l i a n c e upon h e u r i s t i c s and biases i s due t o psycholog i c a l f a c t o r s and how much i s due t o i n e x p e r i e n c e w i t h randomness and ig nora nc e o f s t a t i s t i c s 1 p r i n c i p l e s ? Can i n s t r u c t i o n overcome p e o p l e ' s tendency t o i g n o r e base r a t e d a t a o r r e l y upon h e u r i s t i c p r i n c i p l e s ? F o r example, what would be t h e e f f e c t o f p l a c i n g emphasis upon model b u i l d i n g and s i m u l a t i o n i n i n s t r u c t i o n ? 2 ) Is r e l i a n c e upon causal schemas and d e t e r m i n i s t i c reasoning a phenomenon o f western c u l t u r e , o r c o u l d s i m i l a r r e s u l t s be o b t a i n e d i n c r o s s - c u l t u r a l studies? 3)

When groups make'decisions, a r e t h e y s u b j e c t t o t h e same h e u r i s t i c s and bias es as i n d i v i d u a l s ? I f n o t , what a r e t h e i m p l i c a t i o n s f o r making judgments o r f o r t h e t e a c h i n g o f s t a t i s t i c s ?

4) What i s t h e r o l e of t h e t a s k i t s e l f upon t h e judgmental process? I t i s suggested t h a t c o o p e r a t i v e r e s e a r c h e f f o r t s between p s y c h o l o g i s t s and mathematics educ at o r s be conducted i n o r d e r t o f u r t h e r i n v e s t i g a t e t h e s e quest ions .

434

Werner t i . Tack:

ABSTRACTS

CONDITIONS OF VIOLATING INDIVIDUAL RATIONALITY (pp. 211-227)

In the context of conflicts of i n t e r e s t representable by n-person games in normal form, an outcome a i s called individuuZZy rationu2 i f no player can enforce another outcome b which he prefers over a . The idea that any player can enforce certain outcomes by taking appropriate actions i s based on the concept of control; different forms of control lead t o different controZ systems represented as sets of restricted outcome s e t s . Since most empirica research on experimental cooperative side-payment games uses an experimenta paradigm in which an isolated player can enforce only such outcomes which are of no value t o Him, violations of individual rationality are possible only in a very small fraction of a l l relevant experiments. Here - and in some experiments on non-side-payment games, too - t h e i r relative frequencies are in the magnitude order of 2 t o 7 percent only. Despite t h i s f a c t , they cannot be p u t aside as manifestations of otherwise unexplainable f a u l t s and errors, since there i s some evidence t h a t t h e i r probability depends on spec i f i c characteristics of the bargainiing situation. More violations in situations with marked differences between the individually enforcable outcomes in connection w i t h a high correlation between violations of individual rat i o n a l i t y and equal share solutions lead t o the hypothesis that they serve the function of diminishing existing interindividual differences.

Thomas S. Wallsten: (pp. 21-37)

THE THEORETICAL STATUS OF JUDGMENTAL HEURISTICS

Past empirical research on judgmental heuristics and biases has focussed on questions of classification, with relatively l i t t l e attention given t o the development of general theoretical principles. I t i s the l a t t e r , however, t h a t ultimately will lead t o conclusions of greater generality and usefulness. A selected review of the l i t e r a t u r e on representativeness indicates some of the effects f o r which any complete theory must account and some of the limitations in many of the current experimental designs. The manner in which people use probabilistic information depends, in part, on ( i ) information specificity, ( i i ) information salience, ( i i i ) individual differences, and ( i v ) problem wording. Theories are needed t o elucidate these phenomena, and experimental designs are required t h a t pay more attention t o them. Alf C. Zimmer: VERBAL VS. NUMERICALPROCESSINGOF SUBJECTIVE PROBABILITIES (pp. 159-182) Ubiquitously people have t o address the problem of uncertainty. The main focus of t h i s a r t i c l e i s an.investigation of how people represent their knowledge about the uncertainty o f events or regularities, and how they process t h i s knowledge l'n order t o make decisions i n order t o share t h e i r knowledge. In a f i r s t experiment i t i s asked how people gather information about f r e quencies of events and how this knowledge i s interfered w i t h the response mode, numeri'cal vs. verbal estimates, they are required t o use. The l e a s t Interference occurs i f the subjects a r e allowed t o give verbal answers. From t h i s i t i s concluded t h a t processing knowledge about uncertainty by means of

