Decision Modeling and Behavior in Complex and Uncertain Environments

DECISION MODELING AND BEHAVIOR IN COMPLEX AND UNCERTAIN ENVIRONMENTS

Springer Optimization and Its Applications VOLUME 21 Managing Editor Panos M. Pardalos (University of Florida) Editor—Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University)

Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences. The series Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multiobjective programming, description of software packages, approximation techniques and heuristic approaches.

DECISION MODELING AND BEHAVIOR IN COMPLEX AND UNCERTAIN ENVIRONMENTS Edited By TAMAR KUGLER University of Arizona, Tucson, AZ J. COLE SMITH University of Florida, Florida, FL TERRY CONNOLLY University of Arizona, Tucson, AZ YOUNG-JUN SON University of Arizona, Tucson, AZ

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Editors Tamar Kugler University of Arizona Management and Organizations Eller College Tucson AZ 85721 [email protected]

Terry Connolly University of Arizona Eller College of Management 1130 E. Helen St. Tucson AZ 85721 430Z McClelland Hall [email protected]

J. Cole Smith University of Florida Dept. Industrial & Systems Engineering 303 Weil Hall P.O. Box 116595 Gainesville FL 32611-6595 [email protected]

Young-Jun Son University of Arizona College of Engineering Dept. Systems & Industrial Engineering 1127 E. James E. Rogers Way P.O. Box 210020 Tucson AZ 85721-0020 [email protected]

Managing Editor: Panos M. Pardalos Department of Industrial and Systems Engineering University of Florida Gainesville, FL 32611 [email protected] @

Editor/Combinatorial Optimization. Ding-Zhu Du Department of Computer Science and Engineering University of Texas at Dallas RRichardson, TX 75083 [email protected]

ISSN: 1931-6828 ISBN: 978-0-387-77130-4 DOI: 10.1007/978-0-387-77131-1

e-ISBN: 978-0-387-77131-1

Library of Congress Control Number: 2008926556 Mathematics Subject Classification (2000): 91A30, 91B16, 91A35, 62Cxx, 91B06, 90B50, 91A80, 91E10, 91E40, 68T05, 91E45 © 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

To our spouses and families, who patiently tolerated the inevitable demands that the preparation of this book made on time that was rightfully theirs.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Part I Integrating Decision Analysis and Behavioral Models Improving and Measuring the Effectiveness of Decision Analysis: Linking Decision Analysis and Behavioral Decision Research Robert T. Clemen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Reducing Perceptual and Cognitive Challenges in Making Decisions with Models Jenna L. Marquard and Stephen M. Robinson . . . . . . . . . . . . . . . . . . . . . . . 33 Agricultural Decision Making in the Argentine Pampas: Modeling the Interaction between Uncertain and Complex Environments and Heterogeneous and Complex Decision Makers Guillermo Podest´ a, Elke U. Weber, Carlos Laciana, Federico Bert, and David Letson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Part II Innovations in Behavioral Model Analysis On Optimal Satisficing: How Simple Policies Can Achieve Excellent Results J. Neil Bearden and Terry Connolly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 There Are Many Models of Transitive Preference: A Tutorial Review and Current Perspective Michel Regenwetter and Clintin P. Davis-Stober . . . . . . . . . . . . . . . . . . . . . 99

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Integrating Compensatory and Noncompensatory Decision-Making Strategies in Dynamic Task Environments Ling Rothrock and Jing Yin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Repeated Prisoner’s Dilemma and Battle of Sexes Games: A Simulation Study Jijun Zhao, Ferenc Szidarovszky, and Miklos N. Szilagyi . . . . . . . . . . . . . . 143

Part III Innovations in Descriptive Behavior Models Individual Values and Social Goals in Environmental Decision Making David H. Krantz, Nicole Peterson, Poonam Arora, Kerry Milch, and Ben Orlove . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Does More Money Buy You More Happiness? Manel Baucells and Rakesh K. Sarin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Cognitive Biases Affect the Acceptance of Tradeoff Studies Eric D. Smith, Massimo Piatelli-Palmarini, and A. Terry Bahill . . . . . . . 227

Part IV Experimental Studies in Behavioral Research Valuation of Vague Prospects with Mixed Outcomes David V. Budescu and Sara Templin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Collective Search in Concrete and Abstract Spaces Robert L. Goldstone, Michael E. Roberts, Winter Mason, and Todd Gureckis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Braess Paradox in the Laboratory: Experimental Study of Route Choice in Traffic Networks with Asymmetric Costs Amnon Rapoport, Tamar Kugler, Subhasish Dugar, and Eyran J. Gisches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Non-Euclidean Traveling Salesman Problem John Saalweachter and Zygmunt Pizlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Preface

On February 27 and 28 of 2006, the University of Arizona held a workshop entitled, “Decision Modeling and Behavior in Uncertain and Complex Environments,” sponsored by the Air Force Office of Scientific Research under a Multidisciplinary University Research Initiative (MURI) grant. The purpose of the workshop was to assemble preeminent researchers studying problems at the interface of behavioral and cognitive science, decision analysis, and operations research. This book is a compilation of 14 chapters based on the presentations given during the workshop. These contributions are grouped into four general areas, which describe in some detail the challenges in conducting novel research in this field. Part One is concerned with the need for integrating decision analysis and behavioral models. Robert T. Clemen discusses how the fields of behavioral research and decision analysis have diverged over time, and makes a compelling case to establish new links between the disciplines. He recommends leveraging lessons learned from behavioral studies within prescriptive decision analysis studies and evaluating the practical impact of those prescriptive techniques in helping decision makers achieve their objectives. Jenna L. Marquard and Stephen M. Robinson address eleven common “traps” that face decision model analysts and users. An understanding of these traps leads to an understanding of modeling features that either help or hurt the decision-making process. The authors link theory and practice by examining a set of case studies across a diverse array of model scenarios, and provide a checklist of recommendations for analysts confronted by these eleven traps. Guillermo Podest´ a, Elke U. Weber, Carlos Laciana, Federico Bert, and David Letson present a promising illustration of the use of behavioral analysis in prescriptive modeling. The application under consideration is agricultural production in the Argentine pampas region. At issue is the allocation of land for different crops in the region, where the goals to be maximized by the farmers are modeled by expected utility and prospect theories. The authors demonstrate that the optimal actions for landowners and tenants can differ

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significantly according to the objective being modeled and the parameters used for the objective. Part Two of this book examines innovations in behavioral model analysis; particularly with respect to mathematical analyses of models. Neil Bearden and Terry Connolly consider a decision maker who is restricted to analyzing sequential decision problems via “satisficing,” which is a type of thresholdbased decision making that many humans employ in complex scenarios. The chapter indicates that humans who exhibit satisficing behavior in the best possible manner can often perform well in the sense that their decisions are near-optimal. This result is in contradiction to the faulty conventional wisdom that satisficing decision makers are inherently significantly suboptimal. Decision makers are also known to exhibit nonintuitive behavior regarding the transitivity of preference among options. Michel Regenwetter and Clintin P. Davis-Stober examine complex mathematical models that better explain the transitivity (and intransitivity) of preferences. The authors pay particular attention to resolving the difficulties encountered by examining logical transitivity axioms by experimental choice data. Ling Rothrock and Jing Yin summarize an analysis of Brunswik’s lens model for compensatory and noncompensatory decision-making behaviors. The authors discuss a traditional lens model for compensatory behaviors and a rule-based variation for noncompensatory decision making. The classical prisoners’ dilemma and battle of sexes games receive a new analysis from a game theoretical/simulation perspective. Jijun Zhao, Ferenc Szidarovszky, and Miklos N. Szilagyi examine the influence of multiple agents acting according to various profiles in these games, taking into account media influences. Their agent-based simulation approach leads to insights regarding the impact of individual agent personality on overall societal behavior under these games. Part Three of this volume offers new innovations in descriptive behavior models. David H. Krantz, Nicole Peterson, Poonam Arora, Kerry Milch, and Ben Orlove observe that contemporary models that do not account for the presence of “social goals” cannot accurately model certain observed behavioral phenomena. The authors establish the importance of social goals as an element of rational decision making. The authors identify environmental decision-making problems as a key application in which social goals are a significant decision component, an assertion further supported by the Podest´a et al. study developed in Part One. Manel Baucells and Rakesh K. Sarin examine the behavioral characteristics that explain why more money does not in practice reliably buy more happiness. The authors assert that “projection bias,” which is the tendency of people to project into the future their current reference levels, causes people to overestimate the value of additional money and misallocate their budgets. Biases are also identified by Eric D. Smith, Massimo Piatelli-Palmarini, and A. Terry Bahill, as a primary reason why people tend to reject tradeoff studies. The authors demonstrate that tradeoff studies are susceptible to

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human cognitive biases and provide examples of biases that produce variability in decisions made via tradeoff studies. Similar to the purpose of the Marquard and Robinson paper in Part One, a knowledge of these biases permits decision makers to improve their understanding of, and confidence in, tradeoff studies. Part Four of the text focuses on experimental studies in behavioral research. David V. Budescu and Sara Templin examine how humans assess vague prospects having mixed outcomes; that is, prospects having vague probabilities and a risky outcome that could result in either a gain or a loss (as opposed to traditional studies that have only varying levels of losses). Prior research in this area yielded a generalization of prospect theory (PT) for options with vaguely specified attributes, and the current chapter reinforces these findings regarding subjects’ tendencies to seek vagueness in the domain of gains but avoid vagueness in the domain of losses. Robert L. Goldstone, Michael E. Roberts, Winter Mason, and Todd Gureckis present a study of collective search behavior from a complex systems perspective. Their chapter describes an interactive experiment in which subjects perform real-time collective search in order to achieve a certain goal. The authors report two different experiments: one a physical resource searching problem, and the other an abstract search problem concerning the dissemination of innovations in social networks. The chapter then explores the group behavior characteristics observed in these experiments in order to provide a foundation for future collective search task analysis. Amnon Rapoport, Tamar Kugler, Subhasish Dugar, and Eyran J. Gisches examine group behavior arising in traffic networks. It is well known in the routing literature that selfish routing (each agent choosing its own best path) results in longer travel times than centralized routing. The Braess paradox demonstrates that the suboptimality of selfish routing can be observed by adding a zero-delay link to a network, which exploits agents’ selfish behaviors to actually increase average travel times. This chapter demonstrates conclusively by laboratory experiments that the Braess paradox is indeed a real phenomenon, even on relatively simple networks. Finally, as the last chapters of this book move from simple decisions to complex decisions, John Saalweachter and Zygmunt Pizlo examine human decision-making models in solving the traveling salesman problem, which belongs to the most difficult class of combinatorial optimization problems. Although the traveling salesman problem is usually studied on a two-dimensional Euclidean surface, the problem becomes even more difficult when obstacles prohibiting some straight-line directions are added to the problem. The authors show that in their experiments, subjects continue to perform well in solving the traveling salesman problem in the presence of simple obstacles. However, when complex obstacles were present, not only did the subjects’ performance decline, but the complexity of their mental mechanisms increased. The compilation of these chapters provides a basis for understanding several state-of-the-art research avenues in the field of descriptive and prescriptive

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decision making. Our hope is that the book inspires further interdisciplinary research in these areas, and illustrates the benefit of studies that tie together concepts from the diverse fields represented by the contributions of this book.

Tucson, AZ and Gainesville, FL January 2008

Tamar Kugler J. Cole Smith Terry Connolly Young-Jun Son

Improving and Measuring the Effectiveness of Decision Analysis: Linking Decision Analysis and Behavioral Decision Research Robert T. Clemen Fuqua School of Business, Duke University, Durham, NC 27708-1208, USA [email protected] Summary. Although behavioral research and decision analysis began with a close connection, that connection appears to have diminished over time. This chapter discusses how to re-establish the connection between the disciplines in two distinct ways. First, theoretical and empirical results in behavioral research in many cases provide a basis for crafting improved prescriptive decision analysis methods. Several productive applications of behavioral results to decision analysis are reviewed, and suggestions are made for additional areas in which behavioral results can be brought to bear on decision analysis methods in precise ways. Pursuing behaviorally based improvements in prescriptive techniques will go a long way toward re-establishing the link between the two fields. The second way to reconnect behavioral research and decision analysis involves the development of new empirical methods for evaluating the effectiveness of prescriptive techniques. New techniques, including behaviorally based ones such as those proposed above, will undoubtedly be subjected to validation studies as part of the development process. However, validation studies typically focus on specific aspects of the decision-making process and do not answer a more fundamental question. Are the proposed methods effective in helping people achieve their objectives? More generally, if we use decision analysis techniques, will we do a better job of getting what we want over the long run than we would if we used some other decisionmaking method? In order to answer these questions, we must develop methods that will allow us to measure the effectiveness of decision-making methods. In our framework, we identify two types of effectiveness. We begin with the idea that individuals typically make choices based on their own preferences and often before all uncertainties are resolved. A decision-making method is said to be weakly effective if it leads to choices that can be shown to be preferred (in a way that we make precise) before consequences are experienced. In contrast, when the decision maker actually experiences his or her consequences, the question is whether decision analysis helps individuals do a better job of achieving their objectives in the long run. A decisionmaking method that does so is called strongly effective. We propose some methods for measuring effectiveness, discuss potential research paradigms, and suggest possible research projects. The chapter concludes with a discussion of the beneficial interplay between research on specific prescriptive methods and effectiveness studies.

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1 Introduction Decision analysts are quick to point out the distinction between decision process and decision outcome, and that even the best decision process can be derailed by an unlucky outcome (e.g., [8,11,57]). So why should we use decision analysis (DA) techniques? Typical answers include “gaining insight” and “being coherent,” but the best reason to use DA would be that doing so would more likely get us what we want. Are there any results indicating that this is the case? If an individual or group uses DA, will they be more likely to get what they want? In the long run, are users of DA healthier, wealthier, safer, or in general more satisfied with the consequences of their decisions? Certainly DA can show an individual how to be coherent in making inferences and choices. That is, adherence to DA principles can promise that your decisions will not be self-contradictory and that the inferences you make will be consistent with the laws of probability. Thus, DA tells us what one should do on the basis of logical argument. However, behavioral decision research (BDR) has shown that people do not always make coherent decisions and internally consistent inferences (e.g., [28,36,38]). DA and BDR began with rather close ties, with BDR topics arising largely from questions associated with the appropriateness of subjective expected utility as a descriptive model of human behavior. As such, much of the early BDR literature looked at empirical questions about how well people could judge probabilities and the inconsistencies in preference judgments and decisions. In the late 1960s and early 1970s, it was not uncommon to see experiments that evaluated DA methods for assessing subjective probabilities or utilities. However, in reviewing the two literatures over the past 25 years, one comes away with the sense that BDR and DA have taken somewhat different paths. BDR has increasingly focused on psychological processes with less emphasis on helping to improve DA’s prescriptive techniques. On the other hand, although DA learned many important lessons from the early BDR work, much of current practice ignores recent developments in BDR and still relies on methods developed in the 1970s and early 1980s. For example, probability assessment in practice still follows principles laid out by Spetzler and Sta¨el Von Holstein [67] (e.g., see [39,49,50]). In very general terms, DA helps decision makers address their decisions in careful and deliberate ways. Thus, dual-process theories from psychology, especially Sloman [63] and Kahneman [34], can provide a useful framework. Many of the behavioral phenomena studies in BDR can be explained as resulting from an intuitive or nonconscious process, which Kahneman [34] calls System I. DA can be characterized as avoiding System I’s distortions and biases by careful and conscious deliberation (System II). Put simply, DA works to get decision makers past System I and into System II. This chapter has two goals. To begin, we take the position that current best practices in DA are not always successful in avoiding cognitive biases that arise from System I, and that a reasonable strategy is to find specific ways to

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counteract such biases. Thus, our first goal is to show how researchers can use recently developed psychological models from BDR to develop improved and prescriptive methods for DA. Our proposal goes beyond general statements about how awareness of behavioral biases can help decision makers avoid pitfalls. Instead, an appeal is made to take BDR results and models directly into the DA domain and to develop precise prescriptive methods that, according to the proposed theory, should improve judgment and/or decision making in a specific and systematic way. In Section 2 we briefly review research on probability and preference assessment, highlighting several recent examples that have used BDR results as a basis for new DA methods. Suppose that we are indeed able to develop new prescriptive methods. How will we know if they work? Although it may be possible to demonstrate in the lab that a method reduces a particular behavioral effect or bias, the question remains whether the method, applied in real decision settings, will genuinely be of value to decision makers. Put another way, the question is whether specific, BDR-based prescriptive methods will be more effective than current DA practice or unaided intuitive judgment in getting a decision maker what he or she wants. Thus, our second goal in the chapter is to describe research paradigms that could be used to measure the effectiveness of DA methods. Studies of effectiveness may complement the development of prescriptive DA methods by highlighting important behavioral questions about why various methods perform as they do, and behavioral research can in turn suggest specific ways to improve decision analysis. In evaluating DA and other decision-making techniques, the basic research questions are, “Is XYZ decision-making technique effective in getting people what they want? How does XYZ compare with ABC in terms of effectiveness?” Answering such questions is somewhat problematic. One cannot, for example, compare two different alternatives that an individual might have chosen in a particular decision situation; one person can follow only one path in the decision tree, and it is impossible to compare what actually happens with what might have happened under different circumstances. Measuring the effectiveness of decision-making techniques is a research problem fraught with challenges. For some of these challenges, satisfactory approaches are easily identified, but responding to others will require creative new methods. Nevertheless, comparative studies can be performed using readily available approaches, and we describe some ways researchers can carry out such studies. The remainder of this chapter is organized as follows. The next section delves into the relationship between BDR and DA, with particular attention to existing examples of a productive relationship and suggestions for other areas that could yield important improvements in DA methodology. Our review is selective and meant to highlight particular streams of research. In the following two sections, we turn to the chapter’s second goal of measuring the effectiveness of DA methods; we define the concepts of strong and weak effectiveness and describe in some detail a variety of possible studies and a

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general research agenda. The final section concludes with a discussion of the potential benefits that BDR and DA can gain from the development of specific prescriptive methods and studies of effectiveness.

2 Using BDR to Improve Prescriptive DA Methods DA methodology derives from the subjective expected utility paradigm, with a strong focus on subjective judgment of probabilities and assessment of personal preferences. Much of BDR has likewise focused on these issues. There are, of course, many aspects of decision making that fall outside the scope of subjective expected utility, such as generating alternatives, identifying objectives to consider, or identifying and modeling relevant risks [24]. DA has developed methods for dealing with these and other aspects of decision making, and BDR has studied some of them. For our purposes in this chapter, however, we focus on subjective assessment of probabilities and modeling and assessment associated with value and utility functions. 2.1 Probability Assessment Early Work Early work on subjective probability judgments by Kahneman and Tversky and others (see [36]) emphasized how heuristic judgment processes can lead to biases. This work was important for the development of standard DA procedures for eliciting subjective probabilities [67]. For example, the anchor-andadjust heuristic in particular has played an important role in DA, because overconfidence is one of the most persistent biases that decision analysts face. For the assessment of probability distributions for continuous variables, Spetzler and Sta¨el Von Holstein advocated pushing experts to reason about scenarios that could lead to extreme events and to adjust their probabilities after such reasoning. Much of the early behavioral work on probability judgments focused on calibration of subjective probabilities (see [41]), demonstrating the extent to which subjective probability judgments did not match the objective frequency of occurrence of corresponding events. Some efforts were made to find ways to improve calibration. For example, Sta¨el Von Holstein [68,69] and Schaefer and Borcherding [60] reported that short and simple training procedures could improve the calibration of assessed probabilities, although their results did not show an overwhelming improvement in performance. Fischhoff [17] discussed debiasing techniques intended to improve the quality of subjective probability assessments. More recently, specific lines of inquiry have yielded results that are important for DA; we mention two here. The first is from the work of Gigerenzer

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and colleagues [26,27] and includes asking the expert questions that are “ecologically consistent” with those typically encountered in his or her domain of expertise and framing assessment questions in terms of relative frequencies. Both can substantially improve calibration, but neither is a panacea. By their very nature, risk assessments often require experts to go beyond their day-today experience (e.g., “What is the probability of a failure of a newly redesigned system in a nuclear reactor?”). Also, not all assessment tasks can be readily reframed in frequency terms. Consider the nuclear reactor question. Given an entirely new system design, how would the analyst describe an equivalence class for which the expert could make a relative-frequency judgment? The second is the literature on decomposition of probability judgments, the process of breaking down an assessment into smaller and presumably more manageable judgment tasks, making these simpler judgments, and then recombining them using the laws of probability to obtain the overall probability desired. For example, Hora et al. [31] showed that decomposition can improve assessment performance, and Clemen et al. [10] found similar results in the context of aggregating expert judgments. Morgan and Henrion [50] reviewed the empirical support for decomposition in probability judgment. Recent Directions: Underlying Processes More recent BDR work on probability judgment has turned to understanding the processes underlying observed biases. One example is the notion of dual processing systems. For example, Sloman [63] makes the case for associative and rule-based reasoning systems and their impact on judgments of uncertainty. Kahneman and Frederick [35] and Kahneman [34] propose “System I,” the quick, intuitive processing system, and “System II,” the deliberative reasoning system. They argue that when an individual makes a probabilistic judgment, the process begins in System I, which is subject to a variety of nonconscious effects, one of which is the substitution of features of an object for the characteristic being judged. For example, when asked for a probability judgment, an individual may use availability, representativeness, or affect as substitutes for genuine (and more deliberative) judgments of likelihood. Another example that emphasizes psychological process is the affect heuristic (e.g., [65]), whereby an individual makes judgments based on his affective response to the stimulus. The affect heuristic applies to both probability and preference judgments; Slovic et al. discuss the underpinnings of this heuristic, including how it ties into the fundamental workings of memory and emotion. A third example is support theory [58,72], which provides a theoretical framework for understanding the psychological process by which an individual generates probability statements. An important feature of support theory is the support function, a modeling construct that represents how an individual summarizes the recruited evidence in favor of a hypothesis (a particular description of a possible event). The notation s(A) is used to represent the

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support for hypothesis A; that is, it represents the individual’s evaluation regarding the strength of evidence in favor of A. According to support theory, s(A) is not description invariant; different descriptions of the same event do not necessarily carry the same support. For example, suppose A is “precipitation tomorrow,” which can be decomposed into “rain or frozen precipitation tomorrow.” Although the two descriptions are meant to designate the same event, support theory contends that s(precipitation tomorrow) may not be equal to s(rain or frozen precipitation tomorrow). In fact, support theory typically assumes that describing an event A as a union of disjoint events (A1 or A2 ) increases support and the sum of the support for disjoint events is typically larger than the support of the union of those events. In symbols, s(A) ≤ s(A1 or A2 ) ≤ s(A1 ) + s(A2 ) . In turn, differences in support due to different descriptions can lead to different stated probabilities for the same event A, depending on how A is described. Fox and Tversky [23] argue that this process is separate from and prior to the decision-making stage when an individual must make a decision under uncertainty. More recently, Fox and his colleagues [21,22,62] show that judgments of probabilities are subject to a bias they call partition dependence. For example, when experimental participants were asked to assess the probability that the NASDAQ stock index would close in particular intervals, the assessed probabilities depended strongly on the specific intervals that were specified. Fox and colleagues argue that partition dependence stems from a heuristic in which the individual begins with a default prior probability distribution that assigns equal likelihood to each element in the state space; they dub this default distribution the ignorance prior. Because individuals tend not to adjust the ignorance prior sufficiently to account for their information, the result is that judged probabilities can depend strongly on how the state space is partitioned in the first place. In their conclusion, Fox and Clemen [21] connect the idea of partition dependence and the ignorance prior to standard DA practices in probability assessment. They argue that probability assessment occurs in three separate stages, and that different biases are likely to operate in different stages. In particular, they argue that standard DA practice is well suited to reducing biases that occur in the first and second stages (interpretation of categories and assessment of support). For example, working with experts to define and elaborate the interpretation of each event to be judged can reduce effects due to ambiguity, and in the assessment of support the analyst can encourage an expert to fully articulate her reasoning to reduce the effects such as availability or representativeness. However, biases in the third stage, mapping of support into stated probabilities, include the tendency to anchor on the ignorance prior. Fox and Clemen argue that this bias may resist correction because it is not particularly amenable to conscious reflection. They suggest a number of ways in which the analyst can work with the expert to minimize partition

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dependence, including the use of multiple partitions for use in the assessment process. Fox and Clemen [21] make an explicit connection between their behavioral results and DA practice, however, Clemen and Ulu [13] take this connection one step further. Building on the idea of partition dependence, they construct a model of the probability judgment process that is consistent with a number of known properties of subjective probabilities, including partition dependence as well as binary complementarity and subadditivity (e.g., [72]). In addition, they show that their model is consistent with interior additivity, a property observed by Wu and Gonzalez [78]. In one of their experiments, they observed that they could calculate the revealed or “indirect” probability of event A as P 0 (A) = P (A ∪ B) − P (B) for a variety of specifications of the auxiliary event B, and their various calculations of P 0 (A) tended to be highly consistent. Furthermore, an individual’s direct assessment of P (A) tended to differ substantially from the indirect probabilities P 0 (A). Clemen and Ulu [13] present empirical evidence in support of their model. More importantly for our purposes, they show that indirect probabilities, after being normalized to sum to one across the state space, are not biased by the ignorance prior (according to their model) and hence should not display partition dependence. Their empirical results, although preliminary, support this contention. Thus, Clemen and Ulu suggest that decision analysts can use normalized indirect probabilities as a way to counteract partition dependence. Although early empirical BDR work was able to provide good general guidance to decision analysts, Clemen and Ulu’s work shows that it is possible to build on psychological theory to develop a precise prescriptive procedure for DA. Whether the use of normalized indirect probabilities genuinely improves on standard practice will no doubt be the subject of future empirical studies. 2.2 Understanding and Assessing Preferences Assessing Utility Functions for Risky Choices Much of the work on preferences under risk has focused on the extent to which expressed preferences for lotteries are internally consistent, as exemplified by the Allais paradox [1,2] or Tversky and Kahneman’s [70] work on framing. An especially relevant early example is the phenomenon of preference reversal as described by Lichtenstein and Slovic [42]. Stated preferences may reverse depending on response mode (choosing between two risky alternatives versus specifying a value, typically a probability that makes the two alternatives equally preferable). The result is robust, having been demonstrated in many different domains and different forms. This result was then and continues to be important for DA practice, because it shows that preference elicitation methods that are equivalent under subjective expected utility do not necessarily yield consistent responses. Ord´ on ˜ez et al. [53] reviewed the preference

