Fuzzy Choice Functions: A Revealed Preference Approach (Studies in Fuzziness and Soft Computing)

Irina Georgescu Fuzzy Choice Functions Studies in Fuzziness and Soft Computing, Volume 214 Editor-in-chief Prof. Janu...

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Irina Georgescu Fuzzy Choice Functions

Studies in Fuzziness and Soft Computing, Volume 214 Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 198. Hung T. Nguyen, Berlin Wu Fundamentals of Statistics with Fuzzy Data, 2006 ISBN 978-3-540-31695-4 Vol. 199. Zhong Li Fuzzy Chaotic Systems, 2006 ISBN 978-3-540-33220-6 Vol. 200. Kai Michels, Frank Klawonn, Rudolf Kruse, Andreas Nürnberger Fuzzy Control, 2006 ISBN 978-3-540-31765-4 Vol. 201. Cengiz Kahraman (Ed.) Fuzzy Applications in Industrial Engineering, 2006 ISBN 978-3-540-33516-0 Vol. 202. Patrick Doherty, Witold Łukaszewicz, Andrzej Skowron, Andrzej Szałas Knowledge Representation Techniques: A Rough Set Approach, 2006 ISBN 978-3-540-33518-4 Vol. 203. Gloria Bordogna, Giuseppe Psaila (Eds.) Flexible Databases Supporting Imprecision and Uncertainty, 2006 ISBN 978-3-540-33288-6 Vol. 204. Zongmin Ma (Ed.) Soft Computing in Ontologies and Semantic Web, 2006 ISBN 978-3-540-33472-9 Vol. 205. Mika Sato-Ilic, Lakhmi C. Jain Innovations in Fuzzy Clustering, 2006 ISBN 978-3-540-34356-1

Vol. 206. A. Sengupta (Ed.) Chaos, Nonlinearity, Complexity, 2006 ISBN 978-3-540-31756-2 Vol. 207. Isabelle Guyon, Steve Gunn, Masoud Nikravesh, Lotﬁ A. Zadeh (Eds.) Feature Extraction, 2006 ISBN 978-3-540-35487-1 Vol. 208. Oscar Castillo, Patricia Melin, Janusz Kacprzyk, Witold Pedrycz (Eds.) Hybrid Intelligent Systems, 2007 ISBN 978-3-540-37419-0 Vol. 209. Alexander Mehler, Reinhard Köhler Aspects of Automatic Text Analysis, 2007 ISBN 978-3-540-37520-3 Vol. 210. Mike Nachtegael, Dietrich Van der Weken, Etienne E. Kerre, Wilfried Philips (Eds.) Soft Computing in Image Processing, 2007 ISBN 978-3-540-38232-4 Vol. 211. Alexander Gegov Complexity Management in Fuzzy Systems, 2007 ISBN 978-3-540-38883-8 Vol. 212. Elisabeth Rakus-Andersson Fuzzy and Rough Techniques in Medical Diagnosis and Medication, 2007 ISBN 978-3-540-49707-3 Vol. 213. Peter Lucas, José A. Gámez, Antonio Salmerón (Eds.) Advances in Probabilistic Graphical Models, 2007 ISBN 978-3-540-68994-2 Vol. 214. Irina Georgescu Fuzzy Choice Functions, 2007 ISBN 978-3-540-68997-3

Irina Georgescu

Fuzzy Choice Functions A Revealed Preference Approach

ABC

Dr. Irina Georgescu P.O. Box 1-432 014700 Bucharest, Romania E-mail: [email protected]

Library of Congress Control Number: 2006939263

ISSN print edition: 1434-9922 ISSN electronic edition: 1860-0808 ISBN-10 3-540-68997-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-68997-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author and techbooks using a Springer LATEX macro package Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

SPIN: 11598374

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Acknowledgement

This book started as a PhD thesis: “Rational Choice and Revealed Preference: A Fuzzy Approach”. I want to thank my supervisor Professor Christer Carlsson from Turku Centre for Computer Science, Institute for Advanced Management Systems Research, Abo Akademi University, Turku, Finland for his guidance and support that he has given me during four years of PhD studies, 2001-2005. He has shown a large and consistent interest in my research and our discussions have been valuable and constructive. I would like to especially thank Professor Hannu Nurmi from Turku University for advice on the material in the book, for useful comments and criticism and for being a guide in all stages of writing it. He indicated the general line of the book, provided me with reference materials and corrected with patience several inconsistencies of the manuscript. This book would not have been written without his initiative. I am very grateful to Professor Janusz Kacprzyk from the Polish Academy of Sciences, Warsaw for suggestions on extending the approach in the book and for recommending new reference materials. In addition, I acknowledge the eﬀort and support of my colleague Xuemei Qiu from Turku Centre for Computer Science in the application part of the book.

VI

Acknowledgement

I am indebted to the staﬀ of Springer-Verlag, who guided me through the printing stages and provided valuable advice.

Bucharest, November 2006

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1 Choice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2 Revealed Preference from Consumers to Choice Functions . . . . 14 3

Classical Revealed Preference Theory . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Binary Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Crisp Choice Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Revealed Preference and Congruence Axioms. Consistency Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4

Fuzzy Preference Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1 Fuzzy Sets and Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Continuous T-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Operations with Fuzzy Sets and Fuzzy Relations . . . . . . . . . . . . 58 4.4 Properties and Indicators of Fuzzy Relations . . . . . . . . . . . . . . . . 63 4.5 An Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5

Fuzzy Choice Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1 From Fuzzy Preferences to Exact Choices . . . . . . . . . . . . . . . . . . . 76 5.2 Fuzzy Choice Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

VIII

Contents

5.3 Rational and Normal Fuzzy Choice Functions . . . . . . . . . . . . . . . 97 6

Fuzzy Revealed Preference and Consistency Conditions . . . . 107 6.1 A Fuzzy Approach to the Arrow–Sen Theorem . . . . . . . . . . . . . . 109 6.2 Conditions F α and F β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3 Quasi-transitivity and Condition F δ . . . . . . . . . . . . . . . . . . . . . . . 127 6.4 Other Consistency Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.5 The Fuzzy Arrow Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7

General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.1 The Hierarchy of Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.2 A Particular Class of Fuzzy Choice Functions . . . . . . . . . . . . . . . 154 7.3 A Fuzzy Analysis of the Richter Theorem . . . . . . . . . . . . . . . . . . . 158

8

Degree of Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.1 Dominance in Banerjee’s Framework . . . . . . . . . . . . . . . . . . . . . . . 171 8.2 Degree of Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 8.3 New Congruence Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9

Similarity and Rationality Indicators for Fuzzy Choice Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 9.1 Similarity of Fuzzy Choice Functions . . . . . . . . . . . . . . . . . . . . . . . 191 9.2 Rationality Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9.3 Revealed Preference and Congruence Indicators . . . . . . . . . . . . . 211 9.4 The Arrow Index of a Fuzzy Choice Function . . . . . . . . . . . . . . . 220 9.5 Consistency Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 10.1 Fuzzy Choices Support for Agent–based Automated Negotiations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 10.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 10.1.2 Fuzzy Choices in Automated Negotiations . . . . . . . . . . . . 237

Contents

IX

10.1.3 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 10.1.4 Van de Walle et al.’s Approach . . . . . . . . . . . . . . . . . . . . . . 242 10.2 Reducing Adverse Selection by Fuzzy Choices . . . . . . . . . . . . . . . 245 10.2.1 The Harmfulness of Adverse Selection . . . . . . . . . . . . . . . . 246 10.2.2 Conventional Solutions for Reducing the Eﬀects of Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.2.3 A Multiple Criteria Decision Making Model . . . . . . . . . . . 248 10.2.4 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 10.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 10.2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 10.3 Application 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

1 Introduction

A main topic in welfare economics is the rational behaviour of a consumer when, faced with various prices and incomes, he has to make a choice. The theory of consumption establishes the framework in which the rationality of consumers is deﬁned and the principle on which it is based. By [109], “the rationality of a consumer may be described by postulating that a consumer has a deﬁnite preference over all conceivable commodity bundles and that he chooses those commodity bundles that are optimal with respect to his preference subject to budgetary constraints”. Samuelson’s theory of revealed preference expresses the rationality of a consumer in terms of some preference relation associated with a demand function. The foundation of this theory is built on The Weak Axiom of Consumer Behavior [87] and on The Strong Axiom of Consumer Behavior [63]. The second axiom assures that the demand function can be reconstructed from a revealed preference relation. To make a rational choice is a more general problem that goes beyond the theme of consumer. In economics, social life, medicine, psychology, etc. there are several cases when an agent has to make rational decisions. For instance, when the members of a society vote diﬀerent candidates in an election, a plausible hypothesis is that, having a desideratum, each of them is rational in the act of choice.

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 1–8 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

2

1 Introduction

For this reason, even if the revealed preference theory appeared in the context of consumers, its concepts and results are true in a larger context oﬀered by the theory of choice functions. Following a suggestion of Georgescu–Roegen [57], Uzawa [108] and Arrow [8] have developed a revealed preference theory in an abstract context, with no economic interpretation. The work of Uzawa and Arrow was continued by Sen with his fundamental results concentrated in [91], [92], [93] (see also [94]). In this approach there is no demand function and the basic elements are an arbitrary set X of alternatives and a choice function C deﬁned on a family B of available sets of alternatives (B ⊆ P(X)). By the rationality of C we mean to ﬁnd a preference relation Q on X such that for any available S the choice set C(S) coincides with the set of Q-greatest elements of S. To a choice function C there are associated more preference relations, each of them leading to a diﬀerent notion of rationality. In [8] The Weak Axiom of Revealed Preference W ARP and The Strong Axiom of Revealed Preference SARP are formulated. W ARP and SARP are abstract versions of The Weak Axiom of Consumer Behavior and of The Strong Axiom of Consumer Behavior. The results obtained by Uzawa, Arrow, Sen and others are based on the hypothesis that the domain of the choice function C should contain all ﬁnite subsets of the universe X; in fact it sufﬁces to assume that the domain of C contains the pairs and the triples of alternatives. An important contribution to the development of choice function theory belongs to Richter [84] by introducing The Weak Congruence Axiom W CA and The Strong Congruence Axiom SCA and by proving the equivalence between rational and congruous choice functions. For the choice functions whose domain includes the ﬁnite sets of alternatives, Arrow–Sen theorem [8], [91] establishes the equivalence between the congruence axioms W CA, SCA, the revealed preference axioms W ARP , SARP and other four conditions of rationality. Richter’s result does not use the above–mentioned hypothesis.

1 Introduction

3

Richter’s line was followed by Hansson [61] and Suzumura [100] that created a revealed preference theory for arbitrary choice functions. They introduced a new axiom of revealed preference HARP and the equivalence of SCA and HARP was established. The classic theory of revealed preference is founded on two main concepts: the preference relations and the choice functions. The axioms and the main results of the theory express various connections between preferences and choices. On the other hand, in social and economic life there exist complex processes subject to several parameters in which the preferences and the choices of the individuals and groups of individuals can manifest indecisively, without answers of type “yes or no”. In this case we deal with vague preferences and vague choices. The vague preferences will be modelled by fuzzy preference relations: they show the degree to which an alternative x is preferred to an alternative y. A vague preference can lead to an exact choice (from a set S of alternatives a non–empty subset is selected) or to a vague choice (the degree to which an alternative x is chosen from S is speciﬁed). The ﬁrst contribution on fuzzy preference relations belongs to Orlovsky [79] and a vast literature has been dedicated to this subject (for an exhaustive overview see the monograph [40] as well as [86], [69]). Some authors build their results on the thesis that social choice is governed by fuzzy preferences (hence modelled through fuzzy binary relations) but the act of choice is exact (hence choice functions are crisp). They study choice functions generated by fuzzy preference relations [15], [16], [17]. An important direction is the study of the fuzzy preference structures, concept that allows for an axiomatic approach to the fuzzy preference relations (see [32], [33], [34], [35]). Some papers discuss the topic of fuzzy choice functions. Usually fuzzy choice functions are associated with fuzzy preference relations [104], [86] [69]. In this case the fuzzy choices are the consequence of the fuzzy preferences.

4

1 Introduction

In [14] Banerjee studies fuzzy choice functions that are no longer deﬁned by a fuzzy preference relation. In this way the act of choice is primordial and the preferences are deﬁned by choices. The domain of a Banerjee fuzzy choice function C is the family of all non-empty subsets of a universal set X of alternatives and the range of C is a family of non-zero fuzzy subsets of X. Then any non-empty subset S of X is an available set of alternatives. We have no information about the alternatives in S except that they can be chosen. This book has the following goals: • to develop the main themes of revealed preference theory (rationality, revealed preference, congruence, consistency) for a large class of fuzzy choice functions; • to explore new topics (degree of dominance, similarity, indicators of rationality) speciﬁc to a fuzzy approach to choice functions; • to show the manner in which some problems of multicriterial decision making problems can ﬁnd natural solutions in fuzzy revealed preference theory (Chapter 10). This book targets the study of fuzzy choice functions in a very general form. In our approach the domain of a choice function will be a family B of non-zero fuzzy subsets of X; if S ∈ B and x ∈ X then S(x) can be viewed as the availability degree of the alternative x. In this way the alternatives are singled out by their degree of availability. As in [14], the range of a fuzzy choice function contains fuzzy subsets of X. Of course the class of fuzzy choice functions we consider includes that of Banerjee. Banerjee fuzziﬁes only the range of a choice function; in order to obtain a more general context we use a fuzziﬁcation of both the domain and the range of a choice function. In Chapter 2 some choice problems are discussed and an overview of revealed preference theory from consumers ([87], [88], [63], [111]) to an axiomatic approach ([8], [84], [85], [91], [92], [93], [100], [101], [102], [108], [109], etc.) is traced.

1 Introduction

5

Chapter 3 refers to notions and classic results of revealed preference theory (crisp choice functions, G–rationality, M –rationality, revealed preference and congruence axioms, consistency conditions etc.). Chapter 4 discusses continuous t–norms, fuzzy sets and fuzzy relations and their properties. In the last part of the chapter a fuzzy version of the Szpilrajn theorem is proved. Chapter 5 has three sections. The ﬁrst section deals with exact choice functions generated by fuzzy preference relations. Some properties of the Orlovsky function [79] are presented in detail. The second section introduces the fuzzy choice functions, the preference relations associated with them and formulates the axioms of revealed preference W AF RP , SAF RP and congruence W F CA, SF CA. W AF RP and SAF RP (respectively W F CA and SF CA) are fuzzy versions of the classical axioms W ARP and SARP (respectively W CA and SCA). Following [61], [100] we also consider the axioms W AF RP ◦ and HAF RP (a fuzzy version of HARP ). The fuzzy versions of classical axioms establish relationships between the fuzzy choice functions and various preference relations associated with them. These relationships reﬂect the way in which the fuzzy choices determine vague preferences and how in their turn these preference relations inﬂuence the act of choice. The third section of the chapter deals with M -rational, G-rational, M -normal and G-normal fuzzy choice functions, extending some results obtained by Suzumura in [100] for crisp choice functions. Chapter 6 is devoted to a fuzzy revealed preference theory following the line introduced by Uzawa, Arrow and Sen. The class of fuzzy choice functions studied in this chapter is subject to hypotheses (H1) and (H2); they come naturally from the assumption that the domain of choice functions includes all ﬁnite sets of alternatives. In Section 6.1 we study the relations between the axioms W AF RP , SAF RP , W F CA, SF CA and four other rationality conditions. The main result (Theorem 6.7) establishes equivalences or implications between these properties; some are true for an arbitrary continuous t-norm

6

1 Introduction

and other for the G¨ odel or the Lukasiewicz t-norms. Section 6.2 is concerned with consistency conditions F α and F β, which are fuzzy forms of Sen’s properties α and β [91], [92], [93]. We prove that a fuzzy choice function satisﬁes F α and F β if and only if W F CA holds. Section 6.3 deals with condition F δ, a fuzzy form of Sen’s condition δ. The main result (Theorem 6.23) asserts that for a G–normal fuzzy choice function, the fuzzy preference relation R is quasi-transitive if and only if F δ holds. Other consistency conditions (F α2, F γ2, F β(+), path independence) are discussed in Section 6.4. The last section of the chapter introduces the Fuzzy Arrow Axiom F AA and establishes the equivalence between F AA and the full rationality of fuzzy choice functions. Chapter 7 studies the revealed preference properties for arbitrary fuzzy choice functions ignoring the hypotheses (H1) and (H2). The investigation follows the trend of the Richter–Hansson–Suzumura theory [84], [61], [100]. In Section 7.1 we prove two main theorems: (1) the axioms W F CA and W AF RP ◦ are equivalent; (2) the axioms SF CA and HAF RP are equivalent. Section 7.2 is devoted to the analysis of a particular class of fuzzy choice functions for which the equivalence between W AF RP ◦ , G–normality, M –normality and two other algebraic conditions holds. The last section of the chapter investigates how the Richter theorem can be extended to a fuzzy choice function theory. In Chapter 8 a notion of degree of dominance of an alternative with respect to a fuzzy subset is introduced, extending Banerjee’s notion of dominance. In the literature several notions of dominance have been studied, but with respect to a fuzzy preference relation [69]. The degree of dominance proposed here refers to the act of choice, and not to a preference relation. In Section 8.1 we mention brieﬂy the setting in which Banerjee formulated his concept of dominance. In Section 8.2 a concept of dominance for the type of fuzzy choice functions studied in this book is introduced. We prove that the degree of dominance of an alternative x ∈ X with respect to a fuzzy set can be expressed by means of a degree of dominance of an alternative with respect to fuzzy

1 Introduction

7

subsets of type [x, y], y ∈ X. In Section 8.3, new congruence axioms F C ∗ 1, F C ∗ 2, F C ∗ 3 are formulated based on the degree of dominance introduced in the previous section and we prove that F C ∗ 1 implies F C ∗ 3 and F C ∗ 2 implies F C ∗ 3, generalizing the results of Banerjee [14] and Wang [112]. We introduce a new revealed preference axiom W AF RPD and we prove the equivalence W AF RPD ⇔ F C ∗ 3. Chapter 9 treats new concepts: similarity and various indicators of rationality for fuzzy choice functions. The similarity of fuzzy choice functions expresses the degree to which two acts of fuzzy choice are “similar”. The results of Section 9.1 establish the relationship between the similarity of fuzzy choice functions and the similarity of the fuzzy preference relations associated with them. In Section 9.2 two indicators of rationality RatG (C), RatM (C) and two indicators of normality N ormG (C), N ormM (C) are studied. The revealed preference and congruence indicators W AF RP (C), SAF RP (C), W F CA(C) and SF CA(C) are the topic of Section 9.3. They evaluate the degree to which the choice function C veriﬁes the axioms W AF RP , SAF RP , W F CA, respectively SF CA. The last two sections of Section 8 are dedicated to the study of the Arrow index A(C) and to the consistency indicators F α(C), F β(C). Each of these indicators measures the degree to which a choice function veriﬁes a condition of rationality or consistency. Chapter 10 is devoted to the possible applications of fuzzy choice functions and the fuzzy revealed preferences associated with them. Three distinct applications are discussed, showing the importance of the notions introduced in the previous chapters for decision making processes, by ranking the alternatives according to multiple criteria. The applications try to model economic situations in which partial information or human subjectivity generate vague choices and vague preferences. The mathematical modelling of these situations is done by fuzzy choices examples where criteria are represented by available fuzzy sets of alternatives. For each case one builds a fuzzy choice space, determines the fuzzy choice function and computes the degree of dominance of

8

1 Introduction

alternatives for each fuzzy subset of the universe of alternatives. This leads to a ranking of alternatives with respect to each criterion. The decision-maker will rely on the information obtained in such way.

2 Preliminaries

This chapter consists of two sections. The ﬁrst section contains some examples of choice problems (consumption theory, voting theory etc.). In the second section some fundamental ideas of the revealed preference theory and some moments in its development are presented.

2.1 Choice Problems In economic and social life, individuals or groups of individuals face several alternatives from which they have to choose one or more. A set X of alternatives is structured by several criteria or attributes according to whom the choice is made. When the choice is related to a criterion (attribute) S one does not consider the whole set of alternatives, but only those alternatives that are compatible with S. Therefore a criterion (attribute) S induces an available set of X (denoted also by S) consisting of the alternatives compatible with S. From the available set S one chooses one or more elements: the set C(S) containing the chosen elements will be called the choice set. C(S) will always be a non–empty set of S. If an alternative x belongs to S then we say that x can be chosen with respect to S; if x belongs to C(S) then we say that x is chosen with respect to S. In interpretation, the elements of C(S) represent “the best options” with respect to criterion S.

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 9–23 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

10

2 Preliminaries

Then the act of choice is represented by a rule that associates with an available set of alternatives one of its non–empty subsets. The case when the choice set C(S) consists of several alternatives corresponds to the situation when the decision–maker is indiﬀerent between several “best” alternatives. According to [109], p. 31, the choice can be seen as an act in two stages: “However, since a single selection will often have to be made from the alternatives in S, we may consider that the selection is to be made in two stages: First the set C(S) is selected from S, and second, an element is selected from C(S) by some random device. Thus, the concept of a choice on a budget space may be considered, in this context, to formalize the ﬁrst stage, the deterministic part, of choice theory”. Some authors admit the possibility that the choice set C(S) is empty for some S ([85] p. 31): “For some purposes one might want to allow C(S) to be empty. For example, if the problem is choosing how big a piece of cake to eat, when any length less than a foot is possible, then there may well be a superior choice for any length picked”. In such cases, the admissible sets S for which C(S) is non–empty are called decisive. In the following we shall present some examples of choice problems: Example 2.1. Consumer theory In consumer theory there is the question of choosing the commodity bundles according to prices and incomes. For a better understanding of the basic concepts of the consumer theory, we shall consider an economy with one consumer and n commodities (see e. g. [73]). A commodity bundle is a vector x = (x1 , . . . , xn ) where each component xi is a non–negative real number that represents the quantity of commodity i that will be consumed. By the way of deﬁning it, a commodity bundle can be assimilated with a point from the n–dimensional space Rn +. We can consider the space of commodity bundles X = Rn + . Therefore X is the set of commodity bundles from which the consumer has to make a choice.

2.1 Choice Problems

11

A price vector has the form p = (p1 , . . . , pn ) where the component pi ≥ 0 denotes the price of commodity i (i = 1, . . . , n). Given an income m > 0, a n commodity bundle x must verify the budget restriction px = i=1 pi xi ≤ m, i. e. the value of commodities x1 , . . . , xn evaluated at the prices p1 , . . . , pn should be less or equal than m. The inequality px ≤ m is a budget restriction that the consumer has to take into consideration when choosing an alternative x. Then a price vector p and an income m determine the following available set of alternatives: B(p, m) = {x ∈ Rn + |px ≤ m}. A subset of X having the form B(p, m) is called a budget set. By modifying the price vector p and the income m one obtains another available set. Let us denote by B the family of all budget sets. Following [109] a demand function is a function C that veriﬁes the following conditions: (D1) The vector x = C(p, m) is a commodity bundle in X deﬁned for any price vector p and any positive income m; (D2) Any positive commodity bundle x is chosen for a suitable p and a suitable income m, i. e. x = C(p, m); (D3) The commodity bundle x = C(p, m) veriﬁes the budget equation pC(p, m) = m for all p > 0 and m > 0. Condition D1 says that the domain of C consists of all pairs (p, m) where p is a price vector, and m is a positive income. In fact, by identifying the pair (p, m) with the budget set B(p, m), one can say that the domain of C is the family B of all budget sets. Then C can be viewed as the function C : B → X. Condition D2 shows that for any positive x there exists B(p, m) ∈ B, such that x = C(B(p, m)), i.e. any positive commodity bundle is in the range of C. The real number C(p, m) is the only alternative chosen from the available set B(p, m) and by (D3), it veriﬁes the budget equation px = m. Sometimes the demand function C must verify the Lipschitz condition with respect to m (see [109], p. 11).

12

2 Preliminaries

Example 2.2. A voting problem Social choice theory studies the way the individual preferences become decision problems at the level of society. Suppose we have a society composed of the individuals 1, 2, . . . , n whose preferences regard a set X of social states. The preferences of the i-th individual are expressed by a preference relation Qi deﬁned on the set of the states. If x, y ∈ X then xQi y shows that the individual i considers the state x at least as good as the state y. A proﬁle has the form (Q1 , . . . , Qn ), where Q1 , . . . , Qn are the preference relations corresponding to the individuals 1, . . . , n. P rof (X) is the set of proﬁles on X. There are several approaches to the way the social decision can be generated by a proﬁle (Q1 , . . . , Qn ). We shall mention two of these approaches. (I) ([9], [94], [74]) The individual preference relations Q1 , . . . Qn are aggregated in a collective preference relation by which the social decision is made. To each proﬁle (Q1 , . . . Qn ) one associates a preference relation f (Q1 , . . . , Qn ). The assignment (Q1 , Qn ) → f (Q1 , . . . Qn ) determines a function f deﬁned on the set of proﬁles mapped on to the set of preference relations on X. f is called social welfare function. Usually the assumption is that both the individual preference relations Q1 , . . . , Qn and the collective preference relation f (Q1 , . . . , Qn ) have the same properties. In [9], Q1 , . . . , Qn and f (Q1 , . . . , Qn ) are assumed to verify the properties of completeness and transitivity. (II) The social decision based on the individual preference relations Q1 , . . . , Qn is expressed by a function that assigns to any non–empty subset of social states S ⊆ X a non–empty subset of S, F (S, Q1 , . . . , Qn ). This mechanism is expressed by a function (called social choice function) F : (P(X) \ {∅}) × P rof (X) → P(X) deﬁned by the mapping (S, Q1 , . . . , Qn ) → F (S, Q1 , . . . , Qn ). For any set S of social states, F (S, Q1 , . . . , Qn ) will represent “the best” alternatives of the society chosen from S based on the individual preferences Q1 , . . . , Qn .

2.1 Choice Problems

13

Example 2.3. Let us consider the graph from Figure 2.1:

b

e

2

5

2

1 c

6

a

2

f

1 h

1 1

7 d

1

g

Fig. 2.1. An optimization problem

On each edge of the graph a non–negative real number is written down. The length of an edge can have diﬀerent meanings: a distance, a time interval between two events, a cost, etc. From a node x to a node y one or more routes can be traversed (in case they exist); the length of a route is the length of the edges that compose it. The issue here is how to choose the routes of maximum length between any pair of distinct nodes. For any edge (x, y) of the graph let us denote by Sxy the set of the routes between x and y, and by C(Sxy ) the set of routes of maximum length between x and y. For example, between the nodes c and h we have the following routes: (c, e, h) of length 3; (c, d, f, h) of length 4; (c, d, g, h) of length 4.

14

2 Preliminaries

Then, from the set of alternatives: Sch = {(c, e, h), (c, d, f, h), (c, d, g, h)} the subset C(Sch ) = {(c, d, f, h), (c, d, g, h)} will be chosen. We deal with a choice problem. The non–empty subsets Sxy will form the family B of available sets of alternatives, and C(Sxy ) will be the choice set of Sxy . The universe X of all alternatives will be the union of all non–empty sets Sxy . More explicitly, let us suppose that the nodes represent towns and the lengths of the edges represent distances between towns. Then Sxy is the set of the routes from x to y, and C(Sxy ) consists of the routes of maximum length from x to y.

2.2 Revealed Preference from Consumers to Choice Functions One of the dominant paradigms of social choice is the revealed preference theory. The classical economic theory has as a basic assumption the rationality of the consumer behaviour, as an optimizing behavior subject to some budgetary constraints. Revealed preference is a concept introduced by Samuelson in 1938, in the attempt to postulate the rationality of a consumer’s behaviour in terms of a preference relation associated with a demand function. The observable variables regarding the behaviour of a consumer are prices, quantities purchased and income, while the unobservable variables are his preferences. In revealed preference theory ﬁrst choices are given, then preferences are deﬁned by choices. The consumer reveals by choices his/her preferences, hence the term revealed preferences [88]: “By comparing the costs of diﬀerent combinations of goods at diﬀerent relative price situations, we can infer whether a given batch of goods is preferred to another batch; the individual guinea-pig, by his market behaviour, reveals his preference pattern - if there is such a consistent pattern”.

2.2 Revealed Preference from Consumers to Choice Functions

15

Unlike the results of his predecessors Marshall and Hicks, who proposed a parametric analysis of the demand functions, Samuelson’s theory of revealed preference was meant to construct a theory of demand functions based only on observable variables. Samuelson’s non-parametric approach of the theory of demand functions was less subject to errors than the Hicksian and Marshallian parametric approaches to the same theory. The paradigm of revealed preference theory is emphasized by Suzumura [103]: “If the choice behaviour of an agent is guided systematically by some underlying preferences, that fact will infallibly reveal itself in his actual choices, so that by observing his choices under alternative speciﬁcations of environmental conditions, we may possibly reconstruct his underlying preferences. This was indeed the original insight of Samuelson that propelled him to open the door to the splendid ediﬁce of revealed preference theory for a competitive consumer”. In the previous section, Example 1, we have formulated a problem of choosing the commodity bundles taking into consideration the prices and the income. Let X = Rn + be the set of alternatives, B the family of the budget sets and C : B → X a demand function. Suppose p1 , p2 are price vectors and m1 , m2 are incomes. Let x1 , x2 be two distinct commodity bundles. We say that x1 is revealed preferred to x2 (in the sense of Samuelson) if there exist two budget sets B(p1 , m1 ) and B(p2 , m2 ) such that x1 = C(p1 , m1 ), x2 = C(p2 , m2 ) and p1 x1 ≥ p1 x2 . In other terms, x1 is revealed preferred to x2 if x1 is chosen when x2 is available. We can explain this deﬁnition by the fact that if p1 x1 ≥ p1 x2 , then the consumer can buy the commodity bundle x2 at the price p1 , but he has chosen the commodity bundle x1 instead.

16

2 Preliminaries

The notion of revealed preference introduces a binary relation R on the set X of the alternatives. (1) x1 R x2 iﬀ x1 is revealed preferred to x2 . Also, R can be extended to a binary relation R: (2) x1 Rx2 iﬀ x1 = x2 or x1 R x2 . Let us notice that R is irreﬂexive ( not xR x for all x ∈ X), while R is reﬂexive ( xRx for all x ∈ X ). Let x, y be two commodity bundles. We say that x is indirectly revealed preferred to y if there exist the commodity bundles x1 , x2 , . . . , xs such that xR x1 , x1 R x2 , . . . , xs R y. ˆ y. Then R ˆ is a transitive binary relation In this case we shall denote xR ˆ y, y R ˆ z implies xR ˆ z for all x, y, z ∈ X. on X, i. e. xR Proposition 2.4. For the commodity bundles x1 , x2 the following are equivalent: (1) x1 is revealed preferred to x2 (i. e. x1 R x2 ); (2) There exists an available set S ∈ B such that x1 = C(S) and x2 ∈ S\{x1 }. Proof. The following assertions are equivalent: (a) x1 is revealed preferred to x2 ; (b) x1 = x2 and there exist B(p1 , m1 ), B(p2 , m2 ) ∈ B such that x1 = C(p1 , m1 ), x2 = C(p2 , m2 ) and p1 x1 ≥ p1 x2 ; (c) x1 = x2 and there exists B(p1 , m1 ) ∈ B such that x1 = C(p1 , m1 ) and p1 x1 ≥ p1 x2 ; (d) x1 = x2 and there exists B(p1 , m1 ) ∈ B such that x1 = C(p1 , m1 ) and p1 x1 ≥ m1 ; (e) there exists B(p1 , m1 ) ∈ B such that x1 = C(p1 , m1 ) and x2 ∈ B(p1 , m1 )\ {x1 }.

2.2 Revealed Preference from Consumers to Choice Functions

17

Proposition 2.5. For the commodity bundles x1 , x2 the following are equivalent: (1) x1 Rx2 ; (2) There exists S ∈ B such that x1 = C(S) and x2 ∈ S. Proof. By Proposition 2.4, the following equivalences hold: (a) x1 Rx2 ; (b) x1 = x2 or x1 R x2 ; (c) x1 = x2 or there is S ∈ B such that x1 = C(S) and x2 ∈ S \ {x1 }; (d) there is S ∈ B such that x1 = C(S) and [x1 = x2 or x2 ∈ S \ {x1 }]; (e) there is S ∈ B such that x1 = C(S) and x2 ∈ S.

The notion of revealed preference is tightly connected with the rational behaviour of a consumer, that means to reach an optimum with respect to a preference relation. Assume that p is a price vector and m an income. Following [109], a commodity bundle x0 is called optimum with respect to a preference relation Q in a budget set B(p, m) if x0 ∈ B(p, m) and x0 Rx for any commodity bundle x ∈ B(p, m). A demand function C is derived from a preference relation Q on X if for any price vector p and for any positive income m the commodity bundle C(p, m) is optimum with respect to Q in the budget set B(p, m). Samuelson’s theory of revealed preference oﬀers a way of deﬁning the raˆ tional behaviour of a consumer. The preference relations R (or R ) and R deﬁned above reﬂect the preferences of a consumer that result from the demand function C. Beside them, some conditions on these preference relations are needed, that should express a rational behaviour. The ﬁrst condition of this type was introduced by Samuelson [87]. Weak Axiom of Consumer Behaviour W ACB For any distinct commodity bundles x1 , x2 , if x1 is revealed preferred to x2 , then x2 is not revealed preferred to x1 .

18

2 Preliminaries

W ACB says that if x1 = C(p1 , m1 ) and x2 = C(p2 , m2 ) are distinct, then (3) p1 x2 ≤ p1 x1 ⇒ p2 x1 > p2 x2 . In terms of the preference relation R deﬁned above, the axiom W ACB is brieﬂy written: If x1 R x2 then not x2 R x1 . One notices that we have the equivalences not x2 Rx1 iﬀ x = y and not x2 R x1 . Then we can express W ACB in the equivalent form (4) If x1 R x2 then not x2 Rx1 . W ACB is a rationality condition from which several properties of the demand functions are deduced (see [89], p. 107–117). At the same time, it does not ensure that the demand function C is derived from the preference of a consumer. Houthakker [63] formulated a stronger form, called the Strong Axiom of Consumer Behaviour, managing to extend Samuelson’s results for an arbitrary number of commodity bundles. In terms of commodity and prices, this condition can be formulated: Strong Axiom of Consumer Behaviour (SACB) If p0 x0 ≥ p0 x1 , p1 x1 ≥ p1 x2 , . . . , pn−1 xn−1 ≥ pn−1 xn then pn x0 > pn xn . By using the notion of revealed preference, SACB can be expressed like this: If commodity bundle x0 is revealed preferred to commodity bundle x1 , commodity bundle x1 is revealed preferred to commodity bundle x2 , . . . , commodity bundle xn−1 is revealed preferred to commodity bundle xn , then xn is not revealed preferred to x0 , i.e. x0 R x1 , x1 R x2 , . . . , xn−1 R xn ⇒ not xn Rx0 (for x0 = xn ). An equivalent form of SACB is: For all distinct commodity bundles x, y the following implication holds: ˆ y ⇒ not yRx. (5) xR

2.2 Revealed Preference from Consumers to Choice Functions

19

Houthakker [63] proved that any demand function C that veriﬁes the Lipschitz condition is derived from the preference relation R. Then for any price vector p and positive income m, the commodity bundle C(p, m) is optimum with respect to R in the budget set B(p, m). This way the rationality of C is ensured by reaching the optimum for the preference relation R associated with C. It is obvious that SACB implies W ACB. In the presence of the Lipschitz condition, W ACB does not imply SACB [109]. In [109] there are given conditions under which a demand function that veriﬁes W ACB veriﬁes SACB. From the above discussion, one notices that there exist elements that can be extracted from the consumer theory and put in an abstract context. Instead of commodity bundles one can consider an arbitrary set of alternatives and the budget sets can be replaced by an arbitrary family B of available sets. A demand function will correspond in the abstract context to a demand function C : B → X. If S ∈ B then C(S) will be the alternative chosen from S. Following the examples of choice problem from the previous section, we need to enlarge the concept of choice function. We can admit now that we can select a non–empty subset of S. Accordingly, an act of choice will be represented by a choice function C : B → P(X). Once deﬁned this choice problem, we can naturally formulate the revealed preference theory and the notion of rationality in a context more general than the consumer theory. For this it is necessary to establish a criterion of optimality with respect to a preference relation Q on the set X of alternatives. Such criterion will ensure that the act of choice is rational with respect to the preference relation Q. There exist two ways of deﬁning the optimality with respect to Q: the choice of the Q–greatest alternatives and the choice of the Q–maximal alternatives. The ﬁrst one is the generalization of the condition of optimum for a commodity bundle in a budget set.

20

2 Preliminaries

The notion of rationality does not impose any kind of condition on preference relation. At this level of generality, we cannot say much about the rationality. In real situations ( consumer theory, voting theory, etc. ), the preferences should possess some properties: reﬂexivity, transitivity, etc. Propositions 2.4 and 2.5 indicate how the revealed preference relations ˆ can be deﬁned for the case of an arbitrary choice function C R, R and R too. From (4) and (5) we notice how the axioms W ACB and SACB can be formulated for a choice function C. The Weak Axiom of Revealed Preference (W ARP ) corresponds to W ACB, and The Strong Axiom of Revealed Preference (SARP ) corresponds to SACB. With respect to the domain of the choice function, the axiomatic theory of revealed preference followed two main routes: The ﬁrst direction was inaugurated by Uzawa [108], [109] and Arrow [8]. Their axiomatic approach started from the assumption that the domain of the choice function contains all ﬁnite subsets of a universal set of alternatives [8]: “It is the suggestion [...] that the demand-function point of view would be greatly simpliﬁed if the range over which the choice functions are considered to be determined is broadened to include all ﬁnite sets. Indeed, as Georgescu– Roegen has remarked, the intuitive justiﬁcation of such assumptions as the Weak Axiom of Revealed Preference has no relation to the special form of the budget constraint sets but is based rather on implicit consideration of two–element sets”. This direction was continued by Sen [91], [92], [94] and many others ([10] etc.) A remarkable result is Arrow–Sen theorem [8], [91] that establishes the equivalence between the full rationality of a choice function, the revealed preference axioms W ARP , SARP , the congruence axioms W CA, SCA (see [84]) and other conditions. In fact, as Sen noticed [91], the proofs of these results assume that the domain of the choice function includes all pairs and triples of alternatives. The presence of the pairs of alternatives in the domain of the choice function makes it possible the deﬁnition of a new preference relation

2.2 Revealed Preference from Consumers to Choice Functions

21

¯ which equals R under special circumstances on X (called “base relation”)R, and which represents an eﬃcient instrument for proving the equivalence of diﬀerent axioms. The second direction belongs to Richter [84], [85], Hansson [61] and Suzumura [100], [103]. They consider choice functions deﬁned on arbitrary families of sets of alternatives. Suzumura maintains ([103], p. 27) that under the assumption that the domain of the choice functions is a non-empty collection of non–empty subsets of a universe of alternatives X, “the theory of rational choice functions developed on this minimal domain condition applies to whatever choice situations we may care to specify: Choice of consumers in a competitive or noncompetitive market, of government bureaucracies, of voters, and of whatsoever”. Signiﬁcant results can be proved in this general framework too. For example, a theorem belonging to Richter [84] establishes the equivalence between the property of full rationality and the congruence axiom SCA. Also the axioms W ARP and SCA are equivalent. Axioms SARP and SCA are not equivalent in this case. Hansson [61] introduced another axiom of revealed preference HARP that, according to [100] is equivalent to SCA. HARP is formulated in terms of C–connected sequences of available sets ([61], [100]). In the approach of several authors, the preference relations verify the transitivity property. On the other hand, it was noticed that in some cases this condition is too strong. Such example is called the “threshold eﬀect” remarked by Georgescu–Roegen [56]. An individual is indiﬀerent towards two temperatures that are one degree apart, but is not indiﬀerent towards two temperatures that are several degrees apart. It follows that in this case transitivity does not hold. The “threshold eﬀect” illustrates the intransitivity of the indiﬀerence relation. The following example belonging to Luce [72] is very suggestive for the intransitivity of the indiﬀerence relation:

22

2 Preliminaries

“A person may be indiﬀerent between 100 and 101 grains of sugar in his coﬀee, indiﬀerent between 101 and 102, . . ., and indiﬀerent between 4999 and 5000. If indiﬀerence were transitive he would be indiﬀerent between 101 and 5000 grains, and this is probably false”. We notice that the transitivity of the strict preference relation does not imply the transitivity of the preference relation. Therefore the transitivity has been replaced by weaker conditions (quasi–transitivity, semi–transitivity, pseudo–transitivity, etc.) imposed on a preference relation. Theorems of rationality of fuzzy choice functions with respect to preference relations that verify such conditions have been given by several authors [66], [11], [12], [13], etc. Besides the rationality of a choice function, another issue is to study how the act of choice behaves with respect to the modiﬁcations of the admissible sets. These are the consistency conditions that establish how the choices are inﬂuenced by the expansion or the contraction of an available set, and by diﬀerent set–theoretical operations (union, intersection, etc.). We notice that the terms of “expansion” and “contraction” regard the inclusion of the available sets. We shall focus mainly on the consistency conditions α, β, γ and δ of Sen [91], [92], [93]. Consistency condition α known also under the name of Chernoﬀ property [27] is a typical condition of contraction consistency. It says that for a subset S1 of S2 , if an alternative is best in S2 then it is best in S1 . In [92], consistency condition α is illustrated by the following example: “if the world champion in some game is a Pakistani, then he must also be the champion in Pakistan”. Consistency condition β says if alternatives x and y are best in S1 and S1 ⊆ S2 then x must be best in C(S2 ), if y is. By [92], this condition states that “if some Pakistani is a world champion, then all champions of Pakistan must be champions of the world”. Then we have a case of expansion–consistency.

2.2 Revealed Preference from Consumers to Choice Functions

23

Condition γ expresses the idea that if alternative x is best in every member of a family of available sets, then x will be best in the union of these sets. We remark here not only the expansion–consistency, but also the behavior of the choices with respect of the union of available sets. In [8], Arrow stated that for social choice theory, the rationality conditions must be independent from the path of choice. The mathematical formulation of the path independence condition (P I) was given by Plott in [82]. Plott starts from the idea that if we look at the act of choice from a dynamic point of view, then we have a procedure of the type “divide and conquer” (see [82], p. 1079). An available set splits into subsets of alternatives and the choice is made with respect to each of these smaller available sets; we put together the chosen alternatives and then we make a new choice from this union. According to path independence, the ﬁnal result is independent from the way in which the initial available set has been decomposed in available subsets. Plott proved [82] that the transitive rationality of the choice implies the path independence, but the converse implication does not hold. If the choice sets always have just one element, then the path independence is equivalent to the complete and transitive rationality. T. Bandyopadhyay [10] introduced a notion of sequential path independence equivalent to transitive rationality.

3 Classical Revealed Preference Theory

In the previous chapter we sketched out some historical moments of the revealed preference theory from its analytical form in consumption theory to the axiomatic treatment in the abstract context of the choice functions. We also discussed some of the main ideas of this theory. Starting from them, this chapter is a short overview of the revealed preference theory in a mathematical language. The ﬁrst section contains deﬁnitions and basic results on the binary relations. In Section 3.2 a choice problem is mathematically formulated. The (crisp) notion of choice function is introduced and the main preference relations associated with a choice function are deﬁned. The concepts of rationality and normality are discussed. Section 3.3 studies the axioms of revealed preference, the axioms of congruence and the consistency conditions, and some important connections between them are recalled.

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 25–47 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

26

3 Classical Revealed Preference Theory

3.1 Binary Relations Let X be a non–empty set. We denote by P(X) the family of all subsets of X. A binary relation on X is a subset Q of the cartesian product X × X. Then P(X 2 ) is the family of all binary relations on X. Let x, y ∈ X. If (x, y) ∈ Q then we shall denote xQy and we say that x is in relation Q with y. We can consider X a set of alternatives. Then a binary relation Q on X can be interpreted as a preference relation; xQy means that alternative x is at least as good as alternative y. To a binaryrelation Q on X one assigns its χQ : X 2 → {0, 1}:  1 if (x, y) ∈ Q χQ (x, y) =  0 if (x, y) ∈ Q. The mapping Q → χQ is a bijective correspondence between the set P(X 2 ) 2

of binary relations on X and the set {0, 1}X of the functions X 2 → {0, 1}. A binary relation will be identiﬁed with its characteristic function. Let X = {x1 , . . . , xn } and Q a binary relation on an arbitrary set X. We deﬁne the  Boolean matrix MQ = (mij )1≤i,j≤n by 1 if xi Qxj mij =  0 otherwise . Then the binary relation Q is uniquely determined by the matrix MQ , therefore we can identify Q with MQ . If Q1 , Q2 are two binary relations on X then the union Q1 ∪ Q2 and the intersection Q1 ∩ Q2 will be deﬁned by Q1 ∪ Q2 = {(x, y) ∈ X 2 |(x, y) ∈ Q1 or (x, y) ∈ Q2 } Q1 ∩ Q2 = {(x, y) ∈ X 2 |(x, y) ∈ Q1 and (x, y) ∈ Q2 } The complement ¬Q of a binary relation Q is deﬁned by ¬Q = {(x, y) ∈ X 2 |(x, y) ∈ Q}. The inverse relation Q−1 is given by Q−1 = {(x, y) ∈ X 2 |(y, x) ∈ Q}.

3.1 Binary Relations

Let {Qi }i∈I be a family of binary relations of X. The union the intersection

27

Qi and

i∈I

Qi of this family are deﬁned by

i∈I

Qi = {(x, y) ∈ X 2 |(x, y) ∈ Qi for some i ∈ I};

i∈I

Qi = {(x, y) ∈ X 2 |(x, y) ∈ Qi for all i ∈ I}.

i∈I

The product (=composition) of the binary relations Q1 , Q2 is introduced by Q1 ◦ Q2 = {(x, y) ∈ X 2 |(x, z) ∈ Q1 and (z, y) ∈ Q2 for some z ∈ X}. We shall denote Q1 = Q and Qn = Q ◦ Q ◦ . . . ◦ Q for any n ≥ 2.

n−times The diagonal of X is the relation ∆ = {(x, x)|x ∈ X}. Proposition 3.1. Let Q1 , Q2 , Q3 , Q be four binary relations on X. Then (1) Q1 ◦ (Q2 ◦ Q3 ) = (Q1 ◦ Q2 ) ◦ Q3 ; (2) Q ◦ ∆ = ∆ ◦ Q = Q; (3) Q1 ⊆ Q2 implies Q ◦ Q1 ⊆ Q ◦ Q2 and Q1 ◦ Q ⊆ Q2 ◦ Q; (4) (Q−1 )−1 = Q; −1 (5) (Q1 ◦ Q2 )−1 = Q−1 2 ◦ Q1 .

Proposition 3.2. For any family {Qi }i∈I of binary relations on X and for any binary relation Q on X we have (1) (

Qi )−1 =

i∈I

(2) (

Q−1 i ;

i∈I

Qi )−1 =

i∈I

(3) (

Q−1 i ;

i∈I

Qi ) ◦ Q =

i∈I

(4) Q ◦ (

i∈I

Qi ◦ Q;

i∈I

Qi ) =

Q ◦ Qi .

i∈I

Remark 3.3. The composition of the binary relations is not commutative. Indeed, if we consider X = {1, 2, 3}, Q1 = {(1, 2)} and Q2 = {(2, 3)} then Q1 ◦ Q2 = {(1, 3)} and Q2 ◦ Q1 = ∅.

28

3 Classical Revealed Preference Theory

In order to obtain an appropriate description of the individual and collective preferences it is necessary to diﬀerentiate the binary relations according to diﬀerent properties. In this way, according to [96], p. 2, the relation “at least as good as” must fulﬁll the following conditions: “First, it must be “transitive”, i. e. if x is at least as good as y, and y is at least as good at z, then x should be at least as good as “y”. [...] Second, the relation must be “reﬂexive”, i. e., every alternative x must be thought to be at least as good as itself. [...] Third, the relation must be “complete”, i. e. for any pair of alternatives x and y, either x is at least as good as y, or y is at least as good as x (or possibly both). A man with a preference relation that is complete knows his mind in choices over every pair”. A binary relation Q on X is said to be reﬂexive if xQx for all x ∈ X; irreﬂexive if for some x ∈ X we do not have xQx; symmetric if xQy implies yQx, for all x, y ∈ X; antisymmetric if xQy, yQx implies x = y for all x, y ∈ X; transitive if xQy, yQz implies xQz for all x, y, z ∈ X; total if xQy or yQx for all distinct x, y ∈ X; complete if xQy or yQx for all x, y ∈ X. A binary relation is complete if and only if it is reﬂexive and total. A preorder on X is a reﬂexive and transitive binary relation on X. A weak order is a complete and transitive binary relation on X. Of course a weak order is a preorder. According to the terminology of Sen [91], [92], a regular relation is a complete and transitive relation, i.e. a weak order. A binary relation Q on X is said to be a partial order if it is reﬂexive, antisymmetric and transitive. Q is a strict partial order if it is irreﬂexive and transitive. An equivalence relation is a binary relation that is reﬂexive, symmetric and transitive. If Q is an equivalence relation on X and x ∈ X then x/Q = {y ∈ X|xQy} is called the equivalence class of x. The set X/Q of the equivalence classes is called the quotient set of X by the equivalence relation Q.