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verbal expression imposes l e s s mental work load on the organism than does numerical processing. Possibility theory i s used as a framework f o r modelling the indivl'dual usage o f verbal categories. The ' e l a s t i c ' constraints on the verbal expressions f o r every single subject are determined i n a further experiment by means of sequential testing. The r e s u l t s from this experiment a r e used t o suggest a simple mechanism underlying the "availability'' heuristic. I n further experiments i t i s shown that the superiority of the verbal processing o f knowledge about uncertainty quite generally reduces persistent biases reported in the 1 iterature: conservatism and negligence of regression. In a final experiment about predictions i n a r e a l - l i f e situation i t turns out t h a t in a numerical forecasting task subjects r e s t r i c t e d themselves t o those parts of t h e i r knowledge which a r e numerical. On the other hand subjects in a verbal forecasting task accessed verbally as well as numerically stated knowledge. The conjecture i s made t h a t the superiority of the verbal mode in representing and i n processing knowledge about uncertainty i s due t o "sharea b i l i t y " (Freyd, i n press)constraints which have evolved in the history of humankind and might even be phylogenetically determined.

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AUTHOR INDEX

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AUTHOR INDEX

Abelson, R.P. 344, 346 Adam, J.L. 97, 105 Agnew, N. 269 A i n s l i e , G. 67, 83 Ajzen, I. 45, 59, 328, 346 Alarcon, J. 14, 314, 315, 319, 321, 322, 429 Albers, G. 11, 17, 271, 287, 427 Albers, W. 11, 17, 220, 226, 271 282, 287, 403, 427 A l l a i s 192 Amarel, S. 88, 105, 152 Anderson, J.R. 130, 152 Anderson, N.H. 22, 34 Anstadt, T. 226 A r a n o f f , D. 270 Arkes, H.R. 48, 59 Arrow, K.J. 231, 251 Asch, S.E. 22, 34 Atkinson, J. 190, 204 Aumann, R.J. 216, 219, 226 Avis, N.E. 27, 36, 48, 60 Baddeley, A.D. 188, 204 B a r - H i l l e l , M. 8, 24, 26, 30, 31, 34, 39, 43-47, 50-52, 54, 59, 60, 122, 152, 327, 328, 343, 346, 347, 417, 427 B a r r i o s , A.A. 22, 34 Barth, W. 255, 268 Barthes, R. 145, 152 Barton, C. 23, 37 Bauersfeld, H. 417 Beach, L.R. 27, 28, 35, 48, 59, 66, 71, 72, 83-85, 195, 203, 205, 207 Begg, I. 170, 181 Bekemeier, B. 416 Bellows, N. 330, 348 Benz, H.J. 344, 347 Berclmann. A. 2 1 Berkeley; 0. 4, 5, 9, 12, 16, 71, 75, 76, 81, 83, 121, 124, 126-129, 151, 153, 308, 415, 417, 430 B e r n o u i l i , D. 121, 153, 159 Berscheid, E. 404, 408 Beyth-Marom, R. 48, 59 B i e h l e r , R. 418 B i r d s a l l , T.G. 123, 156 Bischof, W.F. 9, 83, 102, 106, 417, 430 B i x e n s t i n e , V.E. 255, 268 Black, 0. 231, 234, 241, 250, 251 Bludszuweit, H. 255, 268