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reversal literature and also studied whether preference reversals can be reduced by “debiasing” [17]. Having subjects perform the two assessment tasks simultaneously yielded little improvement but providing financial incentives for consistency, however, did reduce the reversal rate. Moreover, their results are consistent with the notion that the simultaneous judgment tasks, in the presence of adequate financial incentives, can lead to a merging of the preference patterns displayed in the different tasks. Hershey et al. [30] discussed biases induced by different preference-elicitation approaches in spite of their formal equivalence. One such bias is the certainty effect [70], whereby individuals tend to overweight certain or nearly certain outcomes. Understanding the certainty effect is important for DA, because standard methods for assessing utility functions under risk use reference lotteries that compare a risky lottery to a sure outcome. In order to account for the certainty effect, McCord and de Neufville [48] propose an assessment method in which the individual compares two lotteries. Wakker and Deneffe [76] took this idea one step further with their tradeoff method, in which the individual compares two 2-outcome lotteries at a time, each involving the same probabilities of winning and losing. The individual’s task is to specify one of the four outcomes (xi ) so that she is indifferent between the two lotteries. The assessed value is then used to construct the next pair of lotteries, the assessed value (xi+1 ) from which leads to the next lottery, and so on. Given the way in which the lotteries are constructed, it is possible to show that the assessed values x1 , . . . , xn are equally spaced in terms of their utility values. Moreover, the tradeoff method works for assessing utility functions under uncertainty or for assessing value functions in nonexpected utility models. Wakker and Deneffe’s [76] tradeoff method is a good example of a specific DA method that was developed to account for a particular behavioral phenomenon. Another example comes from Bleichrodt et al. [6], who, like Clemen and Ulu [13], develop a prescriptive method on the basis of a specific behavioral model. Bleichrodt et al. argue that utility assessments are systematically biased in terms of loss aversion and probability weighting as specified by prospect theory [37,71] regardless of the particular assessment method used. They further show how to use the prospect theory model to remove the bias. Doing so, of course, requires estimating the model parameters. Although ideally one would estimate parameters for the particular individual making the assessment, the authors find that even using the aggregate estimates from Tversky and Kahneman [71] can improve consistency across different utility assessment methods. Assessing Multiattribute Preferences Although many different biases have been identified in the DR literature on multiattribute assessment, we focus here on two key issues: the scale compatibility and attribute splitting effects. Slovic et al. [66] define the scale compatibility effect as a bias that occurs when an attribute’s weight is enhanced because the scale on which that

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attribute is measured is compatible with (or easily commensurable with) the scale of the response mode. For example, suppose a decision maker must judge weights for several attributes, some of which are naturally represented by dollars (e.g., profits, cost, or taxes) and some that are not (e.g., lives lost or environmental damage). A typical assessment method requires the decision maker to “price out” the various alternatives, or to identify an amount of each attribute that is consistent with a particular dollar value. Such an approach is relatively common, for example, in contingent valuation methods. However, due to the scale compatibility effect, those attributes that are already in dollar terms or can be easily converted to dollars will tend to be overweighted, whereas those that are not readily represented in dollar terms will tend to be underweighted. Tversky et al. [74] proposed scale compatibility as a key explanation of preference reversals. If the scale compatibility effect stems from choosing a particular attribute to be the numeraire in judging weights, then a reasonable approach is not to identify a single attribute. This is the approach taken by Delqui´e [14], who proposes bidimensional matching as a prescriptive assessment method. Instead of changing only one attribute in order to identify a preferentially equivalent option, Delqui´e suggest changing two attributes at once. The two attributes are varied in a systematic way until indifference is found. Delqui´e’s experiments show that this approach does reduce the scale compatibility effect. Anderson and Hobbs [3] take a different approach. They develop a model in which the scale compatibility effect is represented by a bias parameter in the model. Using Bayesian statistical methods, they show that one can use a set of tradeoff assessments to derive a posterior distribution for the bias parameter and for the individual’s weights. Thus, Anderson and Hobbs’s approach is similar to that of Clemen and Ulu [13] and Bleichrodt et al. [6] in processing an individual’s judgments ex post in order to adjust for anticipated biases. Another behavioral issue in assessing multiattribute weights is the attribute splitting effect [77]. The attribute splitting effect has to do with how attributes are structured in a hierarchy. For example, consider the two hierarchies shown in Figure 1. The task would be to assess global weights wA , wB , and wC for attributes A, B, and C, respectively. In the left-hand panel, the decision maker would judge these three weights directly. In the right-hand panel, the assessment is broken down. The decision maker judges the local weights vA and vA0 for A and A0 separately from the local weights vB and vC for B and C. Combining the local weights to obtain the global weights, we have wA? = vA , wB ? = vA0 vB , and wC ? = vA0 vC . The problem arises from the fact that wA? obtained using the two-level hierarchy tends to be greater than wA obtained from the one-level hierarchy. The attribute splitting effect is especially problematic for decision analysts, because any value hierarchy with more than two attributes can be represented in multiple ways (and hence may lead to different weights), and there is no canonical representation. The attribute splitting effect is similar to the ignorance prior effect discussed above [21,22,62]. A reasonable hypothesis might be that individuals

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Fig. 1. Two value trees for assessing weights for attributes A, B, and C.

begin with equal weights for attributes at the same level in the hierarchy, but then adjust insufficiently. Starting from this premise, Jacobi and Hobbs [33] offer four possible models to account for attribute splitting. They test their four models in the context of an electrical utility firm evaluating environmental and economic attributes of alternative plans to expand generation capacity. The model that performs the best is similar to Clemen and Ulu’s [13] linear model of probability judgment: a weighted linear combination of a default weight (equal for all attributes at the same level of the hierarchy) and the decision-makers’ “true” weight performs the best by a variety of measures. Thus, like Clemen and Ulu [13], Bleichrodt et al. [6], and Anderson and Hobbs [3], Jacobi and Hobbs [33] take the approach of adjusting the decision-maker’s judgments ex post to account for the attribute splitting bias. Constructed Preferences, Emotions, and DA Although ex post adjustment of assessed weights makes some sense as a way to correct for the attribute splitting effect, it is not clear that this is the most appropriate way to frame the problem. An alternative is to think about the effect as a result of the individual’s process of constructing a response to the assessment question or, more fundamentally, from constructing preferences themselves in a situation where the issues, attributes, and tradeoffs are unfamiliar and poorly understood. In such a situation, clear preferences may not be readily articulated and may require careful thinking. The problem, of course, is that the way in which the assessment questions are asked can direct that thinking in particular ways that may affect the eventual preference judgments. Other judgmental phenomena can be viewed in terms of constructed preferences [54,64]. One particularly intriguing example is the role that emotions play in decision making. We mentioned the affect heuristic [65] in the context of probability judgment, but affect plays perhaps a larger role in preference assessment. For example, Loewenstein et al. [43] characterize how individuals respond emotionally to risky situations, and how such responses can have an impact on both judgments and decisions, and Luce et al. [44] show that negative emotion can arise from thinking about tradeoffs in decisions, and that the emotion can affect decisions. Hsee and Rottenstreich [32] show that when

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feelings predominate, judgments of value tend to reflect only the presence or absence of a stimulus, whereas when deliberation predominates, judgments reflect sensitivity to scope of the stimulus. Exploring the interplay between emotions and decision making leads to deep psychological issues. Although the literature in this area is vast, we offer two examples that relate affect and cognitive functioning. First, in studying self-regulation, Baumeister and his colleagues have found that self-regulation, which includes suppressing one’s emotions, generally appears to consume some of the brain’s executive resources [5,51] and can lead to reduced cognitive functioning [61]. Second, Ashby et al. [4] observe that positive affect generally improves cognitive functioning, and they offer a neurochemical explanation. Positive affect is associated with increased levels of dopamine, which in turn has been shown to be associated with various aspects of cognitive function, including improved creative problem solving and cognitive flexibility. If such effects are occurring at the general level of cognitive functioning, it seems evident that emotions, positive or negative, and having to cope with those emotions in a complex decision situation can have substantial impacts on an individual’s ability to think deliberately about tradeoffs. By extension, in a situation where preferences are not well articulated, emotions could have a profound effect on preference construction. Is it possible to develop prescriptive DA methods that respond to the issues raised by the constructed-preference view? Although the idea of constructed preferences has been known to decision analysts for some time (e.g., [16,75]), no specific methodology has been proposed. If anything, the argument has become circular. For example, Gregory et al. [29] critique the contingent valuation methodology, adopting a constructive-preferences view. As a replacement for contingent valuation, they recommend multiattribute utility analysis, a mainstay of DA methods, arguing that the decomposition approach of multiattribute utility can help individuals think deliberately through complex and unfamiliar tradeoffs. Likewise, Payne et al. [55] appeal to many DA techniques, including multiattribute utility assessment, in describing a “building code” for constructed preferences. As we have seen above, however, even standard DA methods can affect the way in which preferences are constructed and expressed. For the time being, finding good prescriptive ways to manage the construction of preferences appears to put BDR and DA at an impasse. Before continuing to an equally difficult topic, measuring the effectiveness of DA methods, we look for hopeful signs. Both behavioral researchers and decision analysts have a growing understanding of constructed preferences. That understanding is undoubtedly the first step. And if DA methods are viewed as a basis for avoiding many of the pitfalls in constructed preferences, it may be time for behavioral researchers and decision analysts to find productive ways to collaborate on this problem.

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3 What Does “Effectiveness Mean? 3.1 Strong and Weak Effectiveness The simple answer to the question, “What does effectiveness mean?” is that DA and other decision-making techniques are effective to the extent that they help us achieve what we want to achieve. Thus, we must measure the quality of the consequences we get—in terms of what we want—as a function of the decision-making method used. This perspective is consequentialist; that is, it embodies the notion that the ultimate value of expending effort on decision making is because doing so can help one to obtain preferred consequences [24]. In particular, a consequentialist perspective does not include any value that might be obtained from the decision-making process itself. In the spirit of Keeney [40], we assume that it is possible to identify a decision-maker’s objectives at the time of a decision. Measuring effectiveness then requires measuring achievement of these objectives. If a decision method tends to lead to consequences that represent a high level of achievement of the decision objectives, we say that the method is strongly effective. In contrast, if a method tends to generate choices that informed judges generally view as preferable at the time the action is taken, then we say that the method is weakly effective. Although weak effectiveness may appear to be trivial (of course the decision maker must prefer the chosen alternative!), it is not when viewed more broadly. Showing weak effectiveness may be accomplished by showing that alternatives chosen by decision makers using a particular technique are judged to be preferred, or even dominant, when compared to alternatives chosen by other methods. The judgment of alternatives is made ex ante (i.e., in the context of making the decision before experiencing the consequence) by an appropriate sample of individuals. We expand on this approach below and make precise what we mean by “an appropriate sample of individuals” and how we can use their judgments to measure weak effectiveness. Does strong effectiveness imply weak effectiveness? It is certainly tempting to answer in the affirmative; if a technique produces alternatives that in turn lead to preferred consequences, would it not be the case that the decision makers would have evaluated those alternatives as having greater expected utility? Unfortunately, no compelling reason exists to believe this would be the case. In fact, one can imagine that a prescriptive technique could mislead a decision maker into thinking that the recommended alternative dominates all others, although the consequence eventually obtained from the recommended alternative would be inferior compared to the consequences from other alternatives. The issue here is not only the distinction between decision utility and experience utility in a specific decision situation, but also the extent to which a decision method itself can lead to a discrepancy between the two.

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3.2 Elements of Value Saying that a technique is effective when it helps one to achieve his objectives to a greater degree begs the question of what those objectives might be. Keeney [40] describes the process of identifying one’s objectives for decisionmaking purposes. Although we can legitimately expect different decision makers to have their own objectives in specific contexts, some basic classes of objectives may be common to certain types of decision-making units. Table 1 lists some generic objectives that may be of interest to individuals, small groups, corporations, or public-policy organizations. In what follows, we use the term “decision maker” to refer to the decision-making unit, regardless of whether that unit consists of one or more individuals. The objectives in Table 1 are intended to be representative, not exhaustive. These objectives describe typical reasons why the decision maker cares about any decisions within its purview. For the individual, we might characterize the objectives as “why we live.” In contrast, a small interest group’s objectives can be said to represent “why we join” voluntarily with others in common endeavors. The objectives of policy makers include notions of fairness, efficiency, and the management of externalities; we might call these objectives “why we govern,” and the corporation’s objectives could be described as “why we engage in economic activity.” Table 1 provides guidance as to what sort of objectives must be measured in order to determine effectiveness. Although specific decision contexts may have specific objectives, that objective is probably related to one of the objectives in Table 1. For example, maximize starting salary in the context of an individual’s job search is related to a wealth objective. Knowing what to measure to determine effectiveness is crucial; we want to be sure that we are concerned with the extent to which the decision maker’s lot is improved, according to his or her perspective, by using one specific decision method or another. Table 1. Typical objectives of different decision makers. • • • •

Individual: Health, wealth, safety, wisdom, love, respect, prestige Small Interest Group: Impact on community, influence, social standing, camaraderie, goals specific to group’s mission Public Policy Maker: Efficient use or allocation of resources, productivity, environmental quality, safety, health, fair decision processes and outcomes Corporation: Profit, market share, stock price (wealth), sales, lower costs, worker satisfaction

Often, adequate measures can be found using standard DA techniques [40], and measures for many of the objectives listed in Table 1 may be found in this way. However, measuring achievement of some objectives may be straightforward whereas others are quite difficult. For example, measuring overall health

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of a group of constituents may be achieved using standard epidemiological survey methods, likewise, wealth as measured in the relevant currency in principle, although there is the typical problem of obtaining truthful responses to questions about private matters. How does one measure wisdom, though, or respect? For a small group, how does one measure its impact? In policy making, fairness depends on perceptions of the distribution of outcomes as well as the process which led to the allocation. Boiney [7] and Fishburn and Sarin [18,19] provide decision-theoretic procedures for evaluating fairness of allocations (including risky allocations) using the concept of envy among stakeholders. Measuring fairness of process is somewhat more problematic, but not impossible. For example, stakeholders who have a “voice” in a public-policy decision often perceive the process to be fairer than those who have no voice in the decision (e.g., [20]). Thus, one possibility for measuring process fairness would be to measure the extent to which stakeholders are given a voice in a decision (and the extent to which the stakeholders perceive themselves as having a voice). Aside from the objectives listed in Table 1, it is also possible for a decision maker or organization to obtain value from the decision-making process itself. For example, an individual may enjoy the process of discovering her values, or may gain useful experience that can be applied to similar problems later. In an organization, improved communication among workers and heightened commitment to a path of action can result from decision making [12]. Although we acknowledge the importance of value that derives from the process itself, in this chapter we focus on consequence-oriented objectives such as those in Table 1 rather than process-oriented objectives.

4 Measuring Effectiveness Virtually no research has been done that compares DA with other decisionmaking techniques in terms of strong effectiveness. Relatively little work has been done to show weak effectiveness. As mentioned above, much of the early work on behavioral decision was motivated by expected utility and implicitly asked whether DA techniques that stem directly from expected utility theory (e.g., probability and utility assessment) were weakly effective in the narrow sense of being able to provide accurate models of a decision-maker’s preferences or beliefs about uncertainty. However, as argued by Frisch and Clemen [24], many elements of decision making fall outside the expected utility paradigm per se. For example, expected utility theory sheds little light on how to identify one’s objectives or how to find new alternatives. In this section, we have two goals. First, we describe some research paradigms that might be used to measure the effectiveness of DA and other decision-making methods. Second, we give examples of specific effectiveness studies.

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4.1 Measuring Strong Effectiveness Longitudinal Studies Studies of strong effectiveness must ultimately embrace the challenge of longitudinal studies. In most important decisions for individuals, small interest groups, corporations, or public-policy decision makers, consequences are experienced over time. Thus, one obvious way to measure effectiveness is to recruit participants, subject them to manipulations regarding the use of particular decision-making methods, and to track over time the extent to which identified objectives are achieved. Aside from the complicated logistics of tracking a group of mobile individuals over long time spans (and of maintaining long-term funding for doing so), an important issue is identifying an appropriate decision situation and an adequate sample of participants who face that situation. For example, consider college graduates making decisions about careers. At a large school with a strong alumni program, it may be possible to keep track of individuals who have gone through a particular manipulation as part of making career choices. In modeling multiattribute preferences for jobs or careers, for instance, their judgments may be taken at face value or adjusted for scale compatibility effects as discussed above [3]. Another example might be upcoming retirees for a large corporation; as employees approach retirement, it may be possible to recruit some individuals as participants in a study that manipulates decision techniques for retirement planning. In this case, one might study the effectiveness of Bleichrodt et al.’s [6] approach for assessing risk-averse utility functions, which are then used for making portfolio allocation decisions. A third group might be entrepreneurs, segregated into subgroups according to decision methods used. For entrepreneurs, appropriate objectives to measure may include number of profitable ventures launched, capital attracted from outside investors, or total return on investment over a specified period of time. Here, one could imagine testing different methods for counteracting partition dependence in the entrepreneurs’ probability judgments, using methods suggested in Fox and Clemen [21] or Clemen and Ulu [13]. One area that seems particularly apt for longitudinal studies is the medical arena. What are needed are clinical trials of decision-making methods; patients with the same condition and treatment options could be randomized into different groups in which individual decisions would be made based on different methods. The study would follow the progress of the patients and compare their conditions after specified periods of time depending on the particular condition being treated. The results would compare the effectiveness of the different decision methods under investigation. Some related work has been done. For example, Clancy et al. [9] and Protheroe et al. [56] showed that using decision analysis can influence individual medical decisions (screening or vaccinating for Hepatitis B in the former, treatment of atrial fibrillation in the latter). However, in neither case were patients followed in order to

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track their health outcomes. Fryback and Thornbury [25] showed that the use of decision analysis, even informally, can affect physicians’ diagnoses when evaluating radiological evidence of renal lesions. Their results showed that the DA-based diagnoses tended to be more accurate. The study was retrospective; the physicians examined existing patient records for which the actual outcome was known (but not to the physicians in the study). Thus, the use of DA should improve the expected outcome for a renal lesion patient. A genuine clinical trial as described above would be needed to confirm this conclusion. Simulation Studies. The logistic difficulties of real-time longitudinal studies and clinical trials reduce their attractiveness as research methods. Simulations may provide a suitable alternative. Corporate and industry simulations, for example, are common fare in business curricula; similar games that would be amenable to manipulations in decision-making techniques could provide a testbed for the effectiveness of those techniques. Such games would have at least two advantages: the time dimension is highly compressed, and the environment (including in part the objectives of the participants) can be tightly controlled. Games could be designed around individual decisions, corporate strategy, or public policy; the main necessary ingredients are realistic decision situations and outcomes, along with appropriate incentives to engage the participants in the exercise. An example might be a game that requires participants to make a series of marketing strategy decisions for their simulated “firms,” which interact as members of an industry. Different groups could use specific techniques (e.g., a particular computer decision aid versus generic use of DA modeling methods, including decision trees, Monte Carlo simulation, and optimization). A control group having no specific training or decisionmaking instructions would provide a benchmark. Each group’s results would be measured in terms of the objectives specified in the game and could be compared across groups. 4.2 Measuring Weak Effectiveness Comparing Expected Values In contrast to strong effectiveness, studies of weak effectiveness need not be designed to track outcomes and consequences over time. The simplest approach, exemplified by Clemen and Kwit [12], is to compare the expected values of alternatives that are analyzed in a series of decisions. Clemen and Kwit make the comparison by calculating the difference between the expected value of the chosen alternative and the average of the other expected values for the other alternatives analyzed. If it is possible to document the strategy that would have been taken without the analysis (sometimes called the “momentum strategy”), then one can calculate the difference between the expected value of the chosen strategy and the expected value of the momentum strategy.

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Regardless of the specific metric used, this approach requires substantial record keeping in a consistent way over many decision-making projects. Results that document positive value of the analysis indicate “bottom line” value added, but do not necessarily document value obtained in other ways. For example, if an organization has an objective of improving communication across functional areas, consistently using DA on projects that cut across such areas may help achieve this objective by imposing a common language for discussing decisions. However, such value is not likely to be documented in the calculation of incremental expected value added by DA. Panel Preferences Because it is not always possible to capture all aspects of value in a bottom-line analysis, we broaden the question to ask whether the alternatives generated by a particular decision-making method are viewed as preferable. To operationalize this notion, we propose using a panel of judges. Because we are concerned here with the notion of decision utility, it would be natural to have a panel of judges (e.g., individuals sampled from the same population as the original decision makers) express their preferences for those alternatives. These preferences could be based on holistic judgments, full-fledged preference models, or something in between. Holistic judgments would appear to be unsatisfactory; the decomposition approach of DA as well as other formal decision-making methods challenges the view that holistic judgments adequately capture an individual’s preferences. On the other hand, forcing an individual into a specific preference model requires selection of a particular structure and possibly a particular modeling or assessment technique. Thus, it would appear that some in-between approach is needed, one that requires the judge to make relatively easy assessments regarding the candidate alternatives. As one possible method, consider the problem of comparing multiattribute alternatives. We can ask each member of a panel of judges to rate each alternative on a set of relevant attributes. With data of this nature, the researcher can explore all of the dimensions of preference. The strongest result would be to show that a particular decision-making technique tends to generate a high proportion of dominant or efficient alternatives. A dominated alternative is one for which another alternative can be found that improves on all of the attributes. The set of nondominated alternatives is often called the efficient frontier, because if one member of this set is chosen, it is not possible to switch to another without reducing achievement of at least one objective. In order to operationalize this approach, we need a way to measure an alternative’s efficiency, its closeness to the efficient frontier. Figure 2 presents an analytical example for measuring the relative efficiency of a set of alternatives evaluated on two attributes. A decision maker has identified two attributes, X and Y , that are important in evaluating the alternatives and has in fact rated alternatives A, B, and C in terms of X and Y using functions U (x) and V (y). From the graph it is clear that C is dominated by A and that neither A nor B is dominated. Because A and B are both

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on the efficient frontier with respect to this particular set of alternatives, we will set their efficiency measures EA and EB to 100%. We desire a measure that yields a value of less than 100% for EC . Assuming an additive value function, C lies on an indifference curve defined by aU (x) + (1 − a)V (y) = t, where a and (1 − a) can be thought of as weights in a two-attribute additive utility function. Using the same a, the greatest utility achievable is t? , represented by the line segment AB, parallel to the indifference curve through C. Thus, we can define EC to be the ratio t/t? . This is equivalent to calculating the ratio of the distance DC to the distance DE in Figure 2.

Fig. 2. Measuring the relative efficiency of an alternative. Assuming an additive utility function, alternative C lies on an indifference curve having utility t. A measure of efficiency for C relative to the alternatives included in the evaluation set is EC = t/t? . Using the same logic, the efficiency measures for A and B would each be 100%.

We can improve this measure of efficiency slightly. Note that we could have used any of the line segments on the efficient frontier. Each one corresponds to a different set of weights for the additive utility function. To make C appear in the best light possible, we choose the efficient frontier segment that maximizes the ratio t/t? . Mathematically, suppose we have a set A of alternatives, and we wish to calculate the efficiency score Ei for alternative Ai ∈ A, where Ai has attribute values xi for attribute X and yi for attribute Y . We define Ei to be " # aU (xi ) + (1 − a)V (yi ) Ei = max . a supAj ∈A [aU (xj ) + (1 − a)V (yj )] For a particular a, the numerator inside the brackets calculates the utility t for Ai , and the denominator finds the largest possible utility t? over all elements of A. The maximization finds the values for a that maximize the ratio t/t?

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for alternative Ai . This formula generalizes easily to more than two attributes and to more general forms of multiattribute utility functions. Why would such an approach be useful? Suppose we have a set of alternatives representing choices made by decision makers in a particular decision situation and using specified decision-making methods. A panel of judges can score the alternatives on a set of attributes (of their own choosing or attributes predetermined by the experimenter), and then for each judge the data can be used to generate an efficiency measure for each alternative. With efficiency measures calculated for all of the judges, statistical analysis can be used to compare the efficiency measures across alternatives. The judgmental inputs satisfy our desire for something between holistic judgments of preference and a full-blown preference model; the ratings for each attribute are straightforward judgments that still capture the richness of the judges’ preferences. No assessment of utility weights by the judges is needed, and the statistical analysis can determine whether alternatives generated by one decision-making method tend to be more efficient than alternatives from another method. Using panel preferences is not without problems. As with all decision makers, panel judges may be susceptible to many of the biases discussed in the first part of this chapter. For example, we described the problem of constructed preferences. Individual preferences can be shaped by aspects of a decision situation and the task-contingent strategies people use in judgment and choice [54,73]. If we are unsure of the quality of our elicited panel preferences, of what scientific value are the calculated efficiency scores? Although we cannot completely escape the implications of this question, we offer two mitigating considerations. First, to the extent possible, judges should be exposed to the same decision-making environment as the decision makers and should be encouraged to deliberate about the value of the alternatives on the various attributes according to standard procedures and following principles consistent with Payne et al.’s [55] “building code” for decision making. Doing so may not remove all biases, but standard procedures should minimize biases and variability across judges. Second, the approach described above calculates an efficiency score specific to each judge, and therefore agreement across judges is not required. In fact, the procedure for calculating efficiency scores for alternatives for each judge appropriately takes into account variability in preferences across judges; the objective is to identify a subset of alternatives that are preferred in the sense of having (statistically) higher efficiency scores than other alternatives. 4.3 Some Research Projects The paragraphs above describe some general experimental paradigms that could be used to measure the effectiveness of DA and other techniques. In this section we speculate on some specific studies that might be performed.