3.1 Binary Relations

29

Proposition 3.4. Let Q be a binary relation on X. Then (1) Q is reﬂexive iﬀ ∆ ⊆ Q iﬀ ∆ ⊆ Q ∩ Q−1 ; (2) Q is irreﬂexive iﬀ Q ∩ Q−1 ⊆ ¬∆; (3) Q is symmetric iﬀ Q = Q−1 ; (4) Q is antisymmetric iﬀ Q ∩ Q−1 ⊆ ∆; (5) Q is transitive iﬀ Q ◦ Q ⊆ Q; (6) Q is total iﬀ Q ∪ Q−1 = X 2 − ∆; (7) Q is complete iﬀ Q ∪ Q−1 = X 2 . The following two propositions express in the language of the characteristic function the composition and some properties of the binary relations. They will be used in the next chapter in order to deﬁne fuzzy versions of these notions. Proposition 3.5. If Q1 , Q2 are two binary relations on X then χQ1 ◦Q2 (x, y) = (χQ1 (x, z) ∧ χQ2 (z, y)). z∈X

Proposition 3.6. Let Q be a binary relation on X. Then (1) Q is reﬂexive iﬀ χQ (x, x) = 1 for all x ∈ X; (2) Q is irreﬂexive iﬀ χQ (x, x) = 0 for some x ∈ X; (3) Q is symmetric iﬀ χQ (x, y) = χQ (y, x) for all x, y ∈ X; (4) Q is transitive iﬀ χQ (x, y) ∧ χQ (y, z) ≤ χQ (x, z) for all x, y, z ∈ X; (5) Q is total iﬀ χQ (x, y) = 1 or χQ (y, x) = 1 for all distinct x, y ∈ X; (6) Q is complete iﬀ χQ (x, y) = 1 or χQ (y, x) = 1 for all x, y ∈ X. One can prove easily that any intersection of transitive binary relations is transitive. For any binary relation Q, the intersection T (Q) of all transitive binary relations that include Q is called the transitive closure of Q. Proposition 3.7. Let Q be a binary relation on X. Then a binary relation Q∗ is the transitive closure of X iﬀ the following conditions are fulﬁlled: (1) Q∗ is transitive;

30

3 Classical Revealed Preference Theory

(2) Q ⊆ Q∗ ; (3) If Q is a transitive binary relation that includes Q then Q∗ ⊆ Q . ∞

Proposition 3.8. For any binary relation Q on X it holds T (Q) =

Qn .

n=1

The following proposition gives an expression of the transitive closure by using its characteristic function. Proposition 3.9. For any x, y ∈ X it holds ∞ [χQ (x, t1 )∧χQ (t1 , t2 )∧. . .∧χQ (tn , y)]. χT (Q) (x, y) = χQ (x, y)∨ n=1 t1 ,...,tn ∈X

Let Q be a binary relation on X. We assign to Q two binary relations on X: I = IQ = Q ∩ Q−1 : the symmetric part of Q; P = PQ = Q ∩ ¬Q−1 = Q \ Q−1 : the asymmetric part of Q; Then we have for any x, y ∈ X: χI (x, y) = χQ (x, y) ∧ χQ (y, x); χP (x, y) = χQ (x, y) ∧ ¬χQ (y, x). If we interpret the binary relations as preference relations on the set of alternatives X then: I is the indiﬀerence relation associated with Q. xIy: x is at least as good as y and y is at least as good as x. and P is the strict preference associated with Q; xP y: x is at least as good as y and y is not at least as good as x. Remark 3.10. The preferences are modelled by an arbitrary binary relation Q and the strict preferences by PQ . In order to stress out the distinction between preferences and strict preferences, for a binary relation some authors use the terminology of weak preference relation. Then xQy spells out “x is weakly preferred to y”. An alternative terminology is: “y is weakly dominated by x” for xQy, and “y is strictly dominated by x” for xPQ y. Proposition 3.11. If a binary relation Q is complete then for all x, y ∈ X, (x, y) ∈ PQ iﬀ (y, x) ∈ Q.

3.2 Crisp Choice Functions

31

Proof. We prove that PQ = ¬Q−1 . By deﬁnition, Q is complete iﬀ Q ∪ Q−1 = X 2 . This is equivalent with the following assertions: ¬Q−1 ∩ (Q ∪ Q−1 ) = ¬Q−1 ∩ X 2 iﬀ (¬Q−1 ∩ Q) ∪ (¬Q−1 ∩ Q−1 ) = ¬Q−1 ∩ X 2 iﬀ (¬Q−1 ∩ Q) ∪ ∅ = ¬Q−1 iﬀ ¬Q−1 ∩ Q = ¬Q−1 iﬀ PQ = ¬Q−1 . Consequently, (x, y) ∈ PQ iﬀ (x, y) ∈ ¬Q−1 iﬀ (y, x) ∈ ¬Q iﬀ (y, x) ∈ Q.

A binary relation Q is called acyclic if for any natural number n ≥ 2 and x1 , . . . , xn ∈ X we have (x1 , x2 ) ∈ PQ , . . . , (xn−1 , xn ) ∈ PQ ⇒ (xn , x1 ) ∈ PQ . If Q is complete then (xn , x1 ) ∈ PQ iﬀ (x1 , xn ) ∈ Q. It follows that a complete binary relation Q is acyclic iﬀ for all x1 , . . . , xn ∈ X it is true that (x1 , x2 ) ∈ PQ , . . . , (xn−1 , xn ) ∈ PQ ⇒ (x1 , xn ) ∈ Q.

3.2 Crisp Choice Functions In social and economic activities, the individuals or groups are often confronted with the situation of choosing between several alternatives. These alternatives can be goods that are going to be purchased, job oﬀers, candidates in elections, etc. We admit that the alternatives are elements of a set X called universe. They are diﬀerentiated by attributes or criteria. Each attribute or criterion determines a non–empty subset S of X; S consists of those alternatives that possess a certain attribute or that verify a certain criterion. According to a certain attribute or criterion, the choice will be made from the corresponding subset S and not from the entire universe of alternatives; the chosen elements form a non–empty subset C(S) of S.

32

3 Classical Revealed Preference Theory

Then a choice problem consists of: a) a universe X of alternatives under the control of an agent; b) a non-empty family B of non–empty subsets of X; the members of X can represent possible decisions for the agent; c) a mechanism by which the agent may choose from an available set S a subset C(S); it is necessary to impose the condition that at least an element of S has to be chosen, therefore C(S) will always be non–empty. A choice space will be a pair (X, B) where X is a non–empty set of alternatives and B is a non–empty family of non–empty subsets of X. A set S ∈ B can be taken as an available set of alternatives. In the terminology of consumer theory, the pair (X, B) is called a budget space; the elements of X are called bundles and the sets of B are called budgets. As we have seen before, any available set S can be considered as representing a criterion or an attribute on the alternatives. A choice space (X, B) is ﬁnite if the set X of alternatives is ﬁnite. In this framework we can formulate a choice problem: given S ∈ B choose one or more elements in S; we can relate it to the notion of choice function or consumer. A choice function or a consumer on (X, B) is a function C : B → P(X) which to any S ∈ B assigns a non-empty subset C(S) of S. C(S) can be taken as the set of bundles or alternatives chosen subject to S; it will be called the choice set of S. If x ∈ S \ C(S) then we shall say that the alternative x is rejected with respect to S. The elements of C(S) are interpreted as “the best alternatives in S”. In accordance with the deﬁnition of the choice function, several elements can be chosen from an available set S ∈ B. This deﬁnition covers several real situations. For instance, in case of parliamentary elections, from a list of candidates of a political party several individuals can be chosen.

3.2 Crisp Choice Functions

33

Let C : B → P(X) be a choice function such that for any S ∈ B, C(S) has exactly one element x, i. e. C(S) = {x}. By identifying the set {x} with its element x, we can consider that X is the range of f , that means f : B → X. By deﬁnition, a choice problem has the form ((X, B), C) where (X, B) is a choice space and C is a choice function on (X, B). Considering the available sets as criteria in decision–making, a choice problem can be viewed as a decision–making problem. A signiﬁcant part of choice function theory [8], [91], [92], [93] has been developed under the following hypothesis: (H) B contains all non-empty ﬁnite subsets of X. It is important to know if the behaviour of the decision–maker is rational . Accordingly, it is necessary to introduce a concept of rationality of the choice functions. This is done by connecting the choices of the decision–maker with his preferences. Mathematically speaking, a preference is represented by a binary relation Q on the set X of alternatives. For each available set S ∈ B one has to specify what “the best alternative” means starting from a preference relation Q. The choice of those elements from S which are “the best alternatives” will ensure the rationality of the act of choice. Let S be an available set. An element x ∈ X is a maximal element of S with respect to Q ( or Q–maximal ) if there is no y ∈ S such that yPQ x. In other words the alternative x ∈ S is Q–maximal if there is no alternative y ∈ S which is strictly preferred to x in terms of Q. The set of the Q–maximal elements in S will be denoted by M (S, Q): M (S, Q) = {x ∈ S|(y, x) ∈ PQ for all y ∈ S}. An element x ∈ S is a greatest element of S with respect to Q (or Q– greatest) if for any y ∈ S we have xQy. Then x ∈ S is Q–greatest in S if for all y ∈ S, the alternative x is at least as good as the alternative y. The set of Q-greatest elements of S will be denoted by G(S, Q): G(S, Q) = {x ∈ S|(x, y) ∈ Q for all y ∈ S}.

34

3 Classical Revealed Preference Theory

We deﬁne two types of rationality of a choice function, corresponding to the two notions of Q–optimality from above. Therefore, the choice of the Q–maximal elements means maximal–rationality, while the choice of the Q– greatest elements means greatest–rationality. Let (X, B) be a choice space and C a choice function on (X, B). C is said to be G–rational (resp. M –rational) if there exists a preference relation Q on X such that for any S ∈ B we have C(S) = G(S, Q) (resp. C(S) = M (S, Q)). In this case we say that C is G–rationalizable (resp. M –rationalizable) by Q and that Q is the G–rationalization (resp. the M –rationalization) of C. The basic idea of greatest–element rationalizability is that the chosen alternatives are as least as good as all alternatives in an available set S, while in the case of maximal–element rationalizability the chosen elements are not strictly dominated by any alternative in S. The concept of rationality can be reﬁned with the respect to the properties of Q. If Q has a property P and Q is the G–rationalization (resp. the M – rationalization) of C then we shall say that C is P –G–rational ( resp. P –M – rational ). For example, if Q is a transitive preference relation then C is said to be transitive G–rational (resp. transitive M –rational). Example 3.12. ([100], p. 158) Let X = {a, b, c} and B = {S1 , S2 } where S1 = {x, y}, S2 = {y, z}. Let us consider the choice function C on (X, B) deﬁned by C(S1 ) = {x}, C(S2 ) = {y, z} and the following binary relations on X: Q1 = Q1 = {(x, y), (y, z), (z, y)}; Q2 = {(x, y), (y, z), (z, y), (x, z), (z, x)}; Q2 = {(x, y)}. It is easy to see that C(S1 ) = G(S1 , Q1 ) = G(S1 , Q2 ); C(S2 ) = G(S2 , Q1 ) = G(S2 , Q2 ); C(S1 ) = M (S1 , Q1 ) = M (S1 , Q2 ); C(S2 ) = M (S2 , Q1 ) = M (S2 , Q2 ). Therefore C has two G–rationalizations Q1 , Q2 and two M –rationalizations Q1

and Q2 .

Lemma 3.13. [100] Let Q be preference relation on X. Then

3.2 Crisp Choice Functions

35

(i) G(S, Q) ⊆ M (S, Q) for any S ∈ B; (ii) If Q is complete then G(S, Q) = M (S, Q) for any S ∈ B. The previous lemma shows that the notions of complete G–rationality and complete M –rationality coincide. Let Q be a regular preference relation (=reﬂexive, transitive and total). If Q G–rationalizes (= M –rationalizes) the choice function C then C is said to be full rational ( or regular rational). For a preference relation Q on X let us consider the functions GQ : B → P(X); MQ : B → P(X) deﬁned by GQ (S) = G(S, Q), resp. MQ (S) = M (S, Q) for any S ∈ B. Sometimes we will use the notation G(., Q) instead of GQ and M (., Q) instead of MQ . In general GQ and MQ are not choice functions. Example 3.14. Let X = {a, b, c}, A = {a, b}, B = {b, c} and B = {A, B}. Consider the binary relation Q on X deﬁned by Q = {(a, a), (a, b), (b, c)}. Then Q−1 = {(a, a), (b, a), (c, b)} ¬Q−1 = {(a, b), (a, c), (b, b), (b, c), (c, a), (c, c)} PQ = Q ∪ Q−1 = {(a, b), (b, c)}. An easy computation gives G(A, Q) = {a}; G(B, Q) = ∅; M (A, Q) = {a}; M (B, Q) = {b}. It follows that M (., Q) is a choice function on (X, B), while G(., Q) is not a choice function. Proposition 3.15. [96] Let (X, P(X) \ { ∅}) be a ﬁnite choice space. If Q is a reﬂexive, transitive and total (=regular) preference relation on X then GQ and MQ are choice functions on (X, B).

36

3 Classical Revealed Preference Theory

Proposition 3.16. [96] Let (X, P(X) \ { ∅}) be a ﬁnite choice space and Q a reﬂexive and complete preference relation on X. Then the following are equivalent: (1) GQ is a choice function; (2) Q is acyclic. Under these circumstances MQ is a choice function too. So far we have presented two ways in which two choice functions correspond to a preference relation Q on X : Q −→ GQ and Q −→ MQ . Conversely, the choice functions can induce preference relations on the set of the alternatives: Let C : B → P(X) be a choice function. Following [91] we shall present some preference relations generated by C: R = {(x, y)|x ∈ C(S) and y ∈ S for some S ∈ B} P = PR = {(x, y)|(x, y) ∈ R and (y, x) ∈ R} I = IR = {(x, y)|(x, y) ∈ R and (y, x) ∈ R} P˜ = {(x, y)|x ∈ C(S) and y ∈ S − C(S) for some S ∈ B} ˜ = P ˜ = {(x, y)|(y, x) ∈ P˜ } R R ˜ and (y, x) ∈ R} ˜ I˜ = {(x, y)|(x, y) ∈ R Assuming that hypothesis (H) holds we also deﬁne ¯ = {(x, y)|x ∈ C({x, y})} R ¯ and (y, x) ∈ R} ¯ P¯ = PR¯ = {(x, y)|(x, y) ∈ R ¯ and (y, x) ∈ R} ¯ I¯ = IR¯ = {(x, y)|(x, y) ∈ R Remark 3.17. In terms of the characteristic function the binary relations R, ¯ can be expressed: P˜ and R

(i) χR (x, y) =

(χC(S) (x) ∧ χS (y));

S∈B

(ii) χP˜ (x, y) =

(χC(S) (x) ∧ χS (y) ∧ ¬χC(S) (y));

S∈B

(iii) χR¯ (x, y) = χC({x,y}) (x).

3.2 Crisp Choice Functions

37

We shall denote by W (resp. P ∗ ) the transitive closure of R (resp. P˜ ). Relation R (called revealed preference in [95]) has been introduced by Samuelson in 1938 (see [87]); he associates R to a demand function. Samuelson’s theory of revealed preference studies the rationality of a consumer in terms of the preference relation R. In an axiomatic context it has been resumed by Uzawa [108] and Arrow [8]. Its intuitive signiﬁcance is obvious: x is weakly revealed preferred to y if x is chosen and y is available. Relation P˜ (called strong revealed preference in [95]) has been deﬁned by Arrow in [8]. It has the meaning: x is strongly revealed preferred to y if there exists an available set from which x is chosen and y is rejected. ¯ appears in [108] and [8] and corresponds to a concept of Base relation R binary choice: x is preferred to y if x is chosen from the available set {x, y}. Besides these the following revealed preference relation R∗ on X is deﬁned [100]: R∗ = {(x, y)|x ∈ S or x ∈ C(S) or y ∈ C(S) for all S ∈ B} Remark 3.18. In terms of characteristic functions we can write χR∗ (x, y) = [(χS (x) ∧ χC(S) (y)) → χC(S) (x)] S∈B

In interpretation, x is R∗ –revealed preferred to y if “there is no choice situation in which y is chosen and x is available but rejected” [100]. Example 3.19. Let X = {a, b, c, d}, A = {a, b, c}, B = {b, c, d}, B = {A, B} and the choice function C on (X, B) deﬁned by C(A) = {a, b} and C(B) = {c, d}. By applying the deﬁnition of R we obtain R

=

{(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, b), (c, c), (c, d), (d, b),

(d, c), (d, d)}. R can be represented as a matrix: 

 1110

    1 1 1 0   R=  0 1 1 1   0111

38

3 Classical Revealed Preference Theory

From this it follows





R−1





1100

0011

         1 1 1 1  0 0 0 0  , ¬R−1 =   =     1 1 1 1 0 0 0 0     0011 1100

from which



 0010

PR = R ∩ ¬R−1

    0 0 0 0   =  0 0 0 0   0100 

 1100

IR = R ∩ R−1

    1 1 1 0  =   1 1 1 0   0011

By applying the deﬁnition of P˜ we obtain: P˜ = {(a, c), (b, c), (c, b), (d, b)} or as a matrix



 0010

    0 0 1 0 . P˜ =    0 1 0 0   0100 As above we obtain



 1111



 1101

        1 1 0 0 1 1 0 0 ˜ ˜ .    R=  , I =  0 0 1 1 0 0 1 1     1011 1111 ¯ cannot be deﬁned. Since B does not contain all two–element sets of X, R

3.2 Crisp Choice Functions

39

Example 3.20. Let X = {a, b, c} and B = {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}}. Let us consider the choice function C deﬁned by C({a}) = C({a, b}) = {a}, C({b}) = C({b, c}) = {b} and C({c}) = C({a, c}) = {c}. It follows immedi¯ is given by ately that R



 110

   ¯= R  0 1 1 .   101 Let us study now the following situation. The choice function C induces the revealed preference relation R; in its turn, R induces the functions GR and MR . If C = GR (resp. C = MR ) then we say that the choice function C is G–normal (resp. M–normal ). The normality is a special case of rationality: the choice function C is rationalized by the revealed preference relation R. We present now some results that connect the G–rationality and the M – rationality: Proposition 3.21. [100] Any M –rational choice function is G–rational. Proposition 3.22. [100] If C is a choice function on (X, B) then C(S) ⊆ GR (S) ⊆ MR (S). The previous proposition shows that GR and MR are always choice functions. Proposition 3.23. [100] For any choice function C the following are equivalent: (1) C is G–rational; (2) C is G–normal. According to the previous proposition, the concepts of G–rationality and M –normality coincide. Proposition 3.24. [100] Any M –normal choice function is complete G– rational (=complete M –rational).

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3 Classical Revealed Preference Theory

3.3 Revealed Preference and Congruence Axioms. Consistency Conditions The revealed preference axioms are conditions on the rationality of a choice function C; they are stated in terms of the revealed preference relations associated with C. The Weak Axiom of Consumer Behavior and The Strong Axiom of Consumer Behavior were introduced in the classic consumer theory by Samuelson [87] and by Houthakker [63], respectively. They establish the behavior of a demand function with respect to a revealed preference relation.The Strong Axiom of Consumer Behavior is a suﬃcient condition such that the demand function of a competitive consumer be determined by a transitive and complete preference relation. The axioms W ARP and SARP are abstract versions of The Weak Axiom of Consumer Behavior and The Strong Axiom of Consumer Behavior, respec˜ and tively. The present form, due to Arrow, uses the preference relations R, R P ∗. Let (X, B) be a choice space and C a choice function on (X, B). W ARP (Weak Axiom of Revealed Preference) If (x, y) ∈ P˜ then (y, x) ∈ R. SARP (Strong Axiom of Revealed Preference) If (x, y) ∈ P ∗ then (y, x) ∈ R. W ARP has the following signiﬁcance: if there exists an available set S from which x has been chosen and y has been rejected, then for any other available set S , if y has been chosen then x cannot be chosen. Remark 3.25. Let C : B → X be a singleton choice function. Then R = {(x, y)|x = C(S) and y ∈ S for some S ∈ B} P˜ = {(x, y)|x = C(S), y ∈ S and x = y for some S ∈ B}.

3.3 Revealed Preference and Congruence Axioms. Consistency Conditions

41

Then, for any distinct x, y ∈ X, (x, y) ∈ R iﬀ (x, y) ∈ P˜ . It follows that for a singleton choice function C : B → X, W ARP and SARP have the following equivalent form: W ARP For any distinct x, y ∈ X, if (x, y) ∈ R then (y, x) ∈ R. SARP For any (x1 , x2 ), . . . , (xn−1 , xn ), (xn , x1 ) ∈ X 2 , if (x1 , x2 ) ∈ R, . . . , (xn−1 , xn ) ∈ R then (xn , x1 ) ∈ R. Example 3.26. ([103], p. 57) Let X = {x, y, z} and B = {S1 , S2 , S3 } where S1 = {x, y}, S2 = {y, z} and S3 = {x, z}. Let us consider the choice function C : B → P(X) deﬁned by C(S1 ) = {x}, C(S2 ) = S2 and C(S3 ) = {z}. An easy computation gives R = {(x, x), (y, y), (z, z), (x, y), (y, z), (z, x), (z, y)} P˜ = {(x, y), (z, x)} P ∗ = T (P˜ ) = {(x, y), (z, x), (z, y)}. We remark that C veriﬁes W ARP ((u, v) ∈ P˜ implies (v, u) ∈ R, but does not verify SARP ((z, y) ∈ P ∗ and (y, z) ∈ R). An equivalent form of these two axioms of revealed preference has been given by Hansson [61] and Suzumura [100] by using the notion of C– connectedness. A sequence (S1 , . . . , Sn ) in B is called C–connected if Sk ∩ C(Sk+1 ) = ∅ for all k ∈ {1, . . . , n} where Sn+1 = S1 . In this context, the two axioms have the following equivalent form: W ARP For any C-connected pair (S1 , S2 ) in B, S1 ∩ C(S2 ) = C(S1 ) ∩ S2 holds. SARP For any C-connected sequence (S1 , . . . , Sn ) in B we have Sk ∩ C(Sk+1 ) = C(Sk ) ∩ Sk+1 for some k = 1, . . . , n − 1. Also in [61], [100] another revealed preference axiom appears: (Hansson’s Axiom of Revealed Preference) For any C-connected sequence (S1 , . . . , Sn ) in B Sk ∩ C(Sk+1 ) = C(Sk ) ∩ Sk+1 for all k = 1, . . . , n − 1.

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3 Classical Revealed Preference Theory

We remark that these forms of the axioms W ARP , SARP as well as the axiom HARP are stated in terms of the choice function C and not using the revealed preference relations associated with C. In [8] it was also studied the following condition (Arrow Axiom) : AA For any S1 , S2 ∈ B, the following implication holds: [S1 ⊆ S2 ] ⇒ [S1 ∩ C(S2 ) = ∅] or [S1 ∩ C(S2 ) = C(S1 )]. Following [8], AA has the following interpretation: “if some elements are chosen out of the set S2 and then the range of alternatives is narrowed to S1 but still contains some previously chosen elements, no previously unchosen element becomes chosen and no previously chosen element becomes unchosen”. The strong congruence axiom was introduced by Richter [85] and its weaker form by Sen [91]: W CA (Weak Congruence Axiom) For any x, y ∈ X and S ∈ B, if x ∈ S, y ∈ C(S) and (x, y) ∈ R then x ∈ C(S). SCA (Strong Congruence Axiom) For any x, y ∈ X and S ∈ B, if x ∈ S, y ∈ C(S) and (x, y) ∈ W then x ∈ C(S). For example, W CA means that if an alternative x is chosen from an available set S and an alternative y can be chosen from S, and if y is revealed preferred to x ( in the sense of R ), then y is also chosen from S. Example 3.27. ([100], p. 154) Let X = {x, y, z} and B = {S1 , S2 , S3 }, where S1 = {x, y}, S2 = {y, z}, S3 = {x, z}. Let us consider the choice function C on (X, B) deﬁned by C(S1 ) = S1 , C(S2 ) = S2 , C(S3 ) = {x}. By computation we get R = {(x, x), (y, y), (z, z), (x, y), (y, x), (y, z), (z, y), (x, z)} W = T (R) = {(x, x), (y, y), (z, z), (x, y), (y, x), (y, z), (z, y), (x, z), (z, x)}. C veriﬁes W CA: for every i ∈ {1, 2, 3} we have u ∈ Si , v ∈ C(Si ), (u, v) ∈ R ⇒ u ∈ C(Si ).

3.3 Revealed Preference and Congruence Axioms. Consistency Conditions

43

We observe that (z, x) ∈ W , x ∈ C(S3 ), z ∈ S3 but z ∈ C(S3 ) therefore C does not verify SCA. Now we shall recall three results from [91]. Proposition 3.28. [91] Assume hypothesis (H) holds. Then W CA implies that R is a regular preference relation; if C is G-normal then the converse is also true. Example 3.29. Let X = {x, y, z} and B = P(X) \ {∅}. Let us consider the preference relation Q = {(x, x), (y, y), (z, z), (x, y), (y, x), (x, z), (z, y)}. Q is reﬂexive and total. Take the choice function C = G(., Q) associated with Q. C is G–rational, therefore G–normal (cf. Proposition 3.23), therefore it veriﬁes consistency condition α (cf. Proposition 1.28). By computations we obtain C({x, y}) = {x, y} and C({x, y, z}) = {x}. We notice that x, y ∈ C({x, y}), {x, y} ⊆ {x, y, z} but “x ∈ C({x, y, z}) iﬀ y ∈ C({x, y, z})” does not hold. Therefore C does not verify consistency condition β. Proposition 3.30. [91] If hypothesis (H) holds then W CA is equivalent to ˜ R = R. The following Arrow-Sen theorem establishes the equivalence between W CA, SCA, W ARP , SARP and other conditions of rationality. Theorem 3.31. [91] (Arrow-Sen) Assume hypothesis (H) holds. For a choice function C : B → P(X) the following assertions are equivalent: (i) R is a regular preference and C is G–normal; ¯ is a regular preference and C is G–normal; (ii) R (iii) C veriﬁes W CA; (iv) C veriﬁes SCA; (v) C veriﬁes W ARP ; (vi) C veriﬁes SARP ; ˜ (vii) R = R;

44

3 Classical Revealed Preference Theory

¯=R ˜ and C is G–normal. (viii) R The previous theorem is stated in the form given in [91]; some implications or equivalences have been established by Arrow in [8]. Remark 3.32. The Arrow–Sen theorem has a remarkable meaning: (a) ﬁrstly, it is established that the axioms W CA, SCA, W ARP and SARP are equivalent ( in the presence of hypothesis H ); (b) secondly, each of these four axioms is equivalent to the best rationality situation expressed by (i) and (ii): the G–normality of C and its full– rationality; (c) thirdly, if one of these four axioms is fulﬁlled then the preference rela¯ and R ˜ associated with C coincide. tions R, R In [8] the following theorem was proved: Theorem 3.33. Assume that the hypothesis H holds. For a choice function C the following assertions are equivalent: (1) C veriﬁes Arrow Axiom; (2) C is full-rational. It is easy to see that conditions (1), (2) in Theorem 3.33 are equivalent to each of the eight conditions in Arrow–Sen theorem [103]. We remark that all these results are valid in the presence of hypothesis H. In fact, it suﬃces to use the weaker hypothesis that the domain of the choice function includes all two-element sets and all three-element sets. Investigation of the properties of rationality, congruence and revealed preference axioms in the general case, without hypothesis (H) is a direction started by Richter [85] and continued by Hansson [61], Suzumura [100] and others. We will recall ﬁrst an important theorem of Richter [85]. Theorem 3.34. Let C : B → P(X) be an arbitrary choice function. Then the following are equivalent:

3.3 Revealed Preference and Congruence Axioms. Consistency Conditions

45

(i) C veriﬁes SCA; (ii) There exists a regular preference relation Q on X such that Q is the G-rationalization of C. This result says that for an arbitrary choice function C, SCA is equivalent to the full–rationality of C. The following two results can be found in [100]: Theorem 3.35. Conditions W ARP , W CA and R ⊆ R∗ are mutually equivalent. Theorem 3.36. Conditions HARP and SCA are equivalent. Besides rationality and normality, a choice function can be related to consistency. Consistency conditions determine the choice by varying between subsets and supersets of available sets. Now we shall recall the consistency conditions α, β, γ, δ introduced by Sen in [91], [92], [93]. Let C : B → P(X) be a choice function. Condition α . For any pair of sets S, T ∈ B and for any x ∈ S, if x ∈ C(T ) and S ⊆ T then x ∈ C(S). That is, “if x is best in a set it is best in all subsets of it to which x belongs” [91]. Condition β . For any pair S, T ∈ B and for any x, y ∈ C(S), if S ⊆ T then x ∈ C(T ) iﬀ y ∈ C(T ). That is, “if x and y are both best in S, a subset of T , then x is best in T if and only if y is best in T ” [91]. Condition γ . Let M ⊆ B, V the union of all sets in M (V =

M) and

x ∈ X. If x ∈ C(S) for all S ∈ M then x ∈ C(V ). That is, “if x is best in each set in a class of sets such that their union is V , then x must be best in V ” [91].

46

3 Classical Revealed Preference Theory

Condition δ . For any pair of ﬁnite sets S, T ∈ B and for any (distinct) elements x, y ∈ C(S), if S ⊆ T then C(T ) = {x}. That is, “if x and y are both best in S, a subset of T , then neither of them can be uniquely best in T ” [91].

We notice that consistency condition α expresses the behaviour of the choice function at the contraction of the available sets of alternatives, while consistency conditions β, γ and δ express the behaviour of the choice function at the expansion of the available sets. Moreover, consistency condition γ shows how the choices are made with respect to the union of the available sets. Besides these properties, in [93] other consistency conditions are studied: Condition α2 . For any S ∈ B and for any x ∈ S, if x ∈ C(S) then x ∈ C({x, y}) for all y ∈ S.

Condition α2 means that if the alternative x is best in S then x is best in {x, y} for all y ∈ S. Condition γ2 . For any S ∈ B and for any x ∈ S, if x ∈ C({x, y}) for all y ∈ S then x ∈ C(S). Condition β(+) . For any pair of sets S, T ∈ B such that S ⊆ T and for any x ∈ C(S) and y ∈ S, if y ∈ C(T ) then x ∈ C(T ). Path Independence (P I) . For any pair of sets S, T ∈ B, C(S ∪ T ) = C(C(S) ∪ C(T )).

Path independence condition, studied by Plott [82] expresses the idea that the act of choice should be independent of the manner the available sets of alternatives split up.

3.3 Revealed Preference and Congruence Axioms. Consistency Conditions

47

We recall from [91], [92], [93] some propositions on consistency conditions. Later they will be analyzed in the fuzzy framework. Proposition 3.37. If hypothesis (H) holds then a G-normal choice function veriﬁes condition α. Theorem 3.38. Assume hypothesis (H) holds. Then a choice function veriﬁes W CA iﬀ it veriﬁes conditions α and β. A preference relation Q is quasi-transitive iﬀ the strict preference relation PQ is transitive. Theorem 3.39. Assume hypothesis (H) holds. For a G–normal choice function the associated preference relation R is quasi-transitive iﬀ condition δ is veriﬁed.

4 Fuzzy Preference Relations

This chapter is devoted to the study of fuzzy relations, a mathematical concept which models fuzzy relations. The ﬁrst section presents the deﬁnition of fuzzy sets and fuzzy relations together with a short discussion on their signiﬁcance. Section 4.2 studies the continuous t–norms and the residuated structure induced by the real interval [0, 1]. In Section 4.3 there are deﬁned operations with fuzzy sets and fuzzy relations, while Section 4.4 deals with the main properties of fuzzy relations and the associated indicators. The last section of the chapter deals with a fuzzy version of a theorem belonging to Szpilrajn. This result will be used in Chapter 7 for a fuzzy approach of a Richter theorem in classical consumer theory.

4.1 Fuzzy Sets and Fuzzy Relations The modelling of precise phenomena in the real world is done in accordance with the laws of bivalent logic: a sentence is true or false. The classical set theory is the theoretical base for modelling the precise phenomena. A set can be described by a sentence in the bivalent logic; consequently, an object belonging or not belonging to a set veriﬁes or not veriﬁes the property expressed by this sentence.

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 49–74 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

50

4 Fuzzy Preference Relations

Meanwhile there exist many real situations (social, economic, etc.) described by imprecise properties. Attributes as “tall men”, “almost true”, “a real number close to 10” lead to sentences out of the area of “true or false”. The membership of an individual to the set of “tall men” does not follow the criterion “yes or not”. These imprecise attributes will be called vague; the precise attributes will be called exact or crisp. Then it arises the necessity to deﬁne a concept of “set” for modelling the vague attributes. The answer was given by L.A.Zadeh in [117] by introducing the concept of fuzzy set. According to Zadeh (see [117], p.339): “The notion of fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the ﬁelds of pattern classiﬁcation and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply deﬁned criteria of class membership rather than the presence of random criteria”. Now we shall introduce the basic notion of fuzzy set. Let X be a non–empty set. For any subset A of X we consider the characteristic function χA : X → {0, 1} deﬁned by   1 if x ∈ A χA (x) =  0 if x ∈ A. A subset A of X is determined by its characteristic function. In fact, the assignment A → χA establishes a bijective correspondence between the set P(X) of the subsets of X and the set {0, 1}X of the functions of the form X → {0, 1}. The membership of an element x to the set A ⊆ X can be expressed in terms of χA :

4.1 Fuzzy Sets and Fuzzy Relations

x ∈ A ⇔ χA (x) = 1

51

(4.1)

A subset A of X is called a crisp set. The crisp sets are determined by bivalent properties; then an element x of X either belongs or does not belong to a crisp set A. The existence of vague properties leads to the necessity of introducing a notion of “vague subset” of X for which the membership is no longer a bivalent property. Starting by the representation of crisp sets by their characteristic functions L. A. Zadeh deﬁned the notion of fuzzy subset of X. By deﬁnition, a fuzzy subset of X is a function A : X → [0, 1]. For x ∈ X, the real number A(x) from the interval [0, 1] is called the degree of membership of x in A and represents the degree of truth of the statement “x belongs to A”. We denote by F(X) = [0, 1]X the family of fuzzy subsets of X. Since {0, 1}X ⊂ F(X) and P(X) and {0, 1}X can be identiﬁed by the bijection A → χA we can consider that P(X) is a subset of F(X); we write then P(X) ⊆ F(X). Example 4.1. ([120] p.12) We intend to represent by a fuzzy set the “the real numbers close to 10”. For this purpose, let us consider the function A : R → [0, 1] deﬁned by A(x) =

1 , x ∈ R. 1 + (x − 10)2

Then A is a fuzzy subset of R and A(x) represents the degree to which the real number x is “close to 10”. In [96] p.16, Sen observed then often the preferences of individuals have no ﬁxed meaning, being interpreted as “satisfactions, desires, values”. There exist several factors that can lead to vague preferences. A large number of attributes can determine the vagueness of preferences; some of the attributes have a more important role in evaluating an option. The criteria for evaluating the alternatives can be to some extent incomparable to one another. We add

52

4 Fuzzy Preference Relations

A( x )

1

5

10

15

x

Fig. 4.1. Real numbers close to 10

to this the partial information on the object of preferences and the human subjectivity as well. All of these are arguments for the study of vague preferences. If the exact preferences are represented by crisp relations on the set of alternatives, then the vague preferences must be modelled by a concept of fuzzy sets. A binary fuzzy relation on a set X is an element of F(X 2 ), i.e. a function Q : X 2 → [0, 1]. For x, y ∈ X the real number Q(x, y) ∈ [0, 1] represents the degree to which x is in relation Q with y. Q(x, y) = 0 means that x and y are not related at all, and Q(x, y) = 1 means full relationship between x and y. If X = {x1 , . . . , xn } then a fuzzy relation Q on X will be represented by a matrix A = (aij ) of dimension n × n deﬁned by aij = Q(xi , xj ) for any i, j = 1, . . . , n. Example 4.2. ([120], p. 62) Let us consider the statement “the real number x is much bigger than the real number y”. We intend to represent this vague property by a fuzzy relation on the set R of the real numbers.

4.2 Continuous T-norms

53

Let Q : R2 → [0, 1] be the fuzzy relation deﬁned by   0 if x ≤ y Q(x, y) =  (1 + (y − x)−2 )−1 if x > y. Then the real number Q(x, y) can be interpreted as “x is much bigger than y”. If X is a set of alternatives then the vague preferences on the elements of X will be represented by a fuzzy relation Q on X: we will call Q a fuzzy preference relation. For x, y ∈ X, the real number Q(x, y) will represent the degree to which the alternative x is at least as good as y.

4.2 Continuous T-norms The notion of t-norm was introduced by Schweizer and Sklar in [90]. The tnorm models the intersection of two fuzzy sets and the conjunction in fuzzy logic [40], [60], [18]. There is a vast literature dedicated to t-norms and to their applications to fuzzy set theory [18], [40], [60], [68]. In this paragraph we shall recall some notions and properties of t-norms. The basic reference for fuzzy preference relations is the monograph [40]. We shall present next some notions and results of fuzzy relations theory with regard to a continuous t-norm. Let [0, 1] be the unit interval. For any a, b ∈ [0, 1] we shall denote a ∨ b = max (a, b) and a ∧ b = min (a, b). More generally, for any {ai }i∈I ⊆ [0, 1] we denote

ai = sup{ai |i ∈ I}

(4.2)

ai = inf {ai |i ∈ I}

(4.3)

i∈I

i∈I

Then ([0, 1], ∨, ∧, 0, 1) becomes a bounded distributive lattice. Furthermore, [0, 1] is a distributive complete lattice.

54

4 Fuzzy Preference Relations

A triangular norm (=t-norm) is a binary operation ∗ on [0, 1] such that for any a, b, c ∈ [0, 1] the following axioms are veriﬁed: (T 1) commutativity: a ∗ b = b ∗ a; (T 2) associativity: a ∗ (b ∗ c) = (a ∗ b) ∗ c; (T 3) monotonicity: If a ≤ b then a ∗ c ≤ b ∗ c; (T 4) neutral element 1: a ∗ 1 = a. An immediate consequence is that a ∗ 0 = 0 for any a ∈ [0, 1]. A t-norm ∗ : [0, 1]2 → [0, 1] is said to be continuous if it is continuous as a function on the unit interval. With any continuous t-norm ∗ we associate a new binary operation → on [0, 1]: a→b=

{c ∈ [0, 1]|a ∗ c ≤ b}

(4.4)

The operation → is called the residuum or the implication associated with ∗. We list here the most well-known continuous t-norms and their corresponding residua: Lukasiewicz t-norm a ∗L b = max (0, a + b − 1); a →L b = min (1, 1 − a + b)   1 if a ≤ b a ∗G b = min (a, b); a →G b =  b if a > b G¨ odel t-norm

Product t-norm a ∗P b = ab; a →P b =

 

1 if a ≤ b  b/a if a > b

Let ∗ be a continuous t-norm. The properties mentioned in the following three lemmas reﬂect the main connections between the t–norm ∗ and its residuum →. Lemma 4.3. [18], [60] For any a, b, c ∈ [0, 1] the following properties hold: (1) a ∗ b ≤ c ⇔ a ≤ b → c; (2) a ∗ (a → b) = a ∧ b;

4.2 Continuous T-norms

55

(3) a ∗ b ≤ a; a ∗ b ≤ b; (4) b ≤ a → b; (5) a ≤ b ⇔ a → b = 1; (6) a = 1 → a; (7) 1 = a → a; (8) 1 = a → 1; (9) a ∗ (b ∨ c) = (a ∗ b) ∨ (a ∗ c); (10) a ≤ b implies b → c ≤ a → c and c → a ≤ c → b; (11) a → (b → c) = b → (a → c) = (a ∗ b) → c; (12) (a → b) ∗ (b → c) ≤ a → c. Deﬁnition 4.4. A residuated lattice is an algebraic structure (A, ∨, ∧, ∗, 0, 1) with four binary operations ∨, ∧, ∗, → and two constants 0 and 1 such that (1) (A, ∨, ∧, 0, 1) is a bounded lattice; (2) (A, ∗, 1) is a commutative monoid; (3) ∗ and → fulﬁll the residuation property, i.e. a ≤ b → c iﬀ a ∗ b ≤ c for all a, b, c ∈ A. In accordance with Lemma 4.3 (1), ([0, 1], ∨, ∧, ∗, →, 0, 1) is a residuated lattice (see [18], [60]). Let us notice that the property (2) of Lemma 4.3 does not hold for any residuated lattice (see [18], Theorem 2.36, p. 44). Lemma 4.5. [18], [60] For any {ai }i∈I , {bi }i∈I ⊆ [0, 1] and a ∈ [0, 1] the following properties hold: (1) a → ( (2) (

ai ) =

i∈I

(3)

(ai → a) ≤ (

ai ) → a;

i∈I

(a → ai ) ≤ a → (

i∈I

(5) (

(ai → a);

i∈I

i∈I

(4)

(a → ai );

i∈I

ai ) → a =

i∈I

i∈I

ai ) ∗ a =

i∈I

ai );

i∈I

(ai ∗ a);

56

4 Fuzzy Preference Relations

(6) (

ai ) ∗ (

i∈I

(7) (

j∈I

ai ) ∗ (

i∈I

bj ) =

(ai ∗ bj );

i,j∈I

bj ) ≤

j∈I

(ai ∗ bj ) ≤

i,j∈I

(ai ∗ bi ).

i∈I

The negation operation ¬ associated with a continuous t-norm ∗ is deﬁned by ¬a = a → 0 =

{c ∈ [0, 1]|a ∗ c = 0}

(4.5)

Recall the negations associated with Lukasiewicz, G¨ odel and product tnorms ([18] p. 31): Lukasiewicz t-norm ¬a = 1 − a G¨ odel  and product t-norms  1 if a = 0 ¬a =  0 if a > 0 Lemma 4.6. [18], [60] For any a, b, c ∈ [0, 1] the following properties hold: (1) a ≤ ¬b ⇔ a ∗ b = 0; (2) a ∗ ¬a = 0; (3) a ≤ ¬¬a; (4) ¬0 = 1, ¬1 = 0; (5) ¬a = ¬¬¬a; (6) a → b ≤ ¬b → ¬a; (7) a ≤ b ⇒ ¬b ≤ ¬a; (8) ¬(a ∨ b) = ¬a ∧ ¬b; (9) If ∗ is the Lukasiewicz t–norm then ¬¬a = a. The biresiduum ↔ of a continuous t-norm ∗ is deﬁned by ρ(a, b) = a ↔ b = (a → b) ∧ (b → a). for any a, b ∈ [0, 1].

(4.6)

4.2 Continuous T-norms

57

The biresidua corresponding to the Lukasiewicz, G¨ odel and product tnorms are ([18], p. 31): Lukasiewicz a ↔ b = 1 − |a − b| G¨ odel   1 if a=b a↔b=  min(a, b) otherwise product a↔b=

min(a,b) max(a,b)

Lemma 4.7. ([18]) For any a, b, c, d ∈ [0, 1] the following properties hold: (1) ρ(a, 1) = a; (2) a = b ⇔ ρ(a, b) = 1; (3) ρ(a, b) = ρ(b, a); (4) ρ(a, b) ≤ ρ(¬a, ¬b); (5) ρ(a, b) ∗ ρ(b, c) ≤ ρ(a, c); (6) ρ(a, b) ∧ ρ(c, d) ≤ ρ(a ∧ c, b ∧ d); (7) ρ(a, b) ∧ ρ(c, d) ≤ ρ(a ∨ c, b ∨ d); (8) ρ(a, b) ∗ ρ(c, d) ≤ ρ(a ∗ c, b ∗ d); (9) ρ(a, b) ∗ ρ(c, d) ≤ ρ(a → c, b → d); (10) ρ(a, b) ∗ a ≤ b; (11) a ∧ b ≤ ρ(a, b); (12) ρ(a, b) ∗ ρ(c, d) ≤ ρ(ρ(a, c), ρ(b, d)). Lemma 4.8. Let X be a non-empty set and f : X → [0, 1], g : X → [0, 1] two arbitrary functions. Then (1) ρ( f (x), g(x)) ≥ ρ(f (x), g(x));

x∈X

(2) ρ(

x∈X

x∈X

f (x),

x∈X

x∈X

g(x)) ≥

x∈X

ρ(f (x), g(x)).

58

4 Fuzzy Preference Relations

4.3 Operations with Fuzzy Sets and Fuzzy Relations The operations of the crisp sets are based on the Boolean structure of {0, 1}. In deﬁning the operations of fuzzy subsets of X one will start from some operations on [0, 1]. For any continuous t-norm ∗, ([0, 1], ∨, ∧, ∗, →, 0, 1) is a residuated lattice. From this algebraic structure one will deﬁne the operations on F(X). Let us ﬁx a continuous t-norm ∗. Let A, B be two fuzzy subsets of X. We deﬁne the fuzzy subsets A ∩ B, A ∪ B, A ∗ B and A → B of X by (A ∩ B)(x) = A(x) ∧ B(x)

(4.7)

(A ∪ B)(x) = A(x) ∨ B(x)

(4.8)

(A ∗ B)(x) = A(x) ∗ B(x)

(4.9)

(A → B)(x) = A(x) → B(x)

(4.10)

for any x ∈ X. Similarly, for any A ∈ F(X) we deﬁne ¬A ∈ F(X) by (¬A)(x) = ¬A(x) for any x ∈ X. 0 and 1 will be the constant fuzzy subsets 0(x) = 0 and 1(x) = 1 for any x ∈ X. Proposition 4.9. (F(X), ∪, ∩, ∗, →, 0, 1) is a residuated lattice. Proof. Straightforward, by using the deﬁnition of the operations of F(X). One notices that by the restriction of the operations ∪, ∩, → of F(X) to its subset P(X) one obtains the Boolean operations ∪, ∩, →.

4.3 Operations with Fuzzy Sets and Fuzzy Relations

59

Let {Ai }i∈I be a family of fuzzy subsets of X. Let us deﬁne the fuzzy subsets Ai , Ai of X by i∈I

i∈I

(

Ai )(x) =

i∈I

(

Ai (x)

(4.11)

Ai (x)

(4.12)

i∈I

Ai )(x) =

i∈I

i∈I

for each x ∈ X. This way F(X) becomes a complete residuated lattice. If A, B ∈ F(X) then let us denote A ⊆ B if A(x) ≤ B(x) for any x ∈ X. This notion extends the inclusion of the crisp sets. For any x1 , . . . , xn ∈ X denote by [x1 , . . . , xn ] the characteristic function of {x1 , . . . , xn }.

  1 if y ∈ {x , . . . , x } 1 n [x1 , . . . , xn ](y) =  0 if otherwise Let A, B ∈ F(X) and a ∈ [0, 1]; we deﬁne the functions A + B : X → R

and aA : X → R by (A + B)(x) = A(x) + B(x) and (aA)(x) = aA(x) for any x ∈ X. If X = {x1 , . . . , xn }, A ∈ F(X) and ai = A(xi ), i = 1, . . . , n then A = a1 χ{x1 } + a2 χ{x2 } + . . . + an χ{xn } . A fuzzy subset A of X is non-zero if A(x) = 0 for some x ∈ X; A is normal if A(x) = 1 for some x ∈ X. The support of A ∈ F(X) is deﬁned by supp A = {x ∈ X|A(x) > 0}. If A, B ∈ F(X) then we denote I(A, B) = (A(x) → B(x));

x∈X

E(A, B) =

(A(x) ↔ B(x)).

x∈X

I(A, B) is called the subsethood degree of A in B and E(A, B) the degree of equality of A and B. Intuitively I(A, B) expresses the truth value of the statement “A is included in B.” and E(A, B) expresses the truth value of the statement “A and B contain the same elements.”(see [18], p. 82 and p. 85).

60

4 Fuzzy Preference Relations

Example 4.10. ([18], p. 83) For Lukasiewicz, G¨odel and product t-norms we have

{1 − A(x) + B(x)|x ∈ X, A(x) > B(x)} (Lukasiewicz) I(A, B) = {B(x)|x ∈ X, A(x) > B(x)} (G¨ odel) I(A, B) = {B(x)/A(x)|x ∈ X, A(x) > B(x)} (product) I(A, B) =

Example 4.11. ([18], p. 85) For Lukasiewicz, G¨odel and product t-norms we have E(A, B) = E(A, B) = E(A, B) =

{1 − |A(x) − B(x)||x ∈ X} (Lukasiewicz) {A(x) ∧ B(x)|x ∈ X, A(x) = B(x)} (G¨ odel) {A(x)/B(x) ∧ B(x)/A(x)|x ∈ X} (product)

Lemma 4.12. ([18], p. 84-85) For any A, B, C ∈ F(X) we have (i) I(A, A) = 1; E(A, A) = 1; (ii) I(A, B) = 1 iﬀ A ⊆ B; (iii) E(A, B) = 1 iﬀ A = B; (iv) E(A, B) = E(B, A); (v) I(A, B) ∗ I(B, C) ≤ I(A, C); (vi) E(A, B) ∗ E(B, C) ≤ E(A, C); (vii) For any x ∈ X, E(A, B) ∗ A(x) ≤ B(x). Proof. We prove only (v). By applying Lemma 4.5 (7) and Lemma 4.3 (12) we have I(A, B) ∗ I(B, C) = [ ≤

(A(x) → B(x))] ∗ [

x∈X

(B(x) → C(x))]

x∈X

(A(x) → B(x)) ∗ (B(x) → C(x))

x∈X

≤

(A(x) → C(x)) = I(A, C)

x∈X

Besides I(., .) there exist plenty of indicators expressing the inclusion of one set into another. A large class of such indicators was axiomatically developed in [98], [29].