B l u n d e l l , H. 255, 268 B l y t h , C.R. 380, 381 Bogarth, R.S. 301, 308 Bolzano, B. 93, 105 Bonczek, R.H. 130, 139, 153, 430 Borcherding, K. 15, 403, 405, 407 Borgida, E. 26, 34, 36, 39, 47, 59, 61, 328, 335, 348 Boring, E.W. 95, 105 Borkan, B. 200, 206 B o x a l l , J. 150, 151, 156 Brainerd, Ch.J. 300, 301, 303, 304, 309 Braunstein, M.L. 203, 206 B r e h l e r , R. 416 Brekke, N. 39, 59 Bromley, D.B. 398 BrOmme, R. 5, 416, 418 Bronner, F. 140, 153 Broussean, G. 417, 418 Brown, A.L. 379, 399 Brown, L. 381 Brown, R.V. 329, 135, 136, 153 Buckley, W.F. 339, 347 Biihler, K. 95, 105 Bunge, M. 134, 153 B u r n e t t , S.A. 129, 154 Bushyhead, J.B. 25, 35 Capon, N. 303, 309 Caramazza, A. 165, 181 Cardano 314 Cardoze, S. 27, 36, 48, 60 Carnap, N. 293 C a r r o l l , J.S. 26, 27, 35, 50, 51, 59, 203, 206, Case, R. 304, 309 C h a i t i n , G.J. 366, 373 Chammah, M.A. 262, 269 Chapman, J.P. 117-119 Chapman, L.J. 117-119 Chi, M.T.H. 33, 35 Chomsky, N. 130, 153, 397, 3Y9 Choo D.T.G. 18 Christensen-Szalanski, J.J.J. 25, 27, 28, 35, 48, 59, 72, 83 Claparede, E. 98, 105 Clark, A. 2 1 Cohen, L.J. 22, 25, 35, 54-56, 59, 63, 83 Cole, M. 399 Coombs, C.H. 191, 204, 241, 251 Corbin, R.M. 202, 206 Corrigan, B. 188, 204 Couturat, L. 89, 91, 105

439

440

AUTHOR INDEX

Cox, J.R. 33, 35 C r a n d a l l , R. 26, 36, 47, 61, 328,

335, 348 Crossniann, H.J. 287 C r o t t , H.W. 11, 229, 234-236, 239

242, 251, 403, 405, 415, 428 Cyert, R.M. 411, 412 D’Al einbert 314 Davis, J.H. 231, 251 Dawes, R.M. 188, 191, 203, 204, 204 DeBono, E. 97, 105 D e F i n e t t i , B. 159, 181, 292, 309 DeGroot, A. 96, 106 DeHoog, R. 140, 153 Delaney, H.D. 22, 35 DeLoache, J.S. 399 Dershowitz, A. 40, 59 Descartes, R. 88, 89, 105, 430 Deutsch, M. 255, 262, 268, 404, 407 Dewey, J. 98, 106 Dinges, H. 13, 16, 368, 373 Dorner, D. 70, 83 Doherty, M.E. 22, 36 Doumouchel, W.H. 343, 347 Dovalina, I. 27, 36, 48, 60 Dubois, D. 165, 181 Duncker, K. 96, 97, 106 Duran, A.S. 34, 36 D u s o i r , A.E. 29, 30, 35 Eddy, D.M. 40, 44, 60 Edgeworth, Y. 376-378, 381 Edwards, W. 71, 83, 127, 135, 151,

153, 177, 178, 182, 192, 204 23, 35, 63, 71, 72, 76, 80-83, 104-106, 191, 194, 204, 205 E l l s b e r g 192 E l s t e r , J. 73, 81, 83, 127, 153 Emory, F. 287 Estes. W.K. 49, 60 Evans, J.St.B.T. 29, 30, 35, 80, 84 F a l k . R. 332. 337, 343. 347 Feather, N.T; 79,-84, i84, 190, 203, 204 Feigenbaum, E. 99, 106 Feldman, J. 150, 154 F e l t o v i c h , P.J. 33, 35 Fermat 314, 357, 358 F e r r e l l , W.R. 181 F e s t i n g e r , L. 237, 251 Feynman, R.P. 367, 373 F i e d l e r , K. 10, 23, 109, 428 Fillenbaum, S. 21 F i l l m o r e , C.J. 130, 154 Einhorn, H.J.