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Probabilistic Forecasting Competition Imagine a complex forecasting situation that involves many kinds and levels of uncertainty, such as forecasting crude oil prices, diagnosing a disease based on patient signs and symptoms, or troubleshooting a computer software installation. A variety of analytical and modeling techniques are available for problems such as these, ranging from the construction of complex belief nets or other artificial intelligence models to decomposed or even holistic probability judgments made by experts. Given one or more prespecified domains, a competition could be held, pitting different probabilistic forecasting techniques against each other. It would be most natural to hold such a competition in real-time, in which case the result can be used to determine strong effectiveness of the techniques. Strictly proper scoring rules provide natural performance measures that can capture both calibration and skill [52]. In a probabilistic forecasting environment, enough outcomes would have to be recorded to calculate meaningful average scores for the various techniques. Competitions such as this have been run in the forecasting field [45–47]. An alternative to a real-time exercise would be to construct a simulation in which participants would have to make probabilistic forecasts and in which their overall performance would be measured by their average scores. For example, a business simulation in which participants must make judgments and take calculated risks could be designed to incorporate participants’ skill in assessing probabilities related to aspects of their business such as marketing, R&D, production, or competitive analysis. Aside from creating such a game, the challenge would be to implement different probability forecasting options within the context of the game in a way that permits experimental manipulation of the techniques. Scoring rules need not be the basis of comparing probability forecasting techniques, especially if the probability forecast can be related to a specific decision context. For example, in the business simulation game, one might want to measure stock price, profits, market share, or some other objective important to a real or fictitious corporation. Other possibilities are to choose a context such as college choice or retirement planning and have proponents of different techniques develop systems that lead users through the necessary uncertainty judgments and modeling before offering alternatives from which to choose. By tracking the experiences of the participants in a longitudinal study, one could measure the effectiveness of the different systems and implicitly of the underlying techniques. Value Structuring and Creativity In his book, Value-Focused Thinking (VFT), Keeney [40] recommends that decision makers should identify their objectives as a first step in the decisionmaking process, if possible before identifying alternatives. VFT provides many

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tools and techniques for identifying, structuring, and using objectives in decision making. The overall process has become an important element in the toolkit of decision analysts who work with decision makers on complex multiobjective problems. In VFT, Keeney stresses that his approach is valuable for many things, not least of which is its potential for generating creative alternatives. How would one determine whether a technique generates creative alternatives? One must start with a relatively unstructured problem that admits the possibility of creative problem solving; highly structured textbooklike problems typically do not provide adequate leeway for the decision maker to find creative answers. But if we want to be able to evaluate creative alternatives, by definition we may be considering alternatives that we have not yet seen, so no scoring system for the alternatives can be established in advance. Fortunately, a procedure such as the one described in the previous subsection for measuring weak effectiveness can be used. Suppose that VFT and other techniques are used to generate alternatives in some decision situation, either real or simulated. A panel of judges can evaluate the alternatives by rating them on each of several dimensions, and the subsequent efficiency analysis can determine whether VFT tends to generate more efficient alternatives than do the other techniques. To the extent that more creative alternatives are also efficient, such a study can indicate the potential of DA techniques to enhance creativity. Decision-Making Tournament Several different decision paradigms exist, among them DA, the analytic hierarchy process [59], and goal programming to name a few. A tournament could be held by having proponents of different methods address a prespecified set of decision problems. The quality of the decisions chosen could be evaluated by a panel of judges or, if suitable, by tracking the downstream consequences to the decision makers either in a real-world or simulated environment. An important issue that must be faced in such a tournament is coming up with decision problems that present a reasonably level playing field for the various techniques to be tested. Such problems would presumably consist of “case studies” that are rich in realistic and detailed information. Care must be taken not to present information in a way that artificially predisposes a decision maker toward a particular technique; for example, expressing uncertainty explicitly in terms of subjectively assessed probability distributions might create a bias toward DA, whereas explicit indication of pairwise comparisons might predispose a decision maker toward the analytic hierarchy process. Canonical Decision Problems Researchers might want to run the tournament just described with technique proponents as participants more than once for each set of cases. After the

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initial run, it would be necessary in subsequent runs to ensure that the participants were not prejudiced one way or the other by the outcomes and decisions of prior runs. An interesting twist on the tournament idea would be to develop a set of canonical decision problems and an experimental procedure that could be used as a way to test new techniques as they are developed. A parallel can be found in the field of mathematical programming, which has adopted a few computational problems that are commonly used as benchmarks for comparing algorithm performance. Although no similar set of canonical problems exists for decision making, the creation of such a collection would facilitate the comparison of decisionmaking techniques. A strict procedure must be established, however, in which “na¨ıve” subjects would be instructed in the use of a particular technique prior to applying it to the canonical problem. The prior knowledge of the subjects, regarding both the problems they would face and previous results on other decision techniques, must be carefully controlled. Finally, because decisions would not necessarily be made contemporaneously, the panel approach to judging the quality of alternatives is not appropriate. Instead, the decision problems must have appropriate built-in measures for determining the quality of the decisions, and those built-in measures must be directly related to the objectives of the roles adopted by the participants. Ethnological Study of Decision Making A final example is to take an ethnological approach. In this case, one would collect accounts of decision-making styles and techniques in different contexts. For example, a database might be developed that contained accounts of individual decision making, including decision context, framing, techniques used, and choices made. Similar databases could be created for corporate or public-policy decisions, although gag rules may render collection of such data quite difficult. To be able to compare effectiveness, such a database must be augmented with either judgments of quality of the alternative chosen (possibly done by a panel reviewing all decisions in the database) or a measure of the consequences to the decision maker (via later reports of performance). A large database could be analyzed to determine the characteristics of the most effective decision makers. It is clear that a program to study the effectiveness of DA techniques has many facets and issues to which the researcher must attend. The first order of business is to create appropriate experimental and analytical paradigms. For example, the analysis of panel judgments as described above must be refined, tested, and extended to decision situations other than multiattribute choices under certainty. We must learn how to handle the ethics and logistics of longitudinal studies. Good case studies must be developed and refined for use in simulation studies or as canonical decision problems for ongoing research on decision effectiveness. Methodological work of this nature may not appear attractive in and of itself unless it can directly address substantive questions

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of interest. Nevertheless, the importance of such work cannot be overstated for the research program described here; these methodological problems must be solved if definitive results on effectiveness are to be obtained. With an array of experimental methods available, research can address many aspects of the effectiveness of decision-making methods. Some general examples include the following. • Compare specific techniques such as DA, analytic hierarchy process, na¨ıve decision making, and so on as overall paradigms for making decisions. • Explore different aspects of decision making: problem structuring, uncertainty modeling and assessment, preference assessment, and so on. • Examine different types of decision-making contexts, such as personal decisions, corporate strategy, public policy, or multiple-stakeholder decisions. • Investigate decisions in specific domains such as environmental risk assessment, college or career choice, consumer product marketing strategy, research and development, or municipal waste facility location. • Study how effectiveness of different methods varies depending on individual characteristics, such as age, education, attitudes (e.g., attitudes toward quantitative analysis or technology), or cognitive abilities (e.g., ability to work with quantitative information). Many other, more specific studies are also possible, and they might stem from the development of new DA techniques based on BDR findings, or from the introduction of new decision methods.

5 Conclusion: The Interplay of BDR and DA In the first part of this chapter, we discussed ways in which BDR can be brought to bear on the development of new DA methods. Recent efforts have shown how to use BDR results and theory as a basis for developing better DA methods. We discussed new directions that might prove fruitful, such as considering the role of emotions in decision making and the implications for DA methods. In the second part of the chapter, we discussed in detail how researchers can evaluate decision-making methods. The concepts of strong and weak effectiveness provide a framework for studying effectiveness of decisionmaking methods, and we considered some generic research approaches and specific projects. Although we have discussed these two main topics as if they are largely separate, they actually come together in two interesting ways. First, methods from BDR may be useful in developing research paradigms and methods to study effectiveness. Second, the two topics can lead to a research cycle. We have seen that BDR can inform the development of new DA methods. When the new methods are evaluated in terms of effectiveness, knowledge from BDR can be used to explain the results of the effectiveness studies and in turn help researchers refine the methods. As the cycle continues, DA methods improve,

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and we would hope that developing and evaluating new DA methods will contribute to the BDR body of knowledge. An important dilemma that researchers will have to face is the problem of preferences that change over time. For example, in longitudinal studies of strong effectiveness, the researcher must follow participants as they mature and experience consequences over time. The reasons for making the original choice may be less compelling at a later date, leading a decision maker to regret the choice (and, perhaps, the decision to be a participant in the study!). Alternatively, the decision maker may have developed a new rationalization for the original choice based on his new preferences. Yet another possibility is that the decision-making method used led the decision maker to construct his preferences in a particular way, but experiencing the consequences leads to somewhat different preferences. Such complications must be dealt with in order to evaluate strong effectiveness. What does it mean to say that a decision technique is effective at getting us what we want if what we want has changed substantially by the time we experience the consequences? Throughout, we have implicitly made three assumptions that some readers may find controversial. First, we have implicitly assumed that deliberative, System II thinking generally leads to better decisions. Some researchers have offered evidence to the contrary. Recently, for example, Dijksterhuis et al. [15] found that, when making consumer decisions, individuals made “better decisions” when they did not process the information consciously compared to when they did. However, their conclusions are not particularly relevant to our assumption for two reasons. First, their experiments were conducted under conditions vastly different from what one would expect when using DA methods. Dijksterhuis et al. presented information about a variety of different product attributes fairly quickly, focusing on each one for a matter of seconds, and providing four minutes for an individual to think about his or her choice. In contrast, we might expect conscious processing to involve considerably more time, including time to examine claims in detail, acquire and assess new information, discuss the matter with others, and so on. Second, their measure of decision quality was subjective in three of their four experiments (either postdecision satisfaction or attitude toward the target object). In the fourth experiment (their Study 1), the experimenters identified the “best decision” as the choice with the most positive aspects, even though different individuals may weight these aspects very differently in their choices. If the “best decision” was a dominant choice—better than any of the other choices on all aspects—then their method seems reasonable, but otherwise does little to suggest either strong or weak effectiveness. Our second implicit assumption has been that DA methods and the underlying subjective expected utility paradigm are the appropriate deliberative basis for making a decision. However, our argument does not rely on the assumption that DA is “the best” or the “only rational” way to make a decision. All of our arguments could be applied to other deliberative decision-making frameworks, such as the analytic hierarchy process [59] or other multicriteria

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methods. In fact, studies of effectiveness could profitably compare different decision-making frameworks. Decision analysts who have held tenaciously to the belief that DA is the best method may be surprised! Third, in suggesting that a decision-maker’s or expert’s judgments or preference statements may be improved by adjusting them ex post, we have implicitly assumed that this is a legitimate thing to do. However, it leads to situations that may appear somewhat paradoxical. For example, take the Bleichrodt et al. [6] approach to adjusting a decision-maker’s stated risk preferences to account for distortions due to prospect theory. One can imagine the decision maker saying, “My risk tolerance is X,” and the analyst saying, “No, your risk tolerance is really Y.” Is it reasonable for the analyst to take such a bold step? We offer two justifications. First, the decision maker can be educated (on the basis of sound research) that his or her statements may indeed be biased and that correcting for the biases can improve judgments and choices. Second, in a public-policy setting, experts and stakeholders may state their judgments or tradeoff weights and certify that those statements are the best representation of their beliefs and preferences that they are capable of making. Similarly, the analyst can certify that any adjustments reflect up-to-date scientific knowledge about how to account for known biases. Such certification is already implicitly practiced, for example, by scientists who certify that a particular complex mathematical model represents up-to-date scientific understanding. Helping individuals and organizations find paths that lead to their objectives is the ultimate goal of research in decision making. Regardless of the issues raised in the paragraphs above, BDR and effectiveness studies can lead to better decision-making methods and thus can ultimately help decision makers achieve their objectives.

Acknowledgments I am grateful to colleagues to the Fuqua School of Business, the University of Texas at Austin, and the Workshop on Decision Modeling and Behavior in Uncertain and Complex Environments (Tucson AZ, 2006) for the opportunity to present this work and for hours of discussion. Thanks especially to Susan Brodt, Ellen Peters, and an anonymous reviewer for their comments and insights. This work was supported in part by the National Science Foundation under Grant No. SES-0317867. Any opinions, findings, conclusions, or recommendations expressed herein are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Reducing Perceptual and Cognitive Challenges in Making Decisions with Models Jenna L. Marquard1 and Stephen M. Robinson2 1

2

Department of Industrial and Systems Engineering, University of Wisconsin-Madison, 1513 University Avenue, Madison, WI 53706, USA [email protected] Department of Industrial and Systems Engineering, University of Wisconsin-Madison, 1513 University Avenue, Madison, WI 53706, USA [email protected]

Summary. Decision makers face perceptual and cognitive challenges in understanding presentations of formal models used to support decisions. These presentations, often made by analysts who created the models, must allow the decision makers not only to acquire information but also to process that information into a form that they can use effectively for decision making. This chapter illuminates the occurrence of eleven well-known perceptual and cognitive challenges that decision makers may face as they receive and use the results of formal models. It also discusses features of the modeling process that have either aided or hindered decision makers in overcoming these perceptual and cognitive challenges, and draws some conclusions about ways in which people might improve the modeling process to reduce the severity of perceptual and cognitive challenges and thereby to improve the decision-maker’s effectiveness. Using retrospective case analysis, with the known set of perceptual and cognitive challenges as themes, we examine the presence and nature of these themes across five modeling projects. The selected projects span diverse disciplines and vary in the type and complexity of the models developed and in the characteristics of the decision makers. Through examination of the case studies, we see evidence of five of the perceptual and cognitive challenges and indication of an additional two challenges. These challenges stem from the nature of the model presentation, from the roles of the analyst and decision maker in the modeling process, or from factors external to that process. From the results of the case analysis we derive a condensed checklist of recommendations for analysts. By identifying perceptual and cognitive traps along with their sources, this work provides insight not only into what challenges are likely to exist when decision makers use models as a part of their decision-making process, but also into how the structure of the modeling process and the model presentation allow those challenges to arise.

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1 Background People often view formal models as useful tools for systematic analysis to support informed decision making. However, creating formal models does not necessarily produce informed decision making. Decision makers face perceptual and cognitive challenges in understanding presentations of these models, with these presentations often made by analysts who created the models. Decision makers must not only acquire information from a model presentation, but also subsequently process that information into a form that they can use effectively for decision making. Perceptual and cognitive challenges in acquiring and processing information from model presentations may take many forms, including mental shortcuts taken by a decision maker to reduce a problem’s complexity. These challenges may lead to serious errors in decision making. In particular, a decision maker may incorrectly assess elements of a model presentation such as uncertainties, assumptions, and biases underlying the model, and may then draw incorrect conclusions from the model presentation. This chapter draws on the analysis of a series of case studies to illuminate the occurrence of these perceptual and cognitive challenges in decision-makers’ reception of model presentations. The case studies described include a personnel model used by the army, a navy shipbuilding model, a simulation model of air quality in the San Joaquin Valley, a modeling project undertaken to support economic development of the Obergurgl region in Austria, and an analysis of water management for the Netherlands. We also discuss features of the modeling process that have either aided or hindered decision makers in overcoming these perceptual and cognitive challenges. From these case studies and discussions we draw some conclusions about ways in which people might improve the modeling process to reduce the severity of perceptual and cognitive challenges and thereby improve the decision-maker’s effectiveness. 1.1 The Judgmental Gap In complex and/or uncertain situations, even experienced decision makers may be unable to produce good decisions using only conventional wisdom or mental models [5]. This inability has produced advocacy for making decisions based on systematic analysis. Formal models can help in constructing these systematic analyses because they allow for comprehensive structuring of a situation, organizing information with which an unaided decision maker may be unable to cope [13]. However, decision makers using such models need both sufficient analytic tools to supplement their decision-making skills and the ability to appropriately use these tools. The mere presence of analytic tools does not suffice to produce informed decision making. There is a body of work from the past forty years describing a disconnect between the tools produced through systematic analysis and what decision makers desire in order to address a problem [13,22,23]. Raiffa [22] postulates

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the presence of a judgmental gap between the outputs of formal models and decision makers embedded in the real world. In [22, Figure 10.3], he shows objective inputs feeding into a formal model that produces output, and on the other side the real world, but with a judgmental gap separating these two sides. He also states that the width of the judgmental gap may result in analysis that is not relevant and does not provide meaningful insights, and that may then be ignored. Although Raiffa describes the potential consequences of the judgmental gap, he does not describe how the gap is produced nor how it might be detected by the decision maker embedded in the real world. The judgmental gap described by Raiffa may render a model unusable, or may severely limit its usefulness to the decision maker. Work on characterizing the factors that produce the judgmental gap, as well as the means by which a decision maker detects and responds appropriately to that gap, might help to increase models’ usefulness in supporting decision making. Factors producing the judgmental gap might include the assumptions upon which a model is built or the biases of the analysts who created the model. Our emphasis in this chapter is on perceptual and cognitive challenges that may affect whether and how a decision maker detects and responds to the judgmental gap. 1.2 Models as Analytical Tools Models are one type of analytical tool, and the term “model” can be used to refer to a broad array of representations of reference systems. A model of whatever type is a representation, or abstraction, aimed at mimicking some real-world phenomenon. Greenberger et al. [13] propose that a model “represents selected features of the reference system so that one can learn about the reference system by working with the model.” They also distinguish between classes of models. At the highest level, formal models differ from mental models through their expression in a formalized language that facilitates examination and revision. Within formal models, Greenberger et al. distinguish among four model classes; schematic models, physical models, symbolic models, and role-playing models [13]. According to their definition of models, one can learn about a building to be constructed by examining its blueprint (schematic), a molecule by handling a physical model of its structure (physical), a manufacturing plant by manipulating a mathematical model of its operations (symbolic), and the real-estate business by playing Monopoly (role-playing). Our work in this chapter focuses, within formal models, specifically on the class of symbolic models. This class includes but is not limited to mathematical models, including those that use computers either to determine the results of calculations or as devices to support simulation or graphical visualization. 1.3 Conveyance: Information versus Models Two important role types, the analyst and the decision maker, participate in using models and other analytical tools to inform decision making. Roughly

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speaking, a decision maker is an individual or group, possibly embedded within a larger set of decision makers, that makes a commitment toward a certain course of action. An analyst is an aide in the decision-making process, eliciting systemic understanding of a problem situation from one or more decision makers or other stakeholders, forming that information into a coherent analysis and then presenting analytical findings to the decision makers or stakeholders. Like the decision maker, the analyst may or may not be embedded within a larger set of analysts. Stakeholders, a superset of decision makers, include those individuals who have an interest in the decision. The simplified diagram in Figure 1 shows these role types and their relation to one another.

Fig. 1. Conveyance of information.

The existence of these roles produces a set of processes for conveying information from one role type to another. The simplified diagram shown in Figure 1 displays two pathways for conveying information between individuals in these roles, with decision makers conveying situational and personal preference information to analysts and analysts conveying analytical findings back to decision makers, in what can be an iterative and tightly coupled process. These conveyance pathways suggest that each role type possesses complementary knowledge sets, both of which are needed to create informed decisions. Entire disciplines are dedicated to the conveyance of technical or complex information to lay people, however, there has not been a similar body of work produced to assist in conveying technical models to lay people. By “lay people” we mean those who lack the particular technical skills for a deep understanding of the model at issue, although they may be highly qualified in other fields.

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The field of risk communication, for instance, focuses on conveying concepts or messages to lay people with the goal of specific understanding or action [5,9]. This field tends to focus on four conveyance elements; the message, the messenger, the audience, and the context [26]. By contrast, the issues in conveying models to lay people are how to convey relationships and structures, with the goal of understanding and appropriate use of the model. Work in this area, including that described in this chapter, thus complements disciplines such as risk communication by addressing issues surrounding the conveyance of model structures and interactions instead of content and messages. Risk communication and model conveyance also differ in the level of understanding and type of action required by the decision maker. Risk communication messages attempt to induce lay people to take some action or gain some understanding, often in a timely manner, but do not necessarily require comprehension of the logic behind that action or understanding [20]. Model conveyance, on the other hand, requires the decision maker to comprehend better the logic underlying the model but may or may not foster a need for her to take action. 1.4 Model Presentation: Gateway to Understanding Significant empirical work has gone into examining and improving the ability of models to represent two things, a particular reference system and how a decision maker would ideally behave in a given situation. Figure 2 shows these two representations. Much of the work in the former area lies in the area of

Fig. 2. Representations: reference system and ideal behavior.

model verification and validation, whereas work in the latter area has included the elicitation of decision-maker preference and utility values [18,19,24]. By contrast, there is a lack of empirical work examining how decision makers understand and use these models. As was previously mentioned, Raiffa conceptualized these areas as a judgmental gap, but he focused on the potential results of the judgmental gap as opposed to its constituent factors [22]. In many cases those who have vast anecdotal evidence of these factors function in roles, such as consulting, in which they may have neither the facilities nor the incentive to illuminate and address these factors.

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The representations shown in Figure 2 can also be viewed as the main directions in which researchers in the field of operations research have moved in trying to close the judgmental gap described in Section 1.1. Work in operations research has included refining the use of decision-maker preferences and utilities in models [5,18], reducing uncertainties in models [21], increasing model complexity, and undertaking extended sensitivity analyses [8]. This focus on building more accurate models as a way to close the judgmental gap may have derived in part from increases in computational power over the past thirty years and a general focus on computation within the operations research and engineering fields. Despite this significant expenditure of work, the problem of fully closing the judgmental gap has gone unsolved for almost forty years. To participate fully in the decision-making process, lay decision makers need to understand the presentation or conveyance, by the analyst, of the formal model that they are using to support their decision. This understanding is important because the analyst’s presentation serves as the gateway to how a decision maker understands and uses a model. Figure 3 shows this view of

Fig. 3. Model presentation.

model presentation as the gateway or conduit from analyst to decision maker. This figure also shows the information flows and avenues of influence when models are involved in the decision-making process. For instance, a decision maker’s influence on the real world is driven by information flows both from the model presentation and from the real world. We have not seen in the existing literature any treatment of the model presentation as central to whatever assistance the model gives to the decision maker. Many articles have dealt with aiding decision makers, but the

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typical view seems to envision the decision maker’s communicating directly with the model and, in many cases, understanding its technical foundations. For example, the article of Borcherding and Schaefer [6] notes the “tendency to use heuristic, ineffective, and often incorrect rules when making intuitive judgments,” and offers procedures and approaches to aid decision makers in complex decision situations. These include multiattribute utility analysis and discussion of issues in the assessment of probabilities. By contrast, we envision a decision maker without a technical understanding of just what the model is doing, for whom the presentation is therefore an essential interface. This chapter focuses specifically on the information flow from model presentation to decision maker. If decision makers can effectively understand a model presentation, they may be able to more appropriately critique and use the underlying model. For instance, based on her understanding of a model presentation, a decision maker may choose not to use a particular model. When decision makers can appropriately calibrate models’ usefulness, they are likely to be better suited to make truly informed decisions. Despite the need for decision makers to better understand the models that they use, via effective model presentations, as far as we are aware no guidelines currently exist to aid analysts in conveying existing and newly developed models to relevant decision makers. Without such guidelines, analysts may create and refine models without understanding, and sometimes in complete ignorance of, key factors that influence how their models are received or used. 1.5 Perception and Cognition in Decision Making To understand how decision makers might make better use of models, it should be beneficial to examine factors that stand in the way of the information flow from model presentation to decision maker, seen in Figure 3. Although many factors probably influence this information flow, we examine here how perceptual and cognitive challenges faced by decision makers may affect how they understand a given model presentation and thereby understand a given model. This work views perception as the process of a decision maker’s acquiring information from a given model presentation, and views cognition as how the decision maker processes information acquired from the model presentation. Perceptual and cognitive challenges, if present, are likely to interfere with the decision-makers’ ability to acquire information from a model presentation and subsequently to process that information into a form that they can use effectively for decision making. Fortunately, we already have much research describing the ways that perceptual and cognitive factors impede decision makers. Much of the well-known work of Tversky and Kahneman examined how heuristics—useful cognitive shortcuts for reducing problem complexity—can lead to “severe and systematic errors” in assessing probabilities and predicting values for uncertain events; see the collection [17]. Hammond et al. [14] also describe what they term “psychological traps,” again addressing the potential errors resulting

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from perceptual and cognitive shortcuts in decision making. Janis and Mann [16] focus on decision makers embedded in complex and stressful situations, examining why certain patterns in decision making “interfere with vigilant information processing” and what the consequences of these patterns are. Although the works of Hammond, Keeney, and Raiffa and of Janis and Mann look at the process of decision making generally, they do not specifically address how these “psychological traps” and patterns of less than vigilant information processing affect decision makers who use formal models as a basis for their decision making. Across the fields of psychology and decision making, the list of potential perceptual and cognitive challenges faced by decision makers is large. A scan of relevant literature resulted in a list of seventy potential challenges. Because this work is based in the decision sciences field, we build on the work of Hammond et al. [14], using their list of “psychological traps” as a foundation for the challenges faced by those using formal models as a basis for decision making. Table 1 shows the list of these perceptual and cognitive challenges, as adapted from [14]. Table 1. Psychological traps, adapted from [14]. Challenge

Description

Anchoring Status quo trap

Disproportionately weighting initial information Bias toward alternatives that perpetuate the current situation Weighing past decisions, or costs, in the current decision Searching for or interpreting information in a way that supports one’s preconceptions Framing of a question or problem to influence the answer; e.g., as gains versus losses or with different reference points Overconfidence in the accuracy of one’s predictions Overestimating the probability of memorable or dramatic events Neglecting a base-rate in an assessment Compounding of error due to multiple “safe” judgments Viewing patterns in random phenomena Failure to recognize reality as sometimes surprising

Sunk-cost trap Confirming-evidence trap Framing trap

Overconfidence trap Recallability trap Base-rate trap Prudence trap Outguessing randomness trap Surprised-by-surprises trap

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2 Methods Case studies form one end of a research methodology spectrum whose other end is experimental design [10]. Bromley [7] defines case study as a research method to be a “systematic inquiry into an event or a set of related events which aims to describe and explain the phenomenon of interest.” Qualitative research such as case studies can be helpful in exploring a complex research area about which one knows little. This way of employing case studies differs strikingly from the use of case studies in teaching, where the studies are exemplars of a teaching lesson and thus where elements of a case may be omitted or exaggerated [28,29]. Teaching case studies, such as those found in the Harvard Business Review, can be entirely fictitious while retaining their usefulness. In teaching case studies, the lesson to be learned prevails over accuracy and completeness. In case study research, the opposite is true. Here we use cases to illustrate ways in which various perceptual and cognitive challenges can affect how a decision maker understands a model presentation and thereby understands and uses a given model. This illustration may then help to suggest methods for mitigating the effects of these perceptual and cognitive challenges. Here we follow Yin’s case study methodology [28,29] in examining multiple cases to address the following questions. 1. How do the known perceptual and cognitive challenges described in Table 1 affect how decision makers understand a model presentation? 2. What are viable means for reducing the impact of these challenges on decision-makers’ understanding of a model presentation? Using the set of perceptual and cognitive challenges shown in Table 1 as themes, we examine the presence and nature of these themes in five case studies. The chosen case studies, described below, span diverse disciplines and vary in the type and complexity of the models developed and in the characteristics of the decision makers using the models. Following this examination, we summarize the results in Table 3 of Section 3.12 below. 2.1 Compensation, Accessions, and Personnel Management (CAPM) Model The compensation, accessions, and personnel management (CAPM)1 model is designed to aid military decision makers in understanding how various personnel policies affect the retention behavior of military personnel [2–4].These policies affect pay scales for military personnel, retirement benefits, bonuses upon reenlistment, and early retirement incentives. CAPM is an econometric 1

This model has no connection with the capital asset pricing model, also known as CAPM, that is ubiquitous in finance. As the latter model plays no role in this chapter, we use the abbreviation CAPM for the compensation, accessions, and personnel management model with no risk of confusion.