4.3 Operations with Fuzzy Sets and Fuzzy Relations

61

We have seen that fuzzy relations model vague preferences: if x, y ∈ X are two alternatives, then the real number Q(x, y) shows the degree to which x is preferred to y. In other words, Q(x, y) is the degree to which x is “at least as good as” y. To a fuzzy relation Q on X one assigns the fuzzy relations PQ and IQ on X deﬁned by PQ (x, y) = Q(x, y) ∗ ¬Q(y, x) and IQ (x, y) = Q(x, y) ∗ Q(y, x) for any x, y ∈ X. PQ is called the strict preference relation associated with Q, and IQ is called the indiﬀerence relation associated with Q. Then the real number PQ (x, y) (resp. IQ (x, y)) means the degree to which the alternative x is preferred to the alternative y (resp. the degree to which the alternatives x and y are equally preferred). We can consider also operations with fuzzy relations: they will be the operations of the complete residuated lattice F(X 2 ) deﬁned above. We deﬁne the fuzzy relation ∆ on X (the diagonal of X) by   1 if x = y , ∆=  0 if x = y for any x, y ∈ X. If Q is a fuzzy relation on X then the fuzzy relation Q−1 (the inverse of Q ) will be deﬁned by Q−1 (y, x) = Q(x, y) for any x, y ∈ X. Let Q1 , Q2 be two fuzzy relations on X. Their product Q1 ◦ Q2 will be a new fuzzy relation deﬁned by (Q1 ◦ Q2 )(x, y) = Q1 (x, z) ∗ Q2 (z, y) for any x, y ∈ X.

z∈X

Proposition 4.13. Let Q, Q1 , Q2 , Q3 be fuzzy relations on X. Then (1) Q1 ◦ (Q2 ◦ Q3 ) = (Q1 ◦ Q2 ) ◦ Q3 ; (2) Q ◦ ∆ = ∆ ◦ Q = Q; (3) Q1 ⊆ Q2 implies Q1 ◦ Q ⊆ Q2 ◦ Q and Q ◦ Q1 ⊆ Q ◦ Q2 ; (4) (Q−1 )−1 = Q; −1 (5) (Q1 ◦ Q2 )−1 = Q−1 2 ◦ Q1 .

62

4 Fuzzy Preference Relations

Proof. To exemplify we shall prove only (1). For any x, y ∈ X (Q1 ◦ (Q2 ◦ Q3 ))(x, y) = Q1 (x, u) ∗ (Q2 ◦ Q3 )(u, y).

u∈X

=

Q1 (x, u) ∗ (

u∈X

=

Q2 (u, v) ∗ Q3 (v, y))

v∈X

Q1 (x, u) ∗ Q2 (u, v) ∗ Q3 (v, y)

u,v∈X

A simple computation shows that ((Q1 ◦Q2 )◦Q3 )(x, y) has the same value.

Let us deﬁne Qn = Q ◦ Q ◦ . . . ◦ Q for any natural number n ≥ 1.

ntimes Lemma 4.14. Let Q be a fuzzy relation on X. Then for any natural number n ≥ 2 and x, y ∈ X the following holds: Qn (x, y) = Q(x, t1 ) ∗ . . . ∗ Q(tn−1 , y). t1 ,...,tn−1 ∈X

Proposition 4.15. Let Q be a fuzzy relation on X and {Qi }i∈I be a family of fuzzy relations on X. Then (1) (

i∈I

(2) (

Qi )−1 =

i∈I

i∈I

Qi ) ◦ Q =

i∈I

(3) (

Q−1 i ; (

Qi )−1 =

i∈I

(Qi ◦ Q); Q ◦ (

i∈I

Qi ) ◦ Q ⊆

Q−1 i

i∈I Qi ) = (Q ◦ Qi );

i∈I

(Qi ◦ Q); Q ◦ (

i∈I

i∈I

Qi ) ⊆

i∈I

(Q ◦ Qi ).

i∈I

Proof. We shall prove only the ﬁrst equality in (2). For any x, y ∈ X we have (( Qi ) ◦ Q)(x, y) = [( Qi )(x, z) ∗ Q(z, y)] i∈I

z∈X

=

i∈I

z∈X

=

[(

Qi (x, z)) ∗ Q(z, y)]

i∈I

Qi (x, z) ∗ Q(z, y)

z∈X i∈I

=

Qi (x, z) ∗ Q(z, y)

i∈I z∈X

=

i∈I

(Qi ◦ Q)(x, y) = (

i∈I

(Qi ◦ Q))(x, y).

4.4 Properties and Indicators of Fuzzy Relations

63

4.4 Properties and Indicators of Fuzzy Relations We have seen that fuzzy relations model vague preferences. The properties of the fuzzy relations studied in this section (reﬂexivity, transitivity, etc.) throw a better light on the way collective or individual preference relations are connected with various alternatives. Beside these properties, several indicators of a fuzzy preference relation are deﬁned, which correspond to the properties of reﬂexivity, transitivity, etc. These indicators have the following signiﬁcance: instead of checking whether a fuzzy relation Q has the property P , we evaluate the degree to which Q veriﬁes P . For instance, the degree of transitivity T rans(Q) will measure the extent to which the fuzzy relation Q is transitive. From the fuzzy logic point of view, T rans(Q) will measure the degree of truth of the statement “the relation Q is transitive”. We ﬁx a continuous t-norm ∗. Let X be a non–empty set and Q a fuzzy relation on X. Q is said to be: reﬂexive if Q(x, x) = 1 for all x ∈ X; irreﬂexive if Q(x, x) = 0 for all x ∈ X; symmetric if Q(x, y) = Q(y, x) for all x, y ∈ X; ∗–transitive if Q(x, y) ∗ Q(y, z) ≤ Q(x, z) for all x, y, z ∈ X; total if Q(x, y) > 0 or Q(y, x) > 0 for all distinct x, y ∈ X; strongly total if Q(x, y) = 1 or Q(y, x) = 1 for all distinct x, y ∈ X; complete if Q(x, y) > 0 or Q(y, x) > 0 for all x, y ∈ X; strongly complete if Q(x, y) = 1 or Q(y, x) = 1 for all x, y ∈ X. The above properties come from the ones in Proposition 3.6. One can see that some crisp notions have more fuzzy versions: total and strongly total for (crisp) total, complete and strongly complete for (crisp) complete. A regular fuzzy relation is a ∗–transitive and strongly complete fuzzy relation. The following proposition is obvious. Proposition 4.16. Let Q be a fuzzy relation on X. Then (a) Q is reﬂexive iﬀ ∆ ⊆ Q;

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4 Fuzzy Preference Relations

(b) Q is symmetric iﬀ Q = Q−1 ; (c) Q is ∗–transitive iﬀ Q ◦ Q ⊆ Q; (d) If Q is ∗–transitive then Qn ⊆ Q for any n ≥ 1; (e) If Q is reﬂexive then Q ⊆ Qn for any n ≥ 1. Proposition 4.17. If Q is a strongly complete fuzzy relation on X then PQ (x, y) = ¬Q(y, x) for all x, y ∈ X. Proof. Let x, y ∈ X. Since Q is strongly complete, Q(x, y) ∨ Q(y, x) = 1 hence by Lemma 4.5 (5) and Lemma 4.6 (2): ¬Q(y, x) = (Q(x, y) ∨ Q(y, x)) ∗ ¬Q(y, x) = (Q(x, y) ∗ ¬Q(y, x)) ∨ (Q(y, x) ∗ ¬Q(y, x)) = Q(x, y) ∗ ¬Q(y, x) = PQ (x, y). Proposition 4.18. Any intersection of ∗–transitive fuzzy relations is ∗– transitive. Proof. Let {Qi }i∈I be a family of ∗–transitive fuzzy relations on X. Denote Q= Qi . For any x, y, z ∈ X we get i∈I

Q(x, y) ∗ Q(y, z) = ( ≤

Qi (x, y)) ∗ (

i∈I

Qi (x, y) ∗ Qi (y, z) ≤

i∈I

Qi (y, z)) ≤

i∈I

Qi (x, z) = Q(x, z).

i∈I

Let Q be a fuzzy relation on X. Denote by T (Q) the intersection of all ∗–transitive fuzzy relations that contain Q: T (Q) = {Q |Q ⊆ Q and Q is transitive}. According to Proposition 4.18, T (Q) ∗– transitive fuzzy relations that contains Q. T (Q) is called the ∗–transitive closure of Q. Remark that T (Q) = Q iﬀ Q is ∗–transitive. Proposition 4.19. Let Q be a fuzzy relation on X. Then T (Q) =

∞ n=1

Qn .

4.4 Properties and Indicators of Fuzzy Relations ∞

Proof. Let Q =

65

Qn . Then Q is the ∗–transitive closure of Q iﬀ the

n=1

following three conditions are fulﬁlled: (a) Q ⊆ Q ; (b) Q is ∗–transitive; (c) If Q1 is a ∗–transitive fuzzy relation that contains Q then Q ⊆ Q1 . Since Q1 = Q we have Q ⊆ Q . One notices that ∞ ∞ ∞ Qn ) ◦ ( Qm ) = Qn+m ⊆ Q , Q ◦ Q = ( n=1

m=1

therefore Q is ∗–transitive.

m,n=1

Let Q1 be a ∗–transitive fuzzy relation such that Q ⊆ Q1 . For any n ≥ 1 ∞ Qn ⊆ Q1 . we have Qn ⊆ Qn1 ⊆ Q1 since Q1 is ∗–transitive. Then Q = n=1

We have proved that T (Q) = Q . Corollary 4.20. For any x, y ∈ X, ∞ T (Q)(x, y) = Q(x, y) ∨

Q(x, t1 ) ∗ . . . ∗ Q(tn , y)

n=1 t1 ,...,tn ∈X

Proof. By Proposition 4.19 we have ∞ ∞ n Q (x, y) = Q(x, y) ∨ Qn+1 (x, y) = T (Q)(x, y) = n=1

= Q(x, y) ∨

∞

n=1

Q(x, t1 ) ∗ . . . ∗ Q(tn , y).

n=1 t1 ,...,tn ∈X

Remark 4.21. If Q is reﬂexive then T (Q) is also reﬂexive. If we denote Q0 = ∆ ∞ Qn . then in this case T (Q) = n=0

In the approximate reasoning we can distinguish variables that are not identical, but have close behaviour becoming “identiﬁable”. Therefore we need some concepts to express situations when fuzzy sets and/ or fuzzy relations are “identiﬁable”. The notions of similarity and δ–equality are very satisﬁable to describe such situations.

66

4 Fuzzy Preference Relations

Let X be a non–empty set. A fuzzy relation Q on X is said to be a similarity relation if it is reﬂexive, symmetric and ∗–transitive. If x, y ∈ X then Q(x, y) will be called the similarity degree of x and y. Let A, B be two fuzzy subsets of X. Recall the deﬁnition of the degree of equality of A and B: E(A, B) = (A(x) ↔ B(x)). The assignment x∈X

(A, B) −→ E(A, B) deﬁnes a fuzzy relation E on the set F(X). By Lemma 4.12 (iii)-(iv), E is a similarity relation on F(X). E(A, B) is the similarity degree of A and B (with respect to the similarity relation E). If A, B are two fuzzy subsets of X and 0 ≤ δ ≤ 1 then we shall say that A, B are (∗, δ)–equal (A =δ B in symbols) if E(A, B) ≥ δ. Remark 4.22. Assume that ∗ is the Lukasiewicz t–norm. It is easy to see that a ↔ b = 1 − |a − b| for any a, b ∈ [0, 1], therefore E(A, B) = (1 − |A(x) − B(x)|) = 1 − |A(x) − B(x)|. x∈X

Thus A =δ B iﬀ

x∈X

|A(x) − B(x)| ≤ 1 − δ. In this way we obtain the

x∈X

notion of Cai δ–equality, studied in [23] and [24]. If Q is a crisp relation on X then the sentence “Q is reﬂexive” is a statement in the bivalent logic: it is true or false. For a fuzzy relation Q on X it is more appropriate to place the sentence “Q is reﬂexive” in the setting of a fuzzy logic [60], [18], etc.. Then instead of saying that Q is reﬂexive or not we shall consider the degree of truth of that statement. This will be a real number in the interval [0, 1] and it will express “how reﬂexive” the fuzzy relation Q is. The indicators introduced by Deﬁnition 4.23 represent degrees of truth which correspond to properties of fuzzy relations such as reﬂexivity, symmetry, etc. Deﬁnition 4.23. [18] Let Q be a fuzzy relation on X. We deﬁne the following indicators: (a) the degree of reﬂexivity of Q: Ref (Q) = Q(x, x); x∈X

4.4 Properties and Indicators of Fuzzy Relations

67

(b) the degree of symmetry of Q: Sym(Q) = (Q(x, y) → Q(y, x)) = (Q(x, y) ↔ Q(y, x)); x,y∈X

x,y∈X

(c) the degree of ∗–transitivity of Q: T rans(Q) = (Q(x, y) ∗ Q(y, z) → Q(x, z)); x,y,z∈X

(d) the degree of strong totality of Q: ST (Q) = (Q(x, y) ∨ Q(y, x)); x=y

(e) the degree of strong completeness of Q: SC(Q) = (Q(x, y) ∨ Q(y, x)). x,y∈X

It is easy to see that the ﬁrst three indicators can be written in the form ([18], p. 191): Ref (Q) = I(∆, Q) Sym(Q) = E(Q, Q−1 ) T rans(Q) = I(Q2 , Q). Lemma 4.24. For any fuzzy relation Q on X the following assertions hold: (a) Ref (Q) = 1 iﬀ Q is reﬂexive; (b) Sym(Q) = 1 iﬀ Q is symmetric; (c) T rans(Q) = 1 iﬀ Q is ∗–transitive; (d) ST (Q) = 1 iﬀ Q is strongly total; (e) SC(Q) = 1 iﬀ Q is strongly complete. Proof. We shall prove (c) for example: T rans(Q) = 1 iﬀ Q(x, y) ∗ Q(y, z) → Q(x, z) = 1 for all x, y, z ∈ X; iﬀ Q(x, y) ∗ Q(y, z) ≤ Q(x, z) for all x, y, z ∈ X; iﬀ Q is ∗–transitive. Remark 4.25. The ﬁve indicators introduced in Deﬁnition 4.23 correspond to the ﬁve properties of fuzzy relations from Proposition 4.16. They express the extent to which a fuzzy relation Q veriﬁes these properties. The following example holds true when ∗ is the G¨ odel t–norm ∧:

68

4 Fuzzy Preference Relations

Example 4.26. (I) Let X = {x, y} and the fuzzy preference relation given by the matrix



 Q=

ab



cd where 0 ≤ a, b, c, d ≤ 1. Then Ref (Q) = a ∧ d; Sym(Q) = b ↔ c; T rans(Q) = 1; ST (Q) = b ∨ c; SC(Q) = a ∨ b ∨ c ∨ d. (II) Let X = {1, 2, 3} and the fuzzy preference relation Q on X deﬁned by the matrix:

 1

1 1 2 3

  Q =  31 1  1 1 2 2

1 3

   . 

1

By computation we get  1

1 1 2 3

  Q2 =  21 1  1 1 2 2

Therefore T rans(Q) = I(Q2 , Q) =

3

1 3

   . 

1

(Q2 (i, j) → Q(i, j)) =

i,j=1

1 1 1 → = . 2 3 3

Proposition 4.27. Let Q be a fuzzy relation on X and x, y, z ∈ X. Therefore (a) Ref (Q) ≤ Q(x, x); (b) Sym(Q) ∗ Q(x, y) ≤ Q(y, x); (c) T rans(Q) ∗ Q(x, y) ∗ Q(y, z) ≤ Q(x, z); (d) If x = y then ST (Q) ≤ Q(x, y) or ST (Q) ≤ Q(y, x); (e) SC(Q) ≤ Q(x, y) or SC(Q) ≤ Q(y, x).

4.4 Properties and Indicators of Fuzzy Relations

69

Proof. To exemplify, we prove (c). By the deﬁnition of T rans(Q) we have T rans(Q) ≤ Q(x, y) ∗ Q(y, z) → Q(x, z) from where, by applying Lemma 4.3 (1), one obtains T rans(Q) ∗ Q(x, y) ∗ Q(y, z) ≤ Q(x, z). The following two propositions show how the ﬁve indicators deﬁned above are preserved by the fuzzy equality E(., .). Proposition 4.28. Let Q1 , Q2 be two fuzzy relations on X. Then (1) Ref (Q1 ) ∗ E(Q1 , Q2 ) ≤ Ref (Q2 ); (2) ST (Q1 ) ∗ E(Q1 , Q2 ) ≤ ST (Q2 ); (3) SC(Q1 ) ∗ E(Q1 , Q2 ) ≤ SC(Q2 ); (4) Sym(Q1 ) ∗ E(Q1 , Q2 ) ≤ Sym(Q2 ); Proof. (1) For any x ∈ X we have Ref (Q1 ) ∗ E(Q1 , Q2 ) ≤ Q1 (x) ∗ (Q1 (x) ↔ Q2 (x)) ≤ ≤ Q1 (x) ∗ (Q1 (x) → Q2 (x)) = Q1 (x) ∧ Q2 (x) ≤ Q2 (x). It follows Ref (Q1 ) ∗ E(Q1 , Q2 ) ≤

Q2 (x, x) = Ref (Q2 ).

x∈X

(2) Let x, y ∈ X such that x = y. Then ST (Q1 ) ∗ E(Q1 , Q2 ) ≤ (Q1 (x, y) ∨ Q1 (y, x)) ∗ E(Q1 , Q2 ) = = Q1 (x, y) ∗ E(Q1 , Q2 ) ∨ Q1 (y, x) ∗ E(Q1 , Q2 ) ≤ ≤ Q1 (x, y) ∗ (Q1 (x, y) → Q2 (x, y)) ∨ Q1 (y, x) ∗ (Q1 (y, x) → Q2 (y, x)) = = (Q1 (x, y) ∧ Q2 (x, y)) ∨ (Q1 (y, x) ∨ Q2 (y, x)) ≤ Q2 (x, y) ∨ Q2 (y, x). From this we have ST (Q1 ) ∗ E(Q1 , Q2 ) ≤

(Q2 (x, y) ∨ Q2 (y, x)) = ST (Q2 ).

x=y

(3) Similarly as (2). (4) Let x, y ∈ X. Then, by Lemma 4.3 (2) Sym(Q1 ) ∗ E(Q1 , Q2 ) ∗ Q2 (x, y) ≤ Sym(Q1 ) ∗ Q1 (x, y) ∗ E(Q1 , Q2 ) ≤ ≤ Q1 (x, y) ∗ (Q1 (x, y) → Q1 (y, x)) ∗ E(Q1 , Q2 ) = = Q1 (x, y) ∗ Q1 (y, x) ∗ E(Q1 , Q2 ) ≤ ≤ Q1 (y, x) ∗ E(Q1 , Q2 ) ≤ Q2 (y, x).

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4 Fuzzy Preference Relations

By Lemma 4.3 (1), Sym(Q1 ) ∗ E(Q1 , Q2 ) ≤ Q2 (x, y) → Q2 (y, x) for any x, y ∈ X, therefore Sym(Q1 ) ∗ E(Q1 , Q2 ) ≤

(Q2 (x, y) → Q2 (y, x)) = Sym(Q2 ).

x,y∈X

Proposition 4.29. Suppose ∗ is the G¨ odel t-norm ∧. Then for any two fuzzy relations Q1 , Q2 on X we have: T rans(Q1 ) ∧ E(Q1 , Q2 ) ≤ T rans(Q2 ). Proof. Let x, y, z ∈ X. Then by Proposition 4.27 (c) T rans(Q1 ) ∧ E(Q1 , Q2 ) ∧ Q2 (x, y) ∧ Q2 (y, z) = E(Q1 , Q2 ) ∧ T rans(Q1 ) ∧ Q1 (x, y) ∧ Q1 (y, z) ≤ ≤ E(Q1 , Q2 ) ∧ Q1 (x, z) ≤ Q2 (x, z). By applying Lemma 4.3 (1), for any x, y, z ∈ X we have T rans(Q1 ) ∧ E(Q1 , Q2 ) ≤ (Q2 (x, y) ∧ Q2 (y, z)) → Q2 (x, z) from where T rans(Q1 ) ∧ E(Q1 , Q2 ) ≤

[(Q2 (x, y) ∧ Q2 (y, z)) → Q2 (x, z)] =

x,y,z∈X

T rans(Q2 ).

Proposition 4.30. Let Q be a fuzzy relation on X. For any x, y ∈ X we have T rans(Q) ∗ T (Q)(x, y) ≤ Q(x, y). Proof. Let n ≥ 1 and x, y, t1 , . . . , tn ∈ X. Applying several times Proposition 4.27, (c) we get T rans(Q) ∗ Q(x, t1 ) ∗ . . . ∗ Q(tn , y) ≤ Q(x, y). Therefore T rans(Q) ∗ T (Q)(x, y) = T rans(Q) ∗ [Q(x, y) ∨

∞

Q(x, t1 ) ∗ . . . ∗

n=1 t1 ,...,tn ∈X

Q(tn , y)] = = T rans(Q) ∗ Q(x, y) ∨

∞

n=1 t1 ,...,tn ∈X

Q(x, y).

T rans(Q) ∗ Q(x, t1 ) ∗ . . . , ∗Q(tn , y) ≤

4.5 An Extension Theorem

71

Remark 4.31. By Proposition 4.30, for all x, y ∈ X we have T rans(Q) ≤ T (Q)(x, y) → Q(x, y) = T (Q)(x, y) ↔ Q(x, y) hence T rans(Q) ≤ (T (Q)(x, y) ↔ Q(x, y)) = E(Q, T (Q)). x,y∈X

4.5 An Extension Theorem A classical Szpilrajn theorem [106] asserts that any strict partial order is a subrelation of a strict total order. The ﬁrst result in formulating a fuzzy version of Szpilrajn theorem was established by Zadeh in [118]. A fuzzy version of Szpilrajn theorem was also obtained by Gottwald in [58] for t-norms without zero divisors. Paper [19] contains an analysis of Szpilrajn theorem in fuzzy orderings context with respect to a t-norm ∗ and a ∗-equivalence E. In particular Szpilrajn theorem is the main tool for proving a well-known theorem of Richter [85] that establishes the equivalence between rational and congruous consumers. In this section we will formulate and prove a fuzzy version of Szpilrajn theorem, that will be later used for a fuzzy analysis of Richter theorem in consumer theory. Theorem 4.32. (Szpilrajn [106]) Any strict partial order can be embedded in a strict total order on X. An important problem is to see to what extent a fuzzy version of Szpilrajn theorem holds true for an arbitrary continuous t-norm. Our extension theorem will be later used for the fuzzy analysis of Richter theorem in consumer theory. Let R, Q be two fuzzy relations on X such that R ⊆ Q. We say that the extension Q of R preserves the irreﬂexivity of R if Q(x, x) = 0 for each x ∈ X such that R(x, x) = 0. A fuzzy relation R on X is a strict partial ∗-order if it is irreﬂexive and ∗-transitive; a fuzzy relation is a total strict ∗-order if it is total, irreﬂexive and ∗-transitive. In this section we shall prove that any ∗-transitive fuzzy relation R on X can be extended to a total ∗-transitive fuzzy relation Q on X preserving the

72

4 Fuzzy Preference Relations

irreﬂexivity of R. From this one infers that any strict partial ∗-order on X can be extended to a total strict ∗-order on X. This result can be viewed as a fuzzy version of Szpilrajn theorem. ¯ will be Let ∗ be a continuous t-norm. If R is a fuzzy relation on X then R the fuzzy relation deﬁned by

¯ b) = R(a,

 

1 if a = b  R(a, b) if a = b

¯ for any a, b ∈ X. It is obvious that R ⊆ R. ¯ is also ∗-transitive. Lemma 4.33. If R is ∗-transitive then R ¯ b) ∗ R(b, ¯ c) ≤ R(a, ¯ c). If a = c Proof. Let a, b, c ∈ X. We shall prove that R(a, ¯ c) = 1 and the inequality is trivially veriﬁed. Assume a = c. If then R(a, ¯ b) ∗ R(b, ¯ c) = R(a, b) ∗ R(b, c) ≤ R(a, c) = R(a, ¯ c). a = b, b = c then R(a, ¯ b) ∗ R(b, ¯ c) = 1 ∗ R(b, c) = R(b, c) = Supposing a = b, b = c we have R(a, ¯ c); the case a = b, b = c follows similarly. If a = b = c then the R(a, c) = R(a, inequality is obvious. Lemma 4.34. Let R be a ∗-transitive fuzzy relation on X and p, q two distinct elements of X such that R(p, q) = R(q, p) = 0. Then there exists a ∗-transitive fuzzy relation R on X fulﬁlling the following conditions: (i) R ⊆ R ; (ii) R (p, q) = 1; (iii) R preserves the irreﬂexivity of R. Proof. The fuzzy relation R is deﬁned by ¯ p) ∗ R(q, ¯ b)] R (a, b) = R(a, b) ∨ [R(a, ¯ p)∗ for all a, b ∈ X. It is obvious that R ⊆ R and R (p, q) = R(p, q)∨[R(p, ¯ q)] = R(p, q) ∨ 1 = 1. R(q, In order to prove the ∗-transitivity of R one remembers that

4.5 An Extension Theorem

73

¯ p) ∗ R(q, ¯ y))] ∗ [R(y, z) ∨ (R(y, ¯ p) ∗ R (x, y) ∗ R (y, z) = [R(x, y) ∨ (R(x, ¯ z))] = [R(x, y) ∗ R(y, z)] ∨ [R(x, y) ∗ R(y, ¯ p) ∗ R(q, ¯ z)] ∨ [R(x, ¯ p) ∗ R(q, ¯ y) ∗ R(q, ¯ z)] ∨ [R(x, ¯ p) ∗ R(q, ¯ y) ∗ R(y, ¯ p) ∗ R(q, ¯ z)] and R(y, ¯ p) ∗ R(q, ¯ z)]. R (x, z) = R(x, z) ∨ [R(x, The following inequalities follow from hypothesis and by Lemma 4.33: R(x, y) ∗ R(y, z) ≤ R(x, z) ≤ R (x, z); ¯ p) ∗ R(q, ¯ z) ≤ R(x, ¯ y) ∗ R(y, ¯ p) ∗ R(q, ¯ z) ≤ R(x, ¯ p) ∗ R(q, ¯ z) ≤ R(x, y) ∗ R(y, R (x, z); ¯ p) ∗ R(q, ¯ y) ∗ R(y, ¯ z) ≤ R(x, ¯ p) ∗ R(q, ¯ z) ≤ R (x, z); R(x, ¯ p) ∗ R(q, ¯ y) ∗ R(y, ¯ p) ∗ R(q, ¯ z) ≤ R(x, ¯ p) ∗ R(p, ¯ q) ∗ R(q, ¯ z) = 0 R(x, ¯ q) = R(p, q) = 0. since R(p, Therefore R (x, y) ∗ R (y, z) ≤ R (x, z) hence R is ∗-transitive. Assume R(x, x) = 0, then, by Lemma 4.33 one gets ¯ p) ∗ R(q, ¯ x)] = R(x, ¯ p) ∗ R(q, ¯ x) = R(q, ¯ x) ∗ R (x, x) = R(x, x) ∨ [R(x, ¯ p) ≤ R(q, ¯ p) = R(q, p) = 0. R(x, Corollary 4.35. Let R be a strict partial ∗-order on X, p, q ∈ X, p = q and R(p, q) = R(q, p) = 0. Then there exists a strict partial ∗-order R such that R ⊆ R and R (p, q) = 1. Theorem 4.36. Let R be a ∗-transitive fuzzy relation on X. Then there exists a total ∗-transitive fuzzy relation R∗ on X such that (i) R ⊆ R∗ ; (ii) R∗ preserves the irreﬂexivity of R. Proof. Let C be the family of ∗-transitive fuzzy relations Q on X such that R ⊆ Q and Q preserves the irreﬂexivity of R. Obviously R ∈ C. C is partially ordered w.r.t. the inclusion ⊆ of fuzzy relations. We shall prove that (C, ⊆) is inductive. i.e. every totally ordered family {Qi }i∈I ⊆ C has an upper bound in C. Let {Qi }i∈I be a totally ordered family in C, i.e. for all i, j ∈ I we have Qi ⊆ Qj or Qj ⊆ Qi . It suﬃces to prove that Q = Qi belongs to C. i∈I

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Let x, y, z ∈ X. According to Lemma 4.5 (6) we have: Q(x, y) ∗ Q(y, z) = [

i∈I

Qi (x, y)] ∗ [

Qj (y, z)] =

j∈I

(Qi (x, y) ∗ Qj (y, z)).

i,j∈I

Let i, j ∈ I so Qi ⊆ Qj or Qj ⊆ Qi . Assume for example that Qi ⊆ Qj , hence Qi (x, y) ∗ Qj (y, z) ≤ Qj (x, y) ∗ Qj (y, z) ≤ Qj (x, z). This inequality holds for any i, j ∈ I, hence Q(x, y) ∗ Q(y, z) ≤ Q(x, z). Thus Q is ∗-transitive. If R(x, x) = 0 then Qi (x, x) = 0 for any i ∈ I hence Q(x, x) = Qi (x, x) = 0. It is obvious that R ⊆ Q hence Q ∈ C and (C, ⊆) i∈I

is inductive. By Zorn’s Lemma there exists a maximal member R∗ in C, i.e. R∗ ⊆ P and P ∈ C implies R∗ = P . According to the deﬁnition of C, R∗ is ∗-transitive, R ⊆ R∗ and R∗ (x, x) = 0 for each x ∈ X such that R(x, x) = 0. Now we shall prove that R∗ is total, i.e. R∗ (x, y) > 0 or R∗ (y, x) > 0 for all distinct x, y ∈ X. By absurdum assume R∗ (p, q) = R∗ (q, p) = 0 for some distinct p, q ∈ X. By Lemma 4.34 there exists a ∗-transitive fuzzy relation R on X such that R∗ ⊆ R , R (p, q) = 1 and R preserves the irreﬂexivity of R∗ . One remarks that R∗ preserves the irreﬂexivity of R: R(x, x) = 0 ⇒ R∗ (x, x) = 0 ⇒ R (x, x) = 0. Hence R∗ R ∈ C. This contradicts the maximality of R∗ , hence R∗ is total. Corollary 4.37. If R is a strict partial ∗-order on X then there exists a total strict ∗-order R∗ on X such that R ⊆ R∗ . This corollary is a fuzzy version of Szpilrajn theorem.

5 Fuzzy Choice Functions

The economic and social life is governed by complex phenomena, that might have exact and/or vague components. According to the relationship exact–vague, there have been studied the following situations: (a) exact preferences and exact choices; (b) vague preferences and exact choices; (c) vague preferences and vague choices. The ﬁrst situation represents the subject of the classic theory of choice functions [9], [94], [96], [103]. In the second situation, even if the preferences are vague, the act of choice is characterized by the precise speciﬁcation of the chosen alternatives [15], [16], [17], [69], etc. There are cases when both the preferences and the choices are vague (e. g. in diﬀerent moments of a negotiation when the option is not deﬁnitive [70], [71], [67]). This is related to the third situation. The ﬁrst section of this chapter concerns the situation (b); exact choice functions associated with various fuzzy preference relations are studied. With the score functions nine such functions are deﬁned [16], but the results of this section are based mainly on the Orlovsky function [79]. A characterization of

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 75–106 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

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the Orlovsky function is presented in terms of some properties of the fuzzy preference relations. With the Section 5.2 the study of the relationship between the fuzzy choice functions and the fuzzy preference relations begins, therefore we discuss the situation (c). We introduce a general class of fuzzy choice functions that includes the ones studied by Banerjee [14]. With a fuzzy choice function several revealed preference relations are associated, and the revealed preference axioms W AF RP , SAF RP , HAF RP and the congruence axioms W F CA, SF CA are formulated. Section 5.3 studies the rationality and the normality of the fuzzy choice functions, notions that have a central role in this book. They represent the mathematical modelling of the rational behaviour of an act of choice. Two concepts of rationality are studied (M –rationality and G–rationality) and two concepts of normality (M –normality and G–normality) as a special case of rationality. In line with Arrow’s impossibility theorem [9], several papers studied the problem of aggregating the individual preferences into collective preferences (e. g. [94], [96]). Some authors studied the way the individual preferences aggregate into a collective choice function ([103]), while other authors ([3], [4], [7]) studied the way the individual choice functions aggregate into a collective choice function. The fuzzy choice functions introduced by us in this chapter could be a propitious framework for the treatment of this problem in fuzzy context.

5.1 From Fuzzy Preferences to Exact Choices In this section we shall consider vague preferences and exact choices. This situation is mathematically modelled by fuzzy preference relations and by

5.1 From Fuzzy Preferences to Exact Choices

77

crisp choice functions. We shall study how with a fuzzy preference relation we can associate various crisp choice functions. Let X be a universe of alternatives and B a family of available subsets of X, i.e. (X, B) is a choice space. Throughout this section H will be a non–empty family of reﬂexive fuzzy preference relations on X. Example 5.1. [79] An agent wants to choose a non-empty set C(S) from each available set S ∈ B. Based on an expertise he has to ﬁnd a way to evaluate how “an alternative x is preferred to an alternative y”. Suppose n experts are consulted in saying without ambiguity that “alternative x is preferred to alternative y” (for any x, y ∈ X). Herewith each expert i produces a crisp preference relation Qi on X. The agent will combine the evaluations of the n experts. Following [79], for any x, y ∈ X denote by Q(x, y) the fraction of the number of experts that preferred x to y in the total number n: Q(x, y) =

card{i|(x,y)∈Qi } . n

For the agent the real number Q(x, y) ∈ [0, 1] represents the degree of preference of x with respect to y. In this way we have obtained a fuzzy preference relation Q : X 2 → [0, 1]. In his choice, the agent will rely on the fuzzy preference relation Q obtained by the aggregation of the results of the n experts. The agent’s choice will be exact and will be modelled by a crisp choice function C associated with the fuzzy preference relation Q. This situation when the exact choices are based on vague preferences leads to the notion of preference–based choice function. A preference-based choice function (P CF ) is a function C : B×H → P(X) such that for any A ∈ B and Q ∈ H, C(A, Q) is a non–empty subset of X. For any fuzzy preference relation Q ∈ H we obtain a crisp choice function C(., Q) : B → P(X). Intuitively, C(A, Q) is the set of alternatives chosen from the available set A when the fuzzy preference relation Q is taken as basis.

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If for any Q ∈ H the crisp choice function C(., Q) veriﬁes a property P then we shall say that the P CF C veriﬁes P . For example, C veriﬁes the consistency condition α if for any Q ∈ H, for any A, B ∈ B and for any x ∈ X we have x ∈ A, x ∈ C(B, Q), A ⊆ B ⇒ x ∈ C(A, Q). In what follows let us suppose that the set X is ﬁnite. We shall present a simple way to obtain a P CF . A function with the form W : X × B × H → R is called score function. By a score function, with every triple (x, A, Q) ∈ X × B × H one can associate a real number W (x, A, Q) named the score of the alternative x based on the action of Q on A. Let W be a score function. For any A ∈ B and Q ∈ H we denote: CW (A, Q) = {x ∈ A|W (x, A, Q) ≥ W (y, A, Q) for all y ∈ A} The assignment (A, Q) → CW (A, Q) deﬁnes a crisp choice function on (X, B) (due to the ﬁnitude of X). CW (A, Q) represents the choice set of the alternatives with higher score. The ﬁrst P CF has been obtained in this way by Orlovsky [79]. The Orlovsky score is deﬁned by OV (x, A, Q) = miny∈A min(1 − Q(y, x) + Q(x, y), 1) for any A ∈ B, Q ∈ H and x ∈ A. Then the P CF Orlovsky COV is deﬁned by COV (A, Q) = {x ∈ A|OV (x, A, Q) ≥ OV (y, A, Q) for all y ∈ A}. Remark 5.2. The Orlovsky score can be written: OV (x, A, Q) = (Q(y, x) →L Q(x, y)) y∈A

for any x ∈ A, where →L is the Lukasiewicz implication. The following score functions are deﬁned by [16]: M F (x, A, Q) = {Q(x, y)|y ∈ A \ {x}}; mF (x, A, Q) = {Q(x, y)|y ∈ A \ {x}}; M A(x, A, Q) = {Q(y, x)|y ∈ A \ {x}};

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79

{Q(y, x)|y ∈ A \ {x}}; SF (x, A, Q) = y∈A\{x} Q(x, y); SA(x, A, Q) = y∈A\{x} Q(y, x); M D(x, A, Q) = {Q(x, y) − Q(y, x)|y ∈ A \ {x}}; mD(x, A, Q) = {Q(x, y) − Q(y, x)|y ∈ A \ {x}}; SD(x, A, Q) = y∈A\{x} (Q(x, y) − Q(y, x)) mA(x, A, Q) =

for any x ∈ A. Remark 5.3. In [16], M F (x, A, Q) is interpreted as “max for x in A, deﬁned in terms of Q”, mF (x, A, Q) as “min for x in A, deﬁned in terms of Q” and M A, mA, SF , SA, M D, mD, SD have the signiﬁcance “max against”, “min against”, “sum for”, “sum against”, “max diﬀerence”, “min diﬀerence” and “sum diﬀerence”, respectively. The P CF ’s associated with these nine score functions will be deﬁned as follows: CM F (A, Q) = {x ∈ A|M F (x, A, Q) ≥ M F (y, A, Q) for all y ∈ A}; CmF (A, Q) = {x ∈ A|mF (x, A, Q) ≥ mF (y, A, Q) for all y ∈ A}; CM A (A, Q) = {x ∈ A|M A(x, A, Q) ≤ M A(y, A, Q) for all y ∈ A}; CmA (A, Q) = {x ∈ A|mA(x, A, Q) ≤ mA(y, A, Q) for all y ∈ A}; CSF (A, Q) = {x ∈ A|SF (x, A, Q) ≥ SF (y, A, Q) for all y ∈ A}; CSA (A, Q) = {x ∈ A|SA(x, A, Q) ≤ SA(y, A, Q) for all y ∈ A}; CM D (A, Q) = {x ∈ A|M D(x, A, Q) ≥ M D(y, A, Q) for all y ∈ A}; CmD (A, Q) = {x ∈ A|mD(x, A, Q) ≥ mD(y, A, Q) for all y ∈ A}; CSD (A, Q) = {x ∈ A|SD(x, A, Q) ≥ SD(y, A, Q) for all y ∈ A}. For A ⊆ X and Q ∈ H we shall deﬁne the set of pairwise dominant alternatives in A given Q by P D(A, Q) = {x ∈ A|Q(x, y) ≥ Q(y, x) for all y ∈ A}. P D(A, Q) can be empty for some A and Q.

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Proposition 5.4. [16] For all A ∈ B and Q ∈ H, COV (A, Q) = CmD (A, Q). Proof. One distinguishes two cases. Case a P D(A, Q) = ∅. Then for any x ∈ A we have (1) x ∈ P D(A, Q) ⇒ x ∈ COV (A, Q); (2) x ∈ P D(A, Q) ⇒ x ∈ COV (A, Q). The properties (1) and (2) follow according to the implications: x ∈ P D(A, Q) ⇒ Q(x, y) ≥ Q(y, x) for all y ∈ A ⇒ OV (x, A, Q) = 1 ⇒ OV (x, A, Q) ≥ OV (y, A, Q) for all y ∈ A ⇒ x ∈ COV (A, Q). Also x ∈ P D(A, Q) ⇒ Q(x, y0 ) < Q(y0 , x) for some y0 ∈ A ⇒ OV (x, A, Q) ≤ 1 − Q(y0 , x) + Q(x, y0 ) < 1 ⇒ OV (x, A, Q) < OV (x0 , A, Q) = 1 for some x0 ∈ A ⇒ x ∈ COV (A, Q) since, for x0 ∈ P D(A, Q) we have OV (x0 , A, Q) = 1. By (1) and (2) it follows P D(A, Q) = COV (A, Q). Also for any x ∈ A we have (3) x ∈ P D(A, Q) ⇒ Q(x, y) ≥ Q(y, x) for all y ∈ A ⇒ mD(x, A, Q) ≥ 0; (4) x ∈ P D(A, Q) ⇒ Q(x, y0 ) < Q(y0 , x) for some y ∈ A ⇒ mD(x, A, Q) < 0 Let us show that CmD (A, Q) ⊆ P D(A, Q). Let x ∈ P D(A, Q), therefore mD(x, A, Q) < 0. Let y ∈ P D(A, Q). Then mD(y, A, Q) ≥ 0, so mD(x, A, Q) < mD(y, A, Q). Therefore x ∈ CmD (A, Q). For proving the converse inclusion consider two subcases: (i) |P D(A, Q)| = 1, hence there exists exactly one element x0 in P D(A, Q). Then by (3) and (4) mD(x0 , A, Q) ≥ 0 and mD(y, A, Q) < 0 for all y ∈ A \ {x0 }, hence mD(x0 , A, Q) ≥ mD(y, A, Q) for all y ∈ A, i.e. x0 ∈ CmD (A, Q).

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(ii) |P D(A, Q)| ≥ 2, then there exist at least two distinct elements in P D(A, Q). Let x1 , x2 ∈ P D(A, Q), x1 = x2 . Then Q(x1 , y) ≥ Q(y, x1 ), Q(x2 , y) ≥ Q(y, x2 ) for all y ∈ A hence Q(x1 , x2 ) = Q(x2 , x1 ). Let x ∈ P D(A, Q), hence there exists x ∈ P D(A, Q) such that x = x , therefore Q(x, x ) = Q(x , x). Thus mD(x, A, Q) = {Q(x, y) − Q(y, x)|y ∈ A \ {x}} ≤ Q(x, x ) − Q(x , x) = 0, i. e. mD(x, A, Q) = 0. For any y ∈ A we have: • if y ∈ P D(A, Q) then mD(y, A, Q) = 0 (by the previous computation) • if y ∈ P D(A, Q) then mD(y, A, Q) < 0 (by (4)). Therefore for any x ∈ P D(A, Q) we get mD(x, A, Q) = 0 ≥ mD(y, A, Q) for all y ∈ A, hence x ∈ CmD (A, Q). In both cases (i) and (ii) we have established the inclusion P D(A, Q) ⊆ CmD (A, Q). It follows P D(A, Q) = CmD (A, Q). Case b P D(A, Q) = ∅. Let x ∈ A. Then Q(x, y0 ) < Q(y0 , x) for some y0 ∈ A, therefore 1 − Q(y0 , x) + Q(x, y0 ) < 1, from where it follows that OV (x, A, Q) ≤ min(1 − Q(y0 , x) + Q(x, y0 ), 1) = 1 − Q(y0 , x) + Q(x, y0 ). Then OV (x, A, Q) = =

min(1 − Q(y, x) + Q(x, y), 1)

y∈A\{x}

(1 − Q(y, x) + Q(x, y))

y∈A\{x}

=1+

(Q(x, y) − Q(y, x))

y∈A\{x}

Let x, z ∈ A. The following assertions are equivalent: • OV (x, A, Q) ≥ OV (z, A, Q) • 1+ (Q(x, y) − Q(y, x)) ≥ 1 + y∈A\{x}

•

(Q(x, y) − Q(y, x)) ≥

y∈A\{x}

(Q(z, y) − Q(y, z))

y∈A\{x}

(Q(z, y) − Q(y, z))

y∈A\{x}

• mD(x, A, Q) ≥ mD(z, A, Q). It follows COV (A, Q) = {x ∈ A|OV (x, A, Q) ≥ OV (z, A, Q) for all z ∈ A}

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= {x ∈ A|mD(x, A, Q) ≥ mD(z, A, Q) for all z ∈ A} = CmD (A, Q). With this, the proposition has been proved. Corollary 5.5. Let A ∈ B and Q ∈ H. If P D(A, Q) = ∅ then P D(A, Q) = COV (A, Q) = CmD (A, Q). Proof. See the case (a) of the proof of Proposition 5.4. By deﬁnition, a fuzzy relation Q is Banerjee–transitive if for any x, y, z ∈ A we have Q(x, y) ≥ Q(y, x), Q(y, z) ≥ Q(z, y) ⇒ Q(x, z) ≥ Q(z, x). Proposition 5.6. For any fuzzy relation Q on X the following are equivalent: (1) Q is Banerjee–transitive; (2) For all x, y, z ∈ X, if Q(x, y) ≥ Q(y, x) and Q(y, z) ≥ Q(z, y), with strict inequality holding at least once, then Q(x, z) > Q(z, x). Proof. (1) ⇒ (2). For example, we assume that Q(x, y) ≥ Q(y, x) and Q(y, z) > Q(z, y). Since Q is Banerjee–transitive, Q(x, z) ≥ Q(z, x). We assume by absurdum that Q(x, z) = Q(z, x). Since Q(z, x) = Q(x, z) and Q(x, y) ≥ Q(y, x), applying again (1), we get Q(z, y) ≥ Q(y, z), contradicting Q(y, z) > Q(z, y). Similarly, we can show that Q(x, y) > Q(y, x) and Q(y, z) ≥ Q(z, y) imply Q(x, z) > Q(z, x). (2) ⇒ (1). Let x, y, z ∈ X such that Q(x, y) ≥ Q(y, x) and Q(y, z) ≥ Q(z, y). We shall prove that Q(x, z) ≥ Q(z, x). Assume by absurdum that Q(x, z) < Q(z, x). In accordance with (2), from the inequalities Q(z, x) > Q(x, z) and Q(x, y) ≥ Q(y, x) one gets Q(z, y) > Q(y, z), contradicting Q(y, z) ≥ Q(z, y). A fuzzy relation Q on X is acyclic if for n ≥ 2 and for x1 , . . . , xn ∈ X the following implication holds:

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83

Q(x1 , x2 ) > Q(x2 , x1 ), Q(x2 , x3 ) > Q(x3 , x2 ), . . . , Q(xn−1 , xn ) > Q(xn , xn−1 ) ⇒ Q(x1 , xn ) ≥ Q(xn , x1 ) To each fuzzy relation Q on X the following crisp relation Q∗ on X is associated: Q∗ = {(x, y) ∈ X 2 |Q(x, y) ≥ Q(y, x)}. One remarks that Q∗ is complete, i.e. (x, y) ∈ Q∗ or (y, x) ∈ Q∗ for all x, y ∈ X. Lemma 5.7. For any fuzzy relation Q on X the equivalences hold: (1) Q is Banerjee–transitive iﬀ Q∗ is transitive; (2) Q is acyclic iﬀ Q∗ is acyclic. Proof. (1) Obvious. (2) Since Q∗ is a complete crisp binary relation for any x, y ∈ X we have (x, y) ∈ PQ∗ iﬀ (y, x) ∈ Q∗ iﬀ Q(y, x) < Q(x, y). Then the following assertions are equivalent: – Q∗ is acyclic; – for any n ≥ 2, for any x1 , . . . , xn ∈ X, (x1 , x2 ) ∈ PQ∗ , . . . , (xn−1 , xn ) ∈ PQ∗ ⇒ (xn , x1 ) ∈ PQ∗ ; –for any n ≥ 2 and for any x1 , . . . , xn ∈ X, Q(x1 , x2 ) > Q(x2 , x1 ), . . . Q(xn−1 , xn ) > Q(xn , xn−1 ) ⇒ Q(x1 , xn ) ≥ Q(xn , x1 ). Lemma 5.8. Let Q be a fuzzy preference relation on X. If Q is Banerjee– transitive then Q∗ is acyclic. Proof. It is obvious that if Q is Banerjee–transitive then Q is acyclic. According to Lemma 5.7 it follows that Q∗ is acyclic. Remark 5.9. For any A ∈ B and for any Q ∈ H we have P D(A, Q) = {x ∈ A|xQ∗ y for all y ∈ A} = G(A, Q∗ ).

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Proposition 5.10. Let Q ∈ H. The following assertions are equivalent: (1) Q is acyclic; (2) P D(A, Q) = ∅ for any non–empty subset A of X. Proof. According to Proposition 3.16, Lemma 5.7 and Remark 5.9, the following assertions are equivalent:

Q is acyclic; Q∗ is acyclic; G(., Q∗ ) is a choice function; G(A, Q∗ ) = ∅ for any non–empty subset A of X; P D(A, Q) = ∅ for any non–empty subset A of X.

For a given P CF C we consider the following conditions: Pairwise StrictDominance (PSD) For all Q ∈ H and x, y ∈ X, Q(x, y) > Q(y, x) ⇒ C({x, y}, Q) = {x}. Pairwise Weak Dominance (PWD) For all Q ∈ H and x, y ∈ X, Q(x, y) ≥ Q(y, x) ⇒ x ∈ C({x, y}, Q). Reward for Pairwise Weak Dominance (RPWD) For all A ∈ B and Q ∈ H and x ∈ A, Q(x, y) ≥ Q(y, x) for all y ∈ A ⇒ x ∈ C(A, Q). Let us notice that P W D is equivalent with: for all Q ∈ H and x, y ∈ X, P D({x, y}, Q) ⊆ C({x, y}, Q). Lemma 5.11. Let C be a P CF . If C veriﬁes P W D and γ2 then P D(A, Q) ⊆ C(A, Q) for all A ∈ B and Q ∈ H.