Fine, T.L. 293, 309, 365, 373 F i s c h b e i n , E. 299, 303, 304-306,

309, 331, 332, 345, 347, 353, 432 F i s c h h o f f , B. 10, 15, 27, 33, 35, 50-54, 58-60, 79, 84, 125, 135, 154, 155, 162, 171, 173, 181-183, 194-196, 199, 200, 205, 206, 326, 328, 347, 349, 387, 429 F i t t i n g , U. 269 Flood, M.M. 295, 309 Foa, E.B. 404, 407, 408 Foa, U.G. 404, 407, 408 F o l t z , A.M. 40, 60 Fong, G.T. 30, 36 Ford, B. 335 F o r s y t h , B. 21 Freedman, D. 40, 60 Freksa, C. 166, 181 F r e u d e n t h a l , H. 305, 309 Freyd, J.J. 393, 394, 435 F r i n d t e , F. 255, 268 Gabor, A. 287 tiahagan, J.P. 255, 270 Garey, M.R. 92, 93, 106 Gasparski, W. 81, 84 Gibson, J.J. 391, 394 Gibson, R.O. 132, 155 Ginosar, 2. 42, 60 Ginsberg, R. 200, 206 Ginzberg, E. 287 Glaeser, G. 14, 313, 417, 42 Glaser, R. 33, 35 G o i t e i n , 8. 10, 79, 84, 183, 429 Goldberg, L.R. 184, 187, 203 tioldman, H.J. 262, 269 Goodnow, J.J. 296, 309 Goodwin, K.S. 299, 309 Goulet, L.R. 299, 309 Grange, J.W.J. 287 Green, D.R. 309, 331, 332, 336, 345, 347, 353, 364, 373 Gregory, R.L. 179, 181, 392, 394 Grether, D.M. 200, 206 Griggs, R.A. 33, 35 Groner, M. 9, 87, 98, 102, 105, 106. 417. 430 Groner; R. 23, 87, 98, 102, 105, 106, 417, 430 Gustafson. D.H. 140. 155 Gyorgi, A: 148, 154G6del , K. 92, 106 Hacking, I. 159, 181, 292, 309

4,

AUTHOR INDEX

Hadamard, J. 98, 106, 430 Halpin, S.M. 255, 268 Hankel, H. 88, 106 Harten v., G. 15, 337, 363, 414, 430 Hassebrauck, M. 404, 407 Hassebrock, F. 34, 36 Hausmann, L. 68, 84, 146, 155 Hayes, J.R. 33, 35 Heerboth, J. 68, 8 5 Helmholtz v., H. 391, 394 Helson, H. 192, 204 Heneman, H.G. 201, 206 Herbrand 92 Hersh, H.M. 165, 181 Hesse, W. 255, 269 Higgins, E.T. 112, 119 Hintzman, U.L. 163, 181 Hoemann, H.W. 303, 305-307, 309 Horz, H. 294, 310 188, 204, 237, 251 Hoffmann, P.J. Hogarth, R.M. 4, 15, 23, 35, 63, 71-76, 80-84, 104, 106, 129, 154, 194, 205, Holmes, P. 336, 347 Holsapple, C.W. 130, 139, 153 Homans, G.C. 403, 407 Hovland, C . I . 126, 154 Howell, W.C. 129, 154 Huber, 0. 104-106 Hubert, S. 22, 34 H u f f , 0. 39, 60, 335, 338, 348 Humphrey, G. 95, 106 Humphreys, P.C. 4, 5, 9, 12, 16, 71, 75, 76, 81, 83, 121, 125-129, 137, 140, 141, 146, 148, 151, 153-155, 308, 415, 417, 430 I l l i n g w o r t h , J. 269 I n h e l d e r , 6. 13, 17, 291, 294, 297-300, 303, 305-307, 310, 432 Irmscher, M. 163, 181 I r w i n , F.W. 22, 35, 36 Isaac, D.J. 155 I s r a e l , R.B. 381 Jacques, E. 134, 137, 147, 155 Jahnke, H.N. 416, 418 James, W. 381 Janis, I.L. 65, 68, 69, 72, 82, 84, 202, 206 Jaques, E. 132 Jean, R.St. 405, 407 Jepson, C. 29, 36 Johnson, D.M. 123, 155