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model based on an assumption of rational decision making, viz. that an individual in the military will determine whether to remain in the service based on a comparison of the economic value of remaining versus the economic value of leaving. The CAPM model was built as a standalone Microsoft Excel-based tool, in which decision makers can manipulate the model and its underlying assumptions directly through the spreadsheet and its associated graphical interface. 2.2 Navy Shipbuilding and Force Structure Analysis Tool The shipbuilding and force structure analysis tool integrates disparate planning tools used by the navy for future force structure decision making. The goal of this integrated tool is to aid navy decision makers in determining how to effectively use the nation’s shipbuilding resources, given limited funding [1]. These decision makers must project future force structure needs, planning for the design and construction of new additions to the fleet given an existing fleet with various capabilities and operational costs. This tool integrates four existing models: the force transition model, industrial base model, operating and support cost model, and financial adjustments and assumptions model. The tool was built using Microsoft Access as the underlying platform; it allows the user to manipulate all aspects of the model. 2.3 San Joaquin Valley Air Pollution Control In the early 1990s an $18 million project was undertaken to develop SARMAP, a model to aid decision makers in determining how to combat severe air quality problems in the San Joaquin Valley region of California [11]. The air quality problems in this region stem from a combination of topography, meteorology, and human-caused emissions, and the modeling efforts were aimed at understanding how to attain an acceptable air quality level by lowering humancaused emissions. The developed model simulated the effects of various sources of emissions on air quality and, through use of the simulation model, planners created an attainment plan for achieving acceptable air quality levels. Despite the planning and modeling efforts, the goal of attaining federal ozone levels was not reached through the attainment plan and the attainment date was moved from 1999 to 2010. 2.4 Obergurgl: A Microcosm of Economic Development In the 1960s, two researchers from the University of British Columbia began a project to form a more cohesive relationship between analysts and decision makers while addressing problems concerning the economic development of the Obergurgl region of Austria. The project, headed by C. S. Holling and Carl Walters, addressed what they saw as “a lack of congruence or communication between the modeler and intended user” [15,23,27]. Their approach to

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this problem involved bringing modelers together with various experts and lay representatives from the Obergurgl region, and having them jointly build a model of human impact on the alpine ecosystem [15]. These joint development sessions were initially highly problematic, due to variations in participants’ conceptions of the problems and of what building a model meant. Through continued development sessions, participants “began to communicate not via the model but around the model” [23]. In this case, no final model was ever built, but the exercise of jointly working toward a model helped the participants to understand more systematically the problems facing them and to collaborate better in the creation of joint policies. 2.5 POLANO: Protecting an Estuary from Floods After a severe storm caused unprecedented damage to the delta region of the Netherlands in 1953, the Dutch government began extensive planning to protect this region from future flooding [12]. One of the most complex efforts involved evaluating means for protecting the largest estuary in the region, the Oosterschelde. This planning effort, called POLANO, generated a common scorecard framework for comparing the impact of three alternatives: an impermeable dam, a storm surge barrier, and keeping the mouth of the estuary open while building dikes around the perimeter. The major impacts studied were cost, security, ecology, and economic and social impacts. Various modeling efforts were used to generate the impact measure values in the scorecard framework.

3 Results Each of the case studies described above showed evidence of the various perceptual and cognitive challenges described in Table 1, faced by the decision makers upon model presentation. The evidence is organized below for each challenge. 3.1 Anchoring Anchoring occurs when a decision maker gives disproportionate weight to initial or past information in a current situation. In forecasting, for instance, future projections are anchored on past values. Questions can also be anchored by including values in the question itself. Anchoring is not necessarily always harmful, however, it does influence generated answers. Forecasts may vary depending on whether they take past information into account, and the answer to a judgmental question will often vary depending on whether the question was anchored with initial values. In modeling, anchoring is important because initial information provided by the analyst will influence how a decision maker uses or interprets the model.

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In both the CAPM and navy shipbuilding models, many default values are set for the user. In the model presentation, users are then encouraged to alter these default values to match their own situation. The presence of these default values, however, is likely to influence the data that the user will enter into the model. The assumptions used by analysts can also act as anchors, influencing how a decision maker uses a given model. The navy shipbuilding model, for instance, is built on the assumption that the user should first determine force structure needs, with resulting expenditures and the plan for achieving that force structure being the outputs of the model. Another framework for addressing this problem would be to provide budgetary constraints and some ideal force structure, with the model coming as close to the ideal force structure as possible given the budget. This assumption of how the model is to be used heavily anchors and affects how the decision maker views and uses the particular model. 3.2 Status Quo Trap The status quo trap occurs when maintaining a current situation appears easier or less risky than opting for change, creating a bias toward alternatives that perpetuate the current situation. It is often very easy for this to happen, because the consequences of the status quo will generally either be known or will be relatively easy to visualize, whereas it may require a strong imaginative effort to understand the consequences of a change, and even then they may be highly uncertain. None of the case studies showed direct evidence of bias toward the status quo, however, the San Joaquin air quality model decision makers may have been biased in this way through their past tendencies to support business interests over citizen interests [11]. These past tendencies may have influenced the significant amount of industry participation in the modeling process or the apparent leniency toward industry in the modeling recommendations. Users of the CAPM model may also fall into the status quo trap, especially if the personnel changes to be made would decrease wages, retirement bonuses, or reenlistment bonuses. 3.3 Sunk-Cost Trap The sunk-cost trap occurs when past costs or decisions influence a current decision. For instance, if decision makers have spent money on a project that is now not going well, they may be reluctant to stop the project because of the costs already put into the project. Although it was not evident in any of the case studies, the sunk-cost trap is important for decision makers to keep in mind when using models. There may be a tendency to use a model simply because money was spent to create it. In fact, the money spent on building the model is a sunk cost and the assessments of if and how to use the model

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are new decisions that should not be affected by the sunk cost involved in building it. 3.4 Confirming-Evidence Trap The confirming-evidence trap occurs when decision makers search for or interpret information in a way that supports their preconceptions. This error is essentially one of determining a solution before conducting an analysis. Because models of complex situations are laden with uncertainty, analysts must be aware that decision makers may wish to use their model to justify conclusions to which they have already come. In the San Joaquin air quality model, industry representatives were well represented on the planning boards. Although involvement from stakeholders such as industry may be beneficial, other prominent stakeholders such as public interest and community groups were not represented on these boards. When the air quality model was built and policies were recommended, the recommendations appeared lenient toward industry. The Netherlands estuary project provided a novel form of output to decision makers through the use of a scorecard tool. The disaggregated nature of this tool and the fact that no ideal solution was provided to decision makers allowed various stakeholders to interpret the results of the work differently. Those worried about the fishing industry may have favored the solution that had the least impact on the fish population in the estuary and those concerned about budgeting may have leaned toward the solution that cost the least. In a similar manner to the Netherlands estuary project, the Obergurgl project involved a variety of stakeholders and no unique solution. Through joint development of a model, these stakeholders could argue for model inputs and a model structure that benefited their priorities. 3.5 Framing Trap The framing trap occurs when the way a problem is framed influences the answer. A well known example of this trap, taken from [25], deals with two groups faced with the decisions given in Table 2 for a situation in which 600 people were at risk. It is clear from the program descriptions that the effects of Programs A and C are the same, as are those of Programs B and D. In the first group of subjects, where framing was in terms of gains, about three-quarters of the respondents chose the certain program (A) (a risk-averse choice). In the second group, where framing was in terms of losses, about three-quarters chose the risky program (D) (a risk-seeking choice). The Netherlands estuary project showed particularly strong potential vulnerability to framing effects. In this project, analysts determined which impacts to display, their ordering, and their units of expression on the scorecards. On the scorecards, some of the impacts were displayed as positive, or in terms of the gains brought about by an option. Other impacts were displayed as

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Jenna L. Marquard and Stephen M. Robinson Table 2. Framing effect in a risky decision with 600 people at risk, from [25]. Group 1

Group 2

•

•

Program choices

•

If Program A is used, 200 people will be saved If Program B is used, either 600 people will be saved (p = 1/3) or no people will be saved (p = 2/3)

•

If Program C is used, 400 people will die If Program D is used, either nobody will die (p = 1/3) or 600 people will die (p = 2/3)

Choice patterns

72% preferred Program A; 28% preferred Program B

22% preferred Program C; 78% preferred Program D

Expected value of gamble

(1/3)(600) + (2/3)(0) = 200 lives saved

(1/3)(0) + (2/3)(600) = 400 lives lost

Risk preference

Risk-averse

Risk-seeking

negative, or in terms of the detriments brought about by each option. This mixture of positive and negative framing may have influenced how decision makers evaluated the impacts. It should be recalled that this project was conducted before the work of Tversky and Kahneman had produced widespread awareness of framing effects. 3.6 Overconfidence Trap Overconfidence occurs when decision makers strongly believe in the accuracy of their predictions. In the case of modeling, overconfidence occurs when a decision maker strongly believes in the accuracy of an analyst’s predictions. This overconfidence can occur in particular when decision makers view analysts as experts and analysts have not explicitly expressed uncertainties in the communication of their models. This lack of explicit uncertainty shows up in the Netherlands estuary project scorecards, the navy shipyard and CAPM models, and the San Joaquin air quality model. In the description of the San Joaquin air quality model, those involved in the modeling process are described as “highly qualified experts with several decades of experience” using a “robust database for input data.” Decision makers were so confident of the accuracy of the model predictions that they made little or no provision for a margin of error in the predictions. In essence, their action plan relied on the idea that the real world would act exactly as the model predicted.

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3.7 Recallability Trap The recallability trap occurs when the probability of memorable or dramatic events is overestimated. Although not mentioned directly in any of the case studies, this error may have occurred in the Netherlands estuary project. Given the catastrophic nature of the storm that drove the project, decision makers may have given additional weight to solutions that would never allow that type of flooding to happen again. 3.8 Base-Rate Trap The base-rate trap occurs when conclusions are drawn while ignoring a base rate. An example of this given by Hammond et al. [14] asks the question, “A man is shy; is it likely he’s a salesman or a librarian?” In this situation, although librarians may tend to be shyer than salesmen, there are many more male salesmen than male librarians. Given this, the man is more likely to be a salesman. This trap did not appear in any of the case studies examined. 3.9 Prudence Trap The prudence trap occurs when errors are compounded because several “safe” estimates are combined. As a simple example, suppose that one is observing four mutually independent real-valued random variables, each of which can take one of two values, good or bad. If for each variable the probability of good is 90%, then the probability that all four take good values is not 90%; rather, it is (0.9)4 = 0.6561, which is less than 2/3. This is an instance of the well-known principle from reliability theory that a series system may exhibit low reliability even though the reliability of each of its components is high. The case study where this may have occurred is the navy shipbuilding model. Because this model aggregates four existing models, a decision maker who uses conservative estimates in each of the four base models could be faced with serious errors when the models are aggregated. Conversely, the Netherlands estuary project purposefully did not aggregate the various impacts. In this way, any errors are limited to single scorecard elements. 3.10 Outguessing Randomness Trap When decision makers believe they can detect patterns in phenomena that are actually random, this is known as the outguessing randomness trap. This trap did not appear explicitly in any of the case studies. 3.11 Surprised-by-Surprises Trap In some situations seemingly surprising events may in fact have high probability. When a decision maker fails to see these surprises as probabilistic events, this is the surprised-by-surprises trap. This trap did not appear explicitly in any of the case studies.

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3.12 Summary of Observations Table 3 summarizes the observations in the preceding sections, showing which traps occurred in each of the cases we considered. Table 3. Psychological traps observed in various cases. An X in the table indicates that we found evidence of that trap in that case; an I indicates some indication but not conclusive evidence. Trap Anchoring Status quo Sunk cost Confirming evidence Framing Overconfidence Recallability Base-rate Prudence Outguessing randomness Surprised by surprises

CAPM

Navy

X I

X

San Joaquin

POLANO

X

X X X I

I X

X

Obergurgl

X

X

X

4 Discussion Through examination of the case studies, we can see evidence of the various perceptual and cognitive challenges, or psychological traps, defined by Hammond et al. [14]. Interestingly, not all of these traps appeared in the chosen case studies. Possible reasons for nonappearance of a particular trap could be that (1) decision models may in general not be vulnerable to it; (2) some decision models may be vulnerable to it but the cases we studied were not; (3) the trap was actually present in one or more of the studied cases but it did not appear in the documentation that we used as a basis for the case analysis. The first of these explanations would indicate that the modeling process somehow eliminates some of the perceptual and cognitive challenges known to exist in the decision-making process. We have no basis for an opinion as to whether this is or is not true. The second explanation, if correct, would merit further examination of a wider body of case studies. The final explanation rests on the nature of the case study documentation, rather than the case itself. Documentation of models for decision makers is usually focused on the model itself: how to use it and how to draw conclusions from it, as opposed to how the model was received and used by decision makers. This explanation, if

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correct, suggests a need for prospective evaluation of cases where models are developed for decision makers, as opposed to the retrospective method used here. The perceptual and cognitive challenges identified in the various cases are quite interesting, particularly because of the diversity of the areas from which the challenges stem: these include (A) the nature of the model presentation, (B) the roles of the analyst and decision maker in the modeling process, and (C) factors external to the modeling process. One can usefully view these areas in terms of their scope of influence, from micro- to macroinfluencers, as seen in Figure 4.

Fig. 4. Scope of influence: micro to macro.

4.1 Model Presentation Challenges stemming from the nature of the model presentation included both the content of the presentation and method of presentation. Challenges stemming from presentation content arose due to the default values and assumptions embedded in both the CAPM and navy shipbuilding models. Because decision makers interact directly with these modeling tools, the anchoring of the model content may be troublesome. Differences in challenges can also be seen across cases that employed different presentation methods. The CAPM and navy shipbuilding model presentations consisted of decision makers directly interacting with the developed models, whereas the Netherlands estuary project presented models to decision makers through a set of visual and numerical scorecards. Although the CAPM and navy shipbuilding model presentations displayed evidence of anchoring challenges, as described above, the Netherlands estuary project model presentations displayed evidence of

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framing challenges because of the way model outputs were displayed on the scorecards. Similarly, the navy shipbuilding model aggregated four existing models, but allowed the decision maker to generate the inputs for these models. By aggregating these models in the model presentation, this model was subject to prudence challenges arising from the decision maker’s compounding of conservative estimates across the four models. The use of a scorecard presentation method in the Netherlands estuary project purposefully disaggregates errors and thus may combat the prudence trap, but thoughtful design of these scorecards is then necessary to avoid framing challenges. These examples suggest the need for analysts to consider, along with each method of model presentation, the perceptual and cognitive challenges to which that presentation method may be subject. If decision makers are to interact with a model, special attention must be given to the default values and assumptions provided to the decision maker, as well as how the decision maker is likely to make estimates for the data embedded in the model. If decision makers are to be presented with model outputs (e.g. via scorecards), analysts must understand that in the presence of uncertainty the framing of these outputs is likely to influence how the decision maker understands and judges the model outputs. In practice, neither of the authors has ever seen such explicit consideration of perceptual and cognitive traps by working analysts designing model presentations. 4.2 Roles of the Analyst and Decision Maker In some cases, vulnerability to psychological traps might stem from how analysts and decision makers view themselves and their role in the modeling process, how they view each other, and how they interact with each other. If a decision maker views an analyst as an expert, for instance, this could increase vulnerability to the overconfidence trap. Extreme overconfidence occurred in the San Joaquin air quality model where decision makers relied on the real world’s acting exactly as the analysts’ model predicted it would. Another challenge associated with the roles of analysts and decision makers can occur when decision makers do not separate the decision to invest in the modeling process from the decision to use the model, thereby not viewing the investment in the modeling process as a sunk cost. Analysts could aid in combating these challenges by clearly communicating both the uncertainties embedded in their models and the limitations of those models, and by encouraging decision makers’ critical evaluation of both the model presentations and (to the extent feasible) the models themselves. 4.3 Factors External to the Modeling Process Factors external to the modeling process may impose perceptual and cognitive challenges on how decision makers aid in constructing and interpreting models. These factors might include recent or current events, political pressures on

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the decision maker, and expectations that others have of the decision maker. In two of the case studies, lay participants were heavily involved in the model construction process. In these cases, the backgrounds of the participants and their viewpoints may have influenced the construction of the model, leading to challenges associated with building models that confirm participants’ preconceptions or desires. In building the San Joaquin air quality model, industry representatives were much more heavily represented than public interest and community groups. Such a discrepancy in participation may lead to a model whose outputs unfairly favor the interests of the well-represented group. In the Obergurgl project, stakeholders with various backgrounds were in a position to try to structure the model to benefit their own interests. The cross-section of participants in that project, however, is likely to have tamed the impact of any one group’s interests. Analysts need to understand the basis for the information they receive from decision makers and affected stakeholders: if uncritically used, this information may lead to models that simply confirm the conceptions decision makers already had of the situation. The structuring of the Netherlands estuary project model presentation allowed decision makers to view the model outputs in a way that confirmed their preconceptions or desires. Because the model presentation did not aggregate impacts, decision makers could argue for or against solutions based on how they were expected to affect factors of interest to them. In this case, analysts must be aware that certain model presentations may lend themselves to decision makers’ interpreting the presentation to confirm their preconceptions. Decision makers may be influenced by outside pressures to make decisions that do not drastically alter the status quo. Although a model might justify a decision maker’s altering the status quo, the decision maker might have difficulty in choosing such a course of action. For instance, in the CAPM model, any policy change would require some subset of military personnel to receive lowered wages and/or bonuses despite another subset’s having higher wages and/or bonuses. As decision makers can directly manipulate the inputs and structure of the CAPM model, they could be driven to structure the model in such a way that it produces status quo outputs. In such a case, during the initial structuring of the modeling process the participants would do well to consider how much direct control decision makers should have over the model’s inputs. Of course, in many cases such consideration might be subject to the decision maker’s veto, and might therefore be unlikely to happen. Finally, if a modeling project is driven by a dramatic and/or catastrophic event, as in the Netherlands estuary project, model construction and interpretation may be heavily driven by this external event. Because of the nature and recency of the event, modelers may overweight the chances of such an event occurring again and decision makers may seek solutions that would minimize any repeat occurrence of the event. Analysts can best serve the overall goal of informed decisions by understanding and communicating the impact that such an event can have on both model formulation and interpretation, although that may be very difficult to do.

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4.4 Recommendations and Limitations Here is a condensed checklist of recommendation for analysts, based on the above discussion. • Weigh carefully whether to provide default values and assumptions in a model to the user: they can influence the input data and assumptions a decision maker uses when interacting with the model. • Understand that the way model outputs are framed (i.e., positively or negatively) can influence how a decision maker compares alternatives. • If decision makers are to interact with a model, determine the extent and implications of the various uncertainties in the input values they will provide. • When determining the method of model presentation, understand that each method of presentation requires weighing of tradeoffs in the types of challenges to which the method is subject. • Clearly communicate model uncertainties, assumptions, biases, and limitations. • Encourage decision makers to understand and critique the model. • Maintain awareness that decision makers may provide inputs or evaluate model presentations to provide justification for maintaining a status quo situation or to confirm previous beliefs about the problem or system. • Understand that recent or dramatic events can be overestimated in probability or can heavily influence how a model presentation is interpreted. This exploratory work has several limitations. First, the small number of cases examined and the nature of the documentation available almost certainly provide only a partial view of the perceptual and cognitive challenges that affect decision makers’ use of formal models as a part of their decision-making process. To fully understand these challenges, it may be necessary to conduct a series of prospective case study analyses of modeling projects. This prospective type of analysis may then lead to a more informative set of recommendations for analysts. Second, this work looks solely at the perceptual and cognitive challenges that might impede how a decision maker understands and uses a model. The impact of other factors, such as the degree of involvement of the decision maker in the modeling processes, will also be important for further understanding of how decision makers receive and use formal models.

5 Conclusions Formal models can add significant value in assisting informed decision making, but the creation of these models does not necessarily produce informed decisions. Decision makers using such models need sufficient analytic tools— models in this case—to supplement their decision-making skills, as well as the ability to use these tools appropriately. Although there is a significant body of

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work addressing how to build these models, there is a lack of empirical work examining how decision makers understand and use them. To fully participate in the decision-making process, lay decision makers need to understand the presentation or conveyance, by the analyst, of the formal model that is likely to support their decision. Many factors probably influence this understanding, significant among these are perceptual and cognitive challenges to understanding. The “psychological traps” framework of Hammond et al. [14] provides a helpful set of themes for retrospective analysis to understand the role of these challenges. Analysis of the following five case studies disclosed evidence for seven of the eleven psychological traps listed in Table 1. 1. 2. 3. 4. 5.

Compensation, accessions, and personnel management (CAPM) model, Navy shipbuilding and force structure analysis tool, San Joaquin Valley air pollution control, Obergurgl: a microcosm of economic development, and POLANO: protecting an estuary from floods.

From this analysis we concluded that the psychological traps identified in the case studies stemmed from three factors: (A) the nature of the model presentation, (B) the roles of the analyst and decision maker in the modeling process, and (C) factors external to the modeling process. We offered recommendations to help analysts improve the presentation of their models. By identifying perceptual and cognitive traps along with their sources, this work provides insight into not only what challenges are likely to exist when decision makers use models as a part of their decision-making process, but also how the structure of the modeling process and the model presentation allow these challenges to arise. That insight can then aid analysts to combat the traps, and thus to improve the both the effectiveness of the modeling process and the quality of the resulting decisions.

Acknowledgments The work of the first author was supported by a National Science Foundation (NSF) Graduate Research Fellowship, and that of the second author was supported in part by the Air Force Research Laboratory under agreement number FA9550-04-1-0192, and in part by the U.S. Army Research Laboratory and the U.S. Army Research Office under grant number DAAD19-01-1-0502. The U.S. Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the sponsoring agencies or the U.S. Government.