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85

Proof. Let A ∈ B and Q ∈ H. One notices that x ∈ P D(A, Q) iﬀ Q(x, y) ≥ Q(y, x) for all y ∈ A iﬀ x ∈ P D({x, y}, Q) for all y ∈ A. According to P W D we have P D({x, y}, Q) ⊆ C({x, y}, Q) then by applying the equivalences that have been proved and γ2 it follows: x ∈ P D(A, Q) ⇒ x ∈ C({x, y}, Q) for all y ∈ A ⇒ x ∈ C(A, Q). Theorem 5.12. [22] Suppose that H is a non–empty set of fuzzy preference relations reﬂexive and acyclic. Let C : H → P(X) a P CF . The following equivalences hold: (1) C = COV ; (2) C veriﬁes P SD, P W D, α2 and γ2 . Proof. (1) ⇒ (2). We prove that COV veriﬁes the following four conditions. Let Q be a reﬂexive and acyclic fuzzy preference relations and A ∈ B. (a) COV veriﬁes P SD; Q(x, y) > Q(y, x) ⇒ COV ({x, y}, Q) = P D({x, y}, Q) = {z ∈ {x, y}|Q(z, w) ≥ Q(w, z)} = {x} for all w ∈ {x, y}; (b) COV veriﬁes P W D: Q(x, y) ≥ Q(y, x) ⇒ x ∈ COV ({x, y}, Q); (c) COV veriﬁes α2 : x ∈ COV (A, Q) ⇒ x ∈ P D(A, Q) ⇒ Q(x, y) ≥ Q(y, x) for all y ∈ A; ⇒ x ∈ COV ({x, y}, Q) for all y ∈ A (d) C veriﬁes γ2 : if x ∈ COV ({x, y}, Q) for all y ∈ A then Q(x, y) ≥ Q(y, x) for all y ∈ A, i.e. x ∈ COV (A, Q). (2) ⇒ (1). Let A ∈ B and Q ∈ H. We prove that C(A, Q) = COV (A, Q). Since Q is acyclic, according to Proposition 5.10, P D(A, Q) = ∅. Then, by Corollary 5.5, COV (A, Q) = P D(A, Q). It is suﬃcient to prove that C(A, Q) = P D(A, Q). By applying Lemma 5.11, P D(A, Q) ⊆ C(A, Q). Suppose by absurdum that the inclusion C(A, Q) ⊆ P D(A, Q) does not hold;

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then there exists x ∈ C(A, Q) and x ∈ P D(A, Q). Then Q(y0 , x) > Q(x, y0 ) for some y0 ∈ A, hence, by P SD, we get C({x, y0 }, Q) = {y0 }. Since x ∈ C(A, Q) we get by α2 , x ∈ C({x, y0 }, Q) = {y0 }, so x = y0 . This contradicts Q(y0 , x) > Q(x, y0 ). It follows that C(A, Q) ⊆ P D(A, Q). Therefore P D(A, Q) = C(A, Q).

5.2 Fuzzy Choice Functions In [14] Banerjee studied a revealed preference theory for the case of vague preferences and vague choices .His motivation is the following: “If preferences are permitted to be fuzzy, it seems natural to permit the choice function to be fuzzy as well. This also tallies with experience. For instance, a decision–maker, faced with the problem of deciding whether or not to choose an alternative x from a set of alternative A, may feel that he/she is inclined to the extent 0.8 (on, say, a scale from 0 to 1 ) toward choosing it. Moreover, this fuzziness of choice is, at least potentially, observable. For instance, the decision–maker in the example will be able to tell an interviewer the degree of his/her inclinations, or demonstrate these inclinations to an observer by the degree of eagerness or enthusiasm which he/she displays. Hence, while there may be problems of estimations, fuzzy choice functions are, in theory, observable”. Now we shall present the framework in which the theory of Banerjee is developed. Let X be a non-empty set of alternatives, H the family of all non-empty ﬁnite subsets of X and F the family of non-zero fuzzy subsets of X with ﬁnite support. A Banerjee fuzzy choice function is a function C : H → F such that supp C(S) ⊆ S for any S ∈ H. This notion has the following interpretation [14]:

5.2 Fuzzy Choice Functions

87

“For all S ∈ H, C(S)(x) will be taken to represent the extent to which x belongs to the set of chosen alternatives when the available set of alternatives is S”. We remark two main characteristics in the deﬁnition of the Banerjee choice function: (a) the available sets of alternatives are crisp subsets of X; (b) the choice of an alternative x from an available set S is evaluated by a real number C(S)(x) ∈ [0, 1]. In (a) the crisp aspect is preserved: we have no information about the alternatives in S except that they can be chosen. The vagueness of the act of choice appears only in (b). We shall extend the notion of fuzzy choice function by changing the point of view on (a). The available sets will represent the vague attributes or vague criteria. Mathematically, they will be fuzzy subsets of the universe X of alternatives. If a fuzzy subset S of X will represent a vague criterion and x is an alternative, then the real number S(x) means the degree to which the alternative x veriﬁes the criterion S. S(x) will be called the availability degree of x. For example, suppose that X is a set of objects where the green colour appears in four nuances a, b, c, d. The numbers 0, 13 , 23 , 1 indicate the intensity of color green of the nuances a, b, c, d. Then the attribute “green” will be represented by a fuzzy subset V of X whose range is the set {0, 13 , 23 , 1} ⊆ [0, 1]. V (x) =

1 3

indicates that the object x has the nuance b.

The act of choice means that with each available fuzzy set of alternatives S and for each x ∈ X a real number C(S)(x) ∈ [0, 1] is assigned. C(S)(x) is interpreted as the degree to which the alternative x can be chosen with respect to criterion S. Now we shall formalize this thesis. Deﬁnition 5.13. Let us consider a pair (X, B) where X is a non-empty set and B is a non-empty family of non–zero fuzzy subsets of X. The pair (X, B)

88

5 Fuzzy Choice Functions

is called a fuzzy choice space. A fuzzy choice function (=fuzzy consumer) on (X, B) is a function C : B → F(X) such that for each S ∈ B, C(S) is non-zero and C(S) ⊆ S. The same deﬁnition of a fuzzy choice function can be found in [80]. A fuzzy choice problem has the form (X, B, C) where (X, B) is a fuzzy choice space and C is a fuzzy choice function on (X, B). In terms of fuzzy consumers, X is the set of bundles and B is the family of fuzzy budgets ; the pair (X, B) is called a fuzzy budget space . Let x ∈ X be a bundle and S ∈ B be a fuzzy budget; the real number C(S)(x) can be interpreted as the degree to which the bundle x is chosen subject to the fuzzy budget S. The fuzzy set S ∈ B oﬀers an availability degree S(x) for each x ∈ X. The degree C(S)(x) to which x is chosen subject to S naturally belongs to the interval (0, S(x)]. Remark 5.14. Let H be the family of all non-empty ﬁnite subsets of X. H can be identiﬁed with B0 = {χK |K ∈ H} and (X, B0 ) is a fuzzy budget space. A Banerjee fuzzy choice function C : H → F induces a fuzzy choice function C : B0 → F by putting C (χK ) = C(K) for each K ∈ H. Thus C and C can be identiﬁed hence our deﬁnition for fuzzy choice functions includes Banerjee’s. Banerjee fuzziﬁes only the range of a choice function; in our approach both the domain and the range of a choice function are fuzziﬁed. In this case one obtains a very general framework for the fuzzy choice functions and the results have a much deeper meaning. Example 5.15. (I) Generalized Orlovsky function. Consider ∗ an arbitrary continuous t–norm. Let (X, B) be a ﬁnite fuzzy choice space and H a non–empty family of reﬂexive fuzzy preference relations on X. For any A ∈ B, Q ∈ H and x ∈ X we deﬁne the generalized Orlovsky score by

5.2 Fuzzy Choice Functions

OV G(x, A, Q) = A(x) ∗

89

[A(y) → (Q(y, x) → Q(x, y))]

y∈X

= A(x) ∗

[A(y) ∗ Q(y, x) → Q(x, y)]

y∈X

For any A ∈ B and Q ∈ H let us introduce the function COV G (A, Q) : X → [0, 1] by COV G (A, Q)(x) = A(x) ∗

[A(y) → (OV G(y, A, Q) → OV G(x, A, Q))]

y∈X

= A(x) ∗

[A(y) ∗ OV G(y, A, Q) → OV G(x, A, Q)]

y∈X

for any x ∈ X. The assignment (A, Q) −→ COV G (A, Q) deﬁnes a function COV G : B × H → F(X). Next we prove that COV G (., Q) is a fuzzy choice function. Let A ∈ B and Q ∈ H. Then the set U = {x ∈ X|A(x) > 0} is non–empty. Since U is a ﬁnite set, there exists x0 ∈ U such that OV G(y, A, Q) ≤ OV G(x0 , A, Q) for any y ∈ U. Notice that A(y) = 0 for y ∈ U and (A(y) ∗ OV G(y, A, Q)) → OV G(x0 , A, Q) = 1 for y ∈ U , therefore COV G (A, Q)(x0 ) = A(x0 ) > 0. It follows that COV G (., Q) is a fuzzy choice function on (X, B) for any Q ∈ H. In case when B is a family of non–empty crisp subsets of X and ∗ is the Lukasiewicz t–norm, one ﬁnds the Orlovsky function COV . Therefore, COV G is called the generalized Orlovsky function. The computation of the generalized Orlovsky function We will exemplifythe generalized Orlovsky function for the following data:  X = {x, y}, Q = 

1β

, 0 ≤ α, β ≤ 1, A = aχ{x} + bχ{y} , 0 ≤ a, b ≤ 1.

α1 OV G(x, A, Q) = A(x) ∗ [A(x) ∗ Q(x, x) → Q(x, x)] ∗ [A(y) ∗ Q(y, x) →

Q(x, y)] = a ∗ (b ∗ α → β). OV G(y, A, Q) = b ∗ (a ∗ β → α). COV G (A, Q)(x) = A(x) ∗ [A(x) ∗ OV G(x, A, Q) → OV G(x, A, Q)] ∗ [A(y) ∗ OV G(y, A, Q) → OV G(x, A, Q)]

90

5 Fuzzy Choice Functions

= a ∗ (b ∗ b ∗ (a ∗ β → α) → a ∗ (b ∗ α → β)) COV G (A, Q)(y) = b ∗ (a ∗ a ∗ (b ∗ α → β) → b ∗ (a ∗ β → α)). (II) Consider again ∗ an arbitrary continuous t–norm. Let (X, B) be a ﬁnite fuzzy choice space and H a non–empty family of reﬂexive fuzzy preference relations on X. For any A ∈ B, Q ∈ H and x ∈ X we give next the generalized score functions

M F G(x, A, Q) = A(x) ∗

[A(y) ∗ Q(x, y)];

y∈X\{x}

mF G(x, A, Q) = A(x) ∗

[A(y) → Q(x, y)];

y∈X\{x}

M AG(x, A, Q) = A(x) ∗

[A(y) ∗ Q(x, y)];

y∈X\{x}

mAG(x, A, Q) = A(x) ∗

[A(y) → Q(x, y)]

y∈X\{x}

corresponding to the score functions M F , mF , M A, mA introduced in Section 5.1. We also introduce the generalized P CF ’s associated with the generalized score functions from above. CM F G (A, Q)(x) = A(x) ∗ A(x) ∗

[(A(y) ∗ M F (y, A, Q)) → M F (x, A, Q)];

CmF G (A, Q)(x) = A(x) ∗

[A(y) → (mA(y, A, Q) → mA(x, A, Q)] =

[(A(y) ∗ mA(y, A, Q)) → mA(x, A, Q)];

CM AG (A, Q)(x) = A(x) ∗

y∈X

y∈X

A(x) ∗

[A(y) → (mF (y, A, Q) → mF (x, A, Q)] =

[(A(y) ∗ mF (y, A, Q)) → mF (x, A, Q)];

CmAG (A, Q)(x) = A(x) ∗

y∈X

y∈X

A(x) ∗

[A(y) → (M F (y, A, Q) → M F (x, A, Q)] =

y∈X

y∈X

A(x) ∗

[A(y) → (M A(y, A, Q) → M A(x, A, Q)] =

y∈X

[(A(y) ∗ M A(y, A, Q)) → M A(x, A, Q)].

y∈X

(III) In Example 5.1, the evaluation of the n experts was expressed by n crisp preference relations Q1 , . . . , Qn on the set X of the alternatives. Each

5.2 Fuzzy Choice Functions

91

preference relation Qi leads to a crisp choice function Ci : B → P(X) (e.g. by taking Ci = G(., Qi )). If S ∈ B, x ∈ X then x ∈ Ci (S) means that the expert i appreciates that x is chosen from S. In order to combine the evaluations of the n experts, the agent will choose for example, the following expression: for any S ∈ B and x ∈ X, C(S)(x) =

1 n card{i|x

∈ Ci (S)}

C(S)(x) is the fraction of the number of the experts which appreciate that x is chosen from S. One obtains this way a fuzzy choice function C : B → F(X). The domain of this choice function consists of (not necessarily all) non–empty crisp subsets of X and the range of non–zero fuzzy subsets of X. Although a part of fuzzy choice function theory will be developed in this general setting, in order to extend Uzawa–Arrow–Sen theory we need some natural hypotheses. In this book we work mostly with the following hypotheses: (H1) Every S ∈ B and C(S) are normal fuzzy subsets of X; (H2 ) B includes all fuzzy sets [x1 , . . . , xn ], n ≥ 1 and x1 , . . . , xn ∈ X. For the crisp case (B ⊆ P(X)) hypothesis (H1) asserts that any S ∈ B and C(S) are non-empty, hence (H1) is automatically fulﬁlled in accordance with the deﬁnition of choice function; for the same case, (H2) asserts that B includes all non-empty ﬁnite subsets of X. Now we ﬁx a continuous t-norm ∗. The following deﬁnitions will be formulated in the fuzzy set theory associated with the t-norm ∗; we shall use the residuated structure ([0, 1], ∨, ∧, ∗, →, 0, 1). The following three deﬁnitions introduce the main fuzzy revealed preference relations used in fuzzy choice function theory. They generalize to fuzzy sets the crisp revealed preference relations deﬁned in Section 3.2. Deﬁnition 5.16. Let C : B → F(X) be a fuzzy choice function on (X, B). We deﬁne the following fuzzy revealed preference relations R, P, I on X by (i) R(x, y) = (C(S)(x) ∗ S(y)); S∈B

92

5 Fuzzy Choice Functions

(ii) P (x, y) = R(x, y) ∗ ¬R(y, x); (iii) I(x, y) = R(x, y) ∗ R(y, x) for any x, y ∈ X. By interpreting the t–norm ∗ as a conjunction, R(x, y) is the degree of truth of the statement “there exists a criterion S such that alternative x is chosen with respect of S and alternative y veriﬁes S”. Deﬁnition 5.17. Assume hypothesis (H2) holds. Let C be a fuzzy choice function on (X, B). We deﬁne the following fuzzy revealed preference relations ¯ P¯ , I¯ on X by R, ¯ y) = C([x, y])(x); (i) R(x, ¯ y) ∗ ¬R(y, ¯ x); (ii) P¯ (x, y) = R(x, ¯ y) = R(x, ¯ y) ∗ R(y, ¯ x) (iii) I(x, for any x, y ∈ X. ¯ y) is the degree of truth of the statement “at least the alternative x R(x, is chosen from the set {x, y}”. Deﬁnition 5.18. Let C be a fuzzy choice function on (X, B). We deﬁne the ˜ P˜ , I˜ on X by following fuzzy revealed preference relations R, (i) P˜ (x, y) = (C(S)(x) ∗ S(y) ∗ ¬C(S)(y)); S∈B

˜ y) = ¬P˜ (y, x); (ii) R(x, ˜ y) = R(x, ˜ y) ∗ R(y, ˜ x) (iii) I(x, for any x, y ∈ X. P˜ (x, y) is the degree of truth of the statement “there exists a criterion S with respect to which x is chosen and y is rejected”. Preserving the results from the crisp case, we denote by W the transitive closure of R and by P ∗ the transitive closure of P˜ . Easy calculations show that the fuzzy revealed preference relations introduced by these deﬁnitions extend the corresponding crisp revealed preference

5.2 Fuzzy Choice Functions

93

relations. We kept the notations used for crisp choice functions in order to see how Uzawa–Arrow–Sen theory is generalized to fuzzy choice functions. Remark 5.19. Let C : H → F be a Banerjee fuzzy choice function and C : B0 → F the associated fuzzy choice function. For any K ∈ H and u ∈ X we have C (χK )(u) ≤ χK (u) hence u ∈ K ⇒ C (χK )(u) = χK (u) = 0. Assume ∗ is the G¨ odel t-norm ∧. Thus the fuzzy revealed preference relation R associated with C (cf. Deﬁnition 5.16 (i)) has the following form R(x, y) = (C (χK )(x) ∧ χK (y)) K∈H = {C(K)(x)|K ∈ H, x, y ∈ K}. In this case R coincides with the fuzzy revealed preference relation deﬁned by Banerjee in [14], Deﬁnition 2.5. Remark 5.20. Assume the conditions of Remark 5.19 are fulﬁlled but ∗ is the Lukasiewicz t-norm ∗L . According to the deﬁnition of the Lukasiewicz t-norm, for any a, b ∈ [0, 1] we have a ∗L ¬b = max (0, a − b). Let P˜ be the fuzzy revealed preference relation associated with C according to Deﬁnition 5.18 (i). An easy computation shows that for any x, y ∈ X we have P˜ (x, y) =

(C (χK )(x) ∗L χK (y) ∗L ¬C (χK )(y))

K∈H

=

{max(0, C(K)(x) − C(K)(y))|K ∈ H, x, y ∈ K}.

This shows that in this case P˜ coincides with the fuzzy relation deﬁned in [14], Deﬁnition 2.6. Remarks 5.19, 5.20 show that in [14] appear some fuzzy revealed preference relations corresponding to the G¨ odel t-norm and some others corresponding to the Lukasiewicz t-norm, resulting a framework that combines the reasonings and the two t-norms. Deﬁnition 5.21. Let C be a fuzzy choice function on (X, B). We deﬁne the fuzzy preference relations R∗ and R1 on X by

94

5 Fuzzy Choice Functions

(i) R∗ (x, y) =

[(S(x) ∗ C(S)(y)) → C(S)(x)];

S∈B

(ii) R1 (x, y) =

[¬S(x) ∨ C(S)(x) ∨ ¬C(S)(y)]

S∈B

for any x, y ∈ X.

Remark 5.22. By Lemma 4.3 (11), R∗ (x, y) can be written R∗ (x, y) = [S(x) → (C(S)(y) → C(S)(x))]. S∈B

An easy computation shows that both R1 and R∗ are fuzzy generalizations of the crisp relation denoted by R∗ . We observe that for each x ∈ X, R∗ (x, x) = [(S(x) ∗ C(S)(x)) → C(S)(x)] = 1. S∈B

Lemma 5.23. R1 ⊆ R∗ . Proof. Let x, y ∈ X and S ∈ B. By Lemma 4.3 (9) [¬S(x) ∨ C(S)(x) ∨ ¬C(S)(y)] ∗ (S(x) ∗ C(S)(y)) = = [¬S(x) ∗ S(x) ∗ C(S)(y)] ∨ [C(S)(x) ∗ S(x) ∗ C(S)(y)]∨ ∨[¬C(S)(y) ∗ S(x) ∗ C(S)(y)] = = C(S)(x) ∗ S(x) ∗ C(S)(y) ≤ C(S)(x). Therefore by Lemma 4.3 (1) we get ¬S(x) ∨ C(S)(x) ∨ ¬C(S)(y) ≤ (S(x) ∗ C(S)(y)) → C(S)(x). The last inequality holds for any S ∈ B hence by deﬁnitions of R∗ and R1 , we obtain R1 (x, y) ≤ R∗ (x, y). Example 5.24. Assume ∗ is the G¨ odel t-norm ∧. Let us take X = {x, y} and B = {A} with A = 0.2χ{x} + χ{y} and the fuzzy choice function C : B → F(X) deﬁned by C(A) = 0.1χ{x} + χ{y} . In this case R1 (x, x) = ¬A(x) ∨ C(A)(x) ∨ ¬C(A)(x) = ¬0.2 ∨ 0.1 ∨ ¬0.1 = 0.1. Since R∗ (x, x) = 1 it follows that R1 (x, x) = R∗ (x, x), hence R∗ and R1 are distinct. Lemma 5.25. P˜ ⊆ R.

5.2 Fuzzy Choice Functions

95

Proof. For any x, y ∈ X we have: P˜ (x, y) = [C(S)(x)∗S(y)∗¬C(S)(y)] ≤ [C(S)(x)∗S(y)] = R(x, y). S∈B

S∈B

Lemma 5.26. Assume hypotheses (H1), (H2) are fulﬁlled and C : B → F(X) is a fuzzy choice function. ¯ ⊆ R; (i) R ¯ are reﬂexive and strongly total. (ii) R and R ¯ for any x, y ∈ X we have Proof. (i) By the deﬁnition of R and R ¯ y) = C([x, y])(x) = C([x, y])(x) ∗ [x, y](y) ≤ R(x, (C(S)(x) ∗ S(y)) = S∈B

R(x, y). (ii) For any x ∈ X, C([x]) ⊆ [x] hence for each y ∈ X:   1 if y = x C([x])(y) ≤ [x](y) =  0 if y = x. ¯ x) = Since C([x]) is a normal fuzzy set, C([x])(x) = 1. Hence R(x, ¯ is reﬂexive. C([x])(x) = 1 so R Let x, y be two distinct alternatives. For any z ∈ X we have  1 if z ∈ {x, y} C([x, y])(z) ≤ [x, y](z) =  0 otherwise . ¯ y) = C([x, y])(x) = Since C([x, y]) is a normal fuzzy set it follows that R(x, ¯ x) = C([x, y])(y) = 1, hence R ¯ is strongly total. 1 or R(y, ¯ ⊆ R and R ¯ is reﬂexive and strongly total it follows immediately Using R that R is reﬂexive and strongly total. Now we shall consider the following axioms of fuzzy revealed preference:

W AF RP (Weak Axiom of Fuzzy Revealed Preference) P˜ (x, y) ≤ ¬R(y, x) for all x, y ∈ X;

96

5 Fuzzy Choice Functions

SAF RP (Strong Axiom of Fuzzy Revealed Preference) P ∗ (x, y) ≤ ¬R(y, x) for all x, y ∈ X.

Now we shall state two axioms of congruence for fuzzy choice functions:

W F CA (Weak Fuzzy Congruence Axiom) For any S ∈ B and x, y ∈ X the following inequality holds R(x, y) ∗ C(S)(y) ∗ S(x) ≤ C(S)(x).

SF CA (Strong Fuzzy Congruence Axiom) For any S ∈ B and x, y ∈ X the following inequality holds W (x, y) ∗ C(S)(y) ∗ S(x) ≤ C(S)(x).

Axiom W AF RP says that the degree to which there exists a criterion S with respect to which x is chosen, y veriﬁes S but is rejected is less or equal than the degree to which for any other criterion S fulﬁlled by x, y cannot be rejected. Axiom SF CA expresses the fact that the degree to which alternative x is chosen with respect to criterion S, y veriﬁes S and x is revealed preferred to y is less or equal than the degree to which y is chosen with respect to S. Similar interpretations can be given to the axioms SAF RP and SF CA. Generalizing [84], a fuzzy choice function is said to be congruous if it veriﬁes SF CA. Remark 5.27. Axioms W AF RP , SAF RP , W F CA, SF CA are fuzzy versions of axioms W ARP , SARP , W CA, SCA in crisp choice function theory. Fuzzy versions of the above axioms are found in [14] in the form imposed by the context. Remark 5.28. Since P˜ (x, y) ≤ P ∗ (x, y) and R(x, y) ≤ W (x, y) for any x, y ∈ X the following implications hold true for any fuzzy choice function C: SAF RP ⇒ W AF RP ; SF CA ⇒ W F CA.

5.3 Rational and Normal Fuzzy Choice Functions

97

If P˜ (respectively R) is ∗-transitive then P˜ = P ∗ (respectively R = W ), therefore in this case: SAF RP ⇔ W AF RP (respectively SF CA ⇔ W F CA). The next two axioms W AF RP ◦ and SAF RP ◦ extend to fuzzy choice functions the form of W ARP and SARP formulated in terms of C-connected sequences ([100]). W AF RP ◦ For any x, y ∈ X and S1 , S2 ∈ B [S1 (x) ∗ C(S2 )(x)] ∗ [S2 (y) ∗ C(S1 )(y)] ≤ E(S1 ∩ C(S2 ), S2 ∩ C(S1 )). SAF RP ◦ For any x1 , . . . , xn ∈ X and S1 , . . . , Sn ∈ B n n−1 [Sk (xk ) ∗ C(Sk+1 )(xk )] ≤ E(Sj ∩ C(Sj+1 ), Sj+1 ∩ C(Sj )) k=1

j=1

where Sn+1 = S1 . Remark 5.29. In the crisp case W AF RP ◦ has the form: if there exist an element x ∈ S1 ∩ C(S2 ) and an element y ∈ S2 ∩ C(S1 ) then S1 ∩ C(S2 ) = S2 ∩ C(S1 ). This is exactly the axiom W ARP formulated in terms of C-connected sequences, hence W AF RP ◦ is a generalization of W ARP . Similarly, SAF RP ◦ generalizes SARP . We add the axiom HAF RP as a fuzzy extension of HARP : HAF RP For any x1 , . . . , xn ∈ X and S1 , . . . , Sn ∈ B: n n−1 [Sk (xk ) ∗ C(Sk+1 )(xk )] ≤ E(Sj ∩ C(Sj+1 ), Sj+1 ∩ C(Sj )) k=1

j=1

where Sn+1 = S1 . It is clear that HAF RP implies SAF RP ◦ .

5.3 Rational and Normal Fuzzy Choice Functions In the previous section we associated with a fuzzy choice function several fuzzy preference relations. Then vague preferences appear as a consequence

98

5 Fuzzy Choice Functions

of the act of choice. This section deals with the inverse route: the choices are derived from vague preferences. To a fuzzy preference relation Q on the set of alternatives we will assign two constructions: one of them fuzziﬁes the notion of Q–maximal element, the other fuzziﬁes the notion of Q–greatest element. For some fuzzy preference relations, these constructions provide fuzzy choice functions. Then we obtain two correspondences from fuzzy preference relations to fuzzy choice functions. In this way we can introduce two concepts of rationality: M –rationality and G–rationality of fuzzy choice functions. By rationality, a fuzzy choice function C is recoverable from a fuzzy preference relation. A special case of rationality is normality: C is recoverable from the fuzzy revealed preference relation R associated with C. The results proved in this section concern the connections between these four notions: M–rationality, G–rationality, M–normality and G–normality. Assume ∗ is a continuous t–norm. Let Q be a fuzzy preference relation on the universal set of alternatives X and B a non-empty family of non–zero fuzzy subsets of X. Deﬁnition 5.30. For any S ∈ B let us deﬁne the fuzzy subsets M (S, Q) and G(S, Q) of X: (i) M (S, Q)(x) = S(x) ∗

[(S(y) ∗ Q(y, x)) → Q(x, y)]

y∈X

= S(x) ∗ (ii) G(S, Q)(x) = S(x) ∗

[S(y) → (Q(y, x) → Q(x, y))];

y∈X

[S(y) → Q(x, y)].

y∈X

Remark 5.31. Assume Q is a fuzzy preference relation and S = χU with U ∈ P(X). An easy computation shows that   (Q(y, x) → Q(x, y)) if x ∈ U  M (χU , Q)(x) = y∈U   0 if x ∈ U.

5.3 Rational and Normal Fuzzy Choice Functions

99

  Q(x, y) if x ∈ U  G(χU , Q)(x) =

 

y∈U

0 if x ∈ U.

If Q is a crisp relation on X (Q : X 2 → {0, 1} ) then Q(y, x) → Q(x, y) = ¬PQ (y, x) hence we have M (χU , Q)(x) = 1 if and only if ¬PQ (y, x) = 1 for all y ∈ U G(χU , Q)(x) = 1 if and only if Q(x, y) = 1 for all y ∈ U . Therefore Deﬁnition 5.30 generalizes to fuzzy choice functions the Qmaximal point sets and the Q-greatest point sets. The real number M (S, Q)(x) (respectively G(S, Q)(x)) represents the degree to which the alternative x veriﬁes the property of being Q–maximal (respectively Q–greatest) with respect to criterion S. In general the functions M (., Q) and G(., Q) are not fuzzy choice functions; there might be some S ∈ B such that C(S) is the zero fuzzy set. Proposition 5.32. Suppose ∗ is the G¨ odel t-norm. Let Q be a reﬂexive and transitive fuzzy preference relation on a ﬁnite set X and B ⊆ F(X). Then G(., Q) and M (., Q) are fuzzy choice functions on (X, B). Proof. Suppose by absurdum that G(., Q) is not a fuzzy choice function; then there exists S ∈ B such that G(S, Q)(x) = 0 for any x ∈ X. Choose an element x1 ∈ X such that S(x1 ) = 0. Since 0 = G(S, Q)(x1 ) = S(x1 ) ∧ [S(y) → Q(x1 , y)] = 0, y∈X

S(x1 ) = 0 and X is ﬁnite, it follows  that there exists a(x1 ) ∈ X such that  1 if S(a(x1 )) ≤ Q(x1 , a(x1 )) 0 = S(a(x1 )) → Q(x1 , a(x1 )) =  Q(x , a(x )) if S(a(x )) > Q(x , a(x )) 1 1 1 1 1 It follows that Q(x1 , a(x1 )) = 0 and S(a(x1 )) > 0. Since Q is reﬂexive, one obtains x1 = a(x1 ). Denote y1 = x1 , y2 = a(x1 ). Applying the same reasoning to y2 we can ﬁnd y3 ∈ X such that S(y3 ) > 0 and Q(y2 , y3 ) = 0. Consequently one ﬁnds a sequence y1 , y2 , . . . , yk , . . . such that S(yk ) > 0 and Q(yk , yk+1 ) = 0 for any k. Suppose there exist two indices i and j such

100

5 Fuzzy Choice Functions

that i < j and yi = yj ; then, according to the transitivity of Q, Q(yi , yj ) ≤ Q(yi , yi+1 )∧. . .∧Q(yj−1 , yj ) = 0, contradicting the reﬂexivity of Q. Therefore in the sequence y1 , y2 , . . . all the elements are distinct, hence it must be inﬁnite, contradicting the fact that X is ﬁnite. Accordingly, G(., Q) is a fuzzy choice function. Since G(S, Q) ⊆ M (S, Q) for any S ∈ B one obtains that M (., Q) is a fuzzy choice function. Deﬁnition 5.33. A fuzzy choice function C on (X, B) will be called Grational (respectively M -rational) if there is a fuzzy preference relation Q on X such that C(S) = G(S, Q) (respectively C(S) = M (S, Q)) for any S ∈ B. In this case we say that Q is the G–rationalization (respectively the M – rationalization) of C. According to Deﬁnition 5.33 G–rationality and M –rationality express the fulﬁllment of a principle of optimality by the choice function C. G–rationality (respectively M –rationality) means that C evaluates the choice of the Q– greatest (respectively Q–maximal ) alternatives. Lemma 5.34. (i) G(S, Q) ⊆ M (S, Q) for each S ∈ B; (ii) If Q is strongly complete then G(S, Q) = M (S, Q). Proof. (i) Let x ∈ X. Then for each y ∈ X, S(y) ∗ Q(y, x) ≤ S(y) implies S(y) → Q(x, y) ≤ (S(y) ∗ Q(y, x)) → Q(x, y) (by Lemma 4.3 (10)); it follows immediately that G(S, Q)(x) ≤ M (S, Q)(x). (ii) For any x, y ∈ X we have Q(x,  y) = 1 or Q(y, x) = 1. We notice that  1 if Q(x, y) = 1 S(y) → (Q(y, x) → Q(x, y)) =  S(y) → Q(x, y) if Q(y, x) = 1 = S(y) → Q(x, y) then M (S, Q)(x) = G(Q, S)(x) for any x ∈ X.

5.3 Rational and Normal Fuzzy Choice Functions

101

A fuzzy choice function C is totally rational (resp. full rational) if there exists a fuzzy relation Q on X which is reﬂexive, ∗-transitive and total (resp. strongly total) such that C = G(., Q), i.e. C(S)(x) = S(x) ∗ (S(y) → Q(x, y)) y∈X

for any S ∈ B and x ∈ X. Remark 5.35. Let C : B → F(X) be a G–rational fuzzy choice function and Q a fuzzy preference relation such that C(S) = G(S, Q) for each S ∈ B. By Lemma 5.34 (ii), if Q is strongly complete M (S, Q) = G(S, Q) for each S ∈ B; in this case we say that C is strongly complete rational. Lemma 5.36. Any M -rational fuzzy choice function is G-rational. Proof. For any fuzzy relation Q on X let us deﬁne a fuzzy relation Q by Q (x, y) = Q(y, x) → Q(x, y) for any x, y ∈ X. Then for any S ∈ B and x ∈ X: G(S, Q )(x) = S(x) ∗

[S(y) → Q (x, y)]

y∈X

= S(x) ∗

[S(y) → (Q(y, x) → Q(x, y))]

y∈X

= M (S, Q)(x). Then G(S, Q ) = M (S, Q) for any S ∈ B and the lemma follows immediately. For any fuzzy choice function C on (X, B) let us consider the associated revealed preference relation R and deﬁne the functions G∗ : B → F(X) and M ∗ : B → F(X) G∗ (S)(x) = G(S, R)(x) = S(x) ∗

[S(y) → R(x, y)];

y∈X

M ∗ (S)(x) = M (S, R)(x) = S(x) ∗ for any S ∈ B and x ∈ X.

[(S(y) ∗ R(y, x)) → R(x, y)]

y∈X

Lemma 5.37. Let C be a fuzzy choice function. Then (i) G∗ (S) ⊆ M ∗ (S) for any S ∈ B;

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5 Fuzzy Choice Functions

(ii) For any S ∈ B and x ∈ X, C(S)(x) ∗ C(S)(x) ≤ G∗ (S)(x); (iii) If ∗ is the G¨ odel t-norm, then C(S) ⊆ G∗ (S) ⊆ M ∗ (S) for any S ∈ B. Proof. (i) By Lemma 5.34 (i) for Q = R. (ii) Let S ∈ B and x ∈ X. By the deﬁnition of R, C(S)(x) ∗ S(y) ≤ R(x, y) for any y ∈ X. Then C(S)(x) ≤ S(y) → R(x, y) for any y ∈ X, hence C(S)(x) ≤ [S(y) → R(x, y)]. y∈X

Since C(S)(x) ≤ S(x) it follows that C(S)(x) ∗ C(S)(x) ≤ S(x) ∗ [S(y) → R(x, y)] = G∗ (S)(x). y∈X

(iii) By (i) and (ii). According to Lemma 5.37, G∗ and M ∗ are always fuzzy choice functions. Deﬁnition 5.38. A fuzzy choice function C is G-normal (respectively M normal) if C(S) = G∗ (S) (respectively C(S) = M ∗ (S)) for any S ∈ B. Normality is a particular case of rationality. If in the case of rationality, the fulﬁllment of the optimality is done with respect to an arbitrary preference relation Q, in the case of normality this is done with respect to the fuzzy revealed preference relation R. If C is G–normal (respectively M –normal) then it is G-rational (respectively M –rational). Proposition 5.39. Suppose ∗ is the G¨ odel t–norm. Any M -normal fuzzy choice function C is G-normal. Proof. Let S ∈ B. By Lemma 5.37 (iii), C(S) ⊆ G∗ (S) ⊆ M ∗ (S) = C(S) hence C(S) = G∗ (S). The following result emphasizes a remarkable property: G–rationality is equivalent with G–normality. Proposition 5.40. Suppose ∗ is the G¨ odel t–norm. For a fuzzy choice function C the following are equivalent:

5.3 Rational and Normal Fuzzy Choice Functions

103

(1) C is G–rational; (2) C is G–normal. Proof. (1) ⇒ (2) Assume there is a fuzzy relation Q such that C(S) = G(S, Q) for any S ∈ B. For any x, y ∈ X: R(x, y) = [C(S)(x) ∧ S(y)]

S∈B

=

[S(x) ∧ S(y) ∧

S∈B

(S(z) → Q(x, z))].

z∈X

For any S ∈ B we have: S(x) ∧ S(y) ∧ (S(z) → Q(x, z)) ≤ S(x) ∧ S(y) ∧ (S(y) → Q(x, y)) = z∈X

= S(x) ∧ S(y) ∧ Q(x, y) ≤ Q(x, y).

Thus R(x, y) ≤ Q(x, y). We have proved that R ⊆ Q. For any x, y ∈ X we have C(S)(x) ∧ S(y) ≤ R(x, y), hence C(S)(x) ≤ S(y) → R(x, y). Hence C(S)(x) ≤ [S(y) → R(x, y)]. Since C(S)(x) ≤ S(x) it follows

y∈X

that C(S)(x) ≤ S(x) ∧

[S(y) → R(x, y)].

y∈X

Since R(x, y) ≤ Q(x, y) we have S(y) → R(x, y) ≤ S(y) → Q(x, y) for any x, y ∈ X, hence S(x) ∧ (S(y) → R(x, y)) ≤ S(x) ∧ (S(y) → Q(x, y)) = C(S)(x). y∈X It follows that C(S)(x) = S(x) ∧ (S(y) → R(x, y)) for any x ∈ X, y∈X

y∈X

hence C is G-normal. (2) ⇒ (1) Obviously. Proposition 5.41. Suppose ∗ is the G¨ odel t–norm. Let C be a fuzzy choice function on (X, B) such that C(S) is a normal fuzzy subset of X for any S ∈ B. If C veriﬁes W AF RP ◦ then C is M -normal. Proof. By Lemma 5.37 it suﬃces to prove that M ∗ (S)(x) ≤ C(S)(x) for any S ∈ B and x ∈ X. Let y ∈ X such that C(S)(y) = 1; then S(y) = 1. By the deﬁnition of M ∗ (S) and Lemma 4.5 (2) we have M ∗ (S)(x) = S(x) ∧ [(S(z) ∧ R(z, x)) → R(x, z)] ≤ z∈X

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5 Fuzzy Choice Functions

≤ S(x) ∧ [(S(y) ∧ R(y, x)) → R(x, y)] = = S(x) ∧ [R(y, x) → R(x, y)] = = S(x) ∧ [ (C(B)(y) ∧ B(x)) → R(x, y)] =

B∈B

= S(x) ∧

[(C(B)(y) ∧ B(x)) → R(x, y)] ≤

B∈B

≤ S(x) ∧ [(C(S)(y) ∧ S(x)) → R(x, y)] = = S(x) ∧ [S(x) → R(x, y)] = = S(x) ∧ R(x, y) = = S(x) ∧ [C(T )(x) ∧ T (y)] = =

T ∈B

[S(x) ∧ C(T )(x) ∧ T (y)] =

T ∈B

=

[S(x) ∧ T (y) ∧ C(T )(x) ∧ C(S)(y)].

T ∈B

According to W AF RP ◦ , [S(x) ∧ C(T )(x)] ∧ [T (y) ∧ C(S)(y)] ≤ S(x) ∧ C(T )(x) → T (x) ∧ C(S)(x), therefore S(x) ∧ T (y) ∧ C(T )(x) ∧ C(S)(y) ≤ C(S)(x) for any T ∈ B, hence M ∗ (S)(x) ≤ C(S)(x). Corollary 5.42. Suppose ∗ is the G¨ odel t–norm. Let C be a fuzzy choice function on (X, B) such that C(S) is a normal subset of X for any S ∈ B. If C veriﬁes W AF RP ◦ then C is G-rational as well as M -rational. Proof. By Propositions 5.39 and 5.41. Proposition 5.43. Suppose ∗ is the G¨ odel t–norm. Any M -normal fuzzy choice function C is strongly complete rational. Proof. For any S ∈ B, C(S) = M ∗ (S). Let us take the fuzzy relation R◦ on X deﬁned by R◦ (x, y) = R(y, x) → R(x, y) for any x, y ∈ X. If R(y, x) ≤ R(x, y) then R◦ (x, y) = R(y, x) → R(x, y) = 1; if R(y, x) > R(x, y) then R◦ (y, x) = R(x, y) → R(y, x) = 1. Thus R◦ is strongly complete. By a similar argument used in the proof of Lemma 5.36 we have M ∗ (S) = M (S, R) = G(S, R◦ ) and by Lemma 5.34 (ii), G(S, R◦ ) = M (S, R◦ ). Then C(S) = G(S, R◦ ) = M (S, R◦ ) for any S ∈ B, hence C is strongly complete rational.

5.3 Rational and Normal Fuzzy Choice Functions

105

The following ﬁgure summarizes the results of this paragraph.

G

normal

M

normal

M

rational

WAFRP 0 G

rational

Fig. 5.1. Properties of rationality and normality

The following example illustrates a fuzzy choice function which is neither M –rational nor G–rational. Example 5.44. Let X = {x1 , x2 , . . . , xn , . . .} and B = {[x1 ], . . . , [xn ], . . .}. We consider the  fuzzy choice function C on (X, B) deﬁned by  1 if x = x n C([xn ])(x) = n  1 if x = x . n for any n = 1, 2, . . . and for any x ∈ X. We compute the fuzzy preference relation R: ∞  1 if n = m (C([xk ])(xn ) ∧ [xk ](xm )) = n R(xn , xm ) =  0 if n = m k=1 If Q is a fuzzy preference relation on X then for any n = 1, 2 . . . G([xn ], Q)(x) = [xn ](x) ∧ ([xn ](y) → Q(x, y)) y∈X   ([xn ](y) → Q(xn , y)) if x = xn 

=

y∈X

    Q(x , x ) if x = x n n n =  0 if x = x

0 if x = xn

n

By taking Q = Rwe will have for any n = 1, 2, . . .  1 if x = x n G([xn ], R)(x) = n  0 if x = x n

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5 Fuzzy Choice Functions

For any n = 1, 2 . . . we will have M ([xn ], Q)(x) = [xn ](x) ∧ (([xn ](y) ∧ Q(y, x)) → Q(x, y)) y∈X   (([xn ](y) → Q(y, xn )) → Q(xn , y)) if x = xn 

=

=

y∈X

    1 if x = x

0 if x = xn

n

 0 if x = x

n

We notice that C = G(., R) = G(., Q) and C = M (., Q) for any preference relation Q on X, therefore C is neither G–rational nor M –rational. The following example shows that Proposition 5.40 does not hold in case when ∗ is the product t–norm. Example 5.45. Let X = {x, y}, B = {A, B} where A, B ∈ F(X) are given by A = 23 χ{x} + χ{y} , B = χ{x} + 13 χ{y} . Consider the function C : B → F(X) deﬁned by C(A) = 12 χ{x} + χ{y} , C(B) = χ{x} + 13 χ{y} .



We compute the fuzzy preference relation R: R = 

1

1 2

2 3

1

 .

We observe that G(A, R)(x) =

1 3

= C(A)(x) = 12 ,

hence C is not G–normal.



If we take the fuzzy preference relation Q = 

1

3 4

11 G–rational: G(A, Q)(x) = 12 ; G(A, Q)(y) = 1; G(B, Q)(x) = 1; G(B, Q)(y) = 13 .

 , we notice that C is

6 Fuzzy Revealed Preference and Consistency Conditions

In Chapter 5 we formulated the following axioms: W AF RP (Weak Axiom of Fuzzy Revealed Preference) SAF RP (Strong Axiom of Fuzzy Revealed Preference) W F CA (Weak Fuzzy Congruence Axiom) SF CA (Strong Fuzzy Congruence Axiom) These are fuzzy versions of the revealed preference and congruence axioms (W ARP , SARP , W CA, SCA) in classical consumer theory. We work in a fuzzy set theory corresponding to a continuous t-norm; thus the formulation of these axioms depends on the t-norm. Chapter 6 studies various connections between these four axioms and properties of rationality and consistency. Generally we attempt to know to what extent classical consumer theory can be extended to fuzzy choice functions. The results of this chapter follow the line Uzawa–Arrow–Sen and will be obtained under hypotheses (H1) and (H2). We recall that under the hypothesis that the domain of the consumer contains the pairs and the triples of alternatives, the Arrow–Sen theorem asserts the equivalence between W ARP , SARP , W CA, SCA and other four conditions of rationality (see Theorem 3.31). The ﬁrst section of the chapter tries to answer the question to what extent the Arrow-Sen theorem can be extended to fuzzy case. The equivalent

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 107–144 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

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6 Fuzzy Revealed Preference and Consistency Conditions

properties of the Arrow-Sen theorem lead to eight conditions for fuzzy choice functions. The main result of the section (Theorem 6.6) establishes equivalences or implications between these conditions. Some are true for an arbitrary continuous t-norm and others for the G¨ odel or the Lukasiewicz t-norm. Section 6.2 deals with consistency conditions F α, F β, the fuzzy forms of Sen’s properties α, β [91]. The formulation of F α uses the subsethood function I(., .) and F β is stated in terms of ∧ and ↔. We prove that a fuzzy choice function veriﬁes F α, F β if and only if W F CA holds for C. We also study the fuzzy choice functions verifying a special hypothesis (U ); in the crisp case, this hypothesis says that C(S) is a singleton for each subset S. Under this hypothesis we prove that W F CA and F α are equivalent. Section 6.3 is concerned with condition F δ, a fuzzy version of Sen’s condition δ. The main result (Theorem 6.22) shows that for a G–normal choice function, the associated fuzzy preference relation R is quasi-transitive if and only if condition F δ holds. In Section 6.4 we study some other consistency conditions F α2, F β(+), F γ2. We prove that a fuzzy choice function is G–normal if and only if it veriﬁes F α2 and F γ2. Condition F P I is introduced as a fuzzy version of ¯ is the path independence property P I [93]. The fuzzy preference relation R quasi-transitive provided F P I holds. The Arrow Axiom (AA) was formulated by Arrow [8] in order to characterize the full rationality of crisp choice functions. The Fuzzy Arrow Axiom (F AA) which is an extension in a fuzzy context of AA is enunciated in Section 6.5. The formulation of F AA uses the subsethood degree I(., .) to model the inclusion of the fuzzy choice functions and the degree of equality E(., .) to model their equality. The main result of the Section 6.5 (Theorem 6.37) is the proof of the equivalence between F AA and the full rationality of the fuzzy choice functions. We obtain a generalization of Arrow theorem [8]. Further on, we connect Theorem 6.37 and Theorem 5.7 (1) and (2), by establishing the equivalence between the full rationality, F AA, W F CA and SF CA.

6.1 A Fuzzy Approach to the Arrow–Sen Theorem

109

Throughout this chapter assume hypotheses (H1) and (H2) hold. Next ∗ will denote an arbitrary continuous t–norm. In case ∗ is a particular t–norm, this fact will be mentioned.

6.1 A Fuzzy Approach to the Arrow–Sen Theorem In this section we will study various relations between the axioms of revealed preference and congruence. Our aim is to investigate if the Arrow–Sen theorem (Theorem 3.31) is true for fuzzy choice functions. Some equivalences and implications of this theorem hold true in the fuzzy set theory based on a continuous t-norm, others for the G¨ odel or the Lukasiewicz t-norm. Particularly, for the G¨ odel t-norm we prove that a fuzzy choice function C veriﬁes SF CA if and only if the associated fuzzy relation R is a regular preference and C is G–normal. This shows that a strong form of Richter’s Theorem [84] holds if hypotheses (H1), (H2) are fulﬁlled. In the following ∗ will denote an arbitrary continuous t–norm. When a result is proved for a particular t–norm (G¨ odel, Lukasiewicz, etc.), this fact will be speciﬁed. Let C be a fuzzy choice function. In this chapter we shall denote by Cˆ the fuzzy choice function G∗ = G(., R) deﬁned in Section 5.3. By Lemma 5.37 we ˆ have for the G¨ odel t–norm that C(S) ⊆ C(S) for any S ∈ B. Proposition 6.1. If the fuzzy choice function C veriﬁes W F CA then R is a regular preference

1

on X.

Proof. We shall prove the following assertions:

(a) R is reﬂexive.

1

We use “regular preference” instead of “regular fuzzy preference”.

110

6 Fuzzy Revealed Preference and Consistency Conditions

(b) R is strongly total.

These two assertions follow by Lemma 5.26. (c) R is ∗-transitive. Let x, y, z ∈ X. We shall prove that R(x, y) ∗ R(y, z) ≤ R(x, z). Denote T =[x, y, z]. Since 1 if u ∈ {x, y, z} C(T )(u) =  0 otherwise. it follows that C(T )(x) = 1 or C(T )(y) = 1 or C(T )(z) = 1. We study separately these three cases. (c1 ) C(T )(x) = 1. Then

R(x, z) =

(C(S)(x) ∗ S(z)) ≥ C(T )(x) ∗ T (z) = 1.

S∈B

Therefore R(x, z) = 1 and the inequality R(x, y) ∗ R(y, z) ≤ R(x, z) is obviously veriﬁed. (c2 ) C(T )(y) = 1. By W F CA we have R(x, y) ∗ C(T )(y) ∗ T (x) ≤ C(T )(x). Since T (x) = C(T )(y) = 1 we have R(x, y) ≤ C(T )(x) hence R(x, y) ∗ R(y, z) ≤ R(x, y) ≤ C(T )(x) = C(T )(x) ∗ T (z) ≤ R(x, z). (c3 ) C(T )(z) = 1. By W F CA we have R(y, z) = R(y, z) ∗ C(T )(z) ∗ T (y) ≤ C(T )(y) R(x, y) ∗ C(T )(y) = R(x, y) ∗ C(T )(y) ∗ T (x) ≤ C(T )(x) therefore R(x, y) ∗ R(y, z) ≤ R(x, y) ∗ C(T )(y) ≤ C(T )(x) ∗ T (z) ≤ R(x, z).