Johnson, D.S. 92, 93, 106 Johnson, E.J. 72, 84 Johnson, E.M. 140, 155 Johnson, P.E. 34, 36 Johnson-Laird, P.N. 33, 37 Jonas, K. 255, 261, 269 Josephson, J. 296, 297, 310, 398 Jsaac, D.J. 132 Jungermann, H. 3, 9, 63, 68, 84, 146, 155, 431 Kac, M. 364, 373 Kahan, J.P. 220, 227 Kahneman, D. 4, 8, 12, 17, 18, 22-24, 26, 28-32, 36, 37, 39-45, 49, 51, 53, 57, 59-61, 63, 65-68, 74-76, 78, 80-82, 84, 85, 103, 104, 107-109, 118, 119, 122, 123, 125, 147, 150, 152, 155-157, 170, 178, 179, 181, 192, 196, 198, 204-206, 306, 308, 310, 325, 327, 329, 332-334, 345, 348, 350, 382, 391, 394, 397, 398, 415, 417, 428, 430 Kahr, A.S. 136, 153 Kamen, J.M. 287 Kassin, S.M. 39, 60 Katz, N. 200, 206 Kaufmann, A. 181 Kaufmann, M.A. 165, 212, 213, 220, 226 Kauke, M. 255, 268 Kayser, E. 404, 407 Keenev. R,L 403, 407 K e l l G ; H.H. 109, 113, 117, 119, 262. 268 Kelsey, J.L. 40, 6 0 Keynes, J.M. 373 King, G. 112, 119 K i r c h e r , A. 90, 107 K i s t n e r , K. 405, 407 Kleene 92 Kleinmuntz, 8. 105, 106 Kleinmuntz, D.N. 105, 106 Kneppreth, N.P. 140, 155 K n o l l , C. 234, 239, 251 Kobasigawa, A. 304, 310 Kohler, W. 96, 107 Kogan, N. 405, 407 Kohlberg, L. 397, 399 Kolmogorow 314 Konold, C. 329, 350 Krantz, D.H. 29, 30, 36, 125, 155, 343, 347

44 1

442

AUTHOR INDEX

Kreuzig, H.W. 83 K r i v o h l a v y , J. 255, 269 Kruskal, W.H. 348 Kuhn, D. 303, 309 Kuhn, T.S. 329, 348 Kunreuther, H. 200, 201, 206 Kyburg, H.E. 195, 205 La Brecque, M. 334, 348 Laing, J.D. 220, 226 Lakatos, I. 203, 206 Lamm, H. 230, 252 Laplace 159, 292, 314 Larichev, O.J. 154 Lawrence, D.R. 237, 251 Leach, E.R. 145, 155 L e i b n i z , G.W. 88-91, 107, 430 L e i f e r , R.P. 140, 155 Lenat, D.B. 101, 102, 107 Leventhal, G.S. 404, 407 Levy-Bruhl 13 Lichtenberger, J. 212, 219, 222, 226 L i c h t e n s t e i n , S. 27, 33, 35, 53, 58, 60, 135, 155, 162, 171, 173, 181, 182, 187, 188, 194-196, 199, 203, 205, 206, 326, 328, 334, 347, 349 Lietzmann, W. 7, 1 7 L i n d l e y , D.V. 135, 153, 159, 182, 375. 381 Lindman, H. 127, 135, 153 L i n g l e , J.H. 117, 119 L i n k , R.F. 348 L i v e s l e y , W.J. 398 Loomis, J.L. 255, 269 Lopes, L.L. 392, 394 Luce, R.D. 125, 155, 263, 264, 269 L u l l u s , R. 88, 90, 93, 100, 107 Lykken, D.T. 40, 60 Lyon, 0. 54, 60, 328, 348 MacCrimmon, K.R. 406 Mai, N. 269 Manis, M. 27, 36, 48, 51, 52, 60 Mann, L. 65, 68, 69, 72, 82, 84, 202, 206 Manzat, J. 331, 347 March, I.G. 411, 412 March, J.G. 67, 74, 82, 84, 193, 2 04 Markov 92 May, T. 11, 174, 254-256, 259-262, 403, 432 Mayer, A. 95, 107 M c A l l i s t e r , D.W. 72, 84