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References 1. M. V. Arena, J. F. Schank, and M. Abbott. The Shipbuilding and Force Structure Analysis Tool: A User’s Guide. RAND Corporation, Santa Monica, CA, 2004. 2. J. A. Ausink. User’s Guide for the Compensation, Accessions, and Personnel Management (CAPM) Model. RAND Corporation, Santa Monica, CA, 2003. 3. J. A. Ausink, J. A. K. Cave, and M. J. Carrillo. Background and Theory Behind the Compensation, Accessions, and Personnel Management (CAPM) Model. RAND Corporation, Santa Monica, CA, 2003. 4. J. A. Ausink and A. A. Robbert. A Tutorial and Exercises for the Compensation, Accessions, and Personnel Management (CAPM) Model. RAND Corporation, Santa Monica, CA, 2003. 5. P. L. Bernstein. Against the Gods: The Remarkable Story of Risk. John Wiley & Sons, New York, 1996. 6. K. Borcherding and R. E. Schaefer. Aiding decision making and information processing. In M. Irle, editor, Decision Making: Social Psychological and Socioeconomical Analysis, chapter 17, pages 627–673. Walter de Gruyter, Berlin, 1982. 7. D. B. Bromley. Academic contributions to psychological counselling. 1. A philosophy of science for the study of individual cases. Counselling Psychology Quarterly, 3:299–307, 1990. 8. D. G. Cacuci. Sensitivity and Uncertainty Analysis. Chapman & Hall/CRC, Boca Raton, FL, 2003. 9. J. C. Davies. Comparing Environmental Risks: Tools for Setting Government Priorities. Resources for the Future, Washington, DC, 1996. 10. J. R. Feagin, A. M. Orum, and G. Sjoberg. A Case for the Case Study. University of North Carolina Press, Chapel Hill, 1991. 11. J. D. Fine and D. Owen. Technocracy and democracy: Conflicts between models and participation in environmental law and planning. Hastings Law Journal, 56:901–982, 2005. 12. B. F. Goeller, A. Abrahamse, J. H. Bigelow, J. G. Bolten, D. M. DeFerranti, J. C. DeHaven, T. F. Kirkwood, and R. Petruschell. Protecting an estuary from floods. Vol. 1, Summary report: A policy analysis of the Oosterschelde. Technical Report R-2121/1-NETH, RAND Corporation, Santa Monica, CA, 1977. 13. M. Greenberger, M. A. Crenson, and B. L. Crissey. Models in the Policy Process: Public Decision Making in the Computer Era. Russell Sage Foundation, New York, 1976. 14. J. S. Hammond, R. L. Keeney, and H. Raiffa. Smart Choices: A Practical Guide to Making Better Decisions. Harvard Business School Press, Boston, 1999. 15. C. S. Holling. Adaptive Environmental Assessment and Management. John Wiley & Sons, Chichester, England and New York, 1978. 16. I. L. Janis and L. Mann. Decision Making: A Psychological Analysis of Conflict, Choice, and Commitment. Free Press, New York, 1977. 17. D. Kahneman, P. Slovic, and A. Tversky. Judgment Under Uncertainty: Heuristics and Biases. Cambridge University Press, Cambridge, UK; New York, 1982. 18. R. L. Keeney and H. Raiffa. Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge University Press, Cambridge, England, 1993. 19. J. G. March. Bounded rationality, ambiguity, and the engineering of choice. The Bell Journal of Economics, 9:587–608, 1978. 20. National Research Council. Improving Risk Communication. National Academy Press, Washington, DC, 1989.

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21. A. Onatski and N. Williams. Modeling Model Uncertainty. National Bureau of Economic Research, Inc., March 2003. 22. H. Raiffa. Decision Analysis: Introductory Lectures on Choices Under Uncertainty. Addison-Wesley, Reading, MA, 1968. 23. H. Raiffa. The Art and Science of Negotiation. Belknap Press of Harvard University Press, Cambridge, MA, 1982. 24. R. G. Sargent. Verification and validation of simulation models. In S. Chick, P. J. Sanchez, D. Ferrin, and D. J. Morrice, editors, Proceedings of the 2003 Winter Simulation Conference, pages 37–48, 2003. 25. A. Tversky and D. Kahneman. The framing of decisions and the psychology of choice. Science, 211:453–458, 1981. 26. University of Maryland. EnviRN. Online document, July 2005. http://envirn. umaryland.edu/. 27. C. J. Walters. Adaptive Management of Renewable Resources. Macmillan, New York, 1986. 28. R. K. Yin. Applications of Case Study Research, volume 34 of Applied Social Research Methods Series. Sage, Thousand Oaks, CA, 2003. 29. R. K. Yin. Case Study Research: Design and Methods, volume 5 of Applied Social Research Methods Series. Sage, Thousand Oaks, CA, third edition, 2003.

Agricultural Decision Making in the Argentine Pampas: Modeling the Interaction between Uncertain and Complex Environments and Heterogeneous and Complex Decision Makers Guillermo Podest´ a1 , Elke U. Weber2 , Carlos Laciana3 , Federico Bert4 , and 1 David Letson 1

2

3

4

Rosenstiel School of Marine and Atmospheric Science, University of Miami, USA [email protected], [email protected] Columbia University, Department of Psychology and Graduate School of Business, and Center for Research on Environmental Decisions (CRED), USA [email protected] Facultad de Ingenier´ıa, University of Buenos Aires, Argentina [email protected] Facultad de Agronom´ıa, University of Buenos Aires, Argentina [email protected]

Summary. Simulated outcomes of agricultural production decisions in the Argentine Pampas were used to examine “optimal” land allocations among different crops identified by maximization of the objective functions associated with expected utility and prospect theories. We propose a more mathematically tractable formulation for the prospect theory value-function maximization, and explore results for a broad parameter space. Optimal actions differ among some objective functions and parameter values, especially for land tenants, whose enterprise allocation is less constrained by rotations. Our results demonstrate in a nonlaboratory decision context that psychologically plausible deviations from EU maximization matter.

1 Introduction The world faces the dual challenge of feeding a burgeoning 21st century population of perhaps 9 billion, while at the same time not depleting its ecosystems that sustain life and well-being. In recent decades, agricultural output succeeded in outpacing human population growth and has reduced famine. As the food supply must continue to expand, however, it must do so with reduced environmental consequences [29]. New environmental information and its innovative usage will be central to this expansion. Agricultural stakeholders consistently rank climate variability among the top sources of risk to production or profits. The use of climate information

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worldwide is evolving. Whereas decisions used to be based on analysis of historical records, now there is an increasing capability to monitor and predict seasonal regional climate. The increase in scientific and technological capabilities, an increasing appreciation for the importance of climate on human endeavors (including sustainable development, poverty mitigation, and food security), and a greater demand for climate information are all providing greater incentives for the provision of climate services, which can be defined as the timely production and delivery of useful climate data, information, and knowledge to decision makers [7,30]. On seasonal-to-interannual scales, the El Ni˜ no–Southern Oscillation (ENSO) phenomenon is the major single source of climate variability in many parts of the world [38]. The emerging ability to forecast regional climate based on ENSO [2,11,23] offers agricultural decision makers the opportunity to mitigate unwanted impacts and take advantage of expected favorable conditions [15,18,27,28]. However, any efforts to foster effective use of climate information and forecasts in agriculture must be grounded in a firm understanding of the goals, objectives, and constraints of decision makers in the target system, for three reasons. First, climate data, forecasts, and technical assistance with the use of climate information are often publicly provided and highly subsidized. Estimates of the economic value of climate information and forecasts help justify investments in such publicly provided technology and infrastructure by comparing rates of return to those available from investments in other innovations. Research that estimates value of information (VOI) by simulating optimal forecast responses can provide useful insights, but actual use of climate information in agricultural production decisions and the production decisions themselves will most likely deviate from the prescriptions of normative models. Tests and validations of alternative descriptive models of risky decision making and probabilistic information use are thus crucial to obtaining realistic estimates of the value added from climate information. Estimates of the value of climate information should be based on alternative models closely linked to observed decision processes. The impact of alternative assumptions about decision processes and goals needs to be examined. Second, the goals and objectives of farmers’ decisions (i.e., their objective functions, in decision-theoretical terms) influence how climate information (both historical data and forecasts) is used. In turn, this has implications for how climate information should be presented and communicated (i.e., the design of climate forecasts and tutorials on climate information use). Decisions on the current contents and formats of climate forecasts make implicit assumptions about what farmers are trying to achieve and how such information will be used. It will be useful to make these assumptions explicit and put them to test. The probabilistic nature of climate forecasts needs emphasis and explanation for all users, as probabilistic thinking is a relatively recent evolutionary accomplishment [13] and not something that comes naturally to even highly trained professionals [8]. Nevertheless, the expectation of a

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deterministic forecast that will turn out to be either “correct” or “false” is especially damaging in situations where the decision maker will experience postdecisional regret after believing that she acted on a “false” forecast. Better understanding of the outcome variables that matter to farmers also will provide guidelines on whether and how best to “translate” climate forecasts. If, for example, crop yields or the costs of production input get particular attention, it makes sense to “translate” a climate forecast into the agronomic yield and/or cost implications that it holds. Third, decision makers in numerous domains have been shown to have poor insight into their own decision processes and goals and objectives. This offers opportunities for interventions to help farmers to enhance their decisions. When made aware of the objective function and goals implicit in their past decisions, decision makers tend to react in one of two ways. Some are surprised by identified objectives and the associated cues or information they are using in their decisions. Furthermore, once aware of these objectives and cues, these decision makers may wish they were not using them: examples may include unconscious gender discrimination in hiring decisions, or possibly crop yield maximization rather than profit maximization in farm production decisions. Other decision makers may concur with identified goals, objectives, and their associated information cues once made apparent to them, and refuse to give up on them (e.g., greater sensitivity to losses than to gains), even if they violate normative models. Identification of objective functions and decision goals will provide feedback to farmers about their implicit decision processes, which can then be reviewed and either explicitly acknowledged and accepted, or rejected, leading to the modification of decision processes.

2 Background 2.1 Choice Theories: Expected Utility and Prospect Theory The work by von Neumann and Morgenstern [40] provided an explicit formulation of expected utility (EU) and an axiomatic foundation. Subsequent extensions and variations are described by Schoemaker [37]. The EU model has been central in the analysis of choice under risk and uncertainty. It has been successful not only because of its compelling axiomatic foundation and ability to describe economic choices, but also because of its mathematical tractability [41]. Despite its obvious strengths, EU maximization as the (sole) objective of risky choice has encountered some opposition in recent years. There is both experimental and real-world evidence that individuals often do not behave in a manner consistent with EU theory [4,24]. A central assumption of EU theory is that the utility of decision outcomes is determined entirely by the final wealth they generate regardless of context, that is, that it is an absolute or reference-independent construct. Yet, decision-makers’ evaluation of outcomes appears to be influenced by a variety of relative comparisons [19].

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Prospect theory (PT) [20] and its modification, cumulative prospect theory [9,39], currently have become the most prominent alternatives to EU theory. Prospect theory formalizes one type of relative comparison observed when decision makers evaluate the utility of decision outcomes. Its value function V (·) is defined in terms of relative gains or losses, that is, positive or negative deviations from a reference point. Value therefore is determined by changes in wealth, rather than reference-independent states of wealth as in utility theory [19]. Furthermore, the value function for losses is steeper than the value function for gains, resulting in a sharp kink at the reference point. This feature of the value function models the phenomenon of loss aversion, that is, the observation that the negative experience or disutility of a loss of a given magnitude is larger than the positive experience or utility of a gain of the same magnitude. Empirical studies have consistently confirmed loss aversion as an important aspect of human choice behavior [4,5,36]. Rabin [33] emphasized the growing importance of loss aversion as a psychological finding which should be integrated into economic analysis. 2.2 EU Formulation We define a risky prospect q = (p1 , w1 ; . . . ; pn , wn ) as the ensemble of possible wealth/outcome values wi with associated probabilities pi that are nonnegative and add up to one. A common formulation (p. 104 of [16]) states that a decision maker evaluates the expected utility of prospect q as X EU (q) = pi u(wi ) . (1) i

The following real-valued utility function u(·) is given by Pratt [32] as  w1−r if r 6= 1 1−r u(w) ∝ , ln w if r = 1

(2)

where r is the coefficient of constant relative risk aversion (CRRA). CRRA implies that preferences among risky prospects are unchanged if all payoffs are multiplied by a positive constant [16]. The curvature of the utility function, defined by parameter r, captures all information concerning risk attitude. 2.3 Prospect Theory Formulation In prospect theory [20], the subjective value of a prospect is defined as X V (q) = Ω(pi )v(∆wi ) ,

(3)

i

where ∆wi represents the difference between outcome wi and a reference point wref , a free parameter that separates perceived gains from perceived losses.

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The subjective evaluation of this difference can be expressed as suggested by Tversky and Kahneman [39], using for simplicity the same exponent for losses and gains (which tends to be a good approximation based on empirical estimates of the two parameters and which can, of course, be changed to the more general case, if so desired). That assumption allows us to write the PT value function in the following more compact form X v(∆w) = h(∆w)|∆w|α , (4) where function h(∆w) is the step function  1 if ∆w ≥ 0 h(∆w) = , −λ if ∆w < 0

(5)

and λ is a parameter (λ > 1) that reflects the degree of loss aversion. The exponent α in (4) ranges between 0 and 1 and describes the nonlinearity of the value function. Because of the discontinuity at the reference point, the exponent describes the degree of risk aversion (concavity) in the gains region and the degree of risk seeking (convexity) in the losses region. The evaluation of risky prospects is based on subjective probability weights that typically do not correspond to the objective probabilities. Tversky and Kahneman [39] propose the nonlinear function Ω(p), Ω(p) =

pγ 1/γ

(pγ + (1 − pγ ))

,

(6)

to model the subjective weight of event probabilities, which overweights objective probabilities for outcomes at the extremes of the distribution of possible outcomes and underweights outcomes in the middle. The value of Ω(p) depends on positive parameter γ, which must be empirically estimated.

3 A Case Study We compare and contrast the objective functions or choice criteria associated with expected utility and prospect theory in a real-world optimization problem in agricultural management. The decisions examined are related to the production of cereals and oilseeds in the pampas region of central-eastern Argentina, one of the most important agricultural regions in the world [14]. In particular, we examine the nature and magnitude of differences among simulated agricultural production decisions identified as “optimal” by maximization of the objective functions associated with EU and PT. EU maximization is a widely used criterion in agricultural economics, and thus is a useful benchmark against which to compare the results of other objective functions. We argue that as proven and mathematically tractable alternatives to the EU model become available, agricultural and resource economists should at least

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begin to consider alternative objective functions and explore how they might improve analysis and insight [41]. The case study is organized as follows. First, we describe the agricultural production systems in the target region. We then define a set of cropping enterprises that encompasses a realistic range of initial soil conditions and management options for the typical crops in the region, namely maize, soybean, and a wheat–soybean doublecrop (wheat followed during the same cropping cycle by a shorter-cycle soybean). Next we describe how yields and economic returns are simulated for each cropping enterprise using historical climate data, biophysical models, and realistic cost estimates. These results are subsequently used as input to optimization procedures. Finally we show and discuss optimal enterprise allocation for the two objective functions considered. 3.1 The Area of Study The geographic focus of this study is the region of central-eastern Argentina known as the pampas, one of the most productive agricultural areas in the world [14] and of major importance to the Argentine economy (51% of exports, and 12% of GDP over 1999–2001, [6]). The climate, soils, and cropping systems of the Argentine pampas have been characterized by Hall et al. [14]. In particular, we focus on the region near Pergamino (33◦ 560 S, 60◦ 330 W), the most productive subregion of the pampas [31]. Two characteristics of agricultural production in the study region have implications for the optimization described below. First, agriculture in the Pampas is marketoriented and technology-intensive. As a consequence, a broad spectrum of agronomic management options exists and can be explored in the optimization process. Second, a considerable proportion of the area currently farmed is not owned by the farmers exploiting it. Very short land leases (usually one year) provide incentives for tenants to maximize short-term profits via highly profitable crops. In contrast, landowners tend to rotate crops to steward longterm sustainability of production and soil quality [22]. Given the differences in decision-making goals and constraints between landowners and tenants, we model the two groups separately. 3.2 Crop Enterprises We defined 64 different cropping enterprises that reflect a realistic range of cultivation options for the study area. Each enterprise involves the combination of (a) a given crop (maize, full-cycle soybean, and wheat–soybean), (b) various agronomic decisions (cultivar/hybrid, planting date, fertilization options), and (c) a set of initial conditions (water and nitrogen in the soil at planting) that result from previous production decisions. That is, several enterprises may be associated with the same crop, although involving different management options.

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3.3 Simulation of Yields: Agronomic Models Yields for each enterprise were simulated using the crop models in the decision support system for agrotechnology transfer (DSSAT) package [17]: genericCERES [34] for maize and wheat, and CROPGRO [3] for soybean. These models have been calibrated and validated under field conditions in several production environments including the pampas [12,25,26]. The information required to run the DSSAT models includes: (i) daily weather data (maximum and minimum temperature, precipitation, solar radiation), (ii) “genetic coefficients” that describe physiological processes and developmental differences among crop hybrids or varieties, (iii) a description of crop management, and (iv) soil parameters, including soil moisture and N content at the beginning of simulations. Historical (1931–2003) daily weather data for Pergamino provided information about category (i). Genetic coefficients, the management options that defined the enterprises, and likely ranges of initial soil conditions were provided by the Asociaci´ on Argentina de Consorcios Regionales de Experimentaci´on Agr´ıcola (AACREA), a nonprofit farmers’ group (similar in goals to the U.S. Agricultural Extension Services) that partnered with us in this study. Simulations assumed no irrigation, a very infrequent practice in the pampas. For each enterprise, 72 simulated yields were obtained (one for each cropping cycle in the 1931–2003 historical weather record used). 3.4 Simulation of Economic Outcomes Economic outcomes were simulated for a hypothetical 600-hectare farm, the median size of AACREA farms in the Pergamino region. We computed net economic returns per hectare πij for year i and enterprise j as the difference between income and costs: πij = Yij Pj − (Fj + Vij + Si + Ti ) .

(7)

Gross incomes per hectare Yij Pj were the product of simulated yield for a year and enterprise (Yij ) and a constant output price for each crop (Pj ). Assumed output prices were the median of 2000–2005 prices during the month when most of the harvest is marketed (April, May, and January for maize, soybean, and wheat, respectively). After deducting export taxes charged by the Argentine government, these prices were 78.9, 166.0, and 112.0 US $ ton−1 for maize, soybean, and wheat, respectively. Four different kinds of costs were involved in the computation of net returns per hectare: (i) Fixed costs Fj for enterprise j are independent of yield. For landowners, fixed costs included: (a) crop production inputs (e.g., fertilizer, seed, field labor), and (b) farmer’s salary, health insurance, and a fixed fiscal contribution. For land tenants, fixed costs also included (c) land rental (assumed to be 232.5 $ ha−1 , equivalent to the price of 1.4 tons of soybean) and (d) management costs (12 $ ha−1 ). (ii) Variable costs Vij are a function

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of yield on year i for enterprise j. These costs included: (a) harvesting costs, estimated as 8% of gross income (Yij Pj ), (b) transportation costs (about 10 $ ton−1 ), and (c) sales tax and commissions, estimated as 8% of gross income. Variable costs were the same for landowners and tenants. (iii) Structural costs Si are applicable only to landowners and covered: (a) maintenance of farm infrastructure, (b) real estate taxes, and (c) management and technical advice. Structural costs are independent of farm activities or enterprise yields. For the sake of simplicity, however, they were approximated following a criterion used by AACREA: they were a percentage (23%, 18%, and 20% for maize, soybean, and wheat–soybean, respectively) of income per ha after subtracting variable costs (Yij Pj − Vij ). Because structural costs are incurred even if part of the farm is not cultivated, an implicit but not unreasonable assumption, given the high costs of land around Pergamino, is that the entire 600-ha area of the hypothetical farm is cultivated. (iv) Income tax Ti applies equally to landowners and tenants and was computed as follows.  b(π − a) + c if π ≥ a T = , (8) c if π < a where a is a threshold income above which farmers pay an average tax rate b = 0.32. Below a, farmers pay a minimum tax assumed to be 59.33 $ ha−1 . To simplify calculations, an average annual income π of 177.5 $ ha−1 (57.6 $ ha−1 ) was assumed for owners (tenants). 3.5 Optimization Procedure A whole farm production model was used to identify optimal decisions for the objective functions associated with EU and prospect theories. The choice variable in the optimization is the vector x = (x1 , . . . , x64 ) that includes the area in the 600-hectare hypothetical farm allocated to each of the 64 alternative cropping enterprises considered. Different land amounts allocated to the 64 enterprises were considered by the optimization of each objective function. The optimization was performed using algorithm MINOS5 in the GAMS software package [10]. For comparability, all objective functions are expressed in terms of a decision-maker’s wealth, either in an absolute sense (for EU and regretadjusted EU), or as a difference from a specified reference level (in prospect theory). The total wealth of a decision maker at the end of cropping year i is wi = w0 + πi ,

(9)

where w0 is the decision-maker’s initial wealth (i.e., prior to production decisions for year i) and πi is the farmwide income during year i, after deducting costs. Farmwide income πi is calculated as πi =

m X j=1

xj πij ,

(10)

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where πij is the net margin for year i and enterprise j (7) and xj is the amount of land allocated to enterprise j (i.e., a component of the land allocation vector x). Expected Utility Optimization The expected utility (1) of final wealth can be expressed as: EU (x) =

n X

pi u [wi (x)] ,

(11)

i=1

where pi is the probability of a given climate scenario for year i. A climate scenario is defined as the climate conditions over an entire production cycle. We assume that all climate scenarios in the historical record have the same probability (i.e., pi = 1/n, where n is the number of cropping cycles in the historical climate data, in this case, 70 years). Therefore, we can write n

EU (x) =

1X u [wi (x)] . n i=1

(12)

The next step is the optimization max EU (x) = EU (x? ) , x

(13)

where x? = (x?1 , . . . , x?64 ) indicates the proportion of land allocated to each enterprise that maximizes the value of EU. Prospect Theory Value Optimization In prospect theory, value is defined by changes in wealth rather than referenceindependent wealth states. Outcomes wi are evaluated as gains or losses with respect to reference value wref : ∆wi = wi − wref .

(14)

One plausible reference value of wealth that determines whether a farmer thinks of another wealth level as a gain or a loss is the income wr that a farmer could achieve with minimal effort (e.g., by renting his land) added to the decision-maker’s initial wealth: wref = w0 + wr .

(15)

Combining (9) and (14) with (15) we obtain: ∆wi = πi − wr .

(16)

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The total value function for prospect theory (3) then can be rewritten as V (x) =

n X

Ω(pi )v[∆wi (x)] .

(17)

i=1

As for EU, all climate scenarios are assumed to have the same probability (i.e., pi = 1/n), therefore Ω(pi ) is independent of i. Rewriting (17), we obtain   n X 1 V (x) = Ω v[∆wi (x)] , (18) n i=1 which indicates that the constant Ω(1/n) is irrelevant for the optimization; thus one need not worry about the functional form of Ω. The optimization is performed in a way analogous to (13): max V (x) = V (x? ) . x

(19)

Optimizing the value function with the GAMS software [10] available to us was problematic because of the discontinuity of function h(·) (defined in (9)) at ∆wi = 0 (where prospect theory’s value function has a sharp kink and is not differentiable). To address the problem, we used a continuous function ˜ that is numerically equivalent to h(·): h(·) ˜ h(x) = 1/2 [1 − λ + (1 + λ) tanh(%x)] ,

(20)

where % is an arbitrary parameter such that % > 1; large values of % (we used % = 10) reproduce function h(·) more closely. 3.6 Optimization Constraints Allocation of land to cropping enterprises differs for landowners and tenants in the Pergamino region. Landowners tend to adhere to a rotation of crops that offers advantages for soil conservation and control of pests and diseases [22]. In contrast, land tenants seek high profits during short leases (usually one year) and thus usually select enterprises with the greatest economic returns. The clear differences in enterprise allocation between land tenure regimes suggest that we explore optimal decisions separately for landowners and tenants. With three major cropping systems (maize, soybean, and a wheat–soybean double crop) the rotation advocated by AACREA allocates about 33.3% of the land to each of these cropping systems in a given year. To allow owners some flexibility in land allocation, we introduced two constraints in the optimization procedure: land assigned to a crop could be no less than 25%, or more than 45% of the farm area. These constraints did not apply to land tenants, who could allocate the entire farmed area to a single crop. The lack of allocation constraints is consistent with the observed increase in monocropping of soybean that has occurred in the pampas in the last few years [35]. A final constraint specified that 100% of the land had to be assigned to some enterprise (i.e., no land could be left without cultivation).

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3.7 Parameter Space Explored for Each Objective Function Each objective function has a set of parameters. In some cases, the value of a given parameter describes a personality characteristic (e.g., degree of risk aversion or loss aversion) that may vary among decision makers. With no widely accepted values for parameters, a broad range of plausible values should be considered. In this section, we describe and justify our choice of central (or nominal) parameter values. Expected Utility The expected utility function has two parameters: (i) the decision-maker’s initial wealth w0 and (ii) the risk-aversion coefficient r. Initial wealth w0 is defined as liquid assets. For landowners, this quantity was estimated as 40% of the value of the farm land. The definition is based on the assumption that a farmer will not sacrifice future income potential by selling crop land, but can borrow up to 40% of her land value. The 1994–2003 average value of land for Pergamino was 3541 $ ha−1 , making w0 equal to 1400 $ ha−1 (3541 $ ha−1 × 0.4). For land tenants, we assumed a w0 value of 1000 $ ha−1 , the liquid assets required to finance two complete cropping cycles (i.e., in case of a total loss in one cycle, the farmer still has capital to fund a second cycle). For the risk-aversion coefficient r, we followed Anderson and Dillon’s [1] classification: 0.5 is hardly risk averse; 1.0, somewhat risk averse (normal); 2.0, rather risk averse; 3.0, very risk averse; and 4.0, extremely risk averse. We also included risk indifference by considering r values of 0.0. The range of r values was the same for owners and tenants. Prospect Theory Value Function The value function is defined by (i) a reference wealth wr that separates outcomes perceived as gains and losses, (ii) a risk preference parameter α, and (iii) a loss aversion parameter λ that quantifies the relative impact of gains and losses. The combination of all three parameters defines risk aversion in PT. For landowners, wr was estimated as the income easily achieved by renting out the land instead of farming it. This value of wr was estimated to be 232.5 $ ha−1 (a rental fee of 1.4 ton ha−1 of soybean times a price of 166 $ ton−1 ). For land tenants, wr was estimated as the income obtained by placing the tenant’s initial wealth (w0 = 1000 $ ha−1 , as described for EU) in a bank for six months (the duration of a cropping season) at an annual interest of 4% (representative of current rates in Argentina). The nominal wr value, then, was 20 $ ha−1 . For the risk-aversion parameter α and the lossaversion parameter λ, we used the values empirically estimated by Tversky and Kahneman [39] of 0.88 and 2.25, respectively, for both owners and tenants, but also explored a broader range of values.