6.1 A Fuzzy Approach to the Arrow–Sen Theorem

111

¯ Lemma 6.2. Let C be a G–normal fuzzy choice function. Then R = R. Proof. For any x, y ∈ X we have ¯ y) = C([x, y])(x) = C([x, ˆ R(x, y])(x) = [x, y](x) ∗

([x, y](z) → R(x, z)) =

z∈X

= ([x, y](x) → R(x, x)) ∧ ([x, y](y) → R(x, y)) = R(x, x) ∧ R(x, y) = R(x, y). Proposition 6.3. Let C be a G–normal fuzzy choice function. If R is ∗transitive then C veriﬁes W F CA. Proof. Let S ∈ B and x, y ∈ X. We must prove that R(x, y)∗C(S)(y)∗S(x) ≤ C(S)(x). Since ˆ C(S)(x) = C(S)(x) = S(x) ∗

(S(z) → R(x, z))

z∈X

it suﬃces to prove that R(x, y) ∗ C(S)(y) ≤ (S(z) → R(x, z)). z∈X

Let z ∈ X. Since C(S)(y) ∗ S(z) ≤ R(y, z) and R is ∗-transitive, then R(x, y) ∗ C(S)(y) ∗ S(z) ≤ R(x, y) ∗ R(y, z) ≤ R(x, z). Thus R(x, y) ∗ C(S)(y) ≤ S(z) → R(x, z) for each z ∈ X, therefore the desired inequality follows. By Proposition 6.3, if C is a G–normal fuzzy choice function and R is a regular preference then C veriﬁes W F CA. Proposition 6.4. Assume that ∗ is the Lukasiewicz t-norm. If C is a fuzzy ˜ then C veriﬁes W F CA. choice function such that R = R Proof. Suppose by absurdum that C does not verify W F CA then there exist S0 ∈ B and x, y ∈ X such that (a) R(x, y) ∗ C(S0 )(y) ∗ S0 (x) ≤ C(S0 )(x).

112

6 Fuzzy Revealed Preference and Consistency Conditions

˜ y) = ¬P˜ (y, x) then we have Since R(x, y) ≤ R(x, 0 = R(x, y) ∗ P˜ (y, x) = R(x, y) ∗ =

(C(S)(y) ∗ ¬C(S)(x) ∗ S(x)) =

S∈B

(R(x, y) ∗ C(S)(y) ∗ ¬C(S)(x) ∗ S(x))

S∈B

therefore R(x, y) ∗ C(S)(y) ∗ S(x) ∗ ¬C(S)(x) = 0 for each S ∈ B. But ¬¬C(S)(x) = C(S)(x) because ∗ is the Lukasiewicz t-norm, hence we get R(x, y) ∗ C(S)(y) ∗ S(x) ≤ C(S)(x), contradicting (a). ˜ ⊆ R. Lemma 6.5. Assume that ∗ is the Lukasiewicz t-norm. Then R Proof. First we shall establish the inequality ˜ y) ≤ C([x, y])(x). (a) R(x, Since C([x, y])(t) ≤ [x, y](t) =

 1

if

t ∈ {x, y}

 0 otherwise

and C([x, y]) is a normal fuzzy set we have C([x, y])(x) = 1 or C([x, y])(y) = 1. If C([x, y])(x) = 1 then (a) is obviously veriﬁed. Assume C([x, y])(y) = 1. Then ˜ y) = ¬P˜ (y, x) = ¬ R(x,

(C(S)(y) ∗ ¬C(S)(x) ∗ S(x)) ≤

S∈B

≤ ¬(C([x, y])(y) ∗ ¬C([x, y])(x) ∗ [x, y](x)) = ¬¬C([x, y])(x) = C([x, y])(x) and (a) is also veriﬁed. We remark that C([x, y])(x) = C([x, y])(x) ∗ [x, y](y) ≤ R(x, y). ˜ y) ≤ R(x, y). Thus, by (a), R(x, Proposition 6.6. Assume that ∗ is the Lukasiewicz t-norm. If C veriﬁes ˜ W F CA then R = R.

6.1 A Fuzzy Approach to the Arrow–Sen Theorem

113

Proof. Using Lemma 4.5 (5) we get R(x, y) ∗ P˜ (y, x) = R(x, y) ∗ =

(C(S)(y) ∗ ¬C(S)(x) ∗ S(x)) =

S∈B

(R(x, y) ∗ C(S)(y) ∗ ¬C(S)(x) ∗ S(x)).

S∈B

By W F CA, R(x, y) ∗ C(S)(y) ∗ S(x) ≤ C(S)(x), hence, for any S ∈ B we have R(x, y) ∗ C(S)(y) ∗ ¬C(S)(x) ∗ S(x) = 0. It follows that R(x, y) ∗ P˜ (y, x) = 0. We have proved that R(x, y) ≤ ˜ y). ¬P˜ (y, x) = R(x, ˜ y) ≤ R(x, y). By Lemma 6.5, R(x, For a fuzzy choice function C let us consider the following statements (i) R is a regular preference and C is G–normal; ¯ is a regular preference and C is G–normal; (ii) R (iii) C veriﬁes W F CA; (iv) C veriﬁes SF CA; (v) C veriﬁes W AF RP ; (vi) C veriﬁes SAF RP ; ˜ (vii) R = R; ¯=R ˜ and C is G–normal. (viii) R We remark that the previous properties are the fuzzy versions of the equivalent conditions of the Arrow–Sen theorem (see Theorem 3.31). A natural problem is to obtain a fuzzy extension of the Arrow-Sen theorem. The following theorem establishes some relations between the assertions (i)(viii). This extends to the fuzzy case a signiﬁcant part of the Arrow–Sen theorem. Theorem 6.7. Let C be a fuzzy choice function. (1) Conditions (i), (ii) are equivalent.

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6 Fuzzy Revealed Preference and Consistency Conditions

(2) Implication (i) ⇒ (iii) holds; if ∗ is the G¨ odel t-norm ∧ then the implication (iii) ⇒ (i) also holds. (3) Conditions (iii), (iv) are equivalent. (4) If ∗ is the Lukasiewicz t-norm then conditions (iii),(v), (vi), (vii) are equivalent; (5) Implication (viii) ⇒ (vii) holds. ¯ hence (i) ⇔ (ii) holds. Proof. (1) By Lemma 6.2, R = R, (2) By Proposition 6.3, if C is G–normal and R is a regular preference then C veriﬁes W F CA, thus (i) ⇒ (iii) holds. Now we shall prove that for the G¨odel t-norm (iii) ⇒ (i) is true. In accordance with Proposition 6.1, R is ˆ a regular preference. By Lemma 5.37 (iii), C(S) ⊆ C(S) for any S ∈ B. Let ˆ ˆ S ∈ B. We shall prove that C(S) ⊆ C(S), i.e. C(S)(x) ≤ C(S)(x) for each x ∈ X. Since C(S) is a normal fuzzy set C(S)(z) = 1 for some z ∈ X; we also have S(z) = 1. Thus ˆ C(S)(x) = S(x) ∧

(S(y) → R(x, y)) ≤ S(x) ∧ (S(z) → R(x, z)) =

y∈X

S(x) ∧ R(x, z) = R(x, z) ∧ C(S)(z) ∧ S(x) ≤ C(S)(x). ˆ The last inequality follows according to W F CA. Therefore C(S) ⊆ C(S) ˆ so C(S) = C(S). (3) By Remark 5.28, the implication (iv) ⇒ (iii) is obvious. Since W F CA implies the ∗-transitivity of R (see Proposition 6.1), it follows that R is equal to its transitive closure W . Thus W F CA implies SF CA, hence the implication (iii) ⇒ (iv) is proved. (4) Assume that ∗ is the Lukasiewicz t-norm. First we will prove that (iii) ⇒ (v) holds. Suppose by absurdum that C does not verify W AF RP , so P˜ (x, y) ≤ ¬R(y, x) for some x, y ∈ X. Then, by Lemma 4.6 (1), P˜ (x, y) ∗ R(y, x) > 0. In accordance with Lemma 4.5 (5)

6.1 A Fuzzy Approach to the Arrow–Sen Theorem

P˜ (x, y) ∗ R(y, x) = [

115

(C(S)(x) ∗ S(y) ∗ ¬C(S)(y))] ∗ R(y, x) =

S∈B

=

(C(S)(x) ∗ S(y) ∗ ¬C(S)(y) ∗ R(y, x))

S∈B

therefore C(S)(x) ∗ S(y) ∗ ¬C(S)(y) ∗ R(y, x) > 0 for some S ∈ B. Since ¬¬C(S)(y) = C(S)(y), by Lemma 4.6 (1) we get R(y, x) ∗ C(S)(x) ∗ S(y) ≤ C(S)(y), contradicting W F CA. In order to prove the converse implication (v) ⇒ (iii) we assume by absurdum that C does not verify W F CA, i.e. there exist S0 ∈ B and x, y ∈ X such that R(y, x) ∗ C(S0 )(x) ∗ S0 (y) ≤ C(S0 )(y). By Lemma 4.6 (1) we obtain R(y, x) ∗ C(S0 )(x) ∗ S0 (y) ∗ ¬C(S0 )(y) > 0. Thus P˜ (x, y) ∗ R(y, x) =

(C(S)(x) ∗ S(y) ∗ ¬C(S)(y) ∗ R(y, x)) ≥

S∈B

≥ C(S0 )(x) ∗ S0 (y) ∗ ¬C(S0 )(y) ∗ R(y, x) > 0, therefore, by Lemma 4.6 (1), P˜ (x, y) ≤ ¬R(y, x), contradicting W AF RP . The implication (vi) ⇒ (v) is always true (see Remark 5.28). We shall establish the implication (v) ⇒ (vi). If we assume (v), then (iii) holds, since we have proved that (iii) and (v) are equivalent. Thus C veriﬁes W F CA. By Proposition 6.1, R is a regular preference. Applying Proposition 6.6, we ˜ hence R ˜ is a regular preference. We shall prove that P˜ is ∗obtain R = R, transitive, i.e. P˜ (x, y) ∗ P˜ (y, z) ≤ P˜ (x, z). In accordance with the deﬁnition ˜ x) ∗ ¬R(z, ˜ y) ≤ ¬R(z, ˜ x). Since ∗ is of P˜ , this inequality can be written ¬R(y, the Lukasiewicz t-norm, by Lemma 4.6 (1), this last inequality is equivalent to ˜ x) ∗ ¬R(y, ˜ x) ∗ ¬R(z, ˜ y) = 0. R(z, ˜ is strongly total, we have R(y, ˜ x) = 1 or R(x, ˜ y) = 1. If R(y, ˜ x) = 1 Since R ˜ x) = 0 and the previous inequality is obvious. Consider the case then ¬R(y, ˜ y) = 1. R ˜ is ∗-transitive then R(z, ˜ x) = R(z, ˜ x) ∗ R(x, ˜ y) ≤ R(z, ˜ y). R(x,

116

6 Fuzzy Revealed Preference and Consistency Conditions

˜ x) ∗ ¬R(y, ˜ x) ∗ ¬R(z, ˜ y) ≤ R(z, ˜ y) ∗ ¬R(y, ˜ x) ∗ ¬R(z, ˜ y) = 0. Therefore R(z, It follows that P˜ is ∗-transitive then P˜ is identical to its ∗-transitive closure P ∗ . Thus it is obvious that W AF RP ⇒ SAF RP . (iii) ⇔ (vii) By Propositions 6.4 and 6.6. ¯ (by Lemma 6.2). Then R ¯=R ˜ implies (5) If C is G–normal then R = R ˜ R = R. The equivalence (i) ⇔ (iii) proved in Theorem 6.7 (2) for the case of the G¨ odel t-norm can be viewed as a fuzzy extension of Richter theorem (assuming hypotheses (H1), (H2)). Example 6.8. As seen above, Theorem 6.7 establishes some connections between conditions (i)-(viii) in Section 6.1. The example presented in this section is constructed with the aim to clarify these connections and the limitations of Theorem 6.7. Let X = {x, y} and B = {[x], [y], [x, y], A} where A ∈ F(X) is deﬁned by A = 0.3χ{x} + χ{y} . Consider function C : B → F(X) deﬁned by

C([x]) = χ{x} ; C([y]) = χ{y} ; C([x, y]) = 0.25χ{x} + χ{y} ; C(A) = 0.25χ{x} + χ{y} . C is a fuzzy choice function fulﬁlling (H1) and (H2). We determine ﬁrst the fuzzy relation R associated with C. According to Deﬁnition 5.16 (i)

R(x, y) =

(C(S)(x) ∗ S(y))=

S∈B

= (C([x])(x) ∗ [x](y)) ∨ (C([y])(x) ∗ [y](y)) ∨ (C([x, y])(x) ∗ [x, y](y)) ∨ (C(A)(x) ∗ A(y)) = = 1 ∗ 0 ∨ 0 ∗ 1 ∨ 0.25 ∗ 1 ∨ 0.25 ∗ 1 = 0.25.

6.1 A Fuzzy Approach to the Arrow–Sen Theorem

117

Analogously R(x, x) = R(y, y) = R(y, x) = 1, thus   1 0.25  R= 1 1 It is clear that R is ∗-transitive, reﬂexive and strongly total. We check now if the fuzzy choice function C veriﬁes W F CA. For all a, b ∈ X and S ∈ B we must have the inequality (a) R(a, b) ∗ C(S)(b) ∗ S(a) ≤ C(S)(a). For a = b this inequality is always true; consider only the cases a = x, b = y and a = y, b = x. Case a = x, b = y. Since R(x, y) = 0.25 we have to prove that for any S ∈ B the following inequality holds: (b) 0.25 ∗ C(S)(y) ∗ S(x) ≤ C(S)(x). −S = [x]. We have C(S)(y) = C([x])(y) = 0, hence (b) is veriﬁed. −S = [y]. We have 0.25 ∗ C([y])(y) ∗ [y](x) = 0, hence (b) is veriﬁed. −S = [x, y]. We have 0.25 ∗ C([x, y])(y) ∗ [x, y](x) = 0.25 = C([x, y])(x). −S = A. We have 0.25 ∗ C(A)(y) ∗ A(x) = 0.25 ∗ 1 ∗ 0.3 ≤ C(A)(x) = 0.25.

The case a = y, b = x follows similarly. In conclusion C veriﬁes W F CA. The fuzzy relation R being ∗-transitive, R = W then C veriﬁes also SF CA. We check now if condition (i) is veriﬁed for C. R is a regular preference, then we investigate if C is G–normal: ˆ C([x])(x) = 1 = C([x])(x) ˆ C([x])(y) = 0 = C([x])(y) ˆ C([y])(x) = 0 = C([y])(x) ˆ C([y])(y) = 1 = C([y])(y)

118

6 Fuzzy Revealed Preference and Consistency Conditions

ˆ C([x, y])(x) = [x, y](x) ∗ [([x, y](x) → R(x, x)) ∧ ([x, y](y) → R(x, y))] = R(x, y) = 0.25 = C([x, y])(x) ˆ C([x, y])(y) = R(y, x) = 1 = C([x, y])(y) ˆ C(A)(y) = A(y) ∗ [((A(x) → R(y, x)) ∧ (A(y) → R(y, y))] = 0.3 → R(y, x) = 1 = C(A)(y) ˆ C(A)(x) = A(x)∗[((A(x) → R(x, x))∧ (A(y) → R(x, y))] = 0.3∗R(x, y) = 0.3 ∗ 0.25; C(A)(x) = 0.25.

For the G¨ odel t-norm we have ˆ C(A)(x) = 0.3 ∧ 0.25 = 0.25 = C(A)(x), hence Cˆ = C. For the Lukasiewicz t-norm ∗L ˆ C(A)(x) = 0.3 ∗L 0.25 = max(0.3 + 0.25 − 1, 0) = 0 = C(A)(x) and for the product t-norm ∗P ˆ C(A)(x) = 0.3 ∗P 0.25 = 0.3 × 0.25 = C(A)(x). This example shows that the implication (iii) ⇒ (i) in Theorem 6.7 is false in the case of the Lukasiewicz or product t-norms; meanwhile it conﬁrms that the equivalence (i) ⇔ (iii) is true for the G¨ odel t-norm . ˜ According to Deﬁnition 5.18 (i) we have We compute now P˜ and R. P˜ (y, x) =

(C(S)(y) ∗ S(x) ∗ ¬C(S)(x)) =

S∈B

= [C([x])(y) ∗ [x](x) ∗ ¬C([x])(x)] ∨ [C([y])(y) ∗ [y](x) ∗ ¬C([y])(x)]∨ ∨[C([x, y])(y) ∗ [x, y](x) ∗ ¬C([x, y])(x)] ∨ [C(A)(y) ∗ A(x) ∗ ¬C(A)(x)] = = (¬0.25) ∨ (0.3 ∗ ¬0.25). Similarly, P˜ (x, y) = 0. Hence  P˜ = 

 0

0

(¬0.25) ∨ (0.3 ∗ ¬0.25) 0

For the Lukasiewicz t-norm ∗L we have



6.2 Conditions F α and F β

119

P˜ (y, x) = (1 − 0.25)∨ max (0.3 + 0.75 − 1, 0) = 0.75, hence     1 0.25 0 0 ˜=  = R. ,R P˜ =  1 1 0.75 0 For G¨ odel and product t-norms    1 00 ˜= ,R P˜ =  1 00

 1 1

 = R.

It follows that for the G¨ odel and product t-norms the equivalence (iv) ⇔ (vii) in Theorem 6.7 does not take place.

6.2 Conditions F α and F β Conditions α and β were introduced by Sen [91] for crisp choice functions. In this section we will consider fuzzy versions F α and F β of these conditions and we will prove that a fuzzy choice function C veriﬁes F α and F β if and only if C veriﬁes W F CA. We consider a class of fuzzy choice functions verifying a new hypothesis (U ). In the crisp case (U ) expresses that C(S) is a singleton for each fuzzy set S. Among results under hypothesis (U ) there is the equivalence between F α and W F CA. Let ∗ be an arbitrary continuous t-norm. First we will recall the (crisp) conditions α and β. Let C : B → P(X) be a crisp choice function. Condition α . For any S, T ∈ B and for any x ∈ X, we have the implication x ∈ S, x ∈ C(T ) and S ⊆ T ⇒ x ∈ C(S). Condition β . For any S, T ∈ B and for any x, y ∈ X, we have the implication x, y ∈ C(S) and S ⊆ T ⇒ x ∈ C(T ) if and only if y ∈ C(T ).

120

6 Fuzzy Revealed Preference and Consistency Conditions

These two conditions can be extended to the fuzzy case. Let C : B → F(X) be a fuzzy choice function on (X, B). Condition F α . For any S, T ∈ B and x ∈ X we have I(S, T ) ∗ S(x) ∗ C(T )(x) ≤ C(S)(x). Condition Fβ . For any S, T ∈ B and x, y ∈ X we have I(S, T ) ∗ C(S)(x) ∗ C(S)(y) ≤ C(T )(x) ↔ C(T )(y) where ↔ is the biresiduum of the t-norm ∧.

It is obvious that conditions F α, F β generalize α, β. A weak form of conditions F α, F β can be given as: Condition F α . For any S, T ∈ B and x ∈ X, if S ⊆ T then S(x) ∗ C(T )(x) ≤ C(S)(x). Condition Fβ . For any S, T ∈ B and x, y ∈ X, if S ⊆ T then C(S)(x) ∗ C(S)(y) ≤ C(T )(x) ↔ C(T )(y). Since S ⊆ T iﬀ I(S, T ) = 1 clearly F α ⇒ F α , F β ⇒ F β . Although F α , F β are closer to α, β, the use of subsethood degree I(A, B) in F α and F β gives a better expression of the behaviour of fuzzy choice functions. Proposition 6.9. If C is a G–normal fuzzy choice function then C veriﬁes F α. Proof. Since C = Cˆ it suﬃces to show that for any S, T ∈ B and x ∈ X the following inequality holds: ˆ )(x) ≤ C(S)(x). ˆ I(S, T ) ∗ S(x) ∗ C(T By Lemma 4.3 (2) we have for any u ∈ X S(u)∗(S(u) → T (u))∗(T (u) → R(x, u)) = S(u)∗T (u)∗(T (u) → R(x, u)) = S(u) ∗ T (u) ∗ R(x, u) ≤ R(x, u).

6.2 Conditions F α and F β

121

Using Lemma 4.3 (1) one infers (S(u) → T (u)) ∗ (T (u) → R(x, u)) ≤ S(u) → R(x, u). Therefore ˆ )(x) = I(S, T ) ∗ S(x) ∗ C(T = (S(u) → T (u)) ∗ S(x) ∗ T (x) ∗ (T (u) → R(x, u)) ≤ u∈X

≤ S(x) ∗

u∈X

[(S(u) → T (u)) ∗ (T (u) → R(x, u))] ≤

u∈X

≤ S(x) ∗

ˆ (S(u) → R(x, u)) = C(S)(x).

u∈X

Assume that ∗ is the G¨ odel t–norm ∧ for the rest of the chapter. Proposition 6.10. If the fuzzy choice function C fulﬁlls condition F α then ¯ R = R. ¯ ⊆ R is always true. We prove that Proof. By Lemma 5.26 (i) the inclusion R ¯ y) for any x, y ∈ X. Let S ∈ B. Then, by F α R(x, y) ≤ R(x, I([x, y], S) ∧ [x, y](x) ∧ C(S)(x) ≤ C([x, y])(x). We remark that I([x, y], S) = ([x, y](u) → S(u)) = u∈X

= ([x, y](x) → S(x)) ∧ ([x, y](y) → S(y)) = = (1 → S(x)) ∧ (1 → S(y)) = S(x) ∧ S(y).

Thus the above inequality becomes S(x) ∧ S(y) ∧ C(S)(x) ≤ C([x, y])(x). Since C(S)(x) ≤ S(x) we have C(S)(x) ∧ S(y) ≤ C([x, y])(x) for each S ∈ B therefore ¯ y) R(x, y) = (C(S)(x) ∧ S(y)) ≤ C([x, y])(x) = R(x, S∈B

122

6 Fuzzy Revealed Preference and Consistency Conditions

Proposition 6.11. If a fuzzy choice function C veriﬁes W F CA then conditions F α, F β are fulﬁlled. Proof. By Theorem 6.7, W F CA implies the G–normality of C, hence, by Proposition 6.9, condition F α is veriﬁed. Assume by absurdum that C does not fulﬁll F β, hence there exist S, T ∈ B and x, y ∈ X such that I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ C(T )(x) ↔ C(T )(y) = = (C(T )(x) → C(T )(y)) ∧ (C(T )(y) → C(T )(x)). Therefore I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ C(T )(x) → C(T )(y) or I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ C(T )(y) → C(T )(x).

Assume the ﬁrst case holds, hence, by Lemma 4.3 (1) (a) I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ∧ C(T )(x) ≤ C(T )(y). We remark that I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ∧ C(T )(x) ≤ C(S)(y) ∧ S(x) ∧ C(T )(x) = = C(S)(y) ∧ S(x) ∧ C(T )(x) ∧ S(y) because C(S)(y) ∧ S(y) = C(S)(y). Since C(S)(y) ∧ S(x) ≤ R(y, x) one obtains (b) I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ∧ C(T )(x) ≤ R(y, x) ∧ C(T )(x) ∧ S(y). By (a) and (b) one infers R(y, x) ∧ C(T )(x) ∧ S(y) ≤ C(T )(y) contradicting W F CA. Proposition 6.12. If C fulﬁlls conditions F α, F β then W F CA holds. Proof. Let S ∈ B and x, y ∈ X. Since I([x, y], S) = S(x) ∧ S(y) we have S(x) ∧ C(S)(y) ∧ R(x, y) = S(x) ∧ S(y) ∧ C(S)(y) ∧ R(x, y) = = I([x, y], S) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ R(x, y) =

6.2 Conditions F α and F β

123

= I([x, y], S) ∧ [x, y](y) ∧ C(S)(y) ∧ S(x) ∧ S(y) ∧ R(x, y) ≤ ≤ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ R(x, y) because I([x, y], S) ∧ [x, y](y) ∧ C(S)(y) ≤ C([x, y])(y), by F α. Replacing R(x, y) with its expression in Deﬁnition 5.16 (i) we obtain S(x) ∧ C(S)(y) ∧ R(x, y) ≤ ≤ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ =

(C(H)(x) ∧ H(y)) =

H∈B

[C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ C(H)(x) ∧ H(y)].

H∈B

In accordance with F α: C(H)(x) ∧ H(y) = C(H)(x) ∧ H(x) ∧ H(y) = I([x, y], H) ∧ [x, y](x) ∧ C(H)(x) ≤ C([x, y])(x).

Therefore, by F β and Lemma 4.3 (2): C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ C(H)(x) ∧ H(y) ≤ ≤ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ C([x, y])(x) = = I([x, y], S) ∧ C([x, y])(x) ∧ C([x, y])(y) ∧ C(S)(y) ≤ ≤ [C(S)(x) ↔ C(S)(y)] ∧ C(S)(y) ≤ ≤ C(S)(y) ∧ [C(S)(y) → C(S)(x)] = C(S)(y) ∧ C(S)(x) ≤ C(S)(x). These inequalities hold for each H ∈ B therefore S(x) ∧ C(S)(y) ∧ R(x, y) ≤ C(S)(x). Hence the fuzzy choice function C veriﬁes W F CA. Theorem 6.13. For a fuzzy choice function C the following are equivalent: (1) C veriﬁes conditions Fα, Fβ; (2) W F CA holds for C. Proof. By Propositions 6.11 and 6.12.

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6 Fuzzy Revealed Preference and Consistency Conditions

Remark 6.14. The previous theorem generalizes to fuzzy choice functions a result of Sen (see [91], (T8)). This is one more argument that F α, F β are the appropriate versions of α, β and not conditions F α , F β . In crisp consumer theory a special case is made by consumers C : B → P(X) with the property that C(S) is a singleton for any S ∈ B. We generalize this case considering fuzzy choice functions C : B → F(X) that verify: (U ) For any S ∈ B, C(S) = [x], for some x ∈ X. For fuzzy choice  functions C that verify (U ) there is a unique x ∈ X such  1 if y = x that C(S)(y) =  0 if y = x. Proposition 6.15. If a G–normal fuzzy choice function C veriﬁes (U ) then R is a regular preference. Proof. Assume C is G–normal. By Lemma 5.26 we know that R is reﬂexive ¯ and strongly total; by Lemma 6.2, R = R. Let x, y ∈ X. Then ¯ y) = C([x, y])(x); R(y, x) = C([x, y])(y). R(x, y) = R(x, But C([x, y]) is a normal fuzzy set so C([x, y])(x) = 1 or C([x, y])(y) = 1 because C([x, y])(t) ≤ [x, y](t) =

 1

if

t ∈ {x, y}

 0 otherwise.

Thus, by (U ), we have exactly one of two cases: R(x, y) = 1 and R(y, x) = 0. R(x, y) = 0 and R(y, x) = 1. Now we prove that R is ∧-transitive. Let x, y, z ∈ X. We will establish the inequality R(x, y) ∧ R(y, z) ≤ R(x, z). The case R(x, z) = 1 is obvious. Consider R(x, z) = 1. Since R(x, z) = C([x, z])(x) and C veriﬁes (U ), we must have R(x, z) = 0.

6.2 Conditions F α and F β

125

Assume by absurdum R(x, y) ∧ R(y, z) = 0 hence R(x, y) = R(y, z) = 1 and R(y, x) = R(z, y) = 0 because of (U ). Therefore, since C is G–normal C([x, y, z])(x) = [x, y, z](x) ∧

([x, y, z](u) → R(x, u)) =

u∈X

= R(x, x) ∧ R(x, y) ∧ R(x, z) = R(x, y) ∧ R(x, z) = 0 and similarly C([x, y, z])(y) = R(y, x) ∧ R(y, z) = 0 C([x, y, z])(z) = R(z, x) ∧ R(z, y) = 0. Since C([x, y, z])(u) ≤ [x, y, z](u) =

 1

if

u ∈ {x, y, z}

 0 otherwise

and C([x, y, z]) is a normal fuzzy set we have C([x, y, z])(x) = 1 or C([x, y, z])(y) = 1 or C([x, y, z])(z) = 1. Contradiction. Then R is transitive. Remark 6.16. Assume C is a G–normal fuzzy choice function fulﬁlling (U ). ¯ Thus P = P¯ . We prove P¯ = R. ¯ By the proof of Proposition 6.15, R = R. For any x, y ∈ X we have ¯ y) ∧ ¬R(y, ¯ x) = C([x, y])(x) ∧ ¬C([x, y])(y) P¯ (x, y) = R(x, ¯ y) = C([x, y])(x). R(x, According to (U ) we have one of the cases (see the proof of Proposition 6.15) C([x, y])(x) = 1, C([x, y])(y) = 0; C([x, y])(x) = 0, C([x, y])(y) = 1. A simple computation shows that each of these cases leads to C([x, y])(x) = C([x, y])(x) ∧ ¬C([x, y])(y), ¯ y) = P¯ (x, y). Hence P = P¯ = R ¯ = R. so R(x, Proposition 6.17. Let C be a fuzzy choice function verifying (U ). The following assertions are equivalent (1) W F CA holds for C;

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6 Fuzzy Revealed Preference and Consistency Conditions

(2) C veriﬁes condition F α. Proof. (1) ⇒ (2) By Theorem 6.7 W F CA implies the G–normality of C. In accordance with Proposition 6.9 C veriﬁes F α. (2) ⇒ (1) Let x, y ∈ X and S ∈ B. We prove that (a) R(x, y) ∧ C(S)(y) ∧ S(x) ≤ C(S)(x). It is easy to see that I([x, y], S) = S(x) ∧ S(y). Applying F α for [x, y] and S we get C(S)(y)∧S(x) = S(x)∧S(y)∧C(S)(y) = I([x, y], S)∧[x, y](y)∧C(S)(y) ≤ C([x, y])(y), hence R(x, y) ∧ C(S)(y) ∧ S(x) ≤ C(S)(y) ∧ S(x) ≤ C([x, y])(y). Let H ∈ B. Applying again F α for [x, y] and H we get C(H)(x) ∧ H(y) = H(x) ∧ H(y) ∧ C(H)(x) = I([x, y], H) ∧ [x, y](x) ∧ C(H)(x) ≤ C([x, y])(x) hence C(H)(x) ∧ H(y) ∧ C(S)(y) ∧ S(x) ≤ C(H)(x) ∧ H(y) ≤ C([x, y])(x). Since these last inequalities hold for each H ∈ B we infer R(x, y) ∧ C(S)(y) ∧ S(x) = [ (C(H)(x) ∧ H(y))] ∧ C(S)(y) ∧ S(x) =

H∈B

[C(H)(x) ∧ H(y) ∧ C(S)(y) ∧ S(x)] ≤ C([x, y])(x).

H∈B

We conclude that R(x, y) ∧ C(S)(y) ∧ S(x) ≤ C([x, y])(x) ∧ C([x, y])(y). If x = y then C([x, y])(x) ∧ C([x, y])(y) = 0 because of (U ), then R(x, y) ∧ C(S)(y) ∧ S(x) = 0 ≤ C(S)(x). If x = y then R(x, y) ∧ C(S)(y) ∧ S(x) = C(S)(x) ∧ S(x) ≤ C(S)(x). Hence the inequality (a) is proved.

6.3 Quasi-transitivity and Condition F δ

127

6.3 Quasi-transitivity and Condition F δ This section deals with a fuzzy form F δ of Sen’s condition δ. For a G–normal fuzzy choice function C we prove that the associated fuzzy preference relation R is quasi-transitive if and only if condition F δ holds. Let Q be a fuzzy relation on X and PQ be the strict preference relation on X deﬁned by PQ (x, y) = Q(x, y) ∧ ¬Q(y, x) for any x, y ∈ X. If R is the fuzzy revealed preference relation associated with a fuzzy choice function C (cf. Deﬁnition 5.16) then PR = P . Let ∗ be a continuous t–norm. A fuzzy relation Q on X is quasi-transitive if PQ (x, y) ∗ PQ (y, z) ≤ PQ (x, z) for any x, y, z ∈ X. Proposition 6.18. Let Q be a reﬂexive and strongly total fuzzy relation on X. If Q is transitive then Q is quasi–transitive. Proof. By the deﬁnition of PQ we have PQ (x, y) ∗ PQ (y, z) = Q(x, y) ∗ ¬Q(y, x) ∗ Q(y, z) ∗ ¬Q(z, y) for all x, y, z ∈ X. Hence PQ (x, y) ∗ PQ (y, z) ≤ Q(x, y) ∗ Q(y, z) ≤ Q(x, z), Q being transitive. We remark that Q(z, x) ∗ Q(x, y) ≤ Q(z, y) hence Q(z, x) ∗ Q(x, y) ∗ ¬Q(z, y) ≤ Q(z, y) ∗ ¬Q(z, y) = 0 so Q(z, x) ∗ Q(x, y) ∗ ¬Q(z, y) = 0. According to Lemma 4.6 (1), Q(x, y) ∗ ¬Q(z, y) ≤ ¬Q(z, x) hence PQ (x, y) ∗ PQ (y, z) ≤ Q(x, y) ∗ ¬Q(z, y) ≤ ¬Q(z, x). Therefore PQ (x, y) ∗ PQ (y, z) ≤ Q(x, z) ∗ ¬Q(z, x) = PQ (x, z), i.e. Q is quasi-transitive. For the rest of the section suppose that ∗ is the G¨ odel t–norm ∧. Let C : B → F(X) be a fuzzy choice function on (X, B). Deﬁnition 6.19. We say that the fuzzy choice function C veriﬁes the condition F δ if for any S = [a1 , . . . , an ], T = [b1 , . . . , bm ] in B and for any distinct x, y ∈ X the following inequality holds

128

6 Fuzzy Revealed Preference and Consistency Conditions

I(S, T ) ≤ (C(S)(x) ∧ C(S)(y)) → ¬(C(T )(x) ∧

¬C(T )(t)).

t=x

Remark 6.20. By Lemma 4.3 (1) the previous inequality is equivalent to ¬C(T )(t)). I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ ¬(C(T )(x) ∧ t=x

An easy computation shows that in case of crisp choice functions condition F δ is exactly condition δ. Proposition 6.21. Assume the fuzzy choice function C is G–normal and veriﬁes condition F δ. Then the associated fuzzy revealed preference relation R is quasi-transitive. Proof. We prove that for all x, y, z ∈ X the following inequality holds (a) P (x, y) ∧ P (y, z) ≤ P (x, z) = R(x, z) ∧ ¬R(z, x). Recall that R is reﬂexive and strongly total in accordance with Lemma 5.26. Since C is G–normal one gets ˆ C([x, y, z])(x) = C([x, y, z])(x) = R(x, y) ∧ R(x, z); ˆ C([x, y, z])(y) = C([x, y, z])(y) = R(y, x) ∧ R(y, z); ˆ C([x, y, z])(z) = C([x, y, z])(z) = R(z, x) ∧ R(z, y). By the deﬁnition of a fuzzy choice function, C([x, y, z])(x) = 1 or C([x, y, z])(y) = 1 or C([x, y, z])(z) = 1. If C([x, y, z])(y) = 1 then 1 = R(y, x) ∧ R(y, z) ≤ R(y, x), R(y, z) hence R(y, x) = R(y, z) = 1. One gets P (x, y) = R(x, y) ∧ ¬R(y, x) = 0 so (a) is trivially veriﬁed. Similarly, if C([x, y, z])(z) = 1 then P (y, z) = 0 and (a) is veriﬁed. Let us consider the case C([x, y, z])(x) = 1 hence R(x, y) = R(x, z) = 1. Then (a) is equivalent to P (x, y) ∧ P (y, z) ≤ ¬R(z, x). By Lemma 4.6 (1), this last inequality is equivalent to (b) P (x, y) ∧ P (y, z) ∧ R(z, x) = 0. Since C is G–normal we have C([x, z])(x) = R(x, z) = 1 and C([x, z])(z) = R(z, x). In accordance with condition F δ and Lemma 4.6 (7), (8) the following

6.3 Quasi-transitivity and Condition F δ

129

hold R(z, x) = R(x, z) ∧ R(z, x) = I([x, z], [x, y, z]) ∧ C([x, z])(x) ∧ C([x, z])(z) ¬C([x, y, z])(t)) ≤ ¬(C([x, y, z])(x) ∧ = ¬(

t=x

¬C([x, y, z])(t))

t=x

= ¬(¬C([x, y, z])(y) ∧ ¬C([x, y, z])(z)) = ¬(¬(R(y, x) ∧ R(y, z)) ∧ ¬(R(z, x) ∧ R(z, y))) ≤ ¬(¬R(y, x) ∧ ¬R(z, y)) = ¬¬R(y, x) ∨ ¬¬R(z, y) Thus one gets P (x, y) ∧ P (y, z) ∧ R(z, x) ≤ P (x, y) ∧ P (y, z) ∧ (¬¬R(y, x) ∨ ¬¬R(z, y)) ≤ ≤ ¬R(y, x) ∧ ¬R(z, y) ∧ (¬¬R(y, x) ∨ ¬¬R(z, y)) = = (¬R(y, x)∧¬R(z, y)∧¬¬R(y, x))∨(¬R(y, x)∧¬R(z, y)∧¬¬R(z, y)) = 0 and the inequality (b) was proved. Proposition 6.22. If C is a G–normal fuzzy choice function and R is quasitransitive then condition F δ holds. Proof. Suppose by absurdum that F δ does not hold, i.e. there exist S = [a1 , . . . , an ], T = [b1 , . . . , bm ] ∈ B and x, y ∈ X such that (a) I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ ¬(C(T )(x) ∧ ¬C(T )(t)). t=x

We observe that (b) C(S)(y) ∧ C(S)(x) ≤ C(S)(y) ∧ S(x) ≤ R(y, x). If R(y, x) = 0 then, by (b), C(S)(x) ∧ C(S)(y) = 0, contradicting (a); hence R(y, x) > 0. Thus ¬R(y, x) = 0 and P (x, y) = R(x, y) ∧ ¬R(y, x) = 0. Since S(t), T (t) ∈ {0, 1} we have S(t) → T (t) ∈ {0, 1} for each t ∈ X hence I(S, T ) = (S(t) → T (t)) ∈ {0, 1}. t∈X

I(S, T ) = 0 contradicts (a), hence I(S, T ) = 1. Then S ⊆ T so {a1 , . . . , an } ⊆ {b1 , . . . , bm }. If x ∈ {a1 , . . . , an } then C(S)(x) ≤ S(x) = 0 contradicting (a); then x ∈ {a1 , . . . , an }. Similarly, y ∈ {a1 , . . . , an }.

130

6 Fuzzy Revealed Preference and Consistency Conditions

By (a), ¬C(T)(t) > 0 for each t = x. Since  1 if C(T )(t) = 0 ¬C(T )(t) =  0 if C(T )(t) > 0 we must have ¬C(T )(t) = 1, hence C(T )(t) = 0 for each t = x. But C is G–normal, so C(T )(y) = T (y) ∧ (T (z) → R(y, z)). z∈X   R(y, z) if z ∈ {b1 , . . . , bm } Since T (y) = 1 and T (z) → R(y, z) =  1 otherwise it follows that 0 = C(T )(y) = R(y, b1 ) ∧ . . . ∧ R(y, bm ). Thus R(y, z1 ) = 0 for some z1 ∈ {b1 , . . . , bm }. R(y, x) > 0 implies z1 = x; of course z1 = y. R being strongly total, R(z1 , y) = 1, therefore P (z1 , y) = R(z1 , y) ∧ ¬R(y, z1 ) = 1. Applying the same procedure with y instead of x and z1 instead of y there exists z2 ∈ {b1 , . . . , bm }\{y, z1 } such that P (z2 , z1 ) = 1. If z2 = x then according to the hypothesis of quasi-transitivity, 1 = P (x, z1 ) ∧ P (z1 , y) ≤ P (x, y). This contradicts P (x, y) = 0, so z2 = x. In conclusion, z2 ∈ {b1 , . . . , bm } \ {x, y, z1 }. By induction there exists a sequence z1 , z2 , . . . , zk , . . . of elements of {b1 , . . . , bm } such that zk ∈ {b1 , . . . , bm } \ {x, y, z1 , . . . , zk−1 } for any k = 1, 2, . . . (z0 is taken y). Thus the terms of the sequence (zk ) will be distinct, contradicting the ﬁnitude of {b1 , . . . , bm }. The contradiction shows that F δ is veriﬁed. Summing the two previous propositions we get Theorem 6.23. If C is a G–normal fuzzy choice function, then R is quasitransitive if and only if condition F δ is veriﬁed. Remark 6.24. Theorem 6.23 is the generalization for fuzzy choice functions of a Sen result (see [91], (T 10)).

6.4 Other Consistency Conditions

131

6.4 Other Consistency Conditions In [93] there exist other consistency conditions besides properties α, β, γ, δ. In this section we study conditions F α2, F β(+) and F γ2, fuzzy versions of conditions α2, β(+) and γ2 for crisp choice functions [93]. Then a fuzzy choice function is G–normal if and only if conditions F α2 and F γ2 are veriﬁed. If a fuzzy choice function C veriﬁes F β(+) then the associated fuzzy preference relation R is transitive. We will prove that for a fuzzy choice function C, the path independence condition C(S ∪ T ) = C(C(S) ∪ C(T )) implies the ¯ quasi-transitivity of R. Throughout this section C will denote a fuzzy choice function on (X, B). We introduce condition F α2, a fuzzy form of the property α2 in [93].

Condition F α2 For any S ∈ B and for any x, y ∈ X, C(S)(x) ∧ S(y) ≤ C([x, y])(x).

The following proposition shows that condition F α2 is obtained by relaxing condition F α. Proposition 6.25. Condition F α implies condition F α2. Proof. By condition F α we have I([x, y], S) ∧ C(S)(x) ∧ [x, y](x) ≤ C([x, y])(x) for any S ∈ B and x, y ∈ X. Observing that I([x, y], S) ∧ C(S)(x) ∧ [x, y](x) = S(x) ∧ S(y) ∧ C(S)(x) = C(S)(x) ∧ S(y) one gets exactly condition F α2. Proposition 6.26. The following statements are equivalent: (a) Condition F α2; (b) For any S ∈ B and x ∈ X, C(S)(x) ≤

y∈X

¯ (c) R = R.

(S(y) → C([x, y])(x));

132

6 Fuzzy Revealed Preference and Consistency Conditions

Proof. (a) ⇔ (b) For any S ∈ B the following conditions are equivalent: • for any x, y ∈ X, C(S)(x) ∧ S(y) ≤ C([x, y])(x); • for any x, y ∈ X, C(S)(x) ≤ S(y) → C([x, y])(x); • for any x ∈ X, C(S)(x) ≤ (S(y) → C([x, y])(x)). y∈X

Thus (a), (b) are equivalent. ¯ y) = C([x, y])(x) and R ¯ ⊆ R (cf. Lemma 5.26 (i)) the (a) ⇔ (c) Since R(x, following properties are equivalent: • Condition F α2; ¯ y); • for any S ∈ B and x, y ∈ X, C(S)(x) ∧ S(y) ≤ R(x, ¯ y); • for any x, y ∈ X, (C(S)(x) ∧ S(y)) ≤ R(x, S∈B

¯ y); • for any x, y ∈ X, R(x, y) ≤ R(x, ¯ y). • for any x, y ∈ X, R(x, y) = R(x, In conclusion, condition F α2 and (c) are equivalent. Now let us consider the following property.

Condition F γ2 For any S ∈ B and x ∈ X, S(x) ∧

(S(y) → C([x, y])(x)) ≤ C(S)(x).

y∈X

Since for any S ∈ B and x ∈ X we have ¯ y)) ¯ (S(y) → R(x, G(S, R)(x) = S(x) ∧

y∈X

= S(x) ∧

(S(y) → C([x, y])(x, y))

y∈X

condition F γ2 can be expressed ¯ • For any S ∈ B and x ∈ X, G(S, R)(x) ≤ C(S)(x). Proposition 6.27. The following properties are equivalent: (a) C is G–normal; (b) C veriﬁes conditions F α2 and F γ2.

6.4 Other Consistency Conditions

133

¯ hence condition Proof. (a) ⇒ (b) Since C is G–normal, by Lemma 6.2 R = R F α2 is veriﬁed. For each S ∈ B and x, y ∈ X we have

S(x) ∧

(S(y) → C([x, y])(x)) = S(x) ∧

y∈X

¯ y)) (S(y) → R(x,

y∈X

= S(x) ∧

(S(y) → R(x, y))

y∈X

ˆ = C(S)(x) = C(S)(x). Then condition F γ2 is veriﬁed. ¯ (by Proposition 6.26) and F γ2 holds we get (b) ⇒ (a) Since R = R

ˆ C(S)(x) = S(x) ∧

(S(y) → R(x, y))

y∈X

= S(x) ∧

¯ y)) ≤ C(S)(x). (S(y) → R(x,

y∈X

ˆ ˆ The inequality C(S)(x) ≤ C(S)(x) always holds, then C(S)(x) = C(S)(x) for each S ∈ B and x ∈ X. Now we consider a fuzzy version of the property β(+) in [93].

Condition F β(+) For any S, T ∈ B and x, y ∈ X, I(S, T ) ∧ C(S)(x) ∧ S(y) ≤ C(T )(y) → C(T )(x).

Proposition 6.28. F β(+) implies F β. Proof. For any S, T ∈ B and x, y ∈ X

134

6 Fuzzy Revealed Preference and Consistency Conditions

I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ I(S, T ) ∧ C(S)(x) ∧ S(y) ≤ C(T )(y) → C(T )(x) and similarly, I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ C(T )(x) → C(T )(y). Then condition F β follows immediately. Proposition 6.29. If F β(+) holds then for any S, T ∈ B and x ∈ X we have C(S)(x) ∧ C(T )(x) ≤ C(S ∪ T )(x). Proof. Let y ∈ X such that C(S ∪ T )(y) = 1; then S(y) ∨ T (y) = 1 hence S(y) = 1 or T (y) = 1. Assume S(y) = 1. Then, by F β(+): C(S)(x) = I(S, S ∪ T ) ∧ C(S)(x) ∧ S(y) ≤ C(S ∪ T )(y) → C(S ∪ T )(x) = 1 → C(S ∪ T )(x) = C(S ∪ T )(x). Similarly, if T (y) = 1, then C(T )(x) ≤ C(S ∪ T )(x); therefore C(S)(x) ∧ C(T )(x) ≤ C(S ∪ T )(x). Proposition 6.30. If C veriﬁes F β(+) then R is transitive. Proof. Let S, T ∈ B and x, y, z ∈ X. Take w ∈ X such that C(S ∪ T )(w) = 1; since C(S ∪ T )(w) ≤ S(w) ∨ T (w) we have S(w) = 1 or T (w) = 1. Assume T (w) = 1. Exactly as in the proof of Proposition 6.29 one gets (a) C(T )(y) ≤ C(S ∪ T )(y). Applying F β(+) it follows that C(S)(x) ∧ S(y) = I(S, S ∪ T ) ∧ C(S)(x) ∧ S(y) ≤ C(S ∪ T )(y) → C(S ∪ T )(x) hence by Lemma 4.3 (1) (b) C(S)(x) ∧ S(y) ∧ C(S ∪ T )(y) ≤ C(S ∪ T )(x). In accordance with (a) and (b) the following inequalities hold:

6.4 Other Consistency Conditions

135

C(S)(x) ∧ S(y) ∧ C(T )(y) ∧ T (z) ≤ C(S)(x) ∧ S(y) ∧ C(S ∪ T )(y) ∧ T (z) ≤ C(S ∪ T )(x) ∧ T (z) ≤ C(S ∪ T )(x) ∧ (S ∪ T )(z) ≤ R(x, z). These inequalities are true for all S, T ∈ B, hence R(x, y) ∧ R(y, z) = [ (C(S)(x) ∧ S(y))] ∧ [ (C(T )(y) ∧ T (z))]= =

S∈B

T ∈B

(C(S)(x) ∧ S(y) ∧ C(T )(y) ∧ T (z)) ≤ R(x, z).

S,T ∈B

For the rest of the section let us assume that B = F(X) \ {∅}. We say that the fuzzy choice function C is path independent if for any S, T ∈ B the following equality holds (F P I) C(S ∪ T ) = C(C(S) ∪ C(T )). Condition F P I extends to fuzzy setting the path independence property P I for crisp choice functions (see [93], p. 68). We recall that the available fuzzy sets can be interpreted as vague criteria. F P I indicates that the fuzzy choice is independent of the manner such criterion can be decomposed into subcriteria with respect to which the choice takes place. ¯ is quasi-transitive. Proposition 6.31. If F P I holds then the fuzzy relation R Proof. Let x, y, z ∈ X. We must prove that P¯ (x, y) ∧ P¯ (y, z) ≤ P¯ (x, z), i.e. ¯ y) ∧ ¬R(y, ¯ x) ∧ R(y, ¯ z) ∧ ¬R(z, ¯ y) ≤ R(x, ¯ z) ∧ ¬R(z, ¯ x). (a) R(x, ¯ x) > 0 or R(z, ¯ y) > 0 then ¬R(y, ¯ x) = 0 or ¬R(z, ¯ y) = 0 so (a) If R(y, ¯ x) = R(z, ¯ y) = 0 then, since R ¯ is strongly is trivially veriﬁed. Assume R(y, ¯ y) = R(y, ¯ z) = 1. Thus total, R(x, ¯ y) = 1; C([x, y])(y) = R(y, ¯ x) = 0 C([x, y])(x) = R(x, hence C([x, y]) = [x]. Similarly, C([y, z]) = [y]. According to F P I we can write

136

6 Fuzzy Revealed Preference and Consistency Conditions

C([x, y, z]) = C([x, y] ∪ [y, z]) = C(C([x, y]) ∪ C([y, z])) = C([x] ∪ [y]) = C([x, y]) = [x]; C([x, y, z]) = C([x, y] ∪ [z]) = C(C([x, y]) ∪ C([z])) = C([x] ∪ [z]) = C([x, z]) hence C([x, z]) = [x]. This yields ¯ z) = C([x, z])(x) = [x](x) = 1 R(x, ¯ x) = C([x, z])(z) = [x](z) = 0. R(z, ¯ z) ∧ ¬R(z, ¯ x) = 1 and the inequality (a) holds. Then R(x, Proposition 6.32. If F α holds then C(S ∪ T ) ≤ C(C(S) ∪ C(T )) for all S, T ∈ B. Proof. Let S, T ∈ B and x ∈ X. First we prove that (a) C(S ∪ T )(x) ≤ C(S)(x) ∨ C(T )(x). Assume T (x) ≤ S(x) then C(S ∪ T )(x) ≤ S(x) ∨ T (x) = S(x). By F α we get C(S ∪ T )(x) = I(S, S ∪ T ) ∧ C(S ∪ T )(x) ∧ S(x) ≤ C(S)(x) ≤ C(S)(x) ∨ C(T )(x). If S(x) ≤ T (x) then (a) follows similarly. We apply again F α:

C(S ∪ T )(x) = C(S ∪ T )(x) ∧ (C(S) ∪ C(T ))(x) = I(C(S) ∪ C(T ), S ∪ T ) ∧ C(S ∪ T )(x) ∧ (C(S) ∪ C(T ))(x) ≤ C(C(S) ∪ C(T ))(x) Thus C(S ∪ T ) ≤ C(C(S) ∪ C(T )). Remark 6.33. In case of crisp choice functions the converse of Proposition 6.32 also holds (see [93], Proposition 17). An open problem is whether the converse of Proposition 6.32 holds true.