McCauley 44 McClintock, C.G. 262, 269 139, 155 McCosh, A.M. McCrimmon, K.R. 262, 269 McFadden, W. 140, 154 McGoey, D.J. 181 McPhillamy, D.J. 199, 206 Meehl, P.E. 40, 6 1 Mery, J.L. 129, 154 Messick, D.M. 262, 269, 406, 407 Messik, S.J. 295 Michaud, C. 129, 154 Mikula, G. 404, 407 M i l l e r , D.W. 72, 85 M i l l e r , G.A. 11, 17, 187, 203 M i l l e r , L. 200, 206 M i l l e r , N.E. 199, 206 Minsky, M. 102 Mises v., R. 292, 310 M i t c h e l l , T.H. 71, 72, 83, 84, 193, 203, 205, 207 M o l l e r , J.H. 34, 36 Monroe, K.B. 287 Montgomery, H. 72, 85 Morgenstern, 0. 10, 356 Morrison, R.J. 220, 226 Moscovici, S. 230, 252 M o s t e l l e r , F. 339, 348 M u l l e r , G.E. 96, 107 Mulholland, J. 392, 394 Murphy, A. 196, 206 Myers, D.G. 230, 231, 252 22, 36 Mynatt, C.R. Nelson, R.R. 411, 412 Neumann v., J. 10, 356 Newell, A. 94, 98-100, 107, 123, 156, 330, 348 Ng K.O. 18 Niessner, M. 226 N i l s s o n , N.J. 101, 107 N i s b e t t , R.E. 12, 17, 26, 29, 30, 34, 36, 47, 61, 63, 65, 85, 123, 156, 165, 166, 182, 326, 328-330, 333-335, 339, 343, 346, 348, 349, 391, 394 N o e l t i n g , G. 304, 310 Nozawa, G. 163, 181 Oesterwalbesloh, K.W. 287 O l i a n - G o t t l i e b , J.U. 201, 206 Olson, C.L. 31, 36 Orth, J. 95, 107 Ostrom, T.M. 117, 119 Otte, M. 416, 418 Oxenfeldt, A.R. 287

AUTHOR INDEX

Pampu, J. 331, 347 P a r k e s , E.M. 127, 156 Pascal 314 Payne, J.S. 203, 206 Payne, J.W. 67, 72, 83, 85, 104, 105, 107 P e e v e r s , B.H. 398 P e i t e r s , R.S. 348 P e l e g , B. 216, 219, 226 P e t e r s o n , C.R. 48, 66, 85, 136, 153, 195, 205 P f e f f e r , J. 193, 205 P f e i f f e r , H. 416, 418 P f l u g , G . 15, 375, 381, 431 P h i l l i p s , L.D. 13, 17, 18, 63, 71, 75, 85, 123, 127, 129, 135, 136, 141, 150, 151, 153, 155, 156, 161, 162, 173, 177, 178, 181, 182, 195, 205 P i a g e t , J. 13, 17, 131, 135-137, 156, 291, 294, 297-300, 303, 305-307, 31U, 331, 397, 3Y9, 400, 432 P i l i s u k , M. 255, 268 P i s a n i , R. 40, 6 0 P i t z , G . F . 68, 79, 85 P l a t t , C.R. 200, 206 P o i n c a r g , H. 98, 107, 43 P o l l a t s e k , A. 329, 350 P o l y a , G . 94, 97, 107, 108, 43 P r a d e , H. 165, 181 Purves, R. 40, 60 P y l y s h y n , Z. 255, 269, 391, 394 Q u i n n , R.P. 196, 205 Rade, L. 339, 349 R a i f f a , H. 403, 407 Rapoport, A. 220, 227 Rapoport, An. 262, 269 Rasch, G . 174, 182 Reed, H. 26, 36, 47, 61, 328, 335, 3 48 R e i n h a r d t , H.E. 342, 343, 349 R e i t h e r , F . 83 Rembde, M. 287 Rescher, N. 393, 395 R e v e n s t o r f f , D. 255, 269 R i k e r , W.H. 220, 227 R i o s , M. 150, 151, 157 R i s i n g , G.R. 348 R o b e r t s , H.W. 351, 43 Konen, J. 73, 85 Rosen, A. 40, 6 1 Rosen, L.D. 105, 108 R O S S , B. 303, 305, 307, 30Y, 310