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4 Results This section describes the land allocations (i.e., the proportion of land assigned to different enterprises) identified as optimal for each objective function. Only seven out of 64 possible cropping enterprises were selected by the various optimizations. 4.1 EU Maximization Landowners The enterprise allocation that maximized expected utility for landowners was constant for the full range of initial wealth and risk aversion values explored (Figure 1). The maximum area allowed for one crop by the optimization constraints defined for owners (45% of total land) was allocated to full-cycle soybean Soy14, the enterprise with the highest average economic returns (x = 188.1 $ ha−1 ) over the 70 simulated cropping cycles. Conversely, the minimum area required by constraints (25%) was for maize, the crop with lowest average profits. Ma23, the enterprise with the highest average profits for this crop (x = 116.5 $ ha−1 ) was selected. The remainder of the area (30%) was allocated to the wheat–soy enterprise SW21, which had average profits between those of full-cycle soy and maize (x = 168.8 $ ha−1 ). The stability of results for all parameter combinations illustrates the importance of ecological or logistic constraints associated with maintaining a crop rotation: these constraints clearly override any financial or personality characteristics of a decision maker. Land Tenants For land tenants, only two enterprises (full-cycle soybean Soy14 and wheat– soybean SW21) were involved in the maximization of expected utility. Because of the markedly lower economic profits of maize (due to higher production costs and low prices) and the lack of areal constraints for tenants, this crop did not appear at all in the optimal land allocations. The relative proportions of the two selected enterprises (Soy14 and SW 21) depended on the combination of parameters. Figure 2 has four panels with increasingly higher levels of risk aversion r (from top to bottom). In each panel, the optimal allocation of land is shown as a function of initial wealth w0 . For a risk-neutral decision maker (r = 0; Figure 2, upper panel), the optimal action was to allocate the entire area to the double crop enterprise SW21; this result is constant for the entire range of values considered for w0 . Because the decision maker is risk-neutral, the selection of SW21 was based only on its higher mean profit relative to Soy14 (77.6 $ ha−1 versus 69.4 $ ha−1 ), and ignored the higher risks associated with the considerably larger dispersion of profits (122.0 $ ha−1 versus 89.0 $ ha−1 for Soy14). For moderate risk aversion

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Fig. 1. Land allocation (as proportion of the hypothetical 600-ha farm) that maximizes expected utility for landowners. The selected combination of enterprises is constant for all initial wealth w0 and risk aversion r.

values (r = 1.0; Figure 2, second panel from top) and low w0 (below about 1100 $ ha−1 ), the optimal action involved about 75% of the land allocated to SW21 and about 25% to Soy14. For higher w0 values, the optimal action was to allocate the entire area to the double crop enterprise SW21. When slightly higher amounts of risk aversion are considered (r = 1.5; Figure 2, third panel from top), the optimal action involved diversification of enterprises for most values of w0 . For low w0 , diversification is highest: 60% of the land was allocated to SW21 and 40% to Soy14. As w0 increases (and, thus, decision makers can afford higher financial risks), the proportion of land assigned to SW21 grew until this enterprise occupied the entire area, resembling results for risk-neutrality. In other words, increasing initial wealth compensates, to some degree, for the effects of risk aversion. Finally, for a highly risk-averse decision maker (r = 3.0; Figure 2, bottom panel), the optimal land allocation was fairly conservative, as maximum crop diversification (comparable proportions of SW21 and Soy14) prevailed throughout the w0 range. 4.2 Prospect Theory Landowners The land allocation that maximized PT’s value for landowners was fairly similar to results for EU. As for EU, full-cycle soybean Soy14 was the enterprise with the largest area (45%). The area allocated to maize (25%) was again the minimum required by optimization constraints. Unlike EU, though, three different maize enterprises (Ma21, Ma23, and Ma24) were selected for

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Fig. 2. Land allocation (as proportion of the hypothetical 600-ha farm) that maximizes expected utility for land tenants. The four panels show the results for risk neutrality (r = 0, upper panel), small risk aversion (r = 1.0), moderate risk aversion (r = 1.5), and pronounced risk aversion (r = 3.0, bottom panel), in each case plotted as a function of initial wealth w0 .

different portions of the parameter space (figure not shown). All three enterprises had very similar average returns (113.2, 116.5, and 116.3 $ ha−1 for Ma21, Ma23, and Ma24, respectively). The maize enterprise with the highest dispersion (Ma21, SD = 106.8 $ ha−1 ) prevailed for risk-seeking parameter combinations, whereas the less variable enterprise (Ma23, SD = 84.1 $ ha−1 ) was characteristic of moderate and high-risk aversion. Ma21 only appeared for intermediate reference wealth and lower risk preferences. Nevertheless, Kolmogorov–Smirnov tests showed that distributions of economic returns for the three maize enterprises were not significantly different from one another, therefore any differences in land allocation to maize can be considered minor. The wheat–soy double crop (in most cases, enterprise SW21, but also SW20)

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occupied the remaining area. As for EU, results are consistent with the relative average profitability of each crop. Furthermore, the similarity with EU results suggests that constraints associated with maintaining the crop rotation prevail over personality characteristics, and thus optimal allocations are similar even for fairly different objective functions. Land Tenants Just as for EU, the land allocation that maximized prospect theory’s value function for tenants involved two enterprises: full-cycle soybean (Soy14) and wheat–soybean (SW21). As for EU, the specific proportions of these enterprises depended on the combination of parameters. The top-left panel of Figure 3 (wr = 10 $ ha−1 and λ = 1.00) can be used as a reference to discuss the consequences of varying prospect theory’s parameters. In this panel, there is no loss aversion. Also, a low level of reference wealth puts most outcomes into the domain of gains, where low α values imply a more risk-averse decision maker. As a result, a diversified land allocation including two enterprises (Soy14 and SW21) is selected. As α increases and the decision maker becomes less risk-averse, the allocation switches toward an increasingly higher proportion of the more profitable but riskier SW21, until monoculture is reached. As we move along the top row of Figure 3, we detect a mixture of the two dominating enterprises in the central and right-top panels. Nevertheless, there is always a higher proportion of the less-risky Soy14. The conservative land allocations reflect the effect of increases in loss aversion. If we move down the left column of Figure 3, the switch from diversification to a monoculture of SW21 begins at progressively lower values of α. This is due to the fact that as wr increases, an increasing proportion of outcomes is perceived as losses, in which case α indicates risk-seeking. Risk-seeking to risk-neutral decision makers choose the riskier option (SW21) in search of higher profitability, and thus enterprise selection in the bottom-left panel is identical to that of riskneutral EU maximizers (top panel in Figure 2). This pattern is also apparent in the middle-bottom panel of Figure 3, where we now also have loss-aversion (λ = 2.25). The high reference wealth (wr = 80 $ ha−1 ) implies that a high proportion of outcomes are perceived as losses. For low α values, the decision maker is more risk-seeking and thus selects riskier SW21 in order to attain higher profits and get out of the domain of losses. As α increases, risk-seeking decreases and the selected allocation becomes diversified, as loss aversion now takes effect. The result is a higher proportion of the less variable enterprise Soy14. When loss aversion is even stronger, as in the right column of Fig 3 (λ = 3.50), this effect takes over and dictates diversification across the whole range of α, and even more so for higher levels of reference wealth, as more outcomes are in the domain of losses and hence subject to loss aversion.

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Fig. 3. Land allocation (as proportion of the hypothetical 600-ha farm) that maximizes prospect theory’s value function for land tenants. The different panels correspond to combinations of wr (reference wealth in $ ha−1 , increasing from top to bottom) and λ (loss aversion parameter, increasing from left to right). In each panel, the optimal allocation of land is shown as a function of the risk aversion coefficient α.

5 Discussion Our results demonstrate in a nonlaboratory decision context that, in some cases, psychologically plausible deviations from EU maximization lead to differences in optimal land allocation decisions. As an example, for nominal or central values of parameters (considered typical of many decision makers), EU maximization generally suggests that tenants should allocate a much greater proportion of land to the riskier SW21 enterprise than in PT value maximization. The loss aversion that is essential in PT’s formulation dictates more conservative strategies (predominance of Soy14) for this objective function. Nevertheless, in situations where there are prescribed constraints to land allocation (e.g., those associated with maintaining an ecologically sound crop rotation), results are very similar for quite different objective functions (EU and PT’s value) and for a broad range of personality and economic characteristics of decision makers. This consistency is an illustration of one of the important goals of institutions and/or social norms, namely, to make behavior more predictable.

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Optimization of any utility or value function reflects a tradeoff between the expected profits of an enterprise and its risk or dispersion of outcomes. It is interesting to see that different objective functions shape the nature of this tradeoff in different ways that are consistent with the characteristics of each function. In EU optimization, more risk-averse land allocation is encouraged by differences in risk-aversion as indicated by parameter r, and by lower initial wealth w0 . In contrast, in PT’s value optimization, risk-averse behavior is encouraged by a lower reference wealth (that divides the perception of returns into gains versus losses) and much more by the individual loss aversion parameter λ than by the individual risk preference parameter α. The importance of loss aversion is not surprising, given the centrality of this process in PT. Similarly, more risk-seeking land allocation is encouraged by different processes and parameters under the different objective functions. For EU maximization, both parameters r (less risk aversion) and w0 (greater initial wealth) are deciding factors (top panel and right end of middle panel of Figure 2). In PT value optimization, on the other hand, less risk-averse land allocations come about when the decision maker has no loss aversion but a high reference value, with the result that most outcomes are in the domain of losses, in which choices are either risk-seeking or at best risk-neutral (bottom left panel of Figure 3). 5.1 Relevance of Results We envision three main applications of the work presented here. First, an improved understanding of individual differences in preferences and objective functions (when they induce different optimal land allocations) may allow the development of agronomic advice tailored to the personality characteristics of different types of farmers. Such advice will be more effective than the common “one size fits all” agronomic recommendations. Second, knowledge of individual preferences may be helpful to guide the framing and to assess the acceptability of regional or national policies of agricultural sustainability (e.g., policies that encourage crop diversification). Finally, an understanding of production decisions in agriculture may contribute to a better understanding and thus better planning and implementation of a range of related issues, such as adoption of technological innovations and adaptation to climate change.

Acknowledgments This work was supported by an NSF Biocomplexity in the Environment grant (BE-0410348), an NSF Center grant under the Decision Making under Uncertainty Initiative (SES-0345840), and a NOAA Human Dimensions grant (GC04-159). F. Bert is supported by a doctoral fellowship from Argentina’s Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET). The authors are grateful to the management, technical advisors, and

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farmer members of the Asociaci´ on Argentina de Consorcios Regionales de Experimentaci´on Agr´ıcola for their commitment to this research.

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On Optimal Satisficing: How Simple Policies Can Achieve Excellent Results J. Neil Bearden and Terry Connolly University of Arizona, Department of Management and Organizations, 405 McClelland Hall, Tucson, AZ 85721, USA [email protected], [email protected] Summary. Herbert Simon introduced the notion of satisficing to explain how boundedly rational agents might approach difficult sequential decision problems. His satisficing decision makers were offered as an alternative to optimizers, who need impressive computational capacities in order to maximize their payoffs. In this chapter, we present a simplified sequential search problem for a satisficing decision maker, and show how to compute its optimal satisficing search policies. Our analyses provide bounds on the performance of satisficing search policies.

1 Introduction The assumptions upon which rational choice theories—including neoclassical economic theory, game theory, and the like—are built are often quite strong, and imply implausibly large computational powers in human decision makers. Simon [9,10] recognized this and proposed an alternative to the maximization objective upon which these theories rest. He suggested that actual agents try to find alternatives that are “good enough” rather than those that maximize their payoffs. He referred to the former objective as satisficing. Superficially, it seems that these objectives are at odds and that they should produce quite different decision outcomes. In the current chapter, we show that these suspicions are not warranted: satisficing, if done well, can produce outcomes on par with those obtained by maximizing. Consider the commonplace problem of trying to find a house to buy. Presumably, one would like to minimize the cost and maximize the quality of the house one chooses. Furthermore, one might wish to maximize the quality of one’s neighbors and of the surrounding schools, and to minimize the distance one will travel to work. And so on and so on. The complexity of optimally selecting a home quickly blows up when one simultaneously considers all of these factors. As traditionally conceived, a satisficing decision maker (DM) need not, for example, solve any difficult nonlinear mathematical programming problems.

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She simply decides what she will find acceptable in a home and takes the first one that meets these standards. For example, she might decide that she cannot spend more than $100,000, that she will not live more than 30 minutes from work, and that she must have a tree in her front yard. She does not worry about making complicated trade-offs among the attributes of the homes. She just tries to find one that meets her aspirations. This procedure simplifies the choice process, and the computations involved in executing it, enormously. Similar processes are commonly used by multiperson decision-making groups to avoid the problem of reaching agreement on a group multiattribute utility function. For example, the committee charged with selecting a new university president typically (and intendedly) includes representatives of a wide range of constituents, and would face a huge challenge if they needed to make explicit all the trade-offs they each made in evaluating each candidate. The task is greatly simplified if the committee can merely list minimal criteria and search for candidates that meet them all. Simon [9] suggested that DMs such as our househunter might, for a number of reasons, use such simplified (0–1) payoff functions in order to make decisions. Most important for us here, he suggested that it is often difficult to map a vector of apples (e.g., price) and oranges (e.g., driving time) to a scalar payoff because making appropriate trade-offs is difficult. To get around this, when setting out to shop, a DM can simply decide on the minimum number of apples and the minimum number of oranges she will find acceptable. It is easy to see why a DM might make decisions in a satisficing manner when alternatives are encountered sequentially. What is not clear is how well a (0–1) payoff function captures individual preferences. Presumably, three apples and three oranges at $x are preferred to two oranges and two apples at $x, all things being equal. When faced with the two alternative bundles simultaneously, the DM would choose the former. For illustration, let us suppose that a DM is faced with a problem involving just two attributes. For each, he sets an aspiration level, and an alternative that does not exceed both of his aspiration levels will be rejected. Those alternatives that are not immediately rejected are acceptable. An acceptable alternative satisfices. Figure 1 shows this graphically. The alternatives that satisfice are those that fall within the shaded region. The arrows represent the gradient of the DM’s true (non-(0–1)) payoff function at various combinations of the two attribute values. According to Simon [9], the DM might sensibly set his aspiration level for an attribute at a point where the returns on its values are rapidly diminishing. But, importantly, even within the satisficing set, the gradient is positive. Some acceptable alternatives would be preferred to others. Should the DM thus set higher aspiration levels? The satisficing DM faces a dilemma in setting aspiration levels. If they are set too high, the satisficing regime will shrink, and, ceteris paribus, the odds of finding an alternative that satisfices will decrease. On the other hand, if they are set too low, the DM will easily find an alternative, but it may be quite poor. Because by definition, a satisficing DM will take the first alternative

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she encounters whose attributes meet all of her aspiration levels, her problem lies in determining how best to set her aspiration levels. In this chapter, we present a procedure for finding aspiration levels for satisficing search policies that maximize expected payoffs. That is, given that one has settled on a satisficing approach, how can the thresholds be best set?

Fig. 1. Partitioning of the feasible decision alternatives into those that satisfice (shaded region) and those that do not (the rest). More is better on each attribute. The arrows represent the gradient field for the DM’s payoff function. Hence, her payoff is increasing in both attributes but is doing so at a decreasing rate as she approaches the acceptable region.

As we conceive it here, a multiattribute satisficing procedure substitutes a simplified, multithreshold decision rule for what is, in fact, a more complex multiattribute utility function. The simplification may be driven by a desire to avoid the complex assessments and computations involved in making explicit multiattribute trade-offs, by political considerations such as opposition to putting explicit dollar values on saving human lives, or as a device to accommodate the desires of multiple actors whose preferences may not be entirely compatible. The simplification will, in general, exact a payoff penalty. A DM might accept an option that narrowly satisfies all her criteria when another available option would have yielded a higher overall payoff—for example, by offering high values on several important attributes with low, perhaps below-threshold values on other, less important ones. It is also clear that some satisficing strategies are better than others. Some will cause the DM to search for a hopelessly ideal alternative; others will cause her to accept inferior alternatives.

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There are thus two components to the overall loss in expected payoff associated with using a satisficing rather than a maximizing strategy. One component, which we refer to as inherent penalty, is the result of substituting binary, good/bad evaluations for what are in reality continuous multiattribute utility functions. The second, which we refer to as aspiration error, is loss of expected payoff resulting from the poor choice of aspiration levels. Assessing aspiration error obviously requires us to specify what good aspiration levels would be. Thus the title, and primary purpose, of our chapter: we want to be able to specify, for one important class of problems, the set of aspiration levels that yields the best possible expected payoff to a DM who is constrained to use a satisficing strategy. We want, in short, a tool that can allow us to distinguish between payoff loss resulting from satisficing and payoff loss resulting from satisficing badly. The rest of the chapter is organized as follows. Section 2 presents a formal description of a satisficing problem involving sequential search through a finite set of decision alternatives, and also a general procedure for computing its optimal policy. In Section 3, we present a number of special cases of the general problem and show the optimal policies for each. The general problem can be specified in a very large number of ways, so we have attempted to select special cases that broadly span the space of possibilities. We examine the inherent penalty for following satisficing policies in Section 4 by presenting a procedure for solving the maximization version of our problem, and then comparing its results to those obtained by satisficing. Section 5 describes a method for computing optimal heuristic policies, which are constrained versions of the policies discussed in Sections 2 and 3. Results from applying the heuristic policies to the problems studied in Section 3 suggest that little is lost when the satisficing heuristics are constrained to be quite simple. An infinitehorizon version of the satisficing problem in which the DM pays search costs is presented in Section 6. We conclude in Section 7 and discuss some basic implications of this work.

2 Problem Statement and Optimal Policy Suppose a DM may examine as many as N decision alternatives. Each alternative n ∈ {1, . . . , N } is represented by a K-dimensional vector of random variables Xn = (Xn1 , Xn2 , . . . , XnK ), the realization of which is denoted xn = (x1n , x2n , . . . , xK n ). The vector elements correspond to the attributes of the decision alternatives. For each attribute k for each alternative n, we assume that the DM has an aspiration level θnk , which determines which of that attribute’s values she finds acceptable; specifically, an attribute is acceptable if and only if xkn ≥ θnk . The set of all stage n aspiration levels is represented by θn = {θn1 , θn2 , . . . , θnK }. An alternative n is acceptable (or satisfices) if and only if xkn ≥ θnk , ∀k. For shorthand, we use the indicator

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 σn =

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if xkn ≥ θnk , ∀k otherwise.

1 0

A DM’s payoff for selecting an alternative n, denoted φ(xn ), is some monotonically increasing function of the alternative’s attribute values. Most consistent with using aspiration levels to make selection decisions, the function should be increasing in its arguments. can be P For example, the payoff function Q additive in its arguments, φ(xn ) = k xkn ; or multiplicative, φ(xn ) = k xkn , with xkn ≥ 0, ∀n, k; or some combination such as φ(xn ) = x1n + x2n x3n , xkn ≥ 0, ∀n, k. The DM sees the N alternatives sequentially and must select one and only one of them. The alternative that is selected is the first one for which σn = 1, that is, the first one that satisfices. Once passed, an alternative cannot be recalled. The DM’s objective is to maximize his expected payoff. He controls his fate only by setting his aspiration levels. In order to set sensible aspiration levels, the DM must know something about the distribution of the alternatives in the world. For tractability, we assume that the DM has full knowledge of the multivariate distribution from which the alternatives are taken, and denote the density for a vector of attributes x at stage n by fn (x). By allowing the density to vary over n, we can capture, for example, situations in which the quality of encountered alternatives is expected to increase E[φ(Xn )] < E[φ(Xn+1 )], decrease E[φ(Xn )] > E[φ(Xn+1 )], or stay the same E[φ(Xn )] = E[φ(Xn+1 )] over time. Pairs of attributes i and j within an alternative may be correlated %i,j 6= 0 or uncorrelated %i,j = 0. But we assume that the alternative vectors themselves are uncorrelated. Given a set of aspiration levels at stage n, the probability that an alternative will be acceptable (i.e., that σn = 1) can be expressed as Z Z P (σn = 1|θn ) = ··· fn (x)dx1 · · · dxK . 1 θn

K θn

When the DM does not know whether the attributes of an alternative are acceptable, the expected payoff for selecting that alternative is denoted E[φ(X)]. When the DM does know that the alternative is acceptable, we denote the expected payoff for selecting it by E[φ(X)|σn = 1]. For an arbitrary payoff function φ, these expectations can be computed at stage n by Z Z E [φ(Xn )] = · · · fn (x)φ(x)dx1 · · · dxK and R E [φ(Xn )|σn = 1] =

1 θn

···

R 1 θn

R K θn

···

fn (x)φ(x)dx1 · · · dxK

R K θn

fn (x)dx1 · · · dxK

.

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Given that the DM will stop on the first alternative that he finds acceptable, how should he set his aspiration levels so that the expected value of his selected alternative will be maximized? Put differently, how can he maximize his expected payoff while satisficing? Note that the DM faces a straightforward multistage decision problem. And, if we can optimally solve the problem at each stage n, we can solve the DM’s full problem, from n = 1 to n = N . Given this property, we can find optimal decision policies for the DM using dynamic programming methods [3]. The line of reasoning we evoke here to solve the satisficing problem is closely related to one typically used to solve related optimal stopping problems (see, e.g., [6,7]). At any stage n in the decision S problem, the expected payoff or value for following some partial policy θn {θn+1 , . . . , θN } is denoted Vn (θn ). Because for all feasible policies the DM must accept the N th alterative, if reached, VN (θN ) = E[φ(X)]. Then, for any feasible set of aspiration levels at n < N , we have Vn (θ) = P (σn = 1|θ)E [φ(Xn )|σn = 1] + P (σn = 0|θ)Vn+1 .

(1)

For each stage n, we wish to find the θ that maximizes Vn (θ), which we denote S ? ? θn? . The value of a partial optimal policy θn? {θn+1 , . . . , θN } is represented ? ? by Vn ≡ Vn (θn ). Formally, at each stage n = N − 1 to n = 1, we must solve θn? = arg max Vn (θ) , θ∈RK

(2)

Optimal policies can, therefore, be found by coupling numerical optimization (to find the θs) with dynamic programming (to update the V s). Note that when K = 1 this problem reduces to a standard full-information optimal stopping problem (e.g., [6]). This equivalence is illustrated below for a special case of the satisficing problem. In the next section, we present optimal policies for a broad range of problems. These problems differ in a number of ways and demonstrate that the nature of the optimal satisficing policies is not always obvious a priori.

3 Some Numerical Examples The problems described here vary in terms of the properties of the distribution from which the attributes are taken, the nature of the payoff function φ, and the number of attributes K. We restrict most of our examples to problems with three or fewer attributes (K ≤ 3), which themselves are computationally quite involved. However, the algorithm described above can be implemented for any arbitrary K. For the examples reported here that required numerical optimization, we used the fmincon function in MATLAB 6.5, which performs constrained nonlinear optimization. Besides the results we report here on the optimal policies

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themselves, we did extensive analyses of the nature of the solution space. Figure 2 shows an instance of typical behavior of Vn as a function of θn for two-dimensional problems. Fortunately, Vn tends to be quite well behaved, making finding the maximum on the surface relatively straightforward.

Fig. 2. Example of a value function surface. These results are based on x1n ∼ N (0, 1), x1n ∼ N (1, 1), % = 0, and φ(xn ) = x1n + x2n .

Example 1 (Additive Payoffs with Uniform Attributes). Suppose N = 5, K = 2, and, for all n, x1n ∼ Uni[0, 1], x2n ∼ Uni[0, 2], % = 0, and φ(xn ) = x1n + x2n . The optimal policy and its values for this case are shown in Table 1. We can see that the optimal aspiration levels for the second attribute, which has a greater mean (µ = 1), are greater than those for the first attribute (µ = 0.50). Also note that the aspiration levels become progressively less strict as the horizon approaches. This is consistent with Simon’s (1955) house-seller who adjusts her aspiration levels downward as time passes and she fails to sell her house. Table 1. Optimal policies for Example 1. n Vn? θn1? θn2?

1

2

3

4

5

2.07

1.99

1.89

1.75

1.50

0.33

0.26

0.17

0.00

—

1.33

1.26

1.16

1.00

—

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Example 2 (Additive Payoffs with Normal Attributes). First, imagine that N = 5, K = 2, and for all n, x1n ∼ N (0, 1), x2n ∼ N (1, 1), % = 0, and φ(xn ) = x1n + x2n . The optimal policy and its values for this case are shown in the center panel of Table 2. Again, the attribute with the greater mean (Attribute 2) has greater aspiration levels. The behavior of the optimal policy for additive payoffs is quite sensible and consistent with our a priori expectations. But what happens if the attributes are correlated? Using the same parameters but increasing the attribute correlation from % = 0 to % = 0.50, we obtain the policy shown in the lower panel of Table 2. The optimal DM’s aspiration levels increase considerably when the attributes are correlated, as do the expected payoffs Vn? . Given that many things in nature will tend to have correlated attributes, this is a desirable consequence. Interestingly, the expected payoff for satisficing in environments with negatively correlated attributes is considerably lower than in the uncorrelated case (see the top panel of Table 2). We further examine the effects of attribute correlations below in Section 4. Table 2. Optimal policies for Example 2 for both uncorrelated and correlated attributes. n

1

2

3

4

5

Vn?