6.5 The Fuzzy Arrow Axiom

137

6.5 The Fuzzy Arrow Axiom In this section we study the Fuzzy Arrow Axiom, a generalization of the Arrow Axiom to fuzzy choice functions. The main result of this section establishes the equivalence of the Fuzzy Arrow Axiom and the full rationality of fuzzy choice functions. This result is connected with Theorem 6.40, resulting that W F CA, SF CA, Fuzzy Arrow Axiom and the full rationality are equivalent (under hypotheses H1 and H2). Throughout this section we shall assume that hypotheses H1, H2 are veriﬁed. Let (X, B) be a (crisp) choice space. Recall that a choice function C on (X, B) veriﬁes the Arrow Axiom (AA) if for any S1 , S2 ∈ B the following implication holds: [S1 ⊆ S2 ] ⇒ [S1 ∩ C(S2 ) = ∅] or [S1 ∩ C(S2 ) = C(S1 )]. In order to obtain the fuzzy form of AA it is convenient to write the above implication in the equivalent form: [S1 ⊆ S2 ] and [S1 ∩ C(S2 ) = ∅] ⇒ [S1 ∩ C(S2 ) = C(S1 )]. This way of expressing the Arrow Axiom leads to the following deﬁnition: Deﬁnition 6.34. Let C be a fuzzy choice function on a fuzzy choice space (X, B). We say that C veriﬁes the Fuzzy Arrow Axiom F AA if for any S1 , S2 ∈ B and x ∈ X we have I(S1 , S2 ) ∗ S1 (x) ∗ C(S2 )(x) ≤ E(S1 ∩ C(S2 ), C(S1 )). Intuitively, this axiom says: The maximum degree that, for some x, S1 and S2 , x is chosen from S2 , x belongs to S1 and S1 is included in S2 , is less than or equal to the degree that the set of alternatives chosen from S2 , and also in S1 is equal to the set of alternatives chosen from S1 . Remark 6.35. If S1 ⊆ S2 then the previous inequality becomes: S1 (x) ∗ C(S2 )(x) ≤ E(S1 ∩ C(S2 ), C(S1 )) hence it is clear that AA is a particular case of F AA.

138

6 Fuzzy Revealed Preference and Consistency Conditions

Remark 6.36. We observe that F AA is formulated in terms of the functions I(., .) and E(., .). From the previous remark we can see the reason why in formulating F AA we started from the second form of AA and not from the ﬁrst. The following results are true for the G¨ odel t–norm. ¯ is transiLemma 6.37. If the fuzzy choice function C veriﬁes F AA then R tive. Proof. Recall that C veriﬁes H1 and H2. Let x, y, z ∈ X and S = [x, y, z] ∈ B. ¯ y) ∧ R(y, ¯ z) ≤ R(x, ¯ z), i.e. We must prove that R(x, C([x, y])(x) ∧ C([y, z])(y) ≤ C([x, z])(x)

(6.1)

If x = y or y = z or x = z then (6.1) is trivial hence we may assume that x, y, z are distinct. First we shall prove that C([x, y])(x) ∧ C([y, z])(y) ≤ C(S)(x)

(6.2)

Suppose that C([x, y])(x) ∧ C([y, z])(y) ≤ C(S)(x). We shall prove that this condition implies C([x, y])(x) ∧ C([y, z])(y) ≤ C(S)(y). In order to obtain this last property we assume by absurdum that C([x, y])(x) ∧ C([y, z])(y) ≤ C(S)(y)

(6.3)

According to F AA we get C(S)(y) = I([x, y], S) ∧ [x, y](y) ∧ C(S)(y) ≤ E([x, y] ∩ C(S), C([x, y])) ≤ ([x, y](x) ∧ C(S)(x)) ↔ C([x, y])(x) = = C(S)(x) ↔ C([x, y])(x) ≤ C([x, y])(x) → C(S)(x). By Lemma 4.3 (1), C(S)(y) ∧ C([x, y])(x) ≤ C(S)(x), therefore by (6.3) C([x, y])(x) ∧ C([y, z])(y) ≤ C(S)(y) ∧ C([x, y])(x) ≤ C(S)(x). Contradiction, hence C([x, y])(x) ∧ C([y, z])(y) ≤ C(S)(y). From this last property one can infer in a similar way that C([x, y])(x) ∧ C([y, z])(y) ≤

6.5 The Fuzzy Arrow Axiom

139

C(S)(z). Thus C(S)(x) = 1, C(S)(y) = 1 and C(S)(z) = 1 contradicting that C(S) is a normal fuzzy subset of X (by H1). Then (6.2) was proved. Applying again F AA one gets C(S)(x) = I([x, z], S) ∧ [x, z](x) ∧ C(S)(x) ≤ E([x, z] ∩ C(S), C([x, z])) ≤ ([x, z](x) ∧ C(S)(x)) ↔ C([x, z])(x) = = C(S)(x) ↔ C([x, z])(x) ≤ C(S)(x) → C([x, z])(x). By Lemma 4.3 (1), C(S)(x) = C(S)(x)∧C(S)(x) ≤ C([x, z])(x). Therefore using (6.2) one obtains the inequality (6.1). Theorem 6.38. If C : B → F(X) is a fuzzy choice function then the following are equivalent: (1) C is full rational; (2) C veriﬁes F AA. Proof. (1) ⇒ (2). Assume C is full rational, i.e. C = G(., Q) for some reﬂexive, transitive and strongly total fuzzy preference relation Q. Let S1 , S2 ∈ B and x ∈ X. We must prove that: I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ E(S1 ∩ C(S2 ), C(S1 ))

(6.4)

For any S ∈ B and s, t ∈ X we have in accordance with C = G(., Q): [S(y) → Q(s, y)] ≤ S(t) → Q(s, t) C(S)(s) ≤ y∈X

hence, by Lemma 4.3 (1) C(S)(s) ∧ S(t) ≤ Q(s, t)

(6.5)

Let z, v ∈ X. By (6.5) we have I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ C(S1 )(z) ∧ S2 (v) ≤ ≤ [C(S1 )(z) ∧ S1 (x)] ∧ [C(S2 )(x) ∧ S2 (v)] ≤ Q(z, x) ∧ Q(x, v) ≤ Q(z, v) because Q is transitive. Hence, by Lemma 4.3 (1) I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ C(S1 )(z) ≤ S2 (v) → Q(z, v). This inequality holds for each v ∈ X hence

140

6 Fuzzy Revealed Preference and Consistency Conditions

I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ C(S1 )(z) ≤

[S2 (v) → Q(z, v)]

(6.6)

v∈X

We also have I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ C(S1 )(z) ≤ C(S1 )(z) ∧ I(S1 , S2 ) ≤ S1 (z) ∧ [S1 (z) → S2 (z)] = S1 (z) ∧ S2 (z) ≤ S2 (z). Then using (6.6)

I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ C(S1 )(z) ≤ S2 (z) ∧

[S2 (v) → Q(z, v)] =

v∈X

C(S2 )(z) In accordance with Lemma 4.3 (1) and Lemma 4.5 (2) we get for any z ∈ X: I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ C(S1 )(z) → C(S2 )(z) = = [C(S1 )(z) → S1 (z)] ∧ [C(S1 )(z) → C(S2 )(z)] = C(S1 )(z) → (S1 (z) ∧ C(S2 )(z)). We have proved the inequality

I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ C(S1 )(z) → (S1 (z) ∧ C(S2 )(z))

(6.7)

Let y ∈ X. Using I(S1 , S2 ) ≤ S1 (y) → S2 (y) and (6.5) we obtain I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ S1 (z) ∧ C(S2 )(z) ∧ S1 (y) ≤ ≤ S1 (y) ∧ [S1 (y) → S2 (y)] ∧ C(S2 )(x) ∧ C(S2 )(z) = = S1 (y) ∧ S2 (y) ∧ C(S2 )(x) ∧ C(S2 )(z) ≤ [C(S2 )(z) ∧ S2 (x)] ∧ [C(S2 )(x) ∧ S2 (y)] ≤ ≤ Q(z, x) ∧ Q(x, y) ≤ Q(z, y). By Lemma 4.3 (1) we get for any y ∈ X: I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ S1 (z) ∧ C(S2 )(z) ≤ S1 (y) → Q(z, y) hence I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ S1 (z) ∧ C(S2 )(z) ≤

y∈X

[S1 (y) → Q(z, y)].

6.5 The Fuzzy Arrow Axiom

Then I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ S1 (z) ∧ C(S2 )(z) ≤ S1 (z) ∧

141

[S1 (y) →

y∈X

Q(z, y)] = C(S1 )(z). According to Lemma 4.3 (1) we get for any z ∈ X

I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ (S1 (z) ∧ C(S2 )(z)) → C(S1 )(z)

(6.8)

From (6.7) and (6.8) it follows that I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ (S1 (z) ∧ C(S2 )(z)) ↔ C(S1 )(z) i.e. the inequality (6.4). ¯ (2) ⇒ (1) Assume C veriﬁes F AA. It suﬃces to prove that C = G(., R). Let S ∈ B and x ∈ X. We must prove that C(S)(x) = S(x) ∧

¯ z)] [S(z) → R(x,

(6.9)

z∈X

Let z ∈ X. According to I([x, z], S) = S(x) ∧ S(z) and F AA we have C(S)(x)∧S(z) = S(x)∧S(z)∧C(S)(x) = I([x, z], S)∧[x, z](x)∧C(S)(x) ≤ ≤ E([x, z] ∩ C(S), C([x, z])) ≤ [x, z](x) ∧ C(S)(x) → C([x, z])(x) = ¯ z). C(S)(x) → R(x, By Lemma 4.3 (1) ¯ z) C(S)(x) ∧ S(z) = C(S)(x) ∧ S(z) ∧ C(S)(x) ≤ R(x, ¯ z) for any z ∈ X. We conclude that hence C(S)(x) ≤ S(z) → R(x, ¯ z)]. C(S)(x) ≤ S(x) ∧ [S(z) → R(x, z∈X

In order to prove that the converse inequality we consider y ∈ X such that C(S)(y) = 1 (because C(S) is a normal fuzzy set), then S(y) = 1. Thus ¯ ¯ z)] ≤ G(S, R)(x) = S(x) ∧ [S(z) → R(x, z∈X

¯ y) = 1 → R(x, ¯ y) = R(x, ¯ y) = C([x, y])(x). ≤ S(y) → R(x, On the other hand, by I([x, y], S) = S(x) ∧ S(y) and F AA ¯ G(S, R)(x) ≤ S(x) = S(x) ∧ S(y) ∧ C(S)(y) = = I([x, y], S) ∧ [x, y](y) ∧ C(S)(y) ≤ E([x, y] ∩ C(S), C([x, y])) ≤

142

6 Fuzzy Revealed Preference and Consistency Conditions

≤ ([x, y](x) ∧ C(S)(x)) ↔ C([x, y])(x) ≤ C([x, y])(x) → C(S)(x). ¯ By Lemmas 5.26 and 6.37 it follows We have proved that C = G(., R). ¯ is reﬂexive, transitive and strongly total, hence C is full rational. that R

Now we shall connect Theorem 6.7 and Theorem 6.38. Lemma 6.39. Let C be a fuzzy choice function on (X, B). If C = G(S, Q) and R is the fuzzy revealed preference relation associated with C then R ⊆ Q. Proof. Let S ∈ B and x, y ∈ X. Then C(S)(x) = S(x) ∧ [S(u) → Q(x, u)] ≤ S(y) → Q(x, y). u∈X

By Lemma 4.3 (1), C(S)(x) ∧ S(y) ≤ Q(x, y) for each S ∈ B, therefore R(x, y) = (C(S)(x) ∧ S(y)) ≤ Q(x, y). S∈B

Theorem 6.40. If C : B → F(X) is a fuzzy choice function then the following are equivalent: (1) C is full rational; (2) C veriﬁes F AA; (3) C is G–normal and R is reﬂexive, transitive and strongly total; (4) C veriﬁes W F CA; (5) C veriﬁes SF CA. Proof. (1) ⇒ (4) Assume C = G(., Q) for some fuzzy relation Q on X which is reﬂexive, transitive and strongly total. Let S ∈ B and x, y ∈ X. We shall establish the inequality: C(S)(y) ∧ S(x) ∧ Q(x, y) ≤ C(S)(x) (6.10) Since C(S)(x) = S(x) ∧ [S(u) → Q(x, u)] it suﬃces to prove that u∈X

C(S)(y) ∧ S(x) ∧ Q(x, y) ≤ S(u) → Q(x, u)

(6.11)

6.5 The Fuzzy Arrow Axiom

143

for each u ∈ X. By Lemma 4.3 (1) this inequality is equivalent to S(u) ∧ C(S)(y) ∧ S(x) ∧ Q(x, y) ≤ Q(x, u) By Lemma 4.3 (2): S(u) ∧ C(S)(y) = S(u) ∧ S(y) ∧

(6.12)

[S(z) → Q(y, z)] ≤

z∈X

≤ S(u) ∧ [S(u) → Q(y, u)] = S(u) ∧ Q(y, u) ≤ Q(y, u). Hence S(u)∧C(S)(y)∧S(x)∧Q(x, y) ≤ Q(x, y)∧Q(y, u) ≤ Q(x, u) because Q is transitive. Then (6.12) is proved. By Lemma 6.39 and (6.12) we obtain C(S)(y) ∧ S(x) ∧ R(x, y) ≤ C(S)(y) ∧ S(x) ∧ Q(x, y) ≤ C(S)(x). Therefore C veriﬁes W F CA. (3) ⇒ (1) Obvious. The other equivalences follow by Theorems 6.7 and 6.38. By establishing the equivalence between the full rationality, (FAA) and the fuzzy congruence axioms W F CA, SF CA, the above theorem represents the maximal form in which the results from crisp revealed preference theory (on the line Uzawa–Arrow–Sen) can be extended to fuzzy choice functions. On the other hand, the above conditions are not equivalent with the axioms of revealed preference W AF RP , SAF RP (see Example 6.8). We introduce now two more axioms:

Fuzzy Chernoﬀ Axiom (F CA) For any S1 , S2 ∈ B and x ∈ X, I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ I(S1 ∩ C(S2 ), C(S1 ))

Fuzzy Dual Chernoﬀ Axiom (F DCA) For any S1 , S2 ∈ B and x ∈ X,

144

6 Fuzzy Revealed Preference and Consistency Conditions

I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ I(C(S1 ), S1 ∩ C(S2 )).

It is obvious that the Fuzzy Arrow Axiom F AA is equivalent to the conjunction of F CA and F DCA. F CA (respectively F DCA) is a fuzzy form of the Chernoﬀ Axiom CA (respectively Dual Chernoﬀ Axiom DCA) (see [103], pp. 31 and 41). Proposition 6.41. For any fuzzy choice function C : B → F(X) the following conditions are equivalent: (1) C veriﬁes F CA; (2) C veriﬁes consistency condition F α. Proof. (1) ⇒ (2). By F CA: I(S1 , S2 )∧S1 (x)∧C(S2 )(x) ≤ I(S1 ∩C(S2 ), C(S1 )) ≤ (S1 (x)∧C(S2 )(x)) → C(S1 )(x). Using Lemma 4.3 (1) we obtain I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ C(S1 )(x). (2) ⇒ (1). Let S1 , S2 ∈ B and x, y ∈ X. Then, by (2): I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ S1 (y) ∧ C(S2 )(y) ≤ ≤ I(S1 , S2 ) ∧ S1 (y) ∧ C(S2 )(y) ≤ C(S1 )(y) hence, using Lemma 4.3 (1) we get I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ (S1 (y) ∧ C(S2 )(y)) → C(S1 )(y). Since this last inequality holds for any y ∈ X it follows that I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ [(S1 (y) ∧ C(S2 )(y)) → C(S1 )(y)] = I(S1 ∩ C(S2 ), C(S1 )).

y∈X

Corollary 6.42. Any full rational fuzzy choice function C veriﬁes condition F α.

7 General Results

In the previous chapter we studied under hypotheses (H1) and (H2) revealed preference axioms W AF RP , SAF RP , congruence axioms W F CA, SF CA and some consistency conditions. This line of enquiry follows Uzawa–Arrow– Sen theory that starts from the assumption that the domain of the choice function contains the ﬁnite sets of alternatives. Richter, Hansson, Suzumura and their followers investigated the choice functions’rationality and revealed preference and congruence conditions without this hypothesis. The purpose of this chapter is a fuzzy approach of revealed preference and congruence axioms for fuzzy choice functions in the general case, following Richter–Hansson–Suzumura theory; we will work ignoring hypotheses (H1) and (H2). As we have seen in Section 5.2, axioms W ARP and SARP written in terms of C-connected sequences lead to new revealed preference axioms W AF RP ◦ , SAF RP ◦ . We also introduced axiom HAF RP . In Section 7.1 we prove two main theorems: 1. The axioms W F CA and W AF RP ◦ are equivalent. 2. The axioms SF CA and HAF RP are equivalent.

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 145–167 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

146

7 General Results

These two theorems extend two results of Hansson [61] and Suzumura [100] from the theory of choice functions: the axioms W CA and W ARP (respectively SCA and HARP ) are equivalent. Another proposition asserts that W F CA ⇒ W AF RP and SF CA ⇒ SAF RP . According to the ﬁrst theorem we have W AF RP ◦ ⇒ W AF RP . These results are summarized in a diagram that illustrates the hierarchy of axioms and other conditions of rationality. Section 7.2 deals with a particular case of fuzzy choice functions. For them we establish the equivalence between W AF RP ◦ , G-normality, M -normality and other two properties expressed by algebraic identities. Particularly the implication W AF RP ⇒ W AF RP ◦ fails hence W AF RP and W AF RP ◦ are not equivalent. A classical Richter theorem [84] asserts that a classical choice function veriﬁes SCA if and only if it is rationalized by a reﬂexive, transitive and total preference relation. Section 7.3 investigates to what extent Richter’s theorem can be extended to fuzzy choice functions. We work here in a fuzzy set theory based on an arbitrary continuous t-norm ∗. We use the notion of totally rational fuzzy choice function, i.e. a G-rational fuzzy choice function which is rationalized by a reﬂexive, ∗-transitive and total fuzzy preference relation. A fuzzy choice function is said to be congruous if it veriﬁes SF CA. First we prove that every totally rational fuzzy choice function is congruous. To prove the converse implication is an open question. We need to deﬁne the notion of semirational fuzzy choice function in order to construct the proof on the fuzzy level. We ﬁnd a surprising result: any fuzzy choice function is semirational; to prove that, we essentially apply our fuzzy version of Szpilrajn theorem (Theorem 4.36).

7.1 The Hierarchy of Axioms

147

7.1 The Hierarchy of Axioms In this section we study arbitrary fuzzy choice functions with respect to the connections between diﬀerent axioms of revealed preference and congruence. The results will be summarized in the ﬁnal diagram. Proposition 7.1. Let ∗ be a continuous t–norm. Then the following hold: (i) W F CA ⇒ W AF RP ; (ii) SF CA ⇒ SAF RP . Proof. (i) Let C : B → F(X) be a fuzzy choice function fulﬁlling W F CA. For any x, y ∈ X we have

P˜ (x, y) ∗ R(y, x) = [

(C(S)(x) ∗ S(y) ∗ ¬C(S)(y)] ∗ R(y, x)

S∈B

=

[R(y, x) ∗ C(S)(x) ∗ S(y) ∗ ¬C(S)(y)]

S∈B

≤

(C(S)(y) ∗ ¬C(S)(y)) = 0

S∈B

because R(y, x) ∗ C(S)(x) ∗ S(y) ≤ C(S)(y), by W F CA. Thus P˜ (x, y) ∗ R(y, x) = 0, hence P˜ (x, y) ≤ ¬R(y, x). (ii) Suppose C veriﬁes SF CA. For any x, y ∈ X: T (P˜ )(x, y) ∗ R(y, x) = ∞ = [P˜ (x, y) ∨

(P˜ (x, t1 ) ∗ . . . ∗ P˜ (tn , y))] ∗ R(y, x) =

n=1 t1 ,...,tn ∈X ∞

= [P˜ (x, y) ∗ R(y, x)] ∨

[P˜ (x, t1 ) ∗ . . . ∗ P˜ (tn , y) ∗ R(y, x)].

n=1 t1 ,...,tn ∈X

But SF CA implies W F CA, hence by (i), P˜ (x, y) ∗ R(y, x) = 0. For any n ≥ 1 and t1 , . . . , tn ∈ X we have, according to Lemma 5.25: P˜ (x, t1 ) ∗ . . . ∗ P˜ (tn , y) ∗ R(y, x) ≤ R(x, t1 ) ∗ . . . ∗ R(tn−1 , tn ) ∗ P˜ (tn , y) ∗ R(y, x) = = R(y, x) ∗ R(x, t1 ) ∗ . . . ∗ R(tn−1 , tn ) ∗ P˜ (tn , y) ≤

148

7 General Results

≤ T (R)(y, tn ) ∗ P˜ (tn , y) = = T (R)(y, tn ) ∗ [C(S)(tn ) ∗ S(y) ∗ ¬C(S)(y)] =

S∈B

[T (R)(y, tn ) ∗ C(S)(tn ) ∗ S(y) ∗ ¬C(S)(y)] ≤

S∈B

≤

(C(S)(y) ∗ ¬C(S)(y)) = 0

S∈B

because T (R)(y, tn ) ∗ C(S)(tn ) ∗ S(y) ≤ C(S)(y) (by SF CA). Thus P˜ (x, t1 ) ∗ . . . ∗ P˜ (tn , y) ∗ R(y, x) = 0 for any n ≥ 1 and t1 , . . . , tn ∈ X, therefore T (P˜ )(x, y) ∗ R(y, x) = 0. By Lemma 4.3 (1), T (P˜ )(x, y) ≤ ¬R(y, x).

Remark 7.2. Let C be a fuzzy choice function, S ∈ B, x, y ∈ X and ∗ a continuous t–norm. By the deﬁnition of R R(x, y) ∗ S(x) ∗ C(S)(y) = [ (C(T )(x) ∗ T (y))] ∗ S(x) ∗ C(S)(y) =

T ∈B

[S(x) ∗ T (y) ∗ C(T )(x) ∗ C(S)(y)].

T ∈B

Then W F CA is equivalent with the following statement • For any S, T ∈ B and x, y ∈ X S(x) ∗ T (y) ∗ C(T )(x) ∗ C(S)(y) ≤ C(S)(x). For the rest of the section assume that ∗ is the G¨ odel t–norm ∧. Theorem 7.3. For a fuzzy choice function C : B → F(X) the following are equivalent: (i) C veriﬁes W F CA; (ii) R ⊆ R∗ ; (iii) C veriﬁes W AF RP ◦ . Proof. (i) ⇔ (ii). The following assertions are equivalent: • R ⊆ R∗ ; • For any x, y ∈ X: (C(S)(x) ∧ S(y)) ≤ [(T (x) ∧ C(T )(y)) → C(T )(x)]; S∈B

T ∈B

7.1 The Hierarchy of Axioms

149

• For any x, y ∈ X and S, T ∈ B : C(S)(x) ∧ S(y) ≤ (T (x) ∧ C(T )(y)) → C(T )(x). • For any x, y ∈ X and S, T ∈ B: C(S)(x) ∧ S(y) ∧ T (x) ∧ C(T )(y) ≤ C(T )(x). In accordance with Remark 7.2, (i) and (ii) are equivalent. (iii) ⇒ (i) Assume that C veriﬁes W AF RP ◦ . Let x, y ∈ X and S, T ∈ B. By W AF RP ◦ one gets S(x) ∧ T (y) ∧ C(T )(x) ∧ C(S)(y) ≤ E(S ∩ C(T ), T ∩ C(S)) = = [(S(u) ∧ C(T )(u)) ↔ (T (u) ∧ C(S)(u))] ≤ u∈X

≤ (S(x) ∧ C(T )(x)) ↔ (T (x) ∧ C(S)(x)) ≤ ≤ (S(x) ∧ C(T )(x)) → (T (x) ∧ C(S)(x)) = = [(S(x) ∧ C(T )(x)) → T (x)] ∧ [(S(x) ∧ C(T )(x)) → C(S)(x)] = = (S(x) ∧ C(T )(x)) → C(S)(x) because (S(x) ∧ C(T )(x)) → T (x) = 1 (by Lemma 4.3 (4)). It follows that S(x) ∧ T (y) ∧ C(T )(x) ∧ C(S)(y) = = [S(x) ∧ T (y) ∧ C(T )(x) ∧ C(S)(y)] ∧ (S(x) ∧ C(T )(x)) ≤ C(S)(x) in accordance with Lemma 4.3 (1). According to Remark 7.2, C veriﬁes W F CA. (i) ⇒ (iii) Assume C fulﬁlls W F CA. By Remark 7.2, for any S, T ∈ B and x, y, u ∈ X we have S(x)∧T (y)∧C(S)(y)∧C(T )(x)∧S(u)∧C(T )(u) ≤ S(u)∧T (y)∧C(T )(u)∧ C(S)(y) ≤ C(S)(u). Thus, by Lemma 4.3 (1): S(x) ∧ T (y) ∧ C(S)(y) ∧ C(T )(x) ≤ (S(u) ∧ C(T )(u)) → C(S)(u) = = (S(u) ∧ C(T )(u)) → (T (u) ∧ C(S)(u)) and similarly, S(x) ∧ T (y) ∧ C(S)(y) ∧ C(T )(x) ≤ (T (u) ∧ C(S)(u)) → (S(u) ∧ C(T )(u)) The last two inequalities give S(x) ∧ T (y) ∧ C(S)(y) ∧ C(T )(x) ≤ (S(u) ∧ C(T )(u)) ↔ (T (u) ∧ C(S)(u)). This inequality is true for each u ∈ X, hence

150

7 General Results

S(x) ∧ T (y) ∧ C(S)(y) ∧ C(T )(x) ≤ E(S ∩ C(T ), T ∩ C(S)) so C veriﬁes the axiom W AF RP ◦ . Corollary 7.4. W AF RP ◦ ⇒ W AF RP . Proof. By Proposition 7.1 and Theorem 7.3. From Theorem 7.3 one obtains as a particular case the result from [100] that says that the axioms W ARP and W CA are equivalent. The axiom W ARP has two fuzzy versions: W AF RP and W AF RP ◦ . According to Corollary 7.4, W AF RP ◦ ⇒ W AF RP . In the next section we obtain that the converse implication W AF RP ⇒ W AF RP ◦ is not true (see Remark 7.8). Theorem 7.5. For any fuzzy choice function C : B → F(X) the following are equivalent: (i) C veriﬁes HAF RP ; (ii) C veriﬁes SF CA. Proof. (i) ⇒ (ii) Assume C veriﬁes HAF RP . For any S ∈ B and x, y ∈ X we shall prove that (a) T (R)(x, y) ∧ S(x) ∧ C(S)(y) ≤ C(S)(x) The left hand-side in (a) can be written T (R)(x, y) ∧ S(x) ∧ C(S)(y) = ∞ = [R(x, y) ∨ (R(x, z1 ) ∧ . . . ∧ R(zn , y))] ∧ S(x) ∧ C(S)(y) = n=1 z1 ,...,zn ∈X

= [R(x, y)∧S(x)∧C(S)(y)]∨

∞

[R(x, z1 )∧. . .∧R(zn , y)∧S(x)∧

n=1 z1 ,...,zn ∈X

C(S)(y)]. Then proving (a) is equivalent to establishing the following two inequalities: (b) R(x, y) ∧ S(x) ∧ C(S)(y) ≤ C(S)(x); (c) For any integer n ≥ 1 and z1 , . . . , zn ∈ X: R(x, z1 ) ∧ R(z1 , z2 ) ∧ . . . ∧ R(zn , y) ∧ S(x) ∧ C(S)(y) ≤ C(S)(x).

7.1 The Hierarchy of Axioms

151

The axiom HAF RP implies W AF RP ◦ then, by Theorem 7.3, C veriﬁes W F CA. Then (b) is veriﬁed. According to the deﬁnition of R R(x, z1 ) ∧ R(z1 , z2 ) ∧ . . . ∧ R(zn , y) ∧ S(x) ∧ C(S)(y) = =[ (C(S1 )(x)∧S1 (z1 ))]∧. . .∧[ (C(Sn+1 )(zn )∧Sn+1 (y))]∧S(x)∧ S1 ∈B

C(S)(y) = =

Sn+1 ∈B

T (S1 , . . . , Sn+1 )

S1 ,...,Sn+1

where T (S1 , . . . , Sn+1 ) = C(S1 )(x) ∧ S1 (z1 ) ∧ C(S2 )(z1 ) ∧ . . . ∧C(Sn+1 )(zn ) ∧ Sn+1 (y) ∧ S(x) ∧ C(S)(y) = = [S(x) ∧ C(S1 )(x)] ∧ [S1 (z1 ) ∧ C(S2 )(z1 )] ∧ . . . ∧ [Sn (zn ) ∧ C(Sn+1 )(zn )]∧ n+1 ∧[Sn+1 (y) ∧ C(S)(y)] = [Sk (zk ) ∧ C(Sk+1 )(zk )] k=0

with S0 = S, z0 = x and zn+1 = y. According to HAF RP : T (S1 , . . . , Sn+1 ) ≤ E(S ∩ C(S1 ), S1 ∩ C(S)) ≤ ≤ (S(x) ∧ C(S1 )(x)) ↔ (S1 (x) ∧ C(S)(x)) ≤ ≤ (S(x) ∧ C(S1 )(x)) → (S1 (x) ∧ C(S)(x)) = = (S(x) ∧ C(S1 )(x)) → C(S)(x). We also have T (S1 , . . . , Sn+1 ) ≤ S(x) ∧ C(S1 )(x) hence by Lemma 4.3 (2) T (S1 , . . . , Sn+1 ) ≤ (S(x) ∧ C(S1 )(x1 )) ∧ [(S(x) ∧ C(S1 )(x)) → C(S)(x)] = = S(x) ∧ C(S1 )(x) ∧ C(S)(x) ≤ C(S)(x). This inequality holds for all S1 , . . . , Sn+1 ∈ B hence R(x, z1 ) ∧ R(z1 , z2 ) ∧ . . . ∧ R(zn , y) ∧ S(x) ∧ C(S)(y) = = T (S1 , . . . , Sn+1 ) ≤ C(S)(x). S1 ,...,Sn+1

Condition (c) was proved so the proof of (i) ⇒ (ii) is ﬁnished. (ii) Assume C fulﬁlls SF CA. Let S1 , . . . , Sn ∈ B and z1 , . . . , zn ∈ X. We shall prove that the following inequality holds: n (d) [Sk (zk ) ∧ C(Sk+1 )(zk )] ≤ E(S1 ∩ C(S2 ), S2 ∩ C(S1 )) k=1

where Sn+1 = S1 . We remark that

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7 General Results

(e)

n

[Sk (zk ) ∧ C(Sk+1 )(zk )] =

k=1

= S1 (z1 )∧[C(S2 )(z1 )∧S2 (z2 )]∧. . .∧[C(Sn )(zn−1 )∧Sn (zn )]∧C(S1 )(zn ) ≤ ≤ S1 (z1 ) ∧ R(z1 , z2 ) ∧ . . . ∧ R(zn−1 , zn ) ∧ C(S1 )(zn ) ≤ ≤ S1 (z1 ) ∧ T (R)(z1 , zn ) ∧ C(S1 )(zn ) because C(S2 )(z1 ) ∧ S2 (z2 ) ≤ R(z1 , z2 ), . . . , C(Sn )(zn−1 ) ∧ Sn (zn ) ≤ R(zn−1 , zn ). Let z ∈ X. We shall establish the inequality n (f) [Sk (zk ) ∧ C(Sk+1 )(zk )] ≤ (S1 (z) ∧ C(S2 )(z)) → C(S1 )(z) k=1

= [S1 (z) ∧ C(S2 )(z)] → [S2 (z) ∧ C(S1 )(z))].

By Lemma 4.3 (1), (f) is equivalent to n (g) [Sk (zk ) ∧ C(Sk+1 )(zk )] ∧ S1 (z) ∧ C(S2 )(z) ≤ C(S1 )(z). k=1

Let us denote by the left hand side member of the inequality (g). It is obvious that ≤ C(S2 )(z1 ) ∧ C(S2 )(z) ≤ S2 (z1 ) ∧ C(S2 )(z) ≤ R(z, z1 ). By (e) we have ≤ T (R)(z1 , zn ) therefore ≤ R(z, z1 ) ∧ T (R)(z1 , zn ) ≤ T (R)(z, z1 ) ∧ T (R)(z1 , zn ) ≤ T (R)(z, zn ) because T (R) is transitive. According to (e), ≤ C(S1 )(zn ). We also have ≤ S1 (z), therefore ≤ T (R)(z, zn ) ∧ S1 (z) ∧ C(S1 )(zn ) ≤ C(S1 )(z), the last inequality following by SF CA. Then (g) was proved. In accordance with (e) we can write n [Sk (zk ) ∧ C(Sk+1 )(zk )] ∧ S2 (z) ∧ C(S1 )(z) ≤ k=1

≤ S1 (z1 ) ∧ T (R)(z1 , zn ) ∧ C(S1 )(zn ) ∧ S2 (z) ∧ C(S1 )(z) ≤ ≤ S2 (z) ∧ (C(S1 )(z) ∧ S1 (z1 )) ≤ R(z, z1 ) ∧ S2 (z). n [Sk (zk ) ∧ C(Sk+1 )(zk )] ≤ C(S2 )(z1 ) hence by using It is obvious that k=1

W F CA n [Sk (zk ) ∧ C(Sk+1 )(zk )] ∧ S2 (z) ∧ C(S1 )(z) ≤ k=1

≤ R(z, z1 ) ∧ S2 (z) ∧ C(S2 )(z1 ) ≤ C(S2 )(z).

By this last inequality one infers

7.1 The Hierarchy of Axioms n

(h)

153

[Sk (zk ) ∧ C(Sk+1 )(zk )] ≤ (S2 (z) ∧ C(S1 )(z)) → C(S2 )(z) =

k=1

= (S2 (z) ∧ C(S1 )(z)) → (S1 (z) ∧ C(S2 )(z)).

From (f) and (h) we get for each z ∈ X: n [Sk (zk ) ∧ C(Sk+1 )(zk )] ≤ (S1 (z) ∧ C(S2 )(z)) ↔ (S2 (z) ∧ C(S1 )(z)). k=1

Then (d) follows immediately. In a similar way we can show that n [Sk (zk ) ∧ C(Sk+1 )(zk )] ≤ E(S2 ∩ C(S3 ), S3 ∩ C(S2 )) k=1

... n

[Sk (zk ) ∧ C(Sk+1 )(zk )] ≤ E(Sn−1 ∩ C(Sn ), Sn ∩ C(Sn−1 ))

k=1

then HAF RP follows. The proof is ﬁnished. Theorem 7.5 is an extension to fuzzy choice functions of a result from [100] that establishes the equivalence between the axioms HARP and SCA. The results of this section can be summarized in Figure 7.1.

HAFRP

SAFRP 0

WAFRP 0

R* includes R *

SFCA

WFCA

SAFRP

WAFRP

Fig. 7.1. Hierarchy of axioms

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7 General Results

7.2 A Particular Class of Fuzzy Choice Functions In this section a particular class of fuzzy choice function is studied with emphasis on the rationality conditions discussed in the previous section. Let X = {x, y} and B = {A, B} where A, B ∈ F(X) are given by A = αχ{x} + χ{y} ; B = χ{x} + βχ{y} (0 ≤ α, β ≤ 1). Consider the function C : B → F(X) deﬁned by: (7.2.1) C(A) = γχ{x} +χ{y} ; C(B) = χ{x} +δχ{y} (0 ≤ γ ≤ α, 0 ≤ δ ≤ β). One notices that C is a fuzzy choice function that depends on the parameters α, β, γ, δ. In this section we shall investigate the way C veriﬁes or not some properties (W AF RP , W AF RP ◦ , G–normality, M –normality) with respect to parameters α, β, γ, δ. From this analysis we obtain that W AF RP does not imply W AF RP ◦ , therefore these two axioms are not equivalent. At the same time we conclude that the axioms SAF RP and HAF RP are not equivalent. First we compute the fuzzy preference relation R: R(x, x) = R(y, y) = 1; R(x, y) = (C(A)(x) ∧ A(y)) ∨ (C(B)(x) ∧ B(y)) = β ∨ γ; R(y, x) = (C(A)(y) ∧ A(x)) ∨ (C(B)(y) ∧ B(x)) = α ∨ δ. Thus

 1

β∨γ



. α∨δ 1 We also compute the fuzzy relation R∗ : (7.2.2) R = 

R∗ (x, x) = R∗ (y, y) = 1; R∗ (x, y) = [(A(x) ∧ C(A)(y)) → C(A)(x)] ∧ [(B(x) ∧ C(B)(y)) → C(B)(x)] = = α → γ; R∗ (y, x) = [(A(y) ∧ C(A)(x)) → C(A)(y)] ∧ [(B(y) ∧ C(B)(x)) → C(B)(y)] = = β → δ. Hence

7.2 A Particular Class of Fuzzy Choice Functions

 (7.2.3) R∗ = 

1

α→γ

β→δ

1

155

 .

Proposition 7.6. The fuzzy choice function C deﬁned by (7.2.1) veriﬁes W AF RP ◦ if and only if β ∧ (α ∨ δ) = δ and α ∧ (β ∨ γ) = γ. Proof. The following conditions are equivalent: • R ⊆ R∗ ; • α ∨ δ ≤ β → δ and β ∨ γ ≤ α → γ; • β ∧ (α ∨ δ) ≤ δ and α ∧ (β ∨ γ) ≤ γ (by Lemma 4.3 (1)); • β ∧ (α ∨ δ) = δ and α ∧ (β ∨ γ) = γ (since γ ≤ α, δ ≤ β). According to Theorem 7.3 the desired equivalence follows. Let us compute the fuzzy relation P˜ : P˜ (x, x) = P˜ (y, y) = 0; P˜ (x, y) = [C(A)(x) ∧ A(y) ∧ ¬C(A)(y)] ∨ [C(B)(x) ∧ B(y) ∧ ¬C(B)(y)] = = β ∧ ¬δ; P˜ (y, x) = [C(A)(y) ∧ A(x) ∧ ¬C(A)(x)] ∨ [C(B)(y) ∧ B(x) ∧ ¬C(B)(x)] = = α ∧ ¬γ. Therefore



(7.2.4) P˜ = 

0

β ∧ ¬δ

α ∧ ¬γ

0

 .

Proposition 7.7. The fuzzy choice function C veriﬁes W AF RP if and only if α ∧ β ∧ ¬δ = α ∧ β ∧ ¬γ = 0. Proof. The following assertions are equivalent: • C veriﬁes W AF RP ; • P˜ (x, y) ≤ ¬R(y, x) and P˜ (y, x) ≤ ¬R(x, y); • β ∧ ¬δ ≤ ¬(α ∨ δ) and α ∧ ¬γ ≤ ¬(β ∨ γ); • α ∧ β ∧ ¬δ = 0 and α ∧ β ∧ ¬γ = 0.

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7 General Results

Remark 7.8. Let us take α = 1, γ =

1 2, β

= δ =

2 3.

Since ¬δ = ¬γ = 0,

by Proposition 7.7 one can infer that C veriﬁes W AF RP . We remark that α ∧ (β ∨ γ) = 1 ∧ ( 23 ∨ 12 ) =

2 3

= γ, hence, by Proposition 7.6, the axiom

W AF RP ◦ does not hold. In conclusion the implication W AF RP ⇒ W AF RP ◦ fails and W AF RP , W AF RP ◦ are not equivalent conditions. Remark 7.9. We observe that   α ∧ β ∧ ¬γ ∧ ¬δ β ∧ ¬δ . (7.2.5) T (P˜ ) =  α ∧ ¬γ α ∧ β ∧ ¬γ ∧ ¬δ Since α ∧ β ∧ ¬δ = 0 and α ∧ β ∧ ¬γ = 0 implies α ∧ β ∧ ¬γ ∧ ¬δ = 0, T (P˜ ) = P˜ , therefore by Proposition 7.7 we also obtain the equivalence C veriﬁes SAF RP if and only if α ∧ β ∧ ¬δ = α ∧ β ∧ ¬γ = 0. In our case B has two members A, B then the axioms W AF RP ◦ , SAF RP ◦ and HAF RP are equivalent. Then using again the argument in Remark 7.8 the implication SAF RP ⇒ HAF RP does not hold. Proposition 7.10. For the fuzzy choice function C deﬁned by (7.2.1) the following are equivalent: (i) C is G-normal; (ii) C is M -normal; (iii) α ∧ β = γ ∧ δ. Proof. We compute the values of G∗ (A) and G∗ (B): G∗ (A)(x) = A(x) ∧ [A(x) → R(x, x)] ∧ [A(y) → R(x, y)] = α ∧ (β ∨ γ) G∗ (A)(y) = A(y) ∧ [A(x) → R(y, x)] ∧ [A(y) → R(y, y)] = α → (α ∨ δ) = 1 and similarly, G∗ (B)(x) = 1 and G∗ (B)(y) = β ∧ (α ∨ δ). Applying Lemma 4.3 we also compute the values of M ∗ (A) and M ∗ (B): M ∗ (A)(x) = A(x) ∧ [(A(x) ∧ R(x, x)) → R(x, x)] ∧ [(A(y) ∧ R(y, x)) → R(x, y)] =

7.2 A Particular Class of Fuzzy Choice Functions

157

= α ∧ [(α ∨ δ) → (β ∨ γ)] = = α ∧ [α → (β ∨ γ)] ∧ [δ → (β ∨ γ)] = α ∧ (β ∨ γ) ∧ (δ → (β ∨ γ)) = α ∧ (β ∨ γ). ∗

M (A)(y) = A(y) ∧ [(A(x) ∧ R(x, y)) → R(y, x)] ∧ [(A(y) ∧ R(y, y)) → R(y, y)] = (α ∧ (β ∨ γ)) → (α ∨ δ) = 1 and similarly M ∗ (B)(x) = 1 and M ∗ (B)(y) = β ∧ (α ∨ δ). It follows that G∗ (A) = M ∗ (A) and G∗ (B) = M ∗ (B) and each of (i), (ii) are equivalent to • α ∧ (β ∨ γ) = γ and β ∧ (α ∨ δ) = δ • (α ∧ β) ∨ (α ∧ γ) = γ and (β ∧ α) ∨ (β ∧ δ) = δ • α ∧ β ≤ γ and α ∧ β ≤ δ • α∧β ≤γ∧δ • α ∧ β = γ ∧ δ. Theorem 7.11. Let C be the fuzzy choice function deﬁned by (7.2.1). Then the following assertions are equivalent (i) C veriﬁes W AF RP ◦ ; (ii) C is G-normal; (iii) C is M -normal; (iv) α ∧ β = γ ∧ δ; (v) α ∧ (β ∨ γ) = γ and β ∧ (α ∨ δ) = δ. Proof. According to Propositions 7.6 and 7.10 it suﬃces to establish the equivalence of (iv) and (v). Suppose ﬁrst α ≤ β. Then α ∧ β = γ ∧ δ if and only if α = γ ∧ δ if and only if (α = γ and α ≤ δ). Since α ≤ β we have α ∧ (γ ∨ β) = (α ∧ β) ∨ (α ∧ γ) = α ∨ γ = α β ∧ (α ∨ δ) = (α ∧ β) ∨ (β ∧ δ) = α ∨ δ

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7 General Results

hence [α ∧ (γ ∨ β) = γ if and only if α = γ] and [β ∧ (α ∨ δ) = δ if and only if α ∨ δ = δ if and only if α ≤ δ]. It follows that for α ≤ β, (iv) if and only if (v); the case β ≤ α is similar.

7.3 A Fuzzy Analysis of the Richter Theorem The goal of this section is to investigate how we can generalize Richter’s theorem to fuzzy choice functions. First we prove that every totally rational fuzzy choice function is congruous, which extends an implication of Richter’s theorem. The attempt to generalize the converse implication of Richter’s theorem leads to the notion of semirationality and to the following result: every fuzzy choice function is semirational. A strict partial ∗-order on X is an irreﬂexive and ∗-transitive fuzzy relation on X. The notion of similarity relation has a crucial role in the analysis of fuzzy phenomena. A ∗-similarity relation E on X is a reﬂexive, symmetric and ∗-transitive fuzzy relation. If E is a ∗-similarity relation on X and Q a fuzzy relation on X then E is called a congruence w.r.t. Q if E(x, u) ∗ E(y, v) ∗ Q(x, y) ≤ Q(u, v) for all x, y, u, v ∈ X. If E is a congruence w.r.t. Q then E(x, u) ∗ Q(x, y) ≤ Q(u, y) and E(y, v) ∗ Q(x, y) ≤ Q(x, v) for all x, y, u, v ∈ X. Let E be a congruence w.r.t. Q. Deﬁne a (crisp) binary relation on X: x ≈ y ⇔ E(x, y) = 1. It is easy to see that if ≈ is an equivalence relation on X, then one can consider the quotient set Y = X/≈ . [x] will denote the equivalence class of x ∈ X. ˜ on Y deﬁned by Let us consider the fuzzy relation Q ˜ (7.3.1) Q([x], [y]) =

(E(u, x) ∗ E(v, y) ∗ Q(u, v)) for any x, y ∈ X.

u,v∈X

˜ is correctly deﬁned, i.e. Lemma 7.12. The fuzzy relation Q

7.3 A Fuzzy Analysis of the Richter Theorem

(7.3.2)

(E(u, x) ∗ E(v, y) ∗ Q(u, v)) =

u,v∈X

159

(E(u, x ) ∗ E(v, y ) ∗

u,v∈X

Q(u, v)) for all x, y, x , y ∈ X such that x ≈ x and y ≈ y . Proof. Let us denote by α and β the left and the right members in (7.3.2) respectively. Let u, v ∈ X. Then E(u, x) = E(u, x) ∗ E(x, x ) ≤ E(u, x ) E(v, y) = E(v, y) ∗ E(y, y ) ≤ E(v, y ) because E(x, x ) = E(y, y ) = 1. Therefore E(u, x) ∗ E(v, y) ∗ Q(u, v) ≤ E(u, x ) ∗ E(v, y ) ∗ Q(u, v) ≤ β. This inequality holds for any u, v ∈ X hence α ≤ β. The converse inequality follows similarly. ˜ is also Proposition 7.13. If Q is a ∗-transitive fuzzy relation on X then Q ∗-transitive. Proof. Let x, y, z ∈ X. By Lemma 4.5 (5) ˜ ˜ Q([x], [y]) ∗ Q([y], [z]) = [

E(t, z)∗Q(s, t))] =

(E(u, x) ∗ E(v, y) ∗ Q(u, v))] ∗ [

u,v∈X

and

(E(s, y) ∗

s,t∈X

(E(u, x)∗E(v, y)∗Q(u, v)∗E(s, y)∗E(t, z)∗Q(s, t))

u,v,s,t∈X

˜ Q([x], [z]) =

(E(u, x) ∗ E(t, z) ∗ Q(u, t)).

u,t∈X

Let u, v, s, t ∈ X. Since E(v, y) ∗ E(s, y) = E(v, y) ∗ E(y, s) ≤ E(v, s) E(v, s) ∗ Q(u, v) ≤ Q(u, s) R(u, s) ∗ Q(s, t) ≤ Q(u, t)

160

7 General Results

it follows that E(u, x) ∗ E(v, y) ∗ Q(u, v) ∗ E(s, y) ∗ E(t, z) ∗ Q(s, t) = E(u, x) ∗ E(v, y) ∗ E(s, y)∗E(t, z)∗Q(u, v)∗Q(s, t) ≤ E(u, x)∗E(v, s)∗E(t, z)∗Q(u, v)∗Q(s, t) ≤ ˜ E(u, x) ∗ Q(u, s) ∗ E(t, z) ∗ Q(s, t) ≤ E(u, x) ∗ E(t, z) ∗ Q(u, t) ≤ Q([x], [z]). But this inequality holds for all u, v, s, t ∈ X therefore ˜ ˜ ˜ Q([x], [y]) ∗ Q([y], [z]) ≤ Q([x], [z]).