R O S S , L. 12, 17, 63, 65, 85, 123, 156, 165, 182, 296, 297, 310, 326, 329, 334, 335, 343, 349, 391, 394, 398, 399 R O S S , M. 109, 117, 119, 349 R o s t , H. 13, 16 R U S S O , J.E. 105, 108 Sachs, N.J. 68, 85 S a g i , P. 200, 206 S a l a n c i k , G. 193, 205 S a n d l e r , A.M. 148, 156 S a n d l e r , J. 148, 156 Sauer, C. 405, 407 Sauermann, H. 11, 17, 287, 411, 412 Savage, L.J. 127, 146, 156, 159, 176, 182, 192 S c a r d a m a l i a , M. 300, 303, 310 S c h a e f e r , H.J. 226 S c h a e f e r , R.E. 256, 269, 405, 408 S c h e l l i n g , T.C. 287 Schermer, Th. 234, 239, 242, 251 S c h l e n k e r , B.R. 262, 269 Schmitz, G . 287 S c h o e n f e l d , A. 330, 349 SChOlz, R.W. 3, 5, 9, 11-13, 17, 21, 33, 36, 291, 344, 349, 375, 381, 413, 418, 428, 432 Schrage, G. 14, 351, 361, 414, 417, 432 Schictt, K.P. 176, 182 SChulZ, U. 11, 254-256, 259-262, 269, 403, 432 Schwab, D.P. 201, 206 Schwinger, T. 404, 407, 408 S c o t t Morton, M.S. 139, 155 S c r i b n e r , S. 399 Secord, P.F. 398 S e l t e n , R. 11, 15, 17, 287, 409, 411, 412 S e l z , 0. 95, 96, 108 Sermat, V. 255, 262, 269, 270 S h a f e r , G. 25, 36, 381 S h a k l e e , H. 397, 399 S h a p i r a , Z. 10, 79, 84, 183, 429 S h a p i r o , U. 33, 37 Shaughnessy, J.M. 14, 325, 327, 333, 337, 345, 349, 353, 354, 360, 361, 415, 417, 433 Shaw, J.C. 99, 107 Shubik, M. 3, 17 Shugan, S.M. 72, 85 S h u l t e , A. 336, 349 S i c o l y , F . 109, 117, 119, 349 S i e g l e r , R.S. 26, 27, 35, 50, 51,

443

444

AUTHOR INDEX

59, 301-304, 311 4, 11, 17, 33, 35, 36, 65, 81, 85, 98, 99, 102, 103, 107, 108, 123, 156, 187, 192, 203, 287, 330, 348, 411, 412, 430 S i x , U. 405, 408 Sjoberg, L. 81, 85 S l o v i c , P. 27, 33, 35, 53, 54, 58, 60, 65, 66, 77, 84, 85, 123, 155, 159, 171, 182, 187, 188, 192, 194, 195, 199, 200, 203-206, 326-328, 334, 346-349, 391, 394, 397 Smokler, H.E. 195, 205 Snodgrass, J. 22, 36 S o l l e y , C.M. 295 Solomon, L. 255, 270 Sommer, L. 153 Sowter, A.P. 287 S r u l l , T.K. 112, 119 S t a u d e l , T. 83 S t a i n e s , G.L. 196, 205 Standel, F. 84 S t a r r , M.K. 72, 85 Staw, B.M. 201, 206 S t e e l e , M.W. 270 S t e i n , C. 375, 378, 382 S t e i n b r i n g , H i 15,-337, 363, 414, 416. 419. 430 S t e i n e k , H.G. 15, 413, 415, 416, 419 S t i e g l e r , S. 382 S t i t t 44 Stoner, J.A.F. 229, 230, 252, 408 Suppes, P. 125, 155, 159, 182, 263, 264. 269 Svenson, 0. 72, 85, 150, 154, 156, 191, 203, 204 Swets. J.A. 123. 156 Tack,-W.H. 11, ill-213, 219, 220, 226, 403, 406, 434 Tan, I. 18 Tanner, W.P. 123, 156 T a y l o r , S.E. 47, 48, 327, 349 Tedeschi, J.T. 255, 270 Terhune, K.W. 255, 270 T h i b a u l t , J.W. 262. 268 Thoene, B. 234, 234, 251 Thompson, S.C. 47, 48, 109, 113, 117, 119 Thorngate, W. 123, 156 Thraem, S. 226 T i e t z , R. 11, 17, 287 Simon, H.A.