1.70

1.59

θn1? θn2?

1.46

1.28

1.00

−0.07 0.93

−0.16

−0.30

−0.54

—

0.84

0.70

0.46

—

1.00

% = −0.50

% = 0.00 Vn? θn1? θn2?

2.14

1.98

1.78

1.49

0.11

−0.01

−0.19

−0.51

—

1.11

0.99

0.81

0.49

—

1.00

% = 0.50 Vn? θn1? θn2?

2.50

2.30

2.03

1.65

0.33

0.19

−0.02

−0.38

—

1.33

1.19

0.98

0.62

—

Example 3 (Multiplicative Payoffs with Uniform Attributes). The previous examples were based on additive payoff functions. We now examine a multiplicative function, which might be taken to represent payoffs for alternatives with noncompensatory (or nonsubstitutable) attributes. Let us see what happens when N = 5, K = 2, and for all n, x1n ∼ Uni[0, 2], x2n ∼ Uni[2, 3], % = 0, and φ(xn ) = x1n x2n . The results are shown in Table 3. The optimal policy is not

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absolutely intuitive: Above all, the DM should ensure that he does not select an alternative with a small value on the first attribute; the aspiration level for the second attribute is near its minimum and therefore does not play much of a role. For example, if a househunter would be miserable in a house near a freeway—no matter how cheap the house—then she will be well-advised to set a high aspiration level for her location attribute. Table 3. Optimal policies for Example 3. n Vn? θn1? θn2?

1

2

3

4

5

3.88

3.71

3.48

3.12

2.50

1.44

1.37

1.25

1.00

—

2.16

2.06

2.00

2.00

—

Example 4 (A Three Attribute Problem). Think about the dilemma a satisficing DM faces as K increases. If he sets her aspiration levels too high, he can expect to find that most alternatives are not acceptable, because it only takes xkn < θnk for a single k to make alternative n unacceptable; and, all things equal, the probability of at least one attribute being unacceptable increases with the number of attributes. On the other hand, if he sets his aspiration levels too low, finding an acceptable alternative will be easy, but he may end up with a relatively undesirable alternative. What to do? Consider a problem in which x1n ∼ N (0, 1), x2n ∼ N (.5, 1), x3n ∼ N (1, 1), % = 0, and φ(xn ) = x1n + x2n + x3n for all n. Results are shown in Table 4. In isolation, the general nature of the optimal policy is consistent with expectations. However, the results are more interesting if we compare them to those presented in Example 2 in which the DM faced a related problem. In fact, if we replace the payoff function for the current problem with φ(xn ) = x1n + 0(x2n ) + x3n , we get the Example 2 problem (with % = 0). Note that the optimal aspiration levels for the K = 3 problem are uniformly smaller than the corresponding aspiration levels from Example 2. The intuition for this is quite simple. Suppose a DM used the same aspiration levels for attributes 1 and 3 in the current problem that she used for attributes 1 and 2 in the Example 2 problem. Wherever she sets −∞ < θn2 for the K = 3 problem, there is a nonzero probability that an alternative with acceptable values on attributes 1 and 3 will be rejected because attribute 2 is not satisfactory. Thus, if the DM did use the same aspiration levels in both problems, she would find herself searching deeper into the alternatives when K = 3, where, because of the finite horizon, she can expect to do worse. To prevent this, she must lower her aspiration levels as K increases. We should comment that the complexity of finding optimal policies grows rapidly in the number of attributes. The time required to solve the K = 3

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J. Neil Bearden and Terry Connolly Table 4. Optimal policies for Example 4. n Vn? θn1? θn2? θn3?

1

2

3

4

5

2.78

2.60

2.37

2.04

1.50

−0.22

−0.33

−0.49

−0.76

—

0.28

0.17

0.01

−0.27

—

0.78

0.67

0.51

0.23

—

case was greater than the time required for the corresponding K = 2 problem by roughly an order of magnitude. There is some irony in the fact that to determine how to optimally satisfice over just five 3-attribute alternatives requires intensive and quite time-consuming computations. Example 5 (A Directly Solvable Problem: Uniform Attributes). In this example, we demonstrate a special case of the satisficing problem in which the optimal policy can be solved analytically, that is, without resorting to numerical optimization. Assume that we have two uncorrelated attributes xkn ∼ Uni[0, 1], k ∈ {1, 2}, and φ(xn ) = x1n + x2n , for all n. Substituting into (1), we get  

1 2 − θn1 − θn2 1 2 2 Vn (θn ) = 1 − θn 1 − θn θn + θn + 2
+ θn1 + θn2 − θn1 θn2 Vn+1 . (3) Our objective is to find the θn1 and θn2 that maximize this expression. The first-order conditions are ∂V 1 2
2 1 = θn + θn1 θn2 − θn1 − θn2 Vn+1 + Vn+1 − = 0 1 ∂θn 2 2 and

∂V 1 1
2 1 = θ + θn1 θn2 − θn2 − θn1 Vn+1 + Vn+1 − = 0 . 2 ∂θn 2 n 2

Solving the system of equations, we conclude that θn1 = θn2 = 23 Vn+1 − 13 is a stationary point. (There is another stationary point at θn1 = θn2 = 1, which can be disregarded because it will always leave Vn = Vn+1 .) Evaluating the second partial derivatives for permissible Vn+1 , we find that this point is indeed a maximum. Hence, starting with VN? = 1 and working from n = N − 1 ? to n = 1, θnk? = 23 Vn+1 − 13 for k ∈ {1, 2}. Results for a problem with N = 5 are shown in Table 5. Obviously, with θn? in hand for the current problem, optimal policies can be obtained for uniformly distributed attributes on any interval [a, b]. To see the relation to standard optimal stopping problems, consider applying the same reasoning to a problem with a single attribute (K = 1). We have

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Table 5. Optimal policies for Example 5. n Vn? θn1? , θn2?

1

2

3

4

5

1.35

1.30

1.24

1.15

1.00

0.54

0.49

0.43

0.33

—

Vn (θ) = 1 − and

θn1






θn1

1 − θn1 + 2



+ θn1 Vn+1 ,

(4)

∂V = Vn+1 − θn1 . ∂θn1

Setting ∂V /∂θn1 = 0 and solving for θn1 , we find that (4) is maximized when θn1 = Vn+1 . This is the solution to the standard full-information optimal stopping problem [6]. It is easy to see that (and why) optimal satisficing and standard optimal stopping are equivalent when K = 1 for arbitrary densities. 0 0 Most generally, assuming that θnk? = θnk ? , ∀k, k ∈ {1, . . . , K}, for independent uniformly distributed attributes on [0, 1], we get   h
K
K i K − Kθnk Vn (θn ) = 1 − θnk Kθnk + + 1 − 1 − θnk Vn+1 2 and θnk? =

? 2Vn+1 +1−K , k ∈ {1, . . . , K} , K +1

where VN? = K 12 . Expressed this way, we can see what happens to the aspiration levels as K grows. Consider stage N −1 aspiration levels; for these we can k? observe that θN −1 → 0 as K → ∞. For large K, it is unlikely that the DM will ever find an alternative that satisfies unless he sets his aspiration levels very low.1 Example 6 (Nonstationary Attributes). Recall that the distributions from which the alternatives are drawn may vary with n. Imagine a house-hunting DM who believes that the quality of the houses she will encounter during her search will tend to decrease in time. Let us assume that she believes that x1n ∼ N (((N − n)/N ), 1) and x2n ∼ N (2((N − n)/N ), 1), with % = 0, and that, for all n, φ(xn ) = x1n + x2n . The optimal policy for N = 5 is shown in the top panel of Table 6. To provide a reference, the bottom panel of the table shows the optimal policies for this problem when x1n ∼ N (1, 1) and x2n ∼ N (2, 1), ∀n. In both cases, the DM’s aspiration levels decrease as the horizon approaches. Most important, for the DM whose alternatives are expected to decrease in 1

We have also obtained the analogous solution for exponentially distributed attributes.

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quality, we observe lower aspiration levels across the range of n. For her, because things are expected to get worse over time, she should be less choosy at all points in time.

Table 6. Optimal policies for Example 6 for nonstationary and stationary distributions. n

1

2

3

4

5

Vn?

2.77

2.15

θn1? θn2?

1.52

0.83

0.00

0.13 0.93

−0.10

−0.35

−0.72

—

0.50

0.05

−0.52

—

Nonstationary

Stationary Vn? θn1? θn2?

3.99

3.79

3.51

3.05

1.00

0.82

0.53

−0.61

1.50 —

2.00

1.82

1.53

0.38

—

4 Comparing “Maximizing” and “Satisficing” Policies We assume that the maximizing DM evaluates the decision alternatives using his multiattribute utility function. He does not select an alternative that is acceptable on each of its attributes; instead, he makes his accept–reject decisions on the basis of the actual payoff φ(x) for selecting an alternative. It turns out, then, that the maximizing problem is just a standard full-information optimal stopping problem (e.g., [6,8]). One of the questions we wish to answer is: how great is the inherent penalty for satisficing? That is, in comparison to maximizing, how much does one lose by (optimally) satisficing? The optimal maximizing policy for each stage n is an acceptability threshold s?n ; under it, the DM stops at the first stage n at which φ(xn ) ≥ s?n . The value of some stage n < N policy s is V˜n (s) = P [φ(Xn ) ≥ s] E [φ(Xn )|φ(xn ) ≥ s] + P [φ(Xn ) < s] V˜n+1 .

(5)

As before, V˜N? = E[φ(X)]. Our objective for n = N − 1 to n = 1 is to find s?n = arg max V˜n (s) . s∈R

(6)

In order to determine the optimal maximizing policy, we must know the density of φ(x), denoted g[φ(x)], which is a function of both φ(x) and f (x). When

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the payoff function is additive in its arguments and the attributes are independent, g[φ(x)] can be obtained by convolution. For instance, if two independent normally distributed attributes have means µ1 and µ2 and variances σ12 and σ22 , then g[φ(x)] is normal with mean µ1 + µ2 and variance σ12 + σ22 . More generally, the convolution can be applied recursively to get the desired density for K normally distributed attributes when the payoff function is additive. However, in general, an analytic expression for g[φ(x)] will be difficult to obtain. Fortunately, in practice, we do not actually need it. The stage n stopping probability for some policy s can be obtained numerically by evaluating Z Z P [φ(X) ≥ s] = · · · I [φ(x), s] fn (x) dx1 · · · dxK , where  I [φ(x), s] =

1 0

if φ (x) ≥ s otherwise.

The expectation E [φ(Xn )|φ(xn ) ≥ s] can be similarly expressed. Given this, (6) can itself be solved quite easily, because by the principle of optimality, we ? know that s?n = V˜n+1 .

Fig. 3. Geometric relationship between optimal maximizing and optimal satisficing policies for a single stage n. The policies are for a problem with xk ∼ Uni[0, 1], k ∈ {1, 2}, and φ(x) = x1 + 2x2 .

Figure 3 graphically depicts the relationship between satisficing and maximizing policies at a single stage n. Under the maximizing policy, the DM takes

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an alternative whose attribute values both fall above the dotted line, which ? represents attribute combinations with φ(xn ) = Vn+1 . In contrast, the satisficing DM accepts only those alternatives for which x1 ≥ θ1? and x2 ≥ θ2? , which fall in the darkened rectangular area in the top right. Note that the satisficing policy both accepts some alternatives that are rejected by the maximizing policy and also rejects some that are accepted by it. It is easy to see that the maximizing policy is more compensatory than the satisficing policy. Under the maximizing policy, the DM will take an alternative with relatively small values on x1 (x2 ) as long as the alternative’s x2 (x1 ) value compensates. Optimal maximizing policies were obtained for a subset of the problems reported in Section 3. Our interest is not so much in the policies themselves but in the payoffs associated with those policies. These are shown in Tables 7 and 8. The most striking finding is that very little is lost by satisficing. In most of the cases, optimal satisficing earns the DM about 99% of what he would earn maximizing. Importantly, though, how much is lost by satisficing depends, among other things, on the nature of the distribution from which the alternatives are taken. Considering those examples with additive payoff functions (Examples 1, 2, and 4), we can see that the relative efficiency of satisficing is worse when the attribute values are taken from normal distributions (Example 2 with % = 0) than when they are taken from uniform ones (Example 1). However, when the attributes are positively correlated, the performance of satisficing improves; and, in the case of normally distributed attribute values, satisficing can earn more than 99% of what is earned by the maximizing policy. Thus, in the wild, where attributes will likely be correlated, the inherent penalty for satisficing may be negligible. Table 8 also displays the efficiency of the optimal satisficing policy for a range of correlations for the problem described in Example 2. Table 7. The efficiency (V1? /V˜1? ) of the optimal satisficing policy for Examples 1, 3, and 4. (Results for Example 2 are displayed below). Example V1? /V˜1?

1

3

4

0.99

0.99

0.99

Table 8. Efficiency of optimal satisficing policies for several values of % for the problem considered in Example 2. % Vn? /V˜n?

−0.95

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

0.95

0.85

0.86

0.89

0.91

0.94

0.95

0.97

0.99

0.99

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5 Optimal “Heuristic” Satisficing In the general formulation of our problem, we allowed the satisficing DM to set aspiration levels separately for each stage n. One might complain that the complexity of the resulting policies, which have K(N − 1) separate aspiration levels, is not in the spirit of satisficing. In anticipation of this objection, we now examine a constrained version of the problem in which the DM must use the 1 K same aspiration levels θH = {θH , . . . , θH } for all n ∈ {1, . . . , N − 1}. To keep our expressions from becoming too cumbersome, let us denote P (σn = 1|θH ), which is constant for all n, by Pσ . Now, assuming that the attribute density function does not vary with n (i.e., fn = f, ∀n), the value for some policy θH can be expressed as VH (θH ) =

N −1 h X

n−1

(1 − Pσ )

i N −1 Pσ E [φ (X) |σn = 1] + (1 − Pσ ) E [φ (X)] .

n=1

(7) The last term on the right side of (7) corresponds to the event that the DM reaches the final alternative and must accept it. We denote the θH that max? 1? K? ? imizes (7) by θH = {θH , . . . , θH }, and let VH? ≡ VH (θH ). How much is optimal satisficing affected by the constraint that the aspiration levels must be fixed across the alternatives? To address this question, we found optimal policies for Examples 1 through 4. The results are summarized in Table 9. Most interestingly, we observe that the expected payoffs for ? following θH are quite close to those for following the optimal unconstrained satisficing policy: In all cases, the heuristic policy earns at least 98% of the expected payoff for the optimal unconstrained policy. Table 9. Optimal heuristic policies for Examples 1–4. VH? /V1? is the ratio of the expected payoffs for of the heuristic policy to those of the unconstrained policy. Example 2a corresponds to the problem in Example 2 with % = 0; 2b corresponds to the one with % = 0.50. Example VH? VH? /V1? 1? θH 2? θH 3? θH

1

2a

2b

3

4

2.05

2.10

2.64

3.84

1.00

0.99

0.98

0.98

0.99

0.99

0.22

−0.09

0.26

1.34

−0.40

1.22

0.91

1.26

2.00

0.10

—

—

—

—

0.60

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6 Infinite-Horizon Satisficing We now present a continuous-time infinite-horizon (control-type) satisficing problem in which the DM must pay a fixed cost c > 0 per unit of time she searches. If the cost were 0, the DM could, if infinitely patient, wait forever to select a “perfect” alternative. To get the value function, let us replace the sum in (7) with an integral. Then with a slight modification to remove the final term, which vanishes, adding the cost term, and replacing n with t (time), we get Z ∞

t

(1 − Pσ ) (Pσ E [φ (X) |σ = 1] − c) dt ,

V (θ∞ ) =

(8)

1 1 K where θ∞ = {θ∞ , . . . , θ∞ } are the aspiration levels for all t. We assume that the attribute density function does not vary with time, ft = f, ∀t. Now our objective is to solve ? θ∞ = arg max V (θ∞ ) . (9) θ∈RK

As with the previous problems, (9) can be solved numerically. Optimal policies for the infinite-horizon instance of the problem in Example 1 for different search costs are shown in Table 10. As the search costs increase, the DM lowers her aspiration levels and can also expect to earn less, as we expected. Table 10. Optimal policies for infinite-horizon version of Example 1 for different search costs. c

0.01

0.10

0.25

0.50

1.00

? V∞ 1? θ∞ 2? θ∞

2.32

1.40

0.84

0.40

0.07

0.73

0.47

0.29

0.11

0.00

1.73

1.47

1.29

1.11

0.69

7 Conclusion In the half century since Simon first proposed “satisficing” as an alternative to the descriptively implausible optimizing model of decision-maker behavior, the term has come to connote a poor, simple-minded, imperfect approach to decisions. We may be constrained, either by situational limits in the task or by our own cognitive limitations, to use satisficing approaches, but we will pay significant penalties for doing so. The primary objective of the present chapter has been to explore this assumption. At least for one general class of decisions, sequential search problems, the assumption seems to be a poor one. Over quite wide ranges of problem characteristics and payoff functions, the penalty for using a satisficing approach was small or negligible.

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We distinguished between two components which together make up the loss in expected payoff a given decision policy incurs compared to an optimal policy. The inherent penalty is the unavoidable (expected) payoff loss a DM incurs for simply employing a satisficing rather than a maximizing policy. The second component, aspiration error, is the additional loss associated with using a satisficing strategy with nonoptimal aspiration levels. Our analyses suggest that, although aspiration errors can be very large, inherent penalties are often quite small.2 A satisficing DM can do quite well compared to a maximizer faced with the same kinds of problems. (Similar points have been made elsewhere (e.g., [1,4]).) This is particulary true in environments in which the attributes that compose the alternatives are correlated. Hence, satisficing policies may even be ecologically rational [5]. It seems to us likely that the class of problems for which optimal satisficing strategies will be of practical decision-aiding value is quite small. Computing optimal satisficing thresholds requires both extensive knowledge of task and DM characteristics, and considerable computation; and little more effort would be required to compute genuinely optimal strategies. Unless special circumstances demand that a satisficing approach be used (as in the group decision process discussed earlier), we thus do not envision widespread use of optimal satisficing as a decision aid. The results presented here offer greater potential as benchmarks for behavioral studies of actual decision makers who may adopt satisficing strategies, either spontaneously or in response to special circumstances. In such studies it would be natural to ask what aspiration levels subjects actually adopt, how these levels diverge from optimal levels and at what cost, whether they change over time within and between choice tasks, and so. In a companion paper [2], we have shown that experimental subjects who are required to set aspiration levels and who only learn whether the attribute values of encountered alternatives are acceptable (by the definition given above in the problem statement) do as well as subjects who can see the actual attribute values before making selection decisions. In other words, the subjects who were forced to behave as satisficers did as well as those who were free to maximize. We found this using standard methods from experimental economics, including, most importantly, incentive compatible payoffs. The subjects’ aspiration levels did depart systematically from optimality; specifically, they tended to set their aspiration levels too high. In sum, those subjects who were free to maximize behaved somewhat suboptimally. Those who were constrained to satisfice satisficed somewhat suboptimally. But, overall, the two groups did equally well. The core message of the present chapter is that we should treat satisficing not as a specific decision strategy but as a class of strategies. As we have shown, some of the strategies in this class are capable of performance 2

We will examine in detail the effects of aspiration error in a future paper. Preliminary results can be found in Bearden [2].

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essentially equivalent to that of the global optimizing (or maximizing) strategy, at least for a broad class of problems. Other satisficing strategies yield quite poor overall performance. In our terminology the inherent penalty associated with the use of a satisficing strategy may often be quite small, although for an ill-chosen satisficing strategy actual losses may be huge. Clearly, given the very large computational costs incurred in optimal satisficing we are not proposing this as a plausible descriptive model of real DMs. Indeed, it was precisely to avoid such costs that Simon originally proposed the satisficing idea. However, the DM who adopts a satisficing strategy in order to avoid the costs of making explicit interattribute trade-offs is not thereby condemned to suffer large utility losses. The procedures outlined here provide a metric by which the loss associated with any given satisficing strategy can be assessed. This opens the way to developing aspiration level setting rules that, if not optimal, can at least keep the expected losses small. Descriptively, it allows investigators to explore the extent to which specific DMs have been able to develop such rules for themselves. It may well be that, in some settings, DMs with or without such help have been able to arrive at satisficing decision strategies that yield excellent overall performance for little effort. By clarifying the notion of optimal satisficing, we hope to have shed some light on the descriptively more realistic notion of acceptable satisficing, surely the notion that Simon originally had in mind.

Acknowledgments We gratefully acknowledge financial support by a contract F49620-03-1-0377 from the AFOSR/MURI to the Department of Systems and Industrial Engineering and the Department of Management and Organizations at the University of Arizona.

References 1. G. A. Akerlof. Irving Fisher on his head: The consequences of constant threshold-target monitoring of money holdings. The Quarterly Journal of Economics, 93:169–187, 1979. 2. J. N. Bearden and T. Connolly. Multiattribute sequential search. Organizational Behavior & Human Decision Processes, 103:147–158, 2007. 3. R. Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, 1957. 4. G. Gallego and G. van Ryzin. Optimal dynamic pricing of inventories with stochastic demand over finite horizons. Management Science, 40:999–1020, 1994. 5. G. Gigerenzer, P. M. Todd, and The ABC Research Group. Simple Heuristics That Make Us Smart. Oxford University Press, New York, 1999. 6. J. Gilbert and F. Mosteller. Recognizing the maximum of a sequence. Journal of the American Statistical Association, 61:35–73, 1966.

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7. D. V. Lindley. Dynamic programming and decision theory. Applied Statistics, 10:39–51, 1961. 8. J. B. MacQueen and R. G. Miller. Optimal persistance policies. Operations Research, 8:362–380, 1960. 9. H. A. Simon. A behavioral model of rational choice. Quarterly Journal of Economics, 69:99–118, 1955. 10. H. A. Simon. Satisficing. In D. Greenwald, editor, The McGraw-Hill Encyclopedia of Economics, pages 881–886. McGraw-Hill, New York, second edition, 1993.

There Are Many Models of Transitive Preference: A Tutorial Review and Current Perspective Michel Regenwetter1 and Clintin P. Davis-Stober 1

2

Department of Psychology, University of Illinois at Urbana-Champaign, Champaign, IL 61820 [email protected] Department of Political Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801 [email protected]

Summary. Transitivity of preference is a fundamental rationality axiom shared by nearly all normative, prescriptive, and descriptive models of preference or choice. There are many possible models of transitive preferences. We review a general class of such models and we summarize a recent critique of the empirical literature on (in)transitivity of preference. A key conceptual hurdle lies in the fact that transitivity is an algebraic/logical axiom, whereas experimental choice data are, by design, the outcomes of sampling processes. We discuss probabilistic specifications of transitivity that can be cast as (unions of) convex polytopes within the unit cube. Adding to the challenge, probabilistic specifications with inequality constraints (including the standard “weak stochastic transitivity” constraint on binary choice probabilities) fall victim to a “boundary problem” where the log-likelihood test statistic fails to have an asymptotic χ2 -distribution. This invalidates many existing statistical analyses of empirical (in)transitive choice in the experimental literature. We summarize techniques to test models of transitive preference based on two key components: (1) we discuss probabilistic specifications in terms of convex polytopes, and (2) we provide the correct asymptotic distributions to test them. Furthermore, we demonstrate these techniques with examples on illustrative sample data.

1 Introduction This chapter reviews recent and ongoing progress in formulating and properly testing probabilistic specifications of the concept of “transitive preference.” We first review models of transitive preference and summarize our recent critique of its nearly ubiquitous operationalization via “weak stochastic transitivity.” We emphasize that there are many ways to model transitive preferences, and weak stochastic transitivity is only one of these models. We move the focus to a more canonical representation of “transitive preference” in view of variable/uncertain preferences, namely “mixture models of transitive preference” (and their “random utility representations,” when they exist).

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In the terminology of Loomes and Sugden [29] we study “random preference models” whose “core theories” are “linear,” “weak,” “semi- ,” “interval,” and “partial orders,” respectively. To put it more formally, we discuss probabilistic choice models that are mathematically equivalent to the “linear,” “weak,” “semi-,” “interval,” and “partial order polytopes” for “binary choice” or “ternary paired comparison probabilities.” These polytopes form a natural hierarchy of natural probabilistic specifications of transitive preference depending on our choice of additional axioms about preference. To stay clear of a combinatorial explosion, we place a special focus on the case of four choice alternatives. We emphasize theoretical work that is closely connected to actual experimental data. More specifically, we highlight conceptually appropriate probabilistic modeling and methodologically sound statistical testing that is compatible with the empirical sample space created by a given experimental or survey setting. Throughout, we assume that the relevant experimental data originate from paired comparisons, either without, or with the option of expressing indifference between two choice alternatives. We call the former “binary choices” and the latter “ternary paired comparisons” (using a term we adapted from [3]).1 Depending on how we group response categories, we furthermore refer to ternary paired comparisons “with” or “without indifference.” We illustrate the theoretical points with example data from an experiment and a survey.

2 Mathematical Models of Transitive Preference Because any real-world experiment can only use a finite collection of choice options, we assume throughout that the set C of choice options (relevant to the experiment) is finite. In this section, we first review two standard mathematical representations of preference in the deterministic realm: binary (preference) relations, and corresponding real-valued (utility) functions that map choice options into utility values (we follow the standard definitions and terminology of [13–15,28,30,31,41,46]). We then summarize known results about probabilistic models that include the above deterministic models as special cases, namely “mixture models” of transitive preference relations and corresponding “random utility” and “random function” representations. We also briefly review the most commonly used probabilistic specification of transitive preference, namely “weak stochastic transitivity” of binary choice probabilities.