˜ Proposition 7.14. If Q(x, x) = 0 then Q([x], [x]) = 0 for all x ∈ X. Proof. For x ∈ X we have ˜ Q([x], [x]) =

(E(u, x) ∗ E(v, x) ∗ Q(u, v)) = 0

u,v∈X

because E(u, x) ∗ E(v, x) ∗ Q(u, v) ≤ Q(x, x) = 0. ˜ is a strict partial Corollary 7.15. If Q is a strict partial ∗-order on X then Q ∗-order on Y . Let C be a fuzzy choice function. Recall that C is totally rational if C = G(., Q) for some reﬂexive, ∗–transitive and total fuzzy preference relation on X. C is congruous if it veriﬁes SF CA, i.e. for any S ∈ B and x, y ∈ X, we have C(S)(x) ∗ S(y) ∗ W (y, x) ≤ C(S)(y). The following result generalizes a part of Richter’s theorem. Theorem 7.16. Every totally rational fuzzy choice function is congruous. Proof. Assume that the fuzzy choice function C : B → F(X) is totally rational, i.e. there exists a fuzzy relation G on X which is reﬂexive, ∗–transitive and total, and such that

7.3 A Fuzzy Analysis of the Richter Theorem

C(S)(x) = S(x) ∗

161

(S(v) → G(x, v))

v∈X

for all S ∈ B and x ∈ X. First we prove that for any x, y ∈ X, the following inequality holds: (a) R(x, y) ≤ G(x, y). Let S ∈ B. The total ∗–rationality of C yields C(S)(x) ≤

(S(v) → G(x, v)),

v∈X

hence C(S)(x) ≤ S(v) → G(x, v) for each v ∈ X. Particularly, C(S)(x) ≤ S(y) → G(x, y), hence, by Lemma 4.16 (1), we get C(S)(x) ∗ S(y) ≤ G(x, y). This last inequality holds for any S ∈ B, therefore

R(x, y) =

(C(S)(x) ∗ S(y)) ≤ G(x, y).

S∈B

Let S ∈ B and x, y ∈ X. We must prove that (b) C(S)(x) ∗ S(y) ∗ W (y, x) ≤ C(S)(y). Using Corollary 4.20 we compute the left term of (b): C(S)(x) ∗ S(y) ∗ W (y, x) = = C(S)(x) ∗ S(y) ∗ [R(y, x) ∨

∞

(R(y, u1 ) ∗ . . . ∗ R(un , x))].

n=1 u1 ,...,un ∈X

In accordance with (a) we get the inequality:

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7 General Results

(c) C(S)(x) ∗ S(y) ∗ W (y, x) ≤ ≤ C(S)(x) ∗ S(y) ∗ [G(y, x) ∨

∞

(G(y, u1 ) ∗ . . . ∗ G(un , x))].

n=1 u1 ,...,un ∈X

Since G is ∗-transitive, G(y, u1 )∗. . .∗G(un , x) ≤ G(y, x) for all u1 , . . . , un ∈ X therefore

(G(y, u1 ) ∗ . . . ∗ G(un , x)) ≤ G(y, x).

u1 ,...,un ∈X

This inequality holds for any n ≥ 1, hence ∞

(G(y, u1 ) ∗ . . . ∗ G(un , x)) ≤ G(y, x).

n=1 u1 ,...,un ∈X

Then G(y, x) ∨

∞

(G(y, u1 ) ∗ . . . ∗ G(un , x)) = G(y, x)

n=1 u1 ,...,un ∈X

hence the inequality (c) becomes (d) C(S)(x) ∗ S(y) ∗ W (y, x) ≤ C(S)(x) ∗ S(y) ∗ G(y, x).

Now we will establish the inequality (e) C(S)(x) ∗ G(y, x) ≤

(S(v) → G(y, v)).

v∈X

It suﬃces to show that for any v ∈ X we have (f) C(S)(x) ∗ G(y, x) ≤ S(v) → G(y, v). Let v ∈ X. We notice that

7.3 A Fuzzy Analysis of the Richter Theorem

163

C(S)(x) ∗ G(y, x) ∗ S(v) = = S(x) ∗ [

(S(u) → G(x, u))] ∗ G(y, x) ∗ S(v) ≤

u∈X

≤ S(x) ∗ [S(v) → G(x, v)] ∗ G(y, x) ∗ S(v) = = S(x) ∗ S(v) ∗ [S(v) → G(x, v)] ∗ G(y, x) ≤ ≤ S(x) ∗ G(x, v) ∗ G(y, x) ≤ ≤ G(y, x) ∗ G(x, v) ≤ G(y, v) because S(v) ∗ [S(v) → G(x, v)] = S(v) ∧ G(x, v) ≤ G(x, v) and G is ∗transitive. In accordance with Lemma 4.3 (1) we obtain the inequality (f). Thus the inequality (e) was proved. From (e) we can infer (g) C(S)(x) ∗ S(y) ∗ G(y, x) ≤ S(y) ∗

(S(v) → G(y, v)) = C(S)(y).

v∈X

Now the desired inequality (b) follows from (d) and (g). Thus C is congruous. Let C be a (crisp) choice function on (X, B) with B ⊆ P(X). We shall relax the criterion in the deﬁnition of the rational choice function. We shall say that C is semirational if there exists a binary relation G on X which is reﬂexive, transitive and total, and such that for any S ∈ B we have (7.3.6) C(S) ⊆ S ∩ {x|(x, y) ∈ G for all y ∈ S}. But C(S) ⊆ S hence the previous relation is equivalent to the condition (7.3.7) C(S) ⊆ {x|(x, y) ∈ G for all y ∈ S}. We shall extend this new concept to the fuzzy setting.

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7 General Results

Let C be a fuzzy choice function on (X, B). C is called ∗–semirational if there exists a fuzzy relation G on X reﬂexive, ∗-transitive and total, and such that for any S ∈ B and x ∈ X we have (7.3.8) C(S)(x) ≤ (S(y) → G(x, y)). y∈X

Recall that ∗L is the Lukasiewicz t–norm. Let us consider the Lukasiewicz t-conorm ⊕ (see [68], p. 11): (7.3.9) a ⊕ b = min(a + b, 1). Then the operation ⊕ is associative, commutative and a ⊕ 0 = 0 ⊕ a = a for any a ∈ [0, 1]. In this case the negation is given by ¬a = 1 − a hence ¬¬a = a for any a ∈ [0, 1]. Lemma 7.17. ([60]) For any a, b, c ∈ [0, 1] the following hold: (a) a ≤ b implies a ⊕ c ≤ b ⊕ c; (b) a ⊕ (b ∧ c) = (a ⊕ b) ∧ (a ⊕ c); (c) a ⊕ (b ∨ c) = (a ⊕ b) ∨ (a ⊕ c); (d) a ⊕ ¬a = 1. The following result seems to be surprising. Theorem 7.18. Assume that ∗ is a continuous t-norm. Then every fuzzy choice function is ∗-semirational. Proof. Assume C : B → F(X) is a fuzzy choice function. Let us consider the fuzzy relation P on X deﬁned by (a) P (x, y) = W (x, y) ∧ ¬W (y, x) for any x, y ∈ X. We shall prove that P is ∗-transitive. Let x, y, z ∈ X. Then P (x, y) ∗ P (y, z) = [W (x, y) ∧ ¬W (y, x)] ∗ [W (y, z) ∧ ¬W (z, y)] ≤ W (x, y) ∗ W (y, z) ≤ W (x, z)

7.3 A Fuzzy Analysis of the Richter Theorem

165

because W is ∗-transitive. We also have P (x, y) ∗ P (y, z) ∗ W (z, x) = [W (x, y) ∧ ¬W (y, x)] ∗ [W (y, z) ∧ ¬W (z, y)] ∗ W (z, x) ≤ W (x, y) ∗ ¬W (z, y) ∗ W (z, x) = W (z, x) ∗ W (x, y) ∗ ¬W (z, y) ≤ W (z, y) ∗ ¬W (z, y) = 0.

In accordance with the property of negation these inequalities infer P (x, y)∗ P (y, z) ≤ ¬W (z, x) hence P (x, y) ∗ P (y, z) ≤ W (x, z) ∧ ¬W (z, x) = P (x, z). Thus P is ∗-transitive. Now we shall deﬁne the fuzzy relation J:   (b) J(x, y) =

1 if x = y

 W (x, y) ∧ W (y, x) if x = y

.

We shall prove that J is a ∗-similarity relation on X. It is clear that J(x, x) = 1 and J(x, y) = J(y, x) for any x, y ∈ X. For any x, y, z ∈ X such that x = y, y = z and x = z we have J(x, y) ∗ J(y, z) = [W (x, y) ∧ W (y, x)] ∗ [W (y, z) ∧ W (z, y)] ≤ W (x, y) ∗ W (y, z) ≤ W (x, z) and similarly, J(x, y)∗J(y, z) ≤ W (z, x). Thus J(x, y)∗J(y, z) ≤ W (x, z)∧ W (z, x) = J(x, z). This inequality is obviously true for the other cases, hence J is ∗-transitive. Then J is a ∗-similarity relation. Now we shall prove that J is a congruence w.r.t. P , i.e. for all x, y, u, v ∈ X, J(x, u) ∗ J(y, v) ∗ P (x, y) ≤ P (u, v). Assume x = u, y = v hence J(x, u) ∗ J(y, v) ∗ P (x, y) = [W (x, u) ∧ W (u, x)] ∗ [W (y, v) ∧ W (v, y)] ∗ [W (x, y) ∧ ¬W (y, x)] ≤ W (u, x) ∗ W (y, v) ∗ W (x, y) = W (u, x) ∗ W (x, y) ∗ W (y, v) ≤ W (u, v), because W is ∗-transitive. We also have

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7 General Results

J(x, u)∗J(y, v)∗P (x, y)∗W (v, u) ≤ W (u, x)∗W (y, v)∗¬W (y, x)∗W (v, u) = W (y, v) ∗ W (v, u) ∗ W (u, x) ∗ ¬W (y, x) ≤ W (y, x) ∗ ¬W (y, x) = 0 hence J(x, u) ∗ J(y, v) ∗ P (x, y) ≤ ¬W (v, u). Thus J(x, u) ∗ J(y, v) ∗ P (x, y) ≤ W (u, v) ∧ ¬W (v, u) = P (u, v). Hence J is a congruence w.r.t. P . Now let us consider the equivalence relation ≈ on X: x ≈ y ⇔ J(x, y) = 1. Let Y = X/≈ be the quotient set of X w.r.t. ≈. By Proposition 7.13 we can consider the following ∗-transitive fuzzy relation P˜ on Y : (c) P˜ ([x], [y]) =

(J(u, x) ∗ J(v, y) ∗ P (u, v))

u,v∈X

for any x, y ∈ X. In accordance with Theorem 4.36 there exists a total ∗transitive fuzzy relation R on Y such that P˜ ⊆ R. Let us deﬁne the following fuzzy relation H on X: (d) H(x, y) = J(x, y) ⊕ R([x], [y]) for any x, y ∈ X. Let G be the ∗-transitive closure of H. Since H(x, x) ≥ J(x, x) = 1 for each x ∈ X, H is reﬂexive so G is also reﬂexive. We also have G(x, y) ∨ G(y, x) ≥ H(x, y) ∨ H(y, x) ≥ R([x], [y]) ∨ R([y], [x]) > 0 because R is total. Hence G is also total. Of course G is ∗-transitive. For any S ∈ B and x ∈ S we shall establish the inequality (e) C(S)(x) ≤

v∈X

(S(v) → G(x, v)).

7.3 A Fuzzy Analysis of the Richter Theorem

167

In order to prove (e) it suﬃces to show that for any v ∈ X the following inequality holds: (f) C(S)(x) ≤ S(v) → G(x, v).

By Lemma 4.3 (1) the inequality (f) is equivalent to the condition (g) C(S)(x) ∗ S(v) ≤ G(x, v). Let v ∈ X. In accordance with the deﬁnition of R we have (h) C(S)(x) ∗ S(v) ≤ R(x, v) ≤ W (x, v).

But H(x, v) = J(x, v)⊕R([x], [v]) ≥ J(x, v)⊕P˜ ([x], [v]) = J(x, v)⊕

[J(x, s)∗

s,t∈X

J(t, v) ∗ P (s, t)] ≥ J(x, v) ⊕ [J(x, x) ∗ J(v, v) ∗ P (x, v)] = J(x, v) ⊕ P (x, v) because J(x, x) = J(v, v) = 1. Using Lemma 7.17 one gets

J(x, v)⊕P (x, v) = [W (x, v)∧W (v, x)]⊕[W (x, v)∧¬W (v, x)] = (W (x, v)⊕ [W (x, v)∧¬W (v, x)])∧(W (v, x)⊕[W (x, v)∧¬W (v, x)]) ≥ W (x, v)∧[W (v, x)⊕ W (x, v)] ∧ [W (v, x) ⊕ ¬W (v, x)] = W (x, v) ∧ [W (v, x) ⊕ W (x, v)] = W (x, v). Thus W (x, v) ≤ J(x, v) ⊕ P (x, v) ≤ H(x, v), hence, by (h), we obtain C(S)(x) ∗ S(v) ≤ W (x, v) ≤ H(x, v) ≤ G(x, v). Hence (g) was proved and C is ∗-semirational.

8 Degree of Dominance

In the literature of fuzzy preference relations there are several ways to deﬁne the dominance (see [40], [69], [86]). In general the dominance is related to a fuzzy preference relation. For a fuzzy preference relation there exist a lot of ways to deﬁne the degree of dominance of an alternative [14], [15], [16], [17], [40], [69], [79], [86]. The concept of dominance in [14] is related to the act of choice and is expressed in terms of the fuzzy choice function. This chapter aims at introducing a notion of degree of dominance of an alternative x with respect to an available fuzzy subset S of the universe X of alternatives. The degree of dominance deﬁned here reﬁnes Banerjee’s notion of dominance [14]. Banerjee’s notion of dominance expresses the dominant position of some alternatives in the set of alternatives. In the decision making processes a diﬀerentiation of the alternatives according to various criteria is most of the times necessary. In the real world there are cases when these criteria are vague due to the partial information that the decision-maker possesses. The representation of these vague criteria within the choice problems is done by the available fuzzy sets. If x is an alternative and S is an available fuzzy set that corresponds to a criterion then the degree of dominance DS (x) is a number that belongs to the unit interval. This number expresses the position of alternative x with respect to the other alternatives as a result of the act of

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 169–187 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

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8 Degree of Dominance

choice. If an alternative has the degree of dominance equal to 1 then it will be dominant with respect to criterion S. With the degree of dominance one can establish a hierarchy of alternatives with respect to the criterion deﬁned by S. The diﬀerence between Banerjee’s notion of dominance and the degree of dominance introduced in this chapter is that our notion takes into account all the alternatives, not only the dominant ones. At the same time, when there are no dominant alternatives (DS (x) = 1 for any x ∈ X) one can select the alternatives with the maximum degree of dominance. Brieﬂy, the motivation of introducing the degree of dominance resides in: • the degree of dominance is a concept that allows for a direct hierarchy of alternatives in accordance with the criteria of choice; • it is an instrument by which all available alternatives, not only the dominant ones can be ranked; • it helps to formulate new axioms of congruence that reﬁne Banerjee’s [14]; • it oﬀers simple computations for the ranking of alternatives in concrete problems (see Chapter 9). The results of the chapter are obtained in the framework of the G¨ odel t-norm. Section 8.1 is an overview of the context in which Banerjee’s concept of degree of dominance was formulated. Banerjee [14] formulates a fuzzy revealed preference theory for his fuzzy choice functions. In this context he studies three congruence axioms F C1, F C2, F C3. In the same setting, Wang [112] establishes deeper connections between F C1, F C2, F C3. These three axioms are formulated in terms of dominance of an alternative x in an available (crisp) set S of alternatives. In Section 8.2 we introduce our notion of degree of dominance. Given a fuzzy choice function C, with each fuzzy available set and each alternative x, one associates the degree of dominance DS (x) of x with respect to S. The real number DS (x) gives us information about the position of x with respect to all

8.1 Dominance in Banerjee’s Framework

171

alternatives y of X. If DS (x) = 1 then we ﬁnd Banerjee’s notion of dominance. Another notion introduced in this section is the degree of dominance DSQ (x) of x in S in terms of a fuzzy preference relation Q. DSQ (x) corresponds to Banerjee’s notion of dominance of S in terms of Q. In Section 8.3, starting from the concept of degree of dominance deﬁned in the previous section, the congruence axioms F C ∗ 1, F C ∗ 2, F C ∗ 3 for the class of fuzzy choice functions deﬁned in this book are formulated. In Banerjee’s context, conditions F C1, F C2, F C3 are implied by these three axioms. We prove that F C ∗ 1 implies F C ∗ 3 and F C ∗ 2 implies F C ∗ 3. The degree of dominance allows us to formulate a new revealed preference axiom W AF RPD . We prove that the axioms W AF RPD and F C ∗ 1 are equivalent. This theorem parallels a result of Theorem 6.1 which asserts that the revealed preference axiom W AF RP ◦ is equivalent to F C ∗ 3. One result of this section shows that under very loose conditions the degree of dominance of x with respect to S can be expressed in function of the degree of dominance of x with respect to fuzzy subsets of type [x, y], y ∈ X. Section 8.3 concludes with an example that shows the relevance of the concept of degree of dominance for the process of decision making, by establishing a ranking of alternatives with respect to multiple criteria.

8.1 Dominance in Banerjee’s Framework The process of decision making deals in real life with vague preferences, modelled by fuzzy relations. Orlovsky initiated a theory of choice based on fuzzy preference relations [79]. He deﬁned a notion of degree of dominance as a mode of selecting the best alternatives. Several authors have proposed other notions that express the dominance of an alternative [15], [16], [17], [40], [69], [79], [86]. These notions start from a fuzzy preference relation. In [14] Banerjee develops a theory of revealed preference for a class of choice functions whose domain is the family of all non-empty ﬁnite subsets

172

8 Degree of Dominance

of a universe of alternatives X and whose range consists of non-zero fuzzy subsets of X. We need to emphasize that Banerjee’s notion of dominance is directly related to the choice function, not to the fuzzy preference relation. The degree of dominance deﬁned in this thesis extends Banerjee’s notion of dominance. In this section we make a short overview of the results in [14], [112]. Let X be a universal set of alternatives, H the family of non-empty ﬁnite subsets of X and F all non-zero fuzzy subsets of X with ﬁnite support. Recall that a Banerjee fuzzy choice function is a function C : H → F such that supp C(S) ⊆ S for any S ∈ H. The fuzzy revealed preference relation R associated with a fuzzy choice function C is deﬁned by ([14]): (7.1.1) R(x, y) = {C(S)(x)|S ∈ H, x, y ∈ S} for any x, y ∈ X. It is obvious that C(S)(x) ≤ R(x, y) for any S ∈ H and x, y ∈ X. Let C be a fuzzy choice function, S ∈ H and x ∈ S. x is said to be dominant in S if C(S)(y) ≤ C(S)(x) for any y ∈ S. The dominance of x in S means that x has a higher potentiality of being chosen than the other elements of S. It is obvious that this deﬁnition of dominance is related to the act of choice, not to a preference relation. Banerjee also considers a second type of dominance, associated with a fuzzy preference relation. Let Q be a fuzzy preference relation on X, S ∈ H and x ∈ X. x is said to be relation dominant in S in terms of Q if Q(x, y) ≥ Q(y, x) for all y ∈ S. Let S ∈ H, S = {x1 , . . . , xn }. The restriction of Q to S is Q|S = n (Q(xi , xj ) ∧ (Q(xi , xj ))n×n . Then we have the composition Q|S ◦ C(S) = j=1

C(S)(xj )). In [14] Banerjee introduced the following congruence axioms for a fuzzy choice function C: F C1 For any S ∈ H and x, y ∈ S, if y is dominant in S then C(S)(x) = R(x, y).

8.2 Degree of Dominance

173

F C2 For any S ∈ H and x, y ∈ S, if y is dominant in S and R(y, x) ≤ R(x, y) then x is dominant in S. F C3 For any S ∈ H, α ∈ (0, 1] and x, y ∈ S, α ≤ C(S)(y) and α ≤ R(x, y) imply α ≤ C(S)(x). In [112], Wang proved that F C3 holds iﬀ for any S ∈ H, R|S ◦ C(S) ⊆ C(S). Then F C3 is equivalent with any of the following statements: ◦ For any S ∈ H and x ∈ S, (R(x, y) ∧ C(S)(y)) ≤ C(S)(x); y∈S

◦ For any S ∈ H and x, y ∈ S, R(x, y) ∧ C(S)(y) ≤ C(S)(x). In [112] it is proved that F C1 implies F C2, F C3 implies F C2 and F C1, F C3 are independent.

8.2 Degree of Dominance In this section we shall deﬁne a notion of degree of dominance in the framework of the fuzzy choice functions introduced in this book. This kind of dominance is attached to a fuzzy choice function and not to a fuzzy preference relation. It shows to what extent, as the result of the act of choice, an alternative has a dominant position among others. We ﬁx a fuzzy choice function C : B → F(X). Recall that the fuzzy revealed preference relation R on X associated with C is deﬁned by R(x, y) = (C(S)(x) ∧ S(y)) S∈B

for any x, y ∈ X. Particularizing this deﬁnition for the case when C is a Banerjee choice function we obtain the fuzzy relation deﬁned by (7.1.1). As seen in the previous section, the concept of dominance appears essentially in the expression of congruence axioms F C1-F C3. We deﬁne now the degree of dominance of an alternative x with respect to a fuzzy subset S. This will be a real number that shows the position of x among the other alternatives.

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8 Degree of Dominance

Deﬁnition 8.1. Let S ∈ B and x ∈ X. The degree of dominance of x in S is given by DS (x) = S(x) ∧ = S(x) ∧ [(

[C(S)(y) → C(S)(x)]

y∈X

C(S)(y)) → C(S)(x)].

y∈X

If DS (x) = 1 then we say that x is dominant in S. Remark 8.2. Let S be a crisp subset of X. Identifying S with its characteristic function we have the equivalences: DS (x) = 1 iﬀ S(x) = 1 and C(S)(y) ≤ C(S)(x) for any y ∈ X iﬀ x ∈ S and C(S)(y) ≤ C(S)(x) for any y ∈ S. This shows that in this case we obtain exactly the notion of dominance of Banerjee. Remark 8.3. In accordance with Deﬁnition 8.1, x is dominant in S iﬀ S(x) = 1 and C(S)(y) = C(S)(x). y∈X

Remark 8.4. Assume that C veriﬁes (H1), i.e. C(S)(y0 ) = 1 for some y0 ∈ X. In this case C(S)(y) = 1 therefore DS (x) = C(S)(x) for any x ∈ X. y∈X

Lemma 8.5. If [x, y] ∈ B then D[x,y] (x) = C([x, y])(y) → C([x, y])(x). Proof. Since C([x, y]) ⊆ [x, y] we have C([x, y])(z) = 0 for z ∈ {x, y}. Then D[x,y] (x) = [x, y](x) ∧ [C([x, y])(z) → C([x, y])(x)] z∈X

= [C([x, y])(x) → C([x, y])(x)] ∧ [C([x, y])(y) → C([x, y])(x)] = C([x, y])(y) → C([x, y])(x). Proposition 8.6. For any S ∈ B and x, y ∈ X we have (i) C(S)(x) ≤ DS (x) ≤ S(x); (ii) S(x) ∧ DS (y) ∧ [C(S)(y) → C(S)(x)] ≤ DS (x).

8.2 Degree of Dominance

Proof. (i) According to Lemma 4.3 (4), C(S)(x) ≤ ( hence C(S)(x) ≤ S(x) ∧ [(

175

C(S)(y)) → C(S)(x)

y∈X

C(S)(y)) → C(S)(x)] = DS (x).

y∈X

(ii) By Lemma 4.3 (12) the following inequality holds for any z ∈ X: [C(S)(z) → C(S)(y)] ∧ [C(S)(y) → C(S)(x)] ≤ C(S)(z) → C(S)(x). Thus [C(S)(y) → C(S)(x)] ∧ DS (y) ∧ S(x) = = [C(S)(y) → C(S)(x)] ∧ [C(S)(z) → C(S)(y)] ∧ S(x) ∧ S(y) = = S(y) ∧ S(x) ∧ ≤ S(x) ∧

z∈X

([C(S)(z) → C(S)(y)] ∧ [C(S)(y) → C(S)(x)]) ≤

z∈X

[C(S)(z) → C(S)(x)] = DS (x).

z∈X

Remark 8.7. By Proposition 8.6, DS (x) > 0 for some x ∈ X. Then the assignment S → DS is a fuzzy choice function D : B → F(X). According to Remark 8.4, if C veriﬁes (H1) then C = D. It implies that the study of the degree of dominance is interesting for the case when hypothesis (H1) does not hold. Remark 8.8. For S ∈ B and x ∈ X we deﬁne the sequence (DSn (x))n≥1 by induction: DS1 (x) = DS (x); DSn+1 (x) = S(x) ∧

[DSn (y) → DSn (x)].

y∈X

By Proposition 8.6 (i) we have C(S)(x) ≤ DS1 (x) ≤ . . . ≤ DSn (x) ≤ . . . ≤ ∞ DSn (x). The assignments S → DSn , n ≥ 1 DS∞ (x) ≤ S(x), where DS∞ (x) = n=1

and S → DS∞ provide new fuzzy choice functions. The following deﬁnition generalizes Banerjee’s notion of dominant relation in S in terms of Q. Deﬁnition 8.9. Let Q be a fuzzy preference relation on X, S ∈ B and x ∈ X. The degree of dominance of x in S in terms of Q is deﬁned by DSQ (x) = S(x) ∧ [(S(y) ∧ Q(y, x)) → Q(x, y)] y∈X

If DSQ (x) = 1 then we say that x is dominant in S in terms of Q .

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8 Degree of Dominance

Example 8.10. Consider the set of alternatives X = {x, y} and the criterion S = aχ{x} + bχ{y} for which the choice function is given by C(S) = αχ{x} + βχ{y} , 0 < α < a < 1, 0 < β < b < 1. We intend to calculate the sequences (DSn (x))n≥1 , (DSn (y))n≥1 . Case β ≤ α DS1 (x) = DS (x) = a ∧ (β → α) = a DS1 (y) = DS (y) = b ∧ (α → β) = b ∧ β = β DS2 (x) = a ∧ (β → a) = a, DS2 (y) = b ∧ (a → β) DS3 (x) = a, DS3 (y) = b ∧ (a → β) In general DSn (x) = a, DSn (y) = b ∧ (a → β) for n ≥ 2. The case α ≤ β is treated analogously.

8.3 New Congruence Axioms The congruence axioms F C1, F C2, F C3 play an important role in Banerjee’s theory of revealed preference. The formulation of F C1, F C2 uses the notion of dominance and F C3 is exactly the Weak Congruence Axiom (W CA). In this section we introduce the congruence axioms F C ∗ 1, F C ∗ 2, F C ∗ 3 which are reﬁnements of axioms F C1, F C2, F C3. Axioms F C ∗ 1 and F C ∗ 2 are formulated in terms of degree of dominance. F C ∗ 3 is Weak Fuzzy Congruence Axiom (W F CA) deﬁned in Section 5.2. F C ∗ 1 For any S ∈ B and x, y ∈ X the following inequality holds: S(x) ∧ DS (y) ≤ R(x, y) → C(S)(x). F C ∗ 2 For any S ∈ B and x, y ∈ X the following inequality holds: S(x) ∧ DS (y) ∧ (R(y, x) → R(x, y)) ≤ DS (x). F C ∗ 3 For any S ∈ B and x, y ∈ X the following inequality holds: S(x) ∧ C(S)(y) ∧ R(x, y) ≤ C(S)(x).

8.3 New Congruence Axioms

177

The form F C ∗ 1 is derived from F C ∗ 3 by replacing DS (y) by C(S)(y). By Remarks 8.4 and 8.7, DS (x) (respectively DS (y)) can be viewed as a substitute of C(S)(x) (respectively C(S)(y)). If hypothesis (H1) holds, then by Remark 8.4, DS (y) = C(S)(y) and by Lemma 4.3 (1) axioms F C ∗ 1 and F C ∗ 3 are equivalent. Remark 8.11. Let S ∈ B and x, y ∈ X. Thus C(S)(x) ∧ S(x) ∧ DS (y) = C(S)(x) ∧ DS (y) = C(S)(x) ∧ S(y) ∧ [C(S)(z) → C(S)(y)] z∈X

≤ C(S)(x) ∧ S(y) ≤ R(x, y) hence, by Lemma 4.3 (1), S(x) ∧ DS (y) ≤ C(S)(x) → R(x, y). Therefore, if F C ∗ 1 holds then S(x) ∧ DS (y) ≤ R(x, y) ↔ C(S)(x). Remark 8.12. Assume F C ∗ 3 holds. Then for any S ∈ B and x ∈ X: [S(x) ∧ C(S)(y) ∧ R(x, y)] ≤ C(S)(x). y∈X

Assume that R is reﬂexive (for example if (H1), (H2) hold). Since C(S)(x) ≤ S(x) ∧ C(S)(x) ∧ R(x, x) it follows that C(S)(x) = [S(x) ∧ C(S)(y) ∧ R(x, y)]. y∈X

Remark 8.13. Notice that F C ∗ 3 appears under the name W F CA (Weak Fuzzy Congruence Axiom). Proposition 8.14. F C ∗ 1 ⇒ F C ∗ 3. Proof. S(x) ∧ C(S)(y) ∧ R(x, y) ≤ S(x) ∧ DS (y) ∧ R(x, y) ≤ C(S)(x), hence F C ∗ 1 ⇒ F C ∗ 3. Example 8.15 shows that F C ∗ 3 does not necessarily imply F C ∗ 1. Example 8.15. Let X = {a, b} and A = aχ{x} + bχ{y} , C(A) = sχ{x} + tχ{y} , B = cχ{x} + dχ{y} , C(B) = uχ{x} + wχ{y} , where 0 < s ≤ a, 0 < t ≤ b, 0 < u ≤ c, 0 < w ≤ d. Then C is a fuzzy choice function on (X, B) where B = {A, B}.

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8 Degree of Dominance

Applying Deﬁnition 5.16 (i) one obtains: R(x, x) = s ∨ u, R(x, y) = (s ∧ b) ∨ (u ∧ d), R(y, x) = (t ∧ a) ∨ (w ∧ c), R(y, y) = t ∨ w. We compute now the degrees of dominance: DA (x) = a ∧ (t → s) DA (y) = b ∧ (s → t) DB (x) = c ∧ (w → u) DB (y) = d ∧ (u → w). If F C ∗ 1 holds then DA (x) ∧ A(x) ∧ R(x, x) ≤ C(A)(x), i.e. a ∧ (t → s) ∧ (s ∨ u) ≤ s. If we assume t = w < s < u < a then a ∧ (t → s) ∧ (s ∨ u) = a ∧ 1 ∧ u = u > s, hence F C ∗ 1 does not hold. The axiom F C ∗ 3 holds iﬀ the following inequalities are veriﬁed: (1) A(x) ∧ C(A)(y) ∧ R(x, y) ≤ C(A)(x) (2) A(y) ∧ C(A)(x) ∧ R(y, x) ≤ C(A)(y) (3) B(x) ∧ C(B)(y) ∧ R(x, y) ≤ C(B)(x) (4) B(y) ∧ C(B)(x) ∧ R(y, x) ≤ C(B)(y). The ﬁrst condition can be written: a ∧ t ∧ [(s ∧ b) ∨ (u ∧ d)] ≤ s. By distributivity this inequality is equivalent to a ∧ t ∧ u ∧ d ≤ s. But s ≤ a hence a ∧ t ∧ u ∧ d ≤ s iﬀ t ∧ u ∧ d ≤ s. We have proved that (1) is equivalent to (1’) t ∧ u ∧ d ≤ s. In a similar way (2), (3) and (4) are equivalent to (2’) s ∧ w ∧ c ≤ t (3’) w ∧ s ∧ b ≤ u (4’) u ∧ t ∧ b ≤ w. If t = w < s < u < a the inequalities (1’)-(4’) are veriﬁed, hence F C ∗ 3 holds. Thus F C ∗ 3 does not necessarily imply F C ∗ 1. Proposition 8.16. F C ∗ 3 ⇒ F C ∗ 2. Proof. Let S ∈ B and x, y ∈ X. By R(y, x) ≥ C(S)(y) ∧ S(x), Lemma 4.3 (10), (2) and F C ∗ 3 we get for any z ∈ X:

8.3 New Congruence Axioms

179

C(S)(z) ∧ S(x) ∧ [C(S)(z) → C(S)(y)] ∧ [R(y, x) → R(x, y)] = = S(x) ∧ C(S)(z) ∧ C(S)(y) ∧ [R(y, x) → R(x, y)] ≤ ≤ C(S)(z) ∧ (C(S)(y) ∧ S(x)) ∧ [(C(S)(y) ∧ S(x)) → R(x, y)] = = C(S)(z) ∧ S(x) ∧ C(S)(y) ∧ R(x, y) ≤ ≤ S(x) ∧ C(S)(y) ∧ R(x, y) ≤ C(S)(x). Hence, by Lemma 4.3 (1): S(x) ∧ [C(S)(z) → C(S)(y)] ∧ [R(y, x) → R(x, y)] ≤ C(S)(z) → C(S)(x). Since this inequality holds for any z ∈ X we obtain: S(x) ∧ DS (y) ∧ [R(y, x) → R(x, y)] = = S(x) ∧ S(y) ∧ [C(S)(z) → C(S)(y)] ∧ [R(y, x) → R(x, y)] ≤ ≤ S(x) ∧

z∈X

(S(x) ∧ [C(S)(z) → C(S)(y)] ∧ [R(y, x) → R(x, y)]) ≤

z∈X

≤ S(x) ∧

[C(S)(z) → C(S)(x)] = DS (x).

z∈X

Proposition 8.17. If F C ∗ 1 holds then DS (x) ≤ DSR (x) for any S ∈ B and x ∈ X. Proof. By absurdum, assume that there exist S ∈ B and x ∈ X such that DS (x) ≤ DSR (x) = S(x) ∧ [(S(y) ∧ R(y, x)) → R(x, y)]. y∈X

Since DS (x) ≤ S(x), there exists y ∈ X such that DS (x) ≤ (S(y) ∧ R(y, x)) → R(x, y) hence DS (x) ∧ S(y) ∧ R(y, x) ≤ R(x, y), i.e. R(x, y) < DS (x) ∧ S(y) ∧ R(y, x). According to F C ∗ 1 we have DS (x) ∧ S(y) ∧ R(y, x) ≤ C(S)(y). By the deﬁnition of DS (x), DS (x) ≤ C(S)(y) → C(S)(x), therefore DS (x) ∧ S(y) ∧ R(y, x) ≤ C(S)(y) ∧ [C(S)(y) → C(S)(x)] = C(S)(x) ∧ C(S)(y) ≤ C(S)(x) ∧ S(y) ≤ R(x, y).

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8 Degree of Dominance

We have obtained the contradiction R(x, y) < R(x, y), hence the proposition is proved. Theorem 8.18. Assume that the fuzzy choice function C fulﬁlls (H2). Then axiom F C ∗ 1 implies that for any S ∈ B and x ∈ X we have DS (x) = S(x) ∧ [S(y) → D[x,y] (x)]. y∈X

Proof. By Proposition 8.17 one gets R D[x,y] (x) ≤ D[x,y] (x) ≤ R(y, x) → R(x, y)

for any x, y ∈ X. Hence, by Lemma 4.3 (10), (11) S(y) → D[x,y] (x) ≤ S(y) → (R(y, x) → R(x, y)) = ((S(y) ∧ R(y, x)) → R(x, y). Thus using Lemma 4.3 (11), (2) C(S)(y) ∧ [S(y) → D[x,y] (x)] ≤ C(S)(y) ∧ [(S(y) ∧ R(y, x)) → R(x, y)] = C(S)(y) ∧ S(y) ∧ [S(y) → (R(y, x) → R(x, y))] = C(S)(y) ∧ S(y) ∧ [R(y, x) → R(x, y)] = C(S)(y) ∧ [R(y, x) → R(x, y)]. Since R(y, x) ≥ C(S)(y)∧S(x) we have S(x)∧C(S)(y) = S(x)∧C(S)(y)∧ R(y, x) hence, by Lemma 4.3 (11), (2) and Proposition 8.14 one gets S(x) ∧ C(S)(y) ∧ [S(y) → D[x,y] (x)] = S(x) ∧ C(S)(y) ∧ R(y, x) ∧ [S(y) → D[x,y] (x)] ≤ S(x) ∧ C(S)(y) ∧ R(y, x) ∧ [R(y, x) → R(x, y)] = S(x) ∧ C(S)(y) ∧ R(y, x) ∧ R(x, y) ≤ S(x) ∧ C(S)(y) ∧ R(x, y) ≤ C(S)(x). By Lemma 4.3 (1) this yields S(x) ∧ [S(y) → D[x,y] (x)] ≤ C(S)(y) → C(S)(x). This last inequality holds for each y ∈ X hence S(x) ∧ [S(y) → D[x,y] (x)] = S(x) ∧ (S(x) ∧ [S(y) → D[x,y] (x)]) ≤

y∈X

≤ S(x) ∧

y∈X

y∈X

[C(S)(y) → C(S)(x)] = DS (x).

8.3 New Congruence Axioms

181

Now we shall establish the converse inequality. We know that C(S)(x) ∧ S(y) ≤ R(x, y). By Proposition 8.17 the following inequalities hold DS (x) ≤ DSR (x) ≤ (S(y) ∧ R(y, x)) → R(x, y) for any y ∈ X. Then by Lemma 4.3 (1): DS (x) ∧ S(y) ∧ R(y, x) ≤ R(x, y). Since C([x, y])(y) ≤ R(y, x) we get DS (x) ∧ S(y) ∧ C([x, y])(y) ≤ DS (x) ∧ S(y) ∧ R(y, x) ≤ R(x, y). Thus by F C ∗ 3 we obtain DS (x)∧S(y)∧C([x, y])(y) ≤ C([x, y])(y)∧R(x, y) = [x, y](x)∧C([x, y])(y)∧ R(x, y) ≤ C([x, y])(x). It follows that DS (x) ∧ S(y) ∧ C([x, y])(y) ≤ C([x, y])(x) hence, by Lemma 4.3 (1) and Lemma 8.5 DS (x) ∧ S(y) ≤ C([x, y])(y) → C([x, y])(x) = D[x,y] (x). Applying again Lemma 4.3 (1) we obtain DS (x) ≤ S(y) → D[x,y] (x) for each y ∈ X hence DS (x) ≤ S(x) ∧

[S(y) → D[x,y] (x)].

y∈X

The formulation of axiom F C ∗ 3 has Lemma 2.1 in [112] as starting point. The following result establishes the equivalence of F C ∗ 3 with a direct generalization of F C3. Proposition 8.19. The following assertions are equivalent: (1) The axiom F C ∗ 3 holds; (2) For any S ∈ B, x, y ∈ X and α ∈ (0, 1], S(x) ∧ S(y) ∧ [α → C(S)(y)] ∧ [α → R(x, y)] ≤ α → C(S)(x). Proof. By Lemma 4.5 (1) S(x) ∧ S(y) ∧ [α → C(S)(y)] ∧ [α → R(x, y)] = S(x) ∧ S(y) ∧ [α → (C(S)(y) ∧ R(x, y))],

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8 Degree of Dominance

hence, by Lemma 4.3 (1) the inequality in (2) is equivalent to S(x) ∧ S(y) ∧ α ∧ [α → (C(S)(y) ∧ R(x, y))] ≤ C(S)(x). According to Lemma 4.3 (2) S(x) ∧ S(y) ∧ α ∧ [α → (C(S)(y) ∧ R(x, y))] = S(x) ∧ S(y) ∧ α ∧ C(S)(y) ∧ R(x, y) = S(x) ∧ C(S)(y) ∧ R(x, y) ∧ α because C(S)(y) ≤ S(y). Thus the inequality in (2) is equivalent to (a) S(x) ∧ C(S)(y) ∧ R(x, y) ∧ α ≤ C(S)(x). Assuming F C ∗ 3 holds, the inequality (a) also holds since α ≤ 1. Conversely, if in (a) one takes α = 1 one obtains F C ∗ 3. Deﬁnition 8.20. Let C be a fuzzy choice function on (X, B). We deﬁne the fuzzy relation R2 on X by R2 (x, y) = [(S(x) ∧ DS (y)) → C(S)(x)]. S∈B

Remark 8.21. Let C be a fuzzy choice function, S ∈ B and x, y ∈ X. By Deﬁnition 5.16 (i) R(x, y) ∧ S(x) ∧ DS (y) = [ =

(C(T )(x) ∧ T (y))] ∧ S(x) ∧ DS (y)

T ∈B

[S(x) ∧ T (y) ∧ C(T )(x) ∧ DS (y)].

T ∈B

Then F C ∗ 1 is equivalent to the following statement • For any S, T ∈ B and x, y ∈ X S(x) ∧ T (y) ∧ C(T )(x) ∧ DS (y) ≤ C(S)(x). In Section 5.2 the following revealed preference axiom was considered: W AF RP ◦ For any S, T ∈ B and x, y ∈ X the following inequality holds: [S(x) ∧ C(T )(x)] ∧ [T (x) ∧ C(S)(x)] ≤ E(S ∩ C(T ), T ∩ C(S)). Theorem 7.3 asserts that W AF RP ◦ and F C ∗ 3 = W F CA are equivalent. A problem is if we can ﬁnd a similar result for condition F C ∗ 1. In order to obtain an answer to this problem we introduce the following axiom: W AF RPD For any x, y ∈ X and S, T ∈ B, [S(x) ∧ C(T )(x)] ∧ [T (y) ∧ DS (y)] ≤ I(S ∩ C(T ), T ∩ C(S)).

8.3 New Congruence Axioms

183

Theorem 8.22. For a fuzzy choice function C : B → F(X) the following are equivalent: (i) C veriﬁes F C ∗ 1; (ii) R ⊆ R2 ; (iii) C veriﬁes W AF RPD . Proof. (i) ⇔ (ii). The following assertions are equivalent: • R ⊆ R2 ; • For any x, y ∈ X: (C(S)(x) ∧ S(y)) ≤ [(T (x) ∧ DT (y)) → C(T )(x)]; S∈B

T ∈B

• For any x, y ∈ X and S, T ∈ B :

C(S)(x) ∧ S(y) ≤ (T (x) ∧ DT (y)) → C(T )(x). • For any x, y ∈ X and S, T ∈ B: C(S)(x) ∧ S(y) ∧ T (x) ∧ DT (y) ≤ C(T )(x). In accordance with Remark 8.21 (i) and (ii) are equivalent. (iii) ⇒ (i) Assume that C veriﬁes W AF RPD . Let x, y ∈ X and S, T ∈ B. By W AF RPD one gets S(x) ∧ T (y) ∧ C(T )(x) ∧ DS (y) ≤ I(S ∩ C(T ), T ∩ C(S)) = = [(S(u) ∧ C(T )(u)) → (T (u) ∧ C(S)(u))] ≤ u∈X

≤ (S(x) ∧ C(T )(x)) → (T (x) ∧ C(S)(x)) = = [(S(x) ∧ C(T )(x)) → T (x)] ∧ [(S(x) ∧ C(T )(x)) → C(S)(x)] = = (S(x) ∧ C(T )(x)) → C(S)(x) because (S(x) ∧ C(T )(x)) → T (x) = 1 (by Lemma 4.3 (5)). It follows that S(x) ∧ T (y) ∧ C(T )(x) ∧ DS (y) = = [S(x) ∧ T (y) ∧ C(T )(x) ∧ DS (y)] ∧ (S(x) ∧ C(T )(x)) ≤ C(S)(x) in accordance with Lemma 4.3 (1). According to Remark 8.21, C veriﬁes F C ∗ 1. (i) ⇒ (iii) Assume C fulﬁlls F C ∗ 1. By Remark 8.21, for any S, T ∈ B and x, y, u ∈ X we have

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8 Degree of Dominance

S(x) ∧ T (y) ∧ DS (y) ∧ C(T )(x) ∧ S(u) ∧ C(T )(u) ≤ S(u) ∧ T (y) ∧ C(T )(u) ∧ DS (y) ≤ C(S)(u). Thus, by Lemma 4.3 (1): S(x) ∧ T (y) ∧ DS (y) ∧ C(T )(x) ≤ (S(u) ∧ C(T )(u)) → C(S)(u) = = (S(u) ∧ C(T )(u)) → (T (u) ∧ C(S)(u)) This inequality is true for each u ∈ X, hence S(x) ∧ T (y) ∧ DS (y) ∧ C(T )(x) ≤ I(S ∩ C(T ), T ∩ C(S)) so C veriﬁes the axiom W AF RPD . Let W be the transitive closure of the fuzzy preference relation R. We notice that axioms F C ∗ 1-F C ∗ 3 are expressed in terms of R. If R is replaced with W the following three congruence axioms are obtained: SF C ∗ 1 For any S ∈ B and x, y ∈ X the following inequality holds: S(x) ∧ DS (y) ≤ W (x, y) → C(S)(x). SF C ∗ 2 For any S ∈ B and x, y ∈ X the following inequality holds: S(x) ∧ DS (y) ∧ (W (y, x) → W (x, y)) ≤ DS (x). SF C ∗ 3 For any S ∈ B and x, y ∈ X the following inequality holds: S(x) ∧ C(S)(y) ∧ W (x, y) ≤ C(S)(x). SF C ∗ 3 is exactly the congruence axiom SF CA (Strong Fuzzy Congruence Axiom) deﬁned in Section 5.2. Proposition 8.23. SF C ∗ 1 ⇒ SF C ∗ 3. Proof. Similar to the proof of Proposition 8.14. Proposition 8.24. SF C ∗ 3 ⇒ SF C ∗ 2. Proof. Let S ∈ B and x, y ∈ X. We observe that W (x, y) ≥ R(x, y) ≥ C(S)(x) ∧ S(y). Therefore, using SF C ∗ 3 we get for any z ∈ X: C(S)(z) ∧ S(x) ∧ [C(S)(z) → C(S)(y)] ∧ [W (y, x) → W (x, y)] = = S(x) ∧ C(S)(z) ∧ C(S)(y) ∧ [W (y, z) → W (x, y)] ≤ ≤ C(S)(z) ∧ (C(S)(y) ∧ S(x)) ∧ [(C(S)(y) ∧ S(x)) → W (x, y)] =

8.3 New Congruence Axioms

185

= C(S)(z) ∧ (C(S)(y) ∧ S(x)) ∧ W (x, y) ≤ ≤ S(x) ∧ C(S)(y) ∧ W (x, y) ≤ C(S)(x). We proceed next as in the proof of Proposition 8.16. Extending our results from Chapter 7 we have introduced new axioms of congruence expressed in terms of the degree of dominance and we have established relationships between them and some axioms of revealed preference. These results can be summarized in Figure 8.1.

SFC *1

SFC * 3

SFC * 2

FC *1

FC * 3

FC * 2

WAFRPD

WAFRP0

?

Fig. 8.1. Axioms of revealed preference and congruence

The bottom line of the diagram contains axioms of revealed preference; the other lines contain axioms of congruence. An open problem is to complete the above diagram with an axiom of revealed preference equivalent to F C ∗ 2. Example 8.25 will clarify the notion of degree of dominance. Given a set of alternatives and a set of criteria we want to establish the hierarchical structure induced by each criterion. Finally we deﬁne an aggregated degree of dominance and we determine the overall hierarchy. Example 8.25. Consider a universe of alternatives X = {x1 , x2 , x3 , x4 , x5 } and a set of criteria B = {S1 , S2 , S3 }, where

S1 = 0.3χ{x1 } + 0.5χ{x2 } + 0.2χ{x3 } + 0.2χ{x4 } + 0.6χ{x5 } , C(S1 ) = 0.1χ{x1 } + 0.2χ{x2 } + 0.2χ{x3 } + 0.1χ{x4 } + 0.3χ{x5 } ; S2 = 0.2χ{x1 } + 0.4χ{x2 } + 0.3χ{x3 } + 0.5χ{x4 } + 0.4χ{x5 } , C(S2 ) = 0.1χ{x1 } + 0.3χ{x2 } + 0.2χ{x3 } + 0.4χ{x4 } + 0.1χ{x5 } ;

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8 Degree of Dominance

S3 = 0.4χ{x1 } + 0.3χ{x2 } + 0.2χ{x3 } + 0.2χ{x4 } + 0.6χ{x5 } , C(S3 ) = 0.3χ{x1 } + 0.2χ{x2 } + 0.2χ{x3 } + 0.1χ{x4 } + 0.4χ{x5 } . The degrees of dominance of alternative xi , i = 1, . . . , 5 with respect to criterion Sj , j = 1, . . . , 3 calculated according to Deﬁnition 8.1 are represented in the following table: DSi (xj ) x1 x2 x3 x4 x5 S1

0.1 0.2 0.2 0.1 0.6

S2

0.1 0.3 0.2 0.5 0.1

S3

0.3 0.2 0.2 0.1 0.6

Figure 8.2 indicates the hierarchy of alternatives for each of the three criteria. According to criterion S1 , the alternative with the greatest potentiality of being chosen is x5 , alternatives x2 and x3 can be equally chosen with the

x4

x5

x5

x2

x1

x2 , x3

x3

x2 , x3

x1 , x 4

x1 , x5

x4

S1

S2

S3

Fig. 8.2. The hierarchy of alternatives induced by criteria S1 , S2 , S3

8.3 New Congruence Axioms

187

degree 0.2 and alternatives x1 and x4 have the least chance of being chosen 0.1, etc.

1 n

Deﬁne the aggregated degree of dominance of an alternative x: D(x) = DS (x), where n=card B. S∈B

In our case: D(x1 ) =

0.5 3 ,

D(x2 ) =

0.7 3 ,

D(x3 ) = 0.2, D(x4 ) =

0.7 3 ,

D(x5 ) =

1.3 3

The hierarchy of alternatives determined by the aggregated degree of dominance is given by D(x1 ) < D(x3 ) < D(x2 ) = D(x4 ) < D(x5 ). Overall, we notice that alternative x5 has the greatest chances of being selected and alternative x1 the least.