Toda, M. 134, 146, 156 Toman, R.J. 287 Trope, Y. 42, 60 T u r i n g 92 Turner, D. 336, 347 T u r o f f , M. 12, 17 Tversky, A. 4, 8, 11, 12, 17, 18,

22-24, 26, 28-32, 36, 37, 39-45, 49, 51, 53, 57, 59-61, 63, 65-68, 74-76, 78, 80-82, 84, 85, 103, 104, 107-109, 118, 119, 122, 123, 125, 147, 150, 152, 155-157, 159, 170, 178, 179, 181, 182, 191, 192, 196, 198, 204, 206, 306, 308, 325, 327, 329, 332-334, 345, 348, 350, 382, 391, 392, 394, 397, 398, 415, 417, 428, 430 Tweney, R.D. 22, 36 U l a r d t v., I. 68, 84, 146, 155 U l v i l a , J.W. 129, 153 V a i h i n g e r , H. 392, 395 V a r i , A. 127, 154, 157 Vecsenyi, J. 127, 154, 157 Venkatesan, M. 287 Waerden van d e r , B.L. 94, 98, 108 Wagner, C.H. 342, 343, 350 Wald, A. 375, 376, 380, 382, 431 Wallach, M.A. 405, 407 Wallas, G. 98, 108 W a l l e r , M. 13, 15, 291, 397, 432 W a l l i s , W.A. 351, 432 W a l l s t e n , ti. 21 W a l l s t e n , H. 21 W a l l s t e n , T.S. 8, 9, 18, 21-23, 25, 35. 37. 167. 434 Wallster,-G.W.~404, 408 Walster, E. 404, 408 Wason, P.C. 33, 37 Watson, S.R. 148, 157 Wegscheider, R. 269 Weinstein, N.D. 190, 204 Weir, M.W. 296, 306, 311 W e l l , A.D. 329, 330, 345, 350 Wertheimer, M. 96, 108 Whalley, P.C. 18 Whinston, A.B. 130, 139, 153 White, S.E. 1Y3, 205 Wickelgren, W.A. 97, 108 Wilks, S.S. 188, 204 Wilson, T.U. 166, 182, 330, 348 W i n k l e r , R.L. 196, 206, 256, 270 Winter, H. 352, 361 Winter, S.G. 411, 412

AUTHOR INDEX

W i n t e r f e l d t v . , 0. 129, 130, 148, 151, 157 Wisudha, A. 18, 137, 154 W i t t e , W. 166, 167, 182 W i t t g e n s t e i n , L. 182 W o l f f , (i. 7, 18 Wooler, S. 141, 155 W r i g h t , G.N. 13, 17, 161, 182 W r i g h t , P. 188, 204 Wyer, R.S. 112, 119, 196, 205 Zadeh, L.A. 161, 182, 393, 395 Z a v a l l o n i , M. 230, 252 Zimmer, A.C. 10, 15, 135, 157, 159, 165, 166, 170, 181, 182, 391, 393, 434 Zuber, J.A. 11, 234, 242, 251, 403, 405, 415, 428 Zwicky, F. 94, 108

445

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