1

B¨ ockenholt [3] introduced the term “trinary paired comparison.” The term “ternary paired comparison” appears to us more natural, according to most dictionary and encyclopedia entries for the words “trinary” and “ternary.”

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2.1 Transitive Binary Relations Definition 1. A binary relation R on a set of choice alternatives C is a subset of the Cartesian product of C with itself, that is, R ⊆ C × C. In other words, a binary relation on C is a collection of ordered pairs of elements in C. In preference/choice theory it is standard to write (x, y) ∈ R as xRy and to read the relationship as “x is preferred to y.” The central mathematical notion of this chapter is the axiom of “transitivity” of binary relations that is a property common to nearly all normative, prescriptive, and descriptive theories of preference and choice, including, most prominently, expected utility theory [43,49] and prospect theory [25]. Definition 2. A binary relation R on C satisfies the axiom of transitivity if and only if ∀x, y, z ∈ C :

[xRy and yRz] ⇒ xRz .

(1)

Before we review the different kinds of transitive binary relations, it is necessary that we state some additional axioms. Definition 3. If R and S are two (binary) relations, we write RS = {(z, y)| ∃x, zRx, xSy} for their relative product.2 Let R−1 = {(x, y)|yRx} denote the inverse of the binary relation R. If R is a binary relation on C, thus R ⊆ C ×C, we write R = (C × C) \ R for the complement of R (with respect to C). The identity relation on C is IdC = {(c, c)|c ∈ C}. A binary relation R on C is reflexive complete strongly complete antisymmetric asymmetric negatively transitive

if if if if if if

IdC ⊆ R, R ∪ R−1 ∪ IdC = C × C, R ∪ R−1 = C × C, R ∩ R−1 ⊆ IdC , R ∩ R−1 = ∅, R R⊆R.

There are many ways to model transitive preference, as transitivity can be combined with axioms in the above list in various coherent ways. We are now ready to define (what we consider to be) the most important families of transitive binary relations. Definition 4. A strict partial order is an asymmetric and transitive binary relation. An interval order is a strict partial order R with the property that RR−1 R ⊆ R. A semiorder is an interval order R with the property that RRR−1 ⊆ R. A strict weak order is an asymmetric and negatively transitive binary relation. A strict linear order is a transitive, asymmetric, and complete 2

This means that a binary relation is transitive if RR ⊆ R.

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binary relation. A partial order is a transitive, reflexive, and antisymmetric binary relation. A weak order is a transitive, reflexive, and strongly complete binary relation. A linear order is a transitive, reflexive, strongly complete, and antisymmetric binary relation. Figure 1 shows the directed graphs for five strict partial orders. A pair (a, b) belongs to the preference relation if and only if there is an arrow from a to b in the directed graph. Only the leftmost relation is a strict linear order. The two leftmost relations are strict weak orders. The center graph (with triangular nodes) illustrates the axiom RRR−1 ⊆ R used in the semiorder definition, which states that, given the solid arrows, at least one of the two dashed arrows must be present. Similarly, the graph with hexagon-shaped nodes illustrates axiom RR−1 R ⊆ R used in the definition of semiorders and interval orders, which states that, given the solid arrows, at least one of the two dashed arrows must be present. The rightmost graph shows a strict partial order that is neither an interval order, semiorder, strict weak order, nor a strict linear order.

Fig. 1. Directed graphs of five strict partial orders. When the relations are taken as states of preference, a directed arrow from one object to another indicates a strict preference of the first over the second object.

Clearly, strict linear orders, strict weak orders, semiorders, and interval orders are special cases of strict partial orders, whereas linear orders and weak orders are special cases of partial orders. All these relation types can model transitive preference. However, not all of these models treat indifference as being transitive. The following definition is useful to specify which models treat indifference as transitive. Definition 5. Let R be a binary relation. Then, define IncR = R ∩ R−1 and IndR = R ∩ R−1 .

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Observation 1. (1) If R is a strict weak order then IncR is also transitive. However, generally, if R is a semiorder, interval order, or other type of strict partial order then IncR need not be transitive. If R is a weak order then IncR = ∅. (2) If R is a strict partial order then IndR = ∅. If R is a partial order then IndR is transitive. It is standard to interpret IndR as denoting a state of indifference, that is, to interpret (a, b) ∈ IndR to mean that a respondent with preference state R is indifferent between a and b. For IncR there are two different standard interpretations. One is to interpret it also as indifference. This is a common interpretation for semiorders and interval orders, whose motivation was to model transitive strict preference combined with intransitive indifference [13,30]. In Figure 1, IncR holds between any pair of objects that are not linked by an arrow at all. The three rightmost graphs illustrate this type of intransitive indifference. Another interpretation of IncR is as denoting incomparability, in which case (a, b) ∈ IncR means that a respondent with preference state R is unable to compare a and b (with respect to R).3 Much of the decision sciences, rather than relying on combinatorial entities such as binary relations, rely on real-valued functions that they call utility functions and which represent preferences using real numbers. These are our next topic. 2.2 Real Representations of Transitive Binary Relations The following real representations, properties and interpretations are well known (see, e.g., [13–15,28,30,31,41,46]). Observation 2. Let R be a binary relation on C. Suppose that there exist real-valued functions u, l on C with u(c) > l(c), ∀c ∈ C, such that ∀a, b ∈ C :

aRb ⇔ l(a) > u(b) .

(2)

Then R is an interval order on C. If u(x) = l(x)+ ε, ∀x ∈ C, for a fixed positive (utility threshold) ε, then R in the real representation (2) is a semiorder on C. If u = l then R in (2) is a strict weak order on C, but if, in addition, u (and thus also l) is an injective mapping then R in (2) is a strict linear order. The functions u, l can be interpreted as the upper and lower (bound on the) utility, respectively. The representation (2) using > on the real numbers implies transitivity of strict preference. However, it is easy to see that indifference need not be transitive when u 6= l. Most contemporary theories of utility and choice, such as expected utility theory and prospect theory, assume the 3

Alternatively, incomparability between a, b could also result from an absence of paired comparison observations regarding a, b.

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existence of a single utility function u = l. Under this assumption, both strict preference and indifference are transitive, as the axiom is directly inherited from the relation > on the real numbers. It is well known that, when R is a strict linear, strict weak, semi, or interval order, then there always exists a corresponding representation (2). We leave out the details for brevity. We also leave out real (vector-valued) representations of strict partial orders, for brevity, as they are more complicated (see, e.g., [40] for details). Observation 3. Let R be a binary relation on C. Suppose that there exists a real-valued function u : C → R such that ∀a, b ∈ C :

aRb ⇔ u(a) ≥ u(b) .

(3)

Then R in (3) is a weak order on C. If u is injective then R in (3) is a linear order. Clearly, the representations (2) and (3) are closely related. Most utility theories use a single utility function u with u(a) > u(b) denoting “strict preference of a over b,” and u(a) ≥ u(b) denoting “preference or indifference of a over b,” equivalently (notice that the definition of IncR using > as a relation R and using the definition of IndR above with ≥ as a relation R, we have Inc> = Ind≥ ). The combinatorial and algebraic models that we have so far reviewed share the characteristic that they are deterministic representations. Yet, it is common knowledge that real decision makers, when faced with the same decision multiple times, are prone to fluctuate in their actual choices (unless the decision is very clear cut). One reason why choices fluctuate is that choices between options with complex trade-offs are difficult, and decision makers may experience uncertainty about their own preferences. A natural way to model preference uncertainty or preference volatility is via “mixture models.” These models allow us to separate variability of preference (e.g., inconsistency of preference over time) or preference uncertainty from intransitivity of preference. 2.3 Mixture Models of Transitive Binary Relations Definition 6. Let R be a collection of transitive relations on C. A mixture model over R is a probability distribution P : R → [0, 1] Pthat maps each R ∈ R into a probability R 7→ PR , where PR ∈ [0, 1] and R∈R PR = 1. A mixture model of transitive relations is a mixture model over a collection R of transitive binary relations. A mixture model of transitive relations can have several different interpretations. It may quantify the variability of preference or the preference uncertainty of a single person by postulating a sample space of potential mental

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states, all of which are transitive. Alternatively, it may quantify the variability of preference across a population of decision makers, all of whom have transitive preferences. In our illustrative examples below, we consider each of these cases in turn. We first analyze one dataset with repeated choices made by an individual and then we analyze a survey dataset with between-subjects data. As we have already indicated, transitive preferences can be modeled in many different ways. Thus, there are many possible selections of R to build a mixture model. We now introduce shorthand notation for the most important families of binary relations. Definition 7. Denote by LO(C) SWO(C) WO(C) SO(C) IO(C) SPO(C) PO(C)

the the the the the the the

collection collection collection collection collection collection collection

of of of of of of of

linear orders on C strict weak orders on C weak orders on C semiorders on C interval orders on C strict partial orders on C partial orders on C

Definition 8. A collection (Pa,b ) a,b∈C is called a system of binary choice a6=b probabilities if ∀a, b ∈ C, with a 6= b, we have 0 ≤ Pa,b ≤ 1 and Pa,b +Pb,a = 1. This system of binary choice probabilities is induced by linear orders if there exists a probability distribution P on LO(C), such that ∀a, b ∈ C, a 6= b, X Pa,b = PR . (4) R∈LO(C) aRb

In other words, a system of binary choice probabilities is induced by linear orders if there exists a probability distribution over linear orders such that each binary choice probability is the marginal probability of all those linear orders that are consistent with that binary choice. Clearly, if empirical data originate from binary choice probabilities induced by linear orders, this does not prove that preferences must be linear orders. Rather, data of that sort are consistent with a model that postulates linear order preferences.4 Definition 9. A collection (Pa,b ) a,b∈C is called a system of ternary paired a6=b comparison probabilities without indifference, if ∀a, b ∈ C, with a 6= b we have 0 ≤ Pa,b ≤ 1 and Pa,b + Pb,a ≤ 1. This system of ternary paired comparison probabilities without indifference is induced by strict weak orders, respectively, 4

We omit the definition and characterization of a system of binary choice probabilities induced by strict linear orders as it completely matches the case of linear orders.

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semiorders, interval orders, strict partial orders, and partial orders if there exists a probability distribution on R such that ∀a, b ∈ C, a 6= b, X Pa,b = PR , (5) R∈R aRb

with R equal to the collection SWO(C), respectively, SO(C), IO(C), SPO(C), PO(C). Definition 10. A collection (Pa,b ) a,b∈C is called a system of ternary paired a6=b comparison probabilities with indifference, if ∀a, b ∈ C, with a 6= b we have 0 ≤ Pa,b ≤ 1 and Pa,b + Pb,a ≥ 1. This system of ternary paired comparison probabilities with indifference is induced by weak orders if there exists a probability distribution on R such that ∀a, b ∈ C, with a 6= b, (5) holds with R = WO(C). To model the different ways in which decision makers may act in accordance with transitive preferences (that may vary within and between decision makers), we investigate in Section 4 the various ways in which binary choice or ternary paired comparison probabilities may be induced by (mixture models over) various families R of binary relations. In other words, we discuss how to characterize properties on binary choice and ternary paired comparison probabilities that allow us to check whether they are induced by various types of preference relations via (4) or (5). First, we review how mixture models may often be recast as equivalent “random utility models.” 2.4 Random Utility Definition 11. Let I be a finite (index) set. Any collection (Uc,i )c∈C,i∈I of jointly distributed real random variables is called a random utility model on C. A random utility model is called noncoincident if ∀i ∈ I, and for any c 6= d, c, d ∈ C, P (Uc,i = Ud,i ) = 0 . When |I| = 1 we omit the index set and the index i. The following theorem combines results from, e.g., [2,22,23,34,36,37]. Theorem 1. A system of binary choice probabilities (Pa,b ) a,b∈C is induced by a6=b linear orders if and only if there exists a noncoincident random utility model (Uc )c∈C such that ∀a, b ∈ C, a 6= b, Pa,b = P (Ua > Ub ) .

(6)

A system of ternary paired comparison probabilities without indifference (Pa,b ) a,b∈C is induced by strict weak orders if and only if there exists a random a6=b

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utility model (Uc )c∈C (not necessarily noncoincident) such that ∀a, b ∈ C, a 6= b, (6) holds. A system of ternary paired comparison probabilities without indifference (Pa,b ) a,b∈C is induced by interval orders if and only if there exists a random a6=b

utility model (Uc,i )c∈C,i∈{1,2} , with Uc,1 < Uc,2 almost everywhere, such that ∀a, b ∈ C, a 6= b, Pa,b = P (Ua,1 > Ub,2 ) .

(7)

A system of ternary paired comparison probabilities without indifference (Pa,b ) a,b∈C is induced by semiorders if and only if there exists a random utila6=b

ity model (Uc,i )c∈C,i∈{1,2} , with Uc,1 = Uc,2 − 1 almost everywhere, such that ∀a, b ∈ C, a 6= b, (7) holds. A system of ternary paired comparison probabilities with indifference (Pa,b ) a,b∈C is induced by weak orders if and only if there exists a random a6=b

utility model (Uc )c∈C such that ∀a, b ∈ C, a 6= b, Pa,b = P (Ua ≥ Ub ) .

(8)

As Regenwetter and Marley [40] discuss, there is another equivalent representation, called random functions, in which a probability measure is imposed on a suitably chosen space of utility functions. We leave out the details of this representation, as well as the partial order case, for brevity. This completes our review of random utility representations. We now sketch weak stochastic transitivity. 2.5 Weak Stochastic Transitivity Definition 12. A system of binary choice probabilities (Pa,b ) a,b∈C satisfies a6=b weak stochastic transitivity if, ∀x, y, z ∈ C distinct,   1 1 1 Pa,b ≥ and Pb,c ≥ ⇒ Pa,c ≥ . (9) 2 2 2 Theorem 2. [2,32] A system of binary choice probabilities (Pa,b ) a,b∈C satisa6=b fies weak stochastic transitivity if and only if there exists a real-valued function u : C → R with Pa,b ≥

1 ⇔ u(a) ≥ u(b) . 2

(10)

Let R ⊆ C × C be defined by aRb ⇔ Pa,b ≥ 12 . Then R is a weak order. We re-emphasize: there are many ways to model transitive preference. This theorem shows that weak stochastic transitivity implies one particular way to be transitive at the aggregate, namely via an aggregate weak order.

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3 Summary Critique of Prior Work Jointly with other authors, we provide a detailed critique of the past approaches to empirically testing “transitive preferences” elsewhere [39]. DavisStober [8] discusses the statistical problems involved in analyzing empirical data. Here, we only summarize the main points. 3.1 Critique of the Standard Empirical Paradigm In two-alternatives forced choice, the standard paradigm in experiments on transitivity, the respondent must choose one and only one of two choice alternatives on each experimental trial. This forces each binary choice to be asymmetric and complete. Thus, at the level of binary choices, the experimental paradigm artificially enforces the additional axioms of asymmetry and completeness. As we have seen above, the combination of transitivity with these two axioms leads to a strict linear order, a very strong mathematical structure that is equivalent to a complete ranking, without ties, from best to worst. 3.2 Critique of Weak Stochastic Transitivity Weak stochastic transitivity holds if and only if majority rule outcomes [7], applied to individual choice probabilities, yield a transitive social welfare relation. This means that the paradoxes and problematic features of majority rule are directly applicable to weak stochastic transitivity. We only provide a summary (see [39] for details). Violations of weak stochastic transitivity could be artifacts of data aggregation: They could be instances of the famous classical Condorcet paradox of social choice theory, in which transitive individual preferences are aggregated into an intransitive social welfare relation. Loomes and Sugden [29] also pointed out this problem previously. Weak stochastic transitivity also ignores the absolute magnitude of binary choice probabilities. Yet, clearly, binary choice probabilities near 21 are conceptually different from binary choice probabilities near 0 or 1. The former might suggest a respondent’s overall indifference or extreme uncertainty of preference, whereas the latter might suggest approximately deterministic choice perturbed by error. Last, but not least, weak stochastic transitivity and the equivalent weak utility model (10) induce an aggregate weak order on the choice alternatives. Thus, weak stochastic transitivity is mathematically equivalent to transitivity and additional axioms, at the aggregate level. In particular, weak stochastic transitivity also requires that indifference be transitive. (Of course, stronger forms of stochastic transitivity further add additional structure.) 3.3 A Volatile Mix Combining two-alternatives forced choice with weak stochastic transitivity can produce artifactual violations of weak stochastic transitivity. Tsetlin

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et al. [48] argue in a social choice context that there are situations in which several binary choice probabilities are equal to 12 (and weak stochastic transitivity holds), but where intransitivities have substantial nonzero probability in samples. A famous example of such a process in social choice theory is called the “impartial culture.” The impartial culture over three objects is a uniform distribution over the six possible linear orders on those three objects. In random samples from this impartial culture, the probability of an intransitivity in a random sample converges to approximately .09 as the sample size goes to infinity. In our context, this means that if some binary choices with difficult tradeoffs lead to choices “by coin flip” then this may, under somewhat contrived assumptions, artificially inflate the chance of observing intransitivities in empirical samples, because the effectual tie by majority rule at the distribution (probability) level will frequently turn into an aggregate cycle (intransitivity) at the sample level. 3.4 Statistical Methodology The canonical empirical sample space for binary choice or ternary paired comparison is given by a product of independent binomials or trinomials, respectively. The basic idea is to assume that empirical data originate from an independent random sample with replacement from the probabilities (Pa,b ) a,b∈C . a6=b Suppose that a respondent evaluated each paired comparison N times. When we treat this task as a binary choice task, we let Na,b denote the number of times the respondent chose a over b and we assume that Na,b is a realization of a binomial random variable. We write B(Pa,b , N ) for the distribution function of a binomial random variable with probability of success Pa,b and with N repetitions. We also assume that the binomials [B(Pa,b , N )] a,b∈C are indepena6=b dent as the different paired comparisons are separated by decoys. When we treat the task as a ternary paired comparison task without indifference, let Na,b denote the number of times the respondent chose a (i.e., neither “indifference” nor b) in the paired comparison among a, b. Here, we assume that Na,b and Nb,a result from a trinomial sampling process with probabilities Pa,b , Pb,a , and 1 − Pa,b − Pb,a for the three outcome categories. We assume again that these trinomials (with N many repetitions) are independent across choices of a and b as the different paired comparisons are separated by decoys. In the case of ternary paired comparison with indifference, we let Na,b count the number of times that the respondent chose either a or “indifference” in the 0 paired comparison among a and b. Define Pa,b = 1 − Pb,a for all distinct a, b 0 0 0 0 and consider the trinomials with probabilities Pa,b , Pb,a , and 1 − Pa,b − Pb,a 0 for the three outcome categories. Similarly, let Na,b = N − Nb,a for all distinct a, b. The likelihood of the observed data in these three scenarios takes the form

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Y

N −Na,b

Pa,b Na,b (1 − Pa,b )

for binary choice,

a,b∈C a6=b

Y

N −Na,b −Nb,a

Pa,b Na,b Pb,a Nb,a (1 − Pa,b − Pb,a )

a,b∈C a6=b

Y

0

for ternary paired comparison without indifference, 0 0 0
N −Na,b −Nb,a 0 0 1 − Pa,b − Pb,a

0 Na,b 0 Nb,a Pa,b Pb,a

a,b∈C a6=b

for ternary paired comparison with indifference. In our first illustration below, N = 30 denotes the number of (repeated) observations per pair from a single participant in an experiment. In our second illustration, N = 303 denotes the number of participants in a survey, with each participant evaluating each paired comparison once. Using standard techniques, we determine the goodness-of-fit of a given model to a given set of data by computing a log-likelihood ratio test statistic. Here, we need to distinguish between unconstrained choice probabilities and choice probabilities that are constrained by a given model. Writing Pa,b and 0 0 Pa,b for the unconstrained probabilities, and writing θa,b and θa,b for the corresponding constrained probabilities, the standard log-likelihood ratio is given by Q  N −Na,b Na,b (1 − Pa,b ) a,b∈C Pa,b , 2 ln  Q a6=b (11) N −Na,b Na,b (1 − θa,b ) a,b∈C θa,b a6=b Q  N −Na,b −Nb,a Na,b Pb,a Nb,a (1 − Pa,b − Pb,a ) a,b∈C Pa,b , 2 ln  Q a6=b (12) N −Na,b −Nb,a Na,b θb,a Nb,a (1 − θa,b − θb,a ) a,b∈C θa,b a6=b Q  0 0  N −Na,b −Nb,a 0 0 0 Na,b 0 Nb,a 0 0 P 1 − P − P a,b∈C P a,b b,a a,b b,a   2 ln  a6=b  , (13) 0 −N 0  N −Na,b Q 0 0 b,a Na,b 0 Nb,a 0 0 0 θ 1 − θ − θ a,b∈C θ a,b b,a a,b b,a a6=b

with (11) denoting the case of binary choices, (12) denoting the case of ternary paired comparisons without indifference, and (13) denoting the case of ternary paired comparisons with indifference. If standard assumptions were to hold, these three test statistics would asymptotically follow χ2 distributions (see, e.g., [1]). However, in every model we discuss in the chapter, these regularity requirements easily fail. We first consider weak stochastic transitivity. Iverson and Falmagne [24] considered weak stochastic transitivity geometrically. The disallowed events in weak stochastic transitivity are sets of binary choice probabilities of the form 0 ≤ (Pa,b ) a,b∈C ≤ 1 with 12 < Px,y , Py,z , Pz,x ≤ 1 for some choice of distinct a6=b x, y, z ∈ C. Geometrically, these disallowed events are half-unit cubes. Thus,

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weak stochastic transitivity equals the unit cube of probabilities minus the disallowed half-unit cubes. In more abstract terms, weak stochastic transitivity is a certain union of convex polytopes embedded in the unit cube.5 When choice proportions lie outside this union of polytopes, then the maximum likelihood estimator, subject to the constraints imposed by weak stochastic transitivity, lies at the boundary of the parameter space. This means that the log-likelihood ratio test statistic does not have an asymptotic χ2 distribution because the required regularity assumptions are violated (for general discussions of this type of problem, see, e.g., [42]). Iverson and Falmagne’s [24] finding invalidates many statistical tests in the existing empirical literature on weak stochastic transitivity. The same statistical constraints also apply to other probabilistic formulations that feature inequality restrictions on parameters [42]. The work by Iverson and Falmagne [24], which provided the correct asymptotic distribution of the log-likelihood ratio test statistic for weak stochastic transitivity, has almost completely been ignored by the empirical literature on testing transitivity and intransitivity of preference. Unfortunately, Iverson and Falmagne [24] were unable to develop their technique beyond a limited set of models, due to limitations in available mathematical and computational tools at the time. Suck [45] discussed statistical testing of random utility representations in a slightly different context, and most recently, Myung et al. [35] developed a general Bayesian approach to test weak stochastic transitivity, as well as more general models that involve order constraints on model parameters. Davis-Stober [8] expanded Iverson and Falmagne’s [24] approach to a broad class of models in which the model parameters of a probabilistic model are constrained by a union of convex polytopes. This technique directly applies to the mixture models we discuss here. The key to analyzing relevant data correctly is to know the proper asymptotic distribution of the log-likelihood ratio test statistic. Davis-Stober [8] uses recent developments in mathematical statistics and convex geometry to derive this distribution. This distribution is a convex combination of χ2 distributions with varying degrees of freedom, commonly referred to as a χ ¯2 (chi-bar squared) distribution. This completes the review of our recent critique on prior work to test transitivity of individual preferences. We refer the reader to Regenwetter et al. [39] and Davis-Stober [8] for further details, including a detailed literature review.

4 Testing For Transitive Preferences We now discuss how to test whether binary choice probabilities or ternary paired comparison probabilities are induced by transitive relations in the sense discussed in Section 2.3. 5

For an example of a convex polytope, see Figure 2, which illustrates a (different) polytope that we discuss in Section 4.1.

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As an illustrating example we use data from a ternary paired comparison pilot study that we recently ran in preparation for a major experiment. In each experimental trial, a subject looked at a pair of gambles on a computer display and had three options; they could choose the left gamble, the right gamble, or express indifference between the two gambles. The subject considered each possible pair of gambles 30 times, separated by substantial numbers of decoys. Table 1 shows, for one subject, the proportions of times, Qi,j , that the subject chose gamble i over gamble j, for all distinct i, j. This particular respondent, although being allowed to express indifference between gambles, in fact never responded with the “indifference” option.6 Thus, for this experimental subject, Qi,j + Qj,i = 1, even though this was not required by the experimental paradigm. This means that the data from this subject can be treated either as binary choice data or as ternary paired comparison data. It is for this convenient reason that we use this dataset for illustrative purposes here. Table 1. Experiment using ternary paired comparisons, proportion of choices out of 30 repetitions, reported for one subject. Gamble pair (i, j)

(a, b)

(a, c)

(a, d)

(b, c)

(b, d)

(c, d)

Proportion i chosen: Qi,j Proportion j chosen: Qj,i

24 30 6 30

24 30 6 30

14 30 16 30

23 30 7 30

15 30 15 30

23 30 7 30

Proportion indifference between i & j

0

0

0

0

0

0

On the surface it is not clear how one would, in general, characterize whether binary choice proportions or ternary paired comparison proportions such as the ones in Table 1, originated from binary choice or ternary paired comparison probabilities that are induced by a given family of binary relations. Even ignoring the fact that we only observe proportions, it is not straightforward how one would, in general, characterize whether a given collection of binary choice/ternary paired comparison probabilities are induced by such a family of relations. The key lies in the geometry of the space of binary choice probabilities (or of ternary paired comparison probabilities) that are induced by such a family of relations. After fixing a family R of binary relations of interest, we write each R ∈ R using 0/1 coordinates in a suitable high-dimensional space as follows. For each ordered pair (x, y) and each binary relation R ∈ R, let Rxy = 1 if xRy, and Rxy = 0, otherwise. Each binary relation is thereby written