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

Similarity relations were introduced by Zadeh [118] as fuzzy generalizations of equivalence relations. The notion of similarity relation was extended by Trillas and Valverde [107] for an arbitrary t-norm. In the literature on the similarity relations diﬀerent notions are used for the same notion: likeness relation, indistinguishability relation, fuzzy equality, etc. (see [68], p. 254). With regard to the fuzzy preference relations the notion of similarity deﬁned in [118], [107] is successfully applied (see [40]). The reasonings based on fuzzy choice functions require an appropriate notion of similarity. The aim of this chapter is to give a suggestive answer to this question. We deﬁne the degree of similarity E(C1 , C2 ) of two fuzzy choice functions on a fuzzy choice space (X, B). The assignment (C1 , C2 ) → E(C1 , C2 ) induces a similarity relation on the set of fuzzy choice functions on (X, B). For 0 ≤ δ ≤ 1 we introduce the (∗, δ)-equality of two fuzzy choice functions C1 , C2 ; this notion extends the Cai δ–equality [23], [24]. A signiﬁcant part of fuzzy revealed preference theory is centered on connecting the fuzzy choice functions and fuzzy revealed preferences associated with them. Some of these connections are expressed by the functions φ1 , φ2 , ψ1 , ψ2 , ψ3 deﬁned in Section 9.1. The main results of Section 9.1 show how the functions φi and ψj connect the similarity of fuzzy choice functions and the similarity of fuzzy preference

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 189–231 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

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9 Similarity and Rationality Indicators for Fuzzy Choice Functions

relations. In particular, these functions translate the (∗, δ)–equality of fuzzy choice functions into the (∗, δ)–equality of fuzzy preference relations ([23], [24], [116]). The rationality indicators RatG (C) and RatM (C) introduced in Section 9.2 give us a measure of the degree of rationality of a fuzzy choice function C. Instead of checking whether C is rational or not, we will compute its degree of rationality. This way we obtain a modality of comparing the fuzzy choice functions with respect to their rationality. The axioms of revealed preference W AF RP , SAF RP and the axioms of congruence W F CA, SF CA have been intensely studied in the previous chapters. Section 9.3 is devoted to new indicators W AF RP (C), SAF RP (C), W F CA(C) and SF CA(C), which express the degree to which the choice function C veriﬁes the corresponding axioms. The main result of the section is Theorem 9.31, which establishes (in the framework of the G¨ odel t–norm) that W F CA(C) = SF CA(C) and also gives their expression that depends on the rationality indicators. We obtain a statement that not only generalizes a part of the fuzzy version of Arrow–Sen Theorem (see Theorem 6.7), but also gives to this a numerical formulation. Another result of this section is Theorem 9.36 which in the context oﬀered by Lukasiewicz t–norm establishes some relations between the four studied indicators. Theorem 9.31 corresponds to the statements (1) and (2) from Theorem 6.7, while Theorem 9.36 corresponds to the statement (4) of Theorem 6.7. The Arrow index of a fuzzy choice function is the subject of Section 9.4. This number measures the degree to which a choice function veriﬁes the Fuzzy Arrow Axiom. The last section of the chapter deals with the consistency indicators F α(C) and F β(C) which numerically evaluate the extent to which the choice function C satisﬁes the consistency axioms F α and F β, respectively. The equality

9.1 Similarity of Fuzzy Choice Functions

191

W F CA(C) =min(F α(C), F β(C)). Theorem 6.13 can be obtained as a particular case of this equality. Summarizing, by considering the indicators of this chapter, one concludes: • the properties of rationality, the axioms of revealed preference and congruence, the consistency conditions and the connections among them are numerically evaluated; • such indicators allow for a hierarchy of a class of fuzzy choice functions and oﬀers the possibility of selecting that fuzzy choice function that expresses more adequately a decision situation.

9.1 Similarity of Fuzzy Choice Functions In this section we shall introduce two new notions: the degree of similarity and the (∗, δ)–equality of fuzzy choice functions. We shall prove two theorems that establish some correspondences between the similarity of fuzzy choice functions and the similarity of fuzzy preference relations. In particular, we analyze how these correspondences translate the (∗, δ)-equality of fuzzy choice functions into the (∗, δ)-equality of fuzzy preference relations. We ﬁx a continuous t-norm ∗. Let X be an arbitrary set. Recall that a similarity relation on X is a reﬂexive, symmetric and ∗–transitive fuzzy relation on X. For x, y ∈ X, Q(x, y) is the degree of similarity of x and y. If Q1 and Q2 are two fuzzy relations on X then the degree of equality E(Q1 , Q2 ) is given by E(Q1 , Q2 ) = (Q1 (x, y) ↔ Q2 (x, y)). According to Lemma 4.12, the asx,y∈X

signment (Q1 , Q2 ) → E(Q1 , Q2 ) deﬁnes a similarity relation on the set of fuzzy preference relations on X and E(Q1 , Q2 ) is the degree of similarity of Q1 and Q2 . If we interpret X as a universe of alternatives and Q1 and Q2 as the preferences of two agents, then the real number E(Q1 , Q2 ) expresses how “similar” these preferences are.

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9 Similarity and Rationality Indicators for Fuzzy Choice Functions

The approximate reasonings on vague choices require a notion of similarity for fuzzy choice functions; if C1 , C2 are two fuzzy choice functions then we need to deﬁne the degree of similarity E(C1 , C2 ) for expressing the extent to which C1 and C2 behave similarly. At the same time we need to deﬁne the (∗, δ)–equality of fuzzy choice functions. We have seen that the main part of fuzzy revealed preference theory concerns the connections between the fuzzy choice functions and the fuzzy preference relations. Therefore we will study the way the similarity and the (∗, δ)–equality of fuzzy choice functions connect with the similarity and the (∗, δ)–equality of fuzzy preference relations. Deﬁnition 9.1. Let C1 , C2 be two fuzzy choice functions on (X, B). The degree of similarity E(C1 , C2 ) of C1 and C2 is a real deﬁned by E(C1 , C2 ) = ρ(C1 (S)(x), C2 (S)(x)). x∈X S∈B

For δ ∈ [0, 1] we say that C1 and C2 are (∗, δ)–equal (C1 = (∗, δ)C2 in symbols) if E(C1 , C2 ) ≥ δ. The above deﬁnition of the degree of similarity of two fuzzy choice functions C1 and C2 takes into consideration the alternatives and the available fuzzy subsets of the same fuzzy choice space (X, B); the biresiduum connects the way C1 and C2 act on these elements. Proposition 9.2. For any fuzzy choice functions C1 , C2 , C3 on (X, B) the following hold (1) C1 = C2 iﬀ E(C1 , C2 ) = 1; (2) E(C1 , C2 ) = E(C2 , C1 ); (3) E(C1 , C2 ) ∗ E(C2 , C3 ) ≤ E(C1 , C3 ). Proof. (1) and (2) Obvious. (3) By Lemma 4.5 (7) and Lemma 4.7 (8) the following inequalities hold E(C1 , C2 ) ∗ E(C2 , C3 ) = (C1 (S)(x) ↔ S∈B x∈X

9.1 Similarity of Fuzzy Choice Functions

C2 (S)(x)) ∗ ≤

193

(C2 (S)(x) ↔ C3 (S)(x))

S∈B x∈X

[((C1 (S)(x) ↔ C2 (S)(x)) ∗ (C2 (S)(x) ↔ C3 (S)(x))] ≤

S∈B x∈X

≤

[(C1 (S)(x) ↔ C3 (S)(x))] = E(C1 , C3 ).

S∈B x∈X

Denote by F CF (X, B) the set of fuzzy choice functions on (X, B) and by F P (X) the set of fuzzy preference relations on X. According to Proposition 9.2, the assignment (C1 , C2 ) → E(C1 , C2 ) deﬁnes a similarity relation on F CF (X, B). If C1 and C2 are two fuzzy choice functions then E(C1 , C2 ) express how “similar” are the acts of choice corresponding to C1 , respectively C2 . Using this notion of similarity, in the next sections we shall deﬁne several rationality indicators of fuzzy choice functions. In this chapter all the fuzzy preference relations associated with a fuzzy ¯ , P˜ , etc. will be denoted RC , R ¯ C , P˜C , etc. choice function C, i. e. R, R Let F PG (X) (resp. F PM (X)) be the subset of F P (X) containing those fuzzy preference relations Q for which G(., Q) (resp. M (., Q)) is a fuzzy choice functions. Let us consider the functions φ1 : F PM (X) → F CF (X), φ2 : F PG (X) → F CF (X) deﬁned by φ1 (Q) = M (., Q) for any Q ∈ F PM (X) φ2 (Q) = G(., Q) for any Q ∈ F PG (X). We also consider the functions ψi : F CF (X, B) → F P (X), i = 1, 2, 3 deﬁned by ¯ C and ψ3 (C) = P˜C ψ1 (C) = RC , ψ2 (C) = R for any fuzzy choice function C. The following result shows to what extent the similarity of the fuzzy preference relations is transformed by the functions φ1 and φ2 into the similarity of the corresponding choice functions. Theorem 9.3. If Q1 , Q2 ∈ F P (X) then

194

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

(1) E(φ1 (Q1 ), φ1 (Q2 )) ≥ E(Q1 , Q2 ) ∗ E(Q1 , Q2 ); (2) E(φ2 (Q1 ), φ2 (Q2 )) ≥ E(Q1 , Q2 ). Proof. (1) According to Lemma 4.7 (2), (8), (9) and Lemma 4.8 (1) it follows that: E(φ1 (Q1 ), φ1 (Q2 )) = =

ρ(φ1 (Q1 (x)), φ1 (Q2 (x))) =

S∈B x∈X

ρ(M (S, Q1 )(x), M (S, Q2 )(x)) =

S∈B x∈X

=

ρ(S(x) ∗

S∈B x∈X

((S(y) ∗ Q1 (y, x)) → Q1 (x, y)), S(x) ∗

y∈X

((S(y) ∗

y∈X

Q2 (y, x)) → Q2 (x, y))) ≥ ≥ ρ( (S(y) ∗ Q1 (y, x)) → Q1 (x, y)), (S(y) ∗ Q2 (y, x)) → S∈B x∈X

y∈X

y∈X

Q2 (x, y))) ≥ ≥ ρ((S(y)∗Q1 (y, x)) → Q1 (x, y), (S(y)∗Q2 (y, x)) → Q2 (x, y)) ≥

S∈B x,y∈X

≥

ρ(S(y) ∗ Q1 (y, x), S(y) ∗ Q2 (y, x)) ∗ ρ(Q1 (x, y), Q2 (x, y)) ≥

S∈B x,y∈X

≥

ρ(Q1 (y, x), Q2 (y, x)) ∗ ρ(Q1 (x, y), Q2 (x, y))

S∈B x,y∈X

≥ E(Q1 , Q2 ) ∗ E(Q1 , Q2 ). (2) According to Lemma 4.7 (2), (8), (9) and Lemma 4.8 (1) we get E(φ2 (Q1 ), φ2 (Q2 )) = ρ(φ1 (Q1 (x)), φ1 (Q2 (x))) = =

S∈B x∈X

ρ(G(S, Q1 )(x), G(S, Q2 )(x)) =

S∈B x∈X

=

ρ(S(x) ∗

S∈B x∈X

≥

ρ(

S∈B x∈X

≥

[S(y) → Q1 (x, y)], S(x) ∗

y∈X

(S(y) → Q1 (x, y)),

y∈X

[S(y) → Q2 (x, y)]) =

y∈X

(S(y) → Q2 (x, y)) ≥

y∈X

ρ(S(y) → Q1 (x, y), S(y) → Q2 (x, y)) ≥

S∈B x,y∈X

≥

ρ(Q1 (x, y), Q2 (x, y)) = E(Q1 , Q2 ).

S∈B x,y∈X

Corollary 9.4. (1) If Q1 = (∗, δ)Q2 then φ1 (Q1 ) = (∗, δ ∗ δ)φ1 (Q2 ) and φ2 (Q1 ) = (∗, δ)φ2 (Q2 ); (2) If ∗ is the G¨ odel t-norm then Q1 = (∗, δ)Q2 implies φ1 (Q1 ) = (∗, δ)φ1 (Q2 ).

9.1 Similarity of Fuzzy Choice Functions

195

The following theorem establishes how the functions ψ1 , ψ2 and ψ3 translate the similarity of the choice functions into the similarity of the corresponding fuzzy preference relations. Theorem 9.5. If C1 , C2 ∈ F CF (X, B) then (1) E(ψ1 (C1 ), ψ1 (C2 )) ≥ E(C1 , C2 ); (2) E(ψ2 (C1 ), ψ2 (C2 )) ≥ E(C1 , C2 ); (3) E(ψ3 (C1 ), ψ3 (C2 )) ≥ E(C1 , C2 ) ∗ E(C1 , C2 ). Proof. (1) According to Lemma 4.7 (1), (8) and Lemma 4.8 (2) it follows that E(ψ1 (C1 ), ψ1 (C2 )) = ρ(ψ1 (C1 )(x, y), ψ1 (C2 )(x, y)) =

x,y∈X

ρ(RC1 (x, y), RC2 (x, y)) =

x,y∈X

=

ρ(

x,y∈X

≥

(C1 (S)(x) ∗ S(y)),

S∈B

(C2 (S)(x) ∗ S(y)) ≥

S∈B

ρ(C1 (S)(x) ∗ S(y), C2 (S)(x) ∗ S(y)) ≥

x,y∈X S∈B

≥

ρ(C1 (S)(x), C2 (S)(x)) = E(C1 , C2 ).

x,y∈X S∈B

(2) By applying directly the deﬁnitions it follows that E(ψ2 (C1 ), ψ2 (C2 )) = ρ(ψ2 (C1 )(x, y), ψ2 (C2 )(x, y)) = =

x,y∈X

¯ C (x, y), R ¯ C (x, y)) = ρ(R 1 2

x,y∈X

=

ρ(C1 ([x, y])(x), C2 ([x, y])(x)).

x,y∈X

The assignment y → Sy = [x, y] leads to ρ(C1 ([x, y])(x, y), C2 ([x, y])(x)) = ρ(C1 (Sy )(x), C2 (Sy )(x)) ≥

x,y∈X

≥

x∈X y∈X

ρ(C1 (S)(x), C2 (S)(x)) = E(C1 , C2 ).

x∈X S∈B

(3) According to Lemma 4.7 (2), (4), (8) and Lemma 4.8 (2) it follows that ρ(ψ3 (C1 )(x, y), ψ3 (C2 )(x, y)) ≥

x,y∈X

≥

ρ(P˜C1 (x, y), P˜C2 (x, y)) ≥

x,y∈X

≥

x,y∈X

ρ(

S∈B

C1 (S)(x)∗S(y)∗¬C1 (S)(y),

S∈B

C2 (S)(x)∗S(y)∗¬C2 (S)(y)) ≥

196

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

≥

ρ(C1 (S)(x), C2 (S)(x)) ∗ ρ(¬C1 (S)(y), ¬C2 (S)(y)) ≥

x,y∈X S∈B

≥

ρ(C1 (S)(x), C2 (S)(x) ∗ ρ(C1 (S)(y), C2 (S)(y))

x,y∈X S∈B

≥

ρ(C1 (S)(x), C2 (S)(x)) ∗

x,y∈X S∈B

E(C1 , C2 ) ∗ E(C1 , C2 ).

ρ(C1 (S)(y), C2 (S)(y)) =

x,y∈X S∈B

Corollary 9.6. (1) If C1 = (∗, δ)C2 then ψ1 (C1 ) = (∗, δ)ψ1 (C2 ), ψ2 (C1 ) = (∗, δ)ψ2 (C2 ) and ψ3 (C1 ) = (∗, δ ∗ δ)ψ3 (C2 ); (2) If ∗ is the G¨ odel t-norm then C1 = (∗, δ)C2 implies ψ3 (C1 ) = (∗, δ)ψ3 (C2 ). Next let us suppose that Cˆ = G(., RC ). ˆ i.e. Recall that a fuzzy choice function C on (X, B) is G–normal if C = C, if the rationality of C is ensured by the fuzzy revealed preference relation RC . ˆ can be considered a measure of the degree The degree of similarity E(C, C) ˆ is, the more rational the choice function of rationality of C: the bigger E(C, C) C is. In the following let us assume that ∗ is the G¨ odel t-norm ∧. Lemma 9.7. Let C, C be two fuzzy choice functions. Then for any S ∈ B and x ∈ X we have (i) E(C, C ) ∧ C(S)(x) ≤ C (S)(x); (ii) E(C, C ) ∧ ¬C(S)(x) ≤ ¬C (S)(x). Proof. (i) E(C, C ) ∧ C(S)(x) = C(S)(x) ∧ ≤ C(S)(x) ∧ [C(S)(x) → C (S)(x)]

[C(T )(y) ↔ C (T )(y)] ≤

y∈X T ∈B

= C(S)(x) ∧ C (S)(x) ≤ C (S)(x). (ii) According to (i) we have E(C, C ) ∧ ¬C(S)(x) ∧ C (S)(x) ≤ C(S)(x) ∧ ¬C(S)(x) = 0. It follows that E(C, C ) ∧ ¬C(S)(x) ∧ C(S )(x) = 0, therefore E(C, C ) ∧ ¬C(S)(x) ≤ ¬C (S)(x).

9.1 Similarity of Fuzzy Choice Functions

197

Lemma 9.8. Let C and C be two fuzzy choice functions and x, y ∈ X. Then (i) E(C, C ) ∧ RC (x, y) ≤ RC (x, y); (ii) E(C, C ) ∧ ¬RC (x, y) ≤ ¬RC (x, y). Proof. (i) By applying Lemma 9.7 (i) we have: E(C, C ) ∧ RC (x, y) = E(C, C ) ∧ [C(S)(x) ∧ S(y)] =

S∈B

[E(C, C ) ∧ C(S)(x) ∧ S(y)] ≤

S∈B

≤

[C (S)(x) ∧ S(y)] = RC (x, y).

S∈B

(ii) By (i) we have E(C, C ) ∧ ¬RC (x, y) ∧ RC (x, y) ≤ RC (x, y) ∧ ¬RC (x, y) = 0, from which it follows that E(C, C ) ∧ ¬RC (x, y) ≤ ¬RC (x, y).

Lemma 9.9. If C and C are two fuzzy choice functions and x, y ∈ X then E(C, C ) ∧ P˜C (x, y) ≤ P˜C (x, y). Proof. According to Lemma 9.7 we have E(C, C ) ∧ P˜C (x, y) = E(C, C ) ∧ [C(S)(x) ∧ S(y) ∧ ¬C(S)(y)] = =

S∈B

[(E(C, C ) ∧ C(S)(x)) ∧ S(y) ∧ (E(C, C ) ∧ ¬C(S)(y))] ≤

S∈B

≤

[C (S)(x) ∧ S(y) ∧ ¬C (S)(y)] = P˜C (x, y).

S∈B

ˆ ≤ Theorem 9.10. For any fuzzy choice function C on (X, B) we have E(C, C) ¯ C ). E(RC , R Proof. According to the deﬁnition of Cˆ we have ˆ = ˆ E(C, C) [C(S)(x) ↔ C(S)(x)] = =

x∈X S∈B

ˆ [C(S)(x) → C(S)(x)]

x∈X S∈B

ˆ since C ⊆ C. ¯ C we get Using the deﬁnition of R

198

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

¯C ) = E(RC , R =

¯ C (x, y)] = [RC (x, y) ↔ R

x,y∈X

¯ C (x, y)] [RC (x, y) → R

x,y∈X

¯ C ⊆ RC . since R Since RC is reﬂexive and [x, y](x) = [x, y](y) = 1, [x, y](z) = 0 for z ∈ {x, z}, one obtains

ˆ C([x, y])(x) = [x, y](x) ∧

([x, y](z) → RC (x, z))

z∈X

= [[x, y](x) → RC (x, x)] ∧ [[x, y](y) → RC (x, y)] = RC (x, x) ∧ RC (x, y) = RC (x, y). Then ¯ C (x, y) = C([x, ˆ ¯ C (x, y) RC (x, y) → R y])(x) → R ˆ = C([x, y])(x) → C([x, y])(x). Therefore ˆ = E(C, C) ≤

ˆ [C(S)(x) → C(S)(x)] ≤

x∈X S∈B

ˆ (C([x, y])(x) → C([x, y])(x)) =

x∈X y∈X

=

¯ C (x, y)] = E(RC , R ¯ C ). [RC (x, y) → R

x,y∈X

Example 9.11. (I) In this example a particular class of fuzzy choice function is studied with emphasis on the degree of similarity. Let X = {x, y} and B = {A, B} where A, B ∈ F(X) are given by A = αχ{x} + χ{y} ; B = χ{x} + βχ{y} (0 ≤ α, β ≤ 1). Consider the function C : B → F(X) deﬁned by: C(A) = γχ{x} + χ{y} ; C(B) = χ{x} + δχ{y} (0 ≤ γ ≤ α, 0 ≤ δ ≤ β). For any α, β, γ, δ as above, C is a fuzzy choice function on (X, B). First we compute the fuzzy preference relation RC : RC (x, x) = RC (y, y) = 1; RC (x, y) = (C(A)(x) ∧ A(y)) ∨ (C(B)(x) ∧ B(y)) = β ∨ γ;

9.1 Similarity of Fuzzy Choice Functions

199

RC (y, x) = (C(A)(y) ∧ A(x)) ∨ (C(B)(y) ∧ B(x)) = α ∨ δ. Thus  RC = 

1

β∨γ

α∨δ

1

 .

ˆ ˆ We compute the values of C(A) and C(B): ˆ C(A)(x) = A(x) ∧ [A(x) → RC (x, x)] ∧ [A(y) → RC (x, y)] = α ∧ (β ∨ γ) ˆ C(A)(y) = A(y) ∧ [A(x) → RC (y, x)] ∧ [A(y) → RC (y, y)] = = α → (α ∨ δ) = 1 ˆ ˆ and similarly, C(B)(x) = 1 and C(B)(y) = β ∧ (α ∨ δ). ˆ of C and C: ˆ Then we compute the degree of similarity E(C, C) ˆ = ˆ E(C, C) (C(z) → C(z)) S∈{A,B} z∈{x,y}

ˆ ˆ = [C(A)(x) → C(A)(x)] ∧ [C(A)(y) → C(A)(y)]∧ ˆ ˆ ∧[C(B)(x) → C(B)(x)] ∧ [C(B)(y) → C(B)(y)] = [(α ∧ (β ∨ γ)) → γ] ∧ [(β ∧ (α ∨ δ)) → δ] = = {[(α ∧ β) ∨ (α ∧ γ)] → γ} ∧ {[(β ∧ α) ∨ (β ∧ δ) → δ]} = = [(α ∧ β) → γ] ∧ [(α ∧ β) → δ] = (α ∧ β) → (γ ∧ δ) =  1 if α ∧ β = γ ∧ δ . =  γ ∧ δ if α ∧ β > γ ∧ δ (II) We use the same data from Example 5.44. Therefore X = {x1 , x2 , . . . , xn , . . .}, B = {[x1 ], . . . , [xn ], . . .} and the fuzzy choice function C deﬁned by  1 if x = x n C([xn ])(x) = n  1 if x = x . n for any n = 1, 2, . . . and for any x ∈ X. We have seen that  1 if x = x n G([xn ], RC )(x) = n  0 if x = x n   1 if x = x n M ([xn ], RC )(x) =  0 if x = x n

Therefore for any n, m ∈ N: G([xn ], RC )(xn ) ↔ C([xn ])(xn ) =

 

1 n

↔

1 n

if n = m  0 ↔ 1 if n = m

200

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

=

  1 if n = m  0 if n = m

.

G([xn ], RC )(xn ) ↔ M ([xm ], RC )(xn ) =   =

1 n

if n = m

 1 if n = m Therefore

1 n

↔ 1 if n = m

 0 ↔ 0 if n = m

.

E(C, G(., RC )) =

∞

(G([xm ], RC )(xn ) ↔ C([xm ])(xn )) = 0

n,m=1

E(G(., RC ), M (., RC )) = ∞ 1 = 0. n n=1

 

∞

(G([xm ], RC )(xn ) ↔ M ([xm ], RC )(xn )) =

n,m=1

9.2 Rationality Indicators In this paragraph we study the indicators of G-rationality and M -rationality of a fuzzy choice function, notions that reﬁne the property of the fuzzy choice function of being G-rational, resp. M -rational. Among them we will introduce the indicators of G-normality and M -normality, corresponding to the case when the G-rationalization (resp. the M -rationalization) of the fuzzy choice function C is exactly the fuzzy preference relation RC . A main question in fuzzy revealed preference theory is if a fuzzy choice function is rational or not. The way of answering by “yes or no” focusses on the class of rational fuzzy choice functions, but completely ignores the others. In accordance with fuzzy logic, we need to have a measure for “the rationality” of any fuzzy choice function. The rationality indicators will deﬁne the degree to which a fuzzy choice function is rational. In this way we are able to compare two fuzzy choice functions with respect to their rationality. Let (X, B) be a fuzzy choice space and C a fuzzy choice function on (X, B). One denotes by R the set of all fuzzy preference relations on X (R = F(X 2 )).

9.2 Rationality Indicators

201

By the G–rationality (resp. M –rationality) of C we mean that there exists Q ∈ R such that C = G(., G) (resp. C = M (., Q). The real number E(C, G(., Q)) (resp. E(C, M (., Q))) shows to which extent C and G(., Q) (resp. C and M (., Q)) have the same values, i. e. to which extent the fuzzy preference relation Q G–rationalizes (resp. M –rationalizes) C. In order to evaluate how G–rational (resp. M –rational) C is, we will take into account the whole set of values {G(.Q)|Q ∈ R} (resp. {M (.Q)|Q ∈ R} and we will retain its supremum. Deﬁnition 9.12. For the fuzzy choice function C let us deﬁne RatG (C) = E(C, G(., Q)); RatM (C) = E(C, M (., Q)); Q∈R

Q∈R

N ormG (C) = E(C, G(., RC )); N ormM (C) = E(C, M (., RC )). The previous deﬁnition is stated in terms of the similarity degree of fuzzy choice functions. Lemma 9.13. For any set X we have (a) N ormG (C) = 1 iﬀ C is G–normal; (b) N ormM (C) = 1 iﬀ C is M –normal. The real number RatG (C) (resp. RatM (C)) is called the indicator of Grationality (resp. M -rationality) of C. Intuitively, RatG (C) (resp. RatM (C)) expresses the degree to which the fuzzy choice function C is G-rational (resp. M -rational). N ormG (C) (resp. N ormM (C)) is called the indicator of Gnormality (resp. M -normality) of C and evaluates the degree to which C is G-normal (resp. M -normal). Each of these indicators allow us to compare any two fuzzy choice functions. For example, if for the fuzzy choice functions C1 and C2 we have RatG (C1 ) ≥ RatG (C2 ), then the fuzzy choice function C1 is “more G– rational” then C2 . This way one obtains a hierarchy of every family of fuzzy choice functions with respect to each of the four indicators. Now we shall study the connections between these rationality indicators.

202

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

Lemma 9.14. (i) N ormG (C) ≤ RatG (C); (ii) N ormM (C) ≤ RatM (C). Proof. Directly from Deﬁnition 9.12. Proposition 9.15. If Q is a fuzzy preference relation on X then SC(Q) ≤ E(G(., Q), M (., Q)). Proof. Let S ∈ B and x ∈ X. According to Lemma 4.3 (2), (4) we get SC(Q) ∧ M (S, Q)(x) ≤ SC(Q) ∧ [(S(y) ∧ Q(y, x)) → Q(x, y)] ≤ ≤ (Q(x, y) ∨ Q(y, x)) ∧ [(S(y) ∧ Q(y, x)) → Q(x, y)] ≤ ≤ [Q(x, y) ∧ [(S(y) ∧ Q(y, x)) → Q(x, y)]] ∨ [Q(y, x) ∧ [(S(y) ∧ Q(y, x)) → Q(x, y)]] ≤ ≤ Q(x, y) ∨ [Q(y, x) ∧ [(S(y) ∧ Q(y, x)) → Q(x, y)]] = Q(x, y) ∨ [Q(y, x) ∧ [Q(y, x) → (S(y) → Q(x, y))]] ≤ Q(x, y) ∨ [Q(y, x) ∧ (S(y) → Q(x, y))] ≤ S(y) → Q(x, y). Therefore SC(Q) ∧ M (S, Q)(x) ≤ S(x) ∧

[S(y) → Q(x, y)] = G(S, Q)(x).

y∈X

By Lemma 4.3 (1) SC(Q) ≤ M (S, Q)(x) → G(S, Q)(x) = M (S, Q)(x) ↔ G(S, Q)(x) because G(S, Q)(x) ≤ M (S, Q)(x). This inequality holds for all S ∈ B and x ∈ X hence SC(Q) ≤

[M (S, Q)(x) ↔ G(S, Q)(x)] = E(G(., Q), M (., Q)).

S∈B x∈X

Proposition 9.16. RatM (C) ≤ RatG (C). Proof. Let Q be a fuzzy preference relation on X. Let us consider the fuzzy preference relation Q on X deﬁned by Q (x, y) = Q(y, x) → Q(x, y) for any x, y ∈ X. Then for all S ∈ B and x ∈ X: G(S, Q )(x) = S(x) ∧ [S(y) → Q (x, y)] = S(x) ∧

y∈X

y∈X

[(S(y) → (Q(y, x)) → Q(x, y)]

9.2 Rationality Indicators

= S(x) ∧

203

[(S(y) ∧ Q(y, x)) → Q(x, y)]

y∈X

= M (S, Q). Then G(S, Q ) = M (S, Q) for each S ∈ B hence G(., Q ) = M (., Q). It follows that E(C, M (., Q)) = E(C, G(., Q )) = RatG (C). This inequality holds for any Q ∈ R hence RatM (C) = E(C, M (., Q)) ≤ RatG (C). Q∈R

Proposition 9.17. N ormM (C) ≤ N ormG (C). Proof. Let S ∈ B and x ∈ X. By Lemma 5.37 (iii) we have C(S)(x) ≤ G(S, RC )(x) ≤ M (S, RC )(x), hence, using Lemma 4.3 (5), (10): C(S)(x) ↔ M (S, RC )(x) = M (S, RC )(x) → C(S)(x) ≤ ≤ G(S, RC )(x) → C(S)(x) = C(S)(x) ↔ G(S, RC )(x). It follows that N ormM (C) = ≤

[C(S)(x) ↔ M (S, RC )(x)] ≤

S∈B x∈X

[C(S)(x) ↔ G(S, RC )(x)] = N ormG (C).

S∈B x∈X

Theorem 9.18. RatG (C) = N ormG (C). Proof. Let Q ∈ R. We shall prove that (a) E(C, G(., Q)) ≤ N ormG (C). Since C(S)(x) ≤ G(S, RC )(x) for any x ∈ X and S ∈ B it follows that N ormG (C) = [C(S)(x) ↔ G(S, RC )(x)] =

S∈B x∈X

[G(S, RC )(x) → C(S)(x)].

S∈B x∈X

Then to prove (a) is equivalent to show that for any S ∈ B and x ∈ X we have (b) E(C, G(., Q)) ≤ G(S, RC )(x) → C(S)(x). Let S ∈ B and x ∈ X. We shall prove that (c) E(C, G(., Q)) ∧ G(S, RC )(x) ≤ C(S)(x).

204

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

For any y ∈ X and T ∈ B the following inequalities hold G(T, Q)(x) ∧ T (y) = T (x) ∧ T (y) ∧ [T (z) → Q(x, z)] ≤ z∈X

≤ T (y) ∧ (T (y) → Q(x, y)) = T (y) ∧ Q(x, y) ≤ Q(x, y). By Lemma 9.7 (i), E(C, G(., Q)) ∧ C(T )(x) ≤ G(T, Q)(x) for any T ∈ B, therefore using the previous inequalities, we get E(C, G(., Q)) ∧ RC (x, y) = E(C, G(., Q)) ∧ [C(T )(x) ∧ T (y)] = =

T ∈B

[E(C, G(., Q)) ∧ C(T )(x) ∧ T (y)]

T ∈B

≤

[G(T, Q)(x) ∧ T (y)] ≤ Q(x, y).

T ∈B

According to Lemma 4.3 (1) it follows that E(C, G(., Q)) ≤ RC (x, y) → Q(x, y). Then, by Lemma 4.3 (7) one gets E(C, G(., Q)) ∧ G(S, RC )(x) = = S(x) ∧ [S(z) → RC (x, z)] ∧ E(C, G(., Q)) ≤

z∈X

≤ S(x) ∧

[(S(y) → RC (x, z)) ∧ (RC (x, y) → Q(x, y)] ≤

y∈X

≤ S(x) ∧

[S(y) → Q(x, y)] = G(S, Q)(x).

y∈X

Then, by Lemma 9.7 (i) it follows E(C, G(., Q)) ∧ G(S, RC )(x) ≤ E(C, G(., Q)) ∧ G(S, Q)(x) ≤ C(S)(x). We have proved (c). By Lemma 4.3 (1) one obtains immediately (b). From (b) and Lemma 9.14 (i) one obtains RatG (C) = N ormG (C). The previous theorem has a remarkable signiﬁcance. It shows that the indicator of G-rationality coincides with the indicator of G-normality. At the same time it oﬀers a simple way of computing the indicator of G-rationality. According to Theorem 9.18 and Lemma 9.13 (a) it follows that RatG (C) = 1 iﬀ C is G–rational. The following theorem establishes in the presence of hypotheses H1 and H2 a much simpler formula for computing RatG (C) than the one given by Theorem 9.18.

9.2 Rationality Indicators

205

Theorem 9.19. If a fuzzy choice function C veriﬁes hypotheses H1 and H2 ¯ C )). then RatG (C) = E(C, G(., R Proof. Let Q be a reﬂexive fuzzy preference relation on X and x, y ∈ X. We shall prove ¯ C (x, y). (a) E(C, G(., Q)) ∧ Q(x, y) ≤ R We remark that G([x, y], Q)(x) = [x, y](x) ∧

([x, y](z) → Q(x, z)) = Q(x, x) ∧ Q(x, y) =

z∈X

Q(x, y). Then E(C, G(., Q)) ≤ C([x, y])(x) ↔ G([x, y], Q)(x) = ¯ C (x, y) ↔ Q(x, y) ≤ Q(x, y) → R ¯ C (x, y). =R According to Lemma 4.3 (2) we get ¯ C (x, y)) E(C, G(., Q)) ∧ Q(x, y) ≤ Q(x, y) ∧ (Q(x, y) → R ¯ C (x, y) ≤ R ¯ C (x, y). = Q(x, y) ∧ R ¯ C we obtain E(C, G(., RC )) ∧ RC (x, y) ≤ R ¯ C (x, y), If we take in (a) Q = R hence by using the deﬁnition of RC and Lemma 4.5 (5), it follows that (E(C, G(., RC )) ∧ C(S)(x) ∧ S(y)) = S∈B

= E(C, G(., RC )) ∧

(C(S)(x) ∧ S(y)) =

S∈B

¯ C (x, y). = E(C, G(., RC )) ∧ RC (x, y) ≤ R By the deﬁnition of the supremum, it follows that for any S ∈ B the following inequality takes place: ¯ C (x, y). E(C, G(., RC )) ∧ C(S)(x) ∧ S(y) ≤ R By Lemma 4.3 (1) we have ¯ C (x, y)). (b) E(C, G(., RC )) ∧ C(S)(x) ≤ S(y) → R But C(S)(x) ≤ S(x) hence E(C, G(., RC )) ∧ C(S)(x) ≤ S(x) ∧

¯ C (x, y)) = (S(y) → R

y∈X

¯ C )(x). = G(S, R By applying again Lemma 4.3 (1) it follows

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9 Similarity and Rationality Indicators for Fuzzy Choice Functions

¯ C )(x). (c) E(C, G(., RC )) ≤ C(S)(x) → G(S, R ¯ C ⊆ RC , hence by Lemma 4.5 (10): According to the deﬁnition, R ¯ C )(x) = S(x) ∧ ¯ C (x, z)) G(S, R (S(z) → R ≤ S(x) ∧

z∈X

(S(z) → RC (x, z)) = G(S, RC )(x).

z∈X

By applying Lemma 9.7 (i) we obtain ¯ C )(x) ≤ E(C, G(., RC )) ∧ G(S, R ≤ E(C, G(., RC )) ∧ G(S, RC )(x) ≤ C(S)(x), from which, according to Lemma 4.3 (1) ¯ C )(x) → C(S)(x). (d) E(C, G(., RC )) ≤ G(S, R From (c) and (d) it follows ¯ C )(x)) = E(C, G(., RC )) ≤ (C(S)(x) ↔ G(S, R ¯ C )). = E(C, G(., R

S∈B x∈X

¯ C )). The According to Theorem 9.18 we obtain RatG (C) ≤ E(C, G(., R converse inequality is true by the deﬁnition of RatG (C). ¯ C has a simpler expression than RC , therefore under hypotheRemark 9.20. R ses H1 and H2 when we compute RatG (C) the formula from Theorem 9.19 is preferred. Lemma 9.21. Let Q be a fuzzy preference relation on X. Then E(C, M (., Q)) ∧ SC(Q) = E(C, G(., Q)) ∧ SC(Q). Proof. By applying Proposition 9.15 and Proposition 9.2 (2) and (3) E(C, M (., Q)) ∧ SC(Q) ≤ E(C, M (., Q)) ∧ E(G(., Q), M (., Q)) ≤ E(C, G(., Q)) from which we get E(C, M (., Q)) ∧ SC(Q) ≤ E(C, G(., Q)) ∧ SC(Q). The inverse inequality follows similarly. Let us deﬁne ScRat(C) = =

[E(C, M (., Q)) ∧ SC(Q)]

Q∈R

[E(C, G(., Q)) ∧ SC(Q)].

Q∈R

9.2 Rationality Indicators

207

ScRat(C) is called the indicator of strongly complete rationality of C. Proposition 9.22. N ormM (C) ≤ ScRat(C). Proof. Let us consider the fuzzy preference relation R◦ on X deﬁned by R◦ (x, y) = RC (y, x) → RC (x, y) for any x, y ∈ X. According to the proof of Proposition 9.16 we have M (., RC ) = G(., R◦ ). If RC (y, x) ≤ RC (x, y) then R◦ (x, y) = RC (y, x) → RC (x, y) = 1; if RC (y, x) > RC (x, y) then R◦ (y, x) = RC (x, y) → RC (y, x) = 1. Thus R◦ is strongly complete hence SC(R◦ ) = 1. Therefore, by Lemma 9.21 N ormM (C) = E(C, M (., RC )) = E(C, G(., R◦ )) = E(C, G(., R◦ )) ∧ SC(R◦ ) = E(C, M (., R◦ )) ∧ SC(R◦ ) ≤ ScRat(C). Theorem 9.23. Let C1 , C2 be two fuzzy choice functions on (X, B). Then (1) RatG (C1 ) ∧ E(C1 , C2 ) ≤ RatG (C2 ); (2) RatM (C1 ) ∧ E(C1 , C2 ) ≤ RatM (C2 ); (3) N ormG (C1 ) ∧ E(C1 , C2 ) ≤ N ormG (C2 ); (4) N ormM (C1 ) ∧ E(C1 , C2 ) ≤ N ormM (C2 ). Proof. (1) Let Q be a fuzzy preference relation on X. We shall prove that (a) E(C1 , G(., Q)) ∧ E(C1 , C2 ) ≤ E(C2 , G(., Q)) Let S ∈ B and x, y ∈ X. Applying Lemma 9.7 (i) and Lemma 4.3 (2) it follows E(C1 , G(., Q)) ∧ E(C1 , C2 ) ∧ C2 (S)(x) ∧ S(y) ≤ ≤ E(C1 , G(., Q)) ∧ C1 (S)(x) ∧ S(y) ≤ G(S, Q)(x) ∧ S(y) = = S(x) ∧ S(y) ∧ [S(z) → Q(x, z)] ≤ S(y) ∧ (S(y) → Q(x, y)) = z∈X

= S(y) ∧ Q(x, y) ≤ Q(x, y). By Lemma 4.3 (1) one obtains from here E(C1 , G(., Q)) ∧ E(C1 , C2 ) ∧ C2 (S)(x) ≤ S(y) → Q(x, y).

208

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

Since this inequality holds for any y ∈ X we have E(C1 , G(., Q)) ∧ E(C1 , C2 ) ∧ C2 (S)(x) ≤ S(x) ∧

[S(y) → Q(x, y)] =

y∈X

G(S, Q)(x). By Lemma 4.3 (1) (b) E(C1 , G(., Q)) ∧ E(C1 , C2 ) ≤ C2 (S)(x) → G(S, Q)(x) Applying twice Lemma 9.7 (i) it follows E(C1 , G(., Q))∧E(C1 , C2 )∧G(S, Q)(x) ≤ C1 (S)(x)∧E(C1 , C2 ) ≤ C2 (S)(x) , hence, by Lemma 4.3 (1) (c) E(C1 , G(., Q)) ∧ E(C1 , C2 ) ≤ G(S, Q)(x) → C2 (S)(x). From (b) and (c) one obtains that for any S ∈ B and x ∈ X E(C, G(., Q)) ∧ E(C1 , C2 ) ≤ G(S, Q)(x) ↔ C2 (S)(x), from where we get (a). By (a) we have RatG (C1 ) ∧ E(C1 , C2 ) = [E(C1 , G(., Q)) ∧ E(C1 , C2 )] ≤

Q∈R

E(C2 , G(., Q)) = RatG (C2 ).

Q∈R

Inequalities (2), (3), (4) can be proved similarly. Remark 9.24. Let δ ∈ [0, 1]. If RatG (C1 ) ≥ δ and C1 =δ C2 then RatG (C2 ) ≥ δ. This shows that the indicator of G-rationality is preserved by the δ-equality of the fuzzy choice functions. Similar remarks can be formulated for the other three indicators. The following example shows that there exist cases when the two notions that express the notion of rationality (G-rationality and M -rationality) can be very far away from one another. Example 9.25. Let X = {x, y} and the fuzzy subsets of X deﬁned by A = χ{x} + 0χ{y} ; B = 0χ{x} + χ{y} .

9.2 Rationality Indicators

209

If B = {A, B} then we obtain a fuzzy choice space (X, B). Consider the fuzzy choice function C on (X, B) deﬁned by C(A) = aχ{x} + 0χ{y} ; C(B) = 0χ{x} + bχ{y} where 0 < a, b ≤ 1. An arbitrary fuzzy preference relation on X has the form Q(x, x) = a , Q(y, y) = b , Q(x, y) = d , Q(y, x) = c where 0 ≤ a , b , c , d ≤ 1. By an easy computation we obtain G(A, Q)(x) = a ; G(A, Q)(y) = G(B, Q)(x) = 0 ; G(B, Q)(y) = b hence G(A, Q) = a χ{x} + 0χ{y} and G(B, Q) = 0χ{x} + b χ{y} . Applying the deﬁnition we get E(C, G(., Q)) = [C(A)(x) ↔ G(A, Q)(x)] ∧ [C(A)(y) ↔ G(A, Q)(y)] ∧ [C(B)(x) ↔ G(B, Q)(x)] ∧ [C(B)(y) ↔ G(B, Q)(y)] = (a ↔ a ) ∧ (b ↔ b ). Thus RatG (C) =

{(a ↔ a ) ∧ (b ↔ b )|a , b ∈ [0, 1]} = 1.

The fuzzy choice function M (., Q) has the form: M (A, Q)(x) = M (B, Q)(y) = 1 M (A, Q)(y) = M (B, Q)(x) = 0. By calculus we get E(C, M (., Q)) = a ∧ b .Since this equality holds for any Q it follows that RatM (C) = a ∧ b. In conclusion, RatG (C) = 1 and RatM (C) = a ∧ b. While C is G–rational for any a and b, the indicator of M –rationality RatM (C) varies with a and b. The more a and b are less, the more the M – rationality of C is less. The previous example illustrates that there exist cases when the G–rationality and the M –rationality of a fuzzy choice function can diﬀer very much. The goal of the following example is to illustrate the way a set of fuzzy choice functions can be ranked with the help of the indicator of G–rationality. Example 9.26. Consider the set of alternatives X = {x, y} and two fuzzy subsets A and B of X given by A = 12 χ{x} + χ{y} , B = χ{x} + 12 χ{y} . If we

210

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

take A and B available sets we obtain the choice space (X, B). Suppose we have a decision making problem for which there are proposed as models 10 fuzzy choice problems (e. g. as a result of 10 independent expertises). The 10 fuzzy choice problems are speciﬁed by the corresponding choice functions: C1 (A) = 13 χ{x} + χ{y} ; C1 (B) = χ{x} + 15 χ{y} C2 (A) = 14 χ{x} + χ{y} ; C2 (B) = χ{x} + 15 χ{y} C3 (A) = 18 χ{x} + χ{y} ; C3 (B) = χ{x} + 13 χ{y} C4 (A) = 15 χ{x} + χ{y} ; C4 (B) = χ{x} + 14 χ{y} C5 (A) = 13 χ{x} + χ{y} ; C5 (B) = χ{x} + 13 χ{y} C6 (A) = 18 χ{x} + χ{y} ; C6 (B) = χ{x} + 19 χ{y} C7 (A) = 17 χ{x} + χ{y} ; C7 (B) = χ{x} + 18 χ{y} C8 (A) = 18 χ{x} + χ{y} ; C8 (B) = χ{x} + 14 χ{y} C9 (A) = 16 χ{x} + χ{y} ; C9 (B) = χ{x} + 13 χ{y} C10 (A) = 13 χ{x} + χ{y} ; C10 (B) = χ{x} + 16 χ{y} . According to the calculations of Example 9.11 (I) and Theorem 9.18: RatG (C1 ) = E(C1 , Cˆ1 ) =min( 13 , 15 ) =

1 5

RatG (C2 ) = E(C2 , Cˆ2 ) =min( 14 , 15 ) =

1 5

RatG (C3 ) = E(C3 , Cˆ3 ) =min( 18 , 13 ) =

1 8

RatG (C4 ) = E(C4 , Cˆ4 ) =min( 15 , 14 ) =

1 5

RatG (C5 ) = E(C5 , Cˆ5 ) =min( 13 , 13 ) =

1 3

RatG (C6 ) = E(C6 , Cˆ6 ) =min( 18 , 19 ) =

1 9

RatG (C7 ) = E(C7 , Cˆ7 ) =min( 17 , 18 ) =

1 8

RatG (C8 ) = E(C8 , Cˆ8 ) =min( 18 , 14 ) =

1 8

RatG (C9 ) = E(C9 , Cˆ9 ) =min( 16 , 13 ) =

1 6

RatG (C10 ) = E(C10 , Cˆ10 ) =min( 13 , 16 ) = 16 . With the help of the G–rationality indicator we obtain the following hierarchy of the 10 fuzzy choice functions (see Figure 9.1). It follows that the fuzzy choice problem given by the fuzzy choice function C5 is the best with respect to G–rationality.

9.3 Revealed Preference and Congruence Indicators

211

C5

C1 , C2 , C4

C9 ,C10

C3 , C7 , C 8

C6

Fig. 9.1. Hierarchy of fuzzy choice functions

9.3 Revealed Preference and Congruence Indicators The fuzzy revealed preference and congruence axioms are conditions on the rationality of vague choices. The indicators studied in the previous section represent a way of evaluating the rationality of a vague act of choice. Following

212

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

this reasoning, we need to associate to the axioms of fuzzy revealed preference and congruence some indicators with the purpose of evaluating how much such condition is veriﬁed by a fuzzy choice function. The goal of this section is to deﬁne and study some indicators of the axioms of revealed preference W AF RP , SAF RP and the axioms of congruence W F CA, SF CA. These indicators express the degree to which the axioms W AF RP , SAF RP , W F CA and SF CA are veriﬁed by a fuzzy choice function. The results proved in this section establish connections between these indicators and the G–rationality indicator. Deﬁnition 9.27. For a fuzzy choice function C on (X, B) we deﬁne the following indicators of the axioms W AF RP , SAF RP , W F CA and SF CA:

(i) W AF RP (C) =

[P˜C (x, y) → ¬RC (y, x)];

x,y∈X

(ii) SAF RP (C) =

[PC∗ (x, y) → ¬RC (y, x)];

x,y∈X

(iii) W F CA(C) =

[S(x) ∗ C(S)(y) ∗ RC (x, y) → C(S)(x)];

x,y∈X S∈B

(iv) SF CA(C) =

[S(x) ∗ C(S)(y) ∗ WC (x, y) → C(S)(x)].

x,y∈X S∈B

Remark 9.28. For a choice function C the following equivalences hold:

W AF RP (C) = 1 iﬀ C veriﬁes W AF RP ; SAF RP (C) = 1 iﬀ C veriﬁes SAF RP ; W F CA(C) = 1 iﬀ C veriﬁes W F CA; SF CA(C) = 1 iﬀ C veriﬁes SF CA. The indicator W AF RP (C) expresses the degree to which the choice function C veriﬁes W AF RP . Similar interpretations can be given to the other three indicators. The following proposition holds for an arbitrary t–norm ∗.

9.3 Revealed Preference and Congruence Indicators

213

Proposition 9.29. If C is a fuzzy choice function on (X, B) then W F CA(C) ≤ T rans(RC ). Proof. Let x, y, z ∈ X. We shall prove the inequality (a) W F CA(C) ∗ RC (x, y) ∗ RC (y, z) ≤ RC (x, z) Let us denote T = [x, y, z]. By (H2) C(T ) is normal hence C(T )(x) = 1 or C(T )(y) = 1 or C(T )(z) = 1. We analyze these three cases. • C(T )(x) = 1 In this case 1 = C(T )(x) ∗ T (z) ≤ RC (x, z), hence RC (x, z) = 1. • C(T )(y) = 1 Thus W F CA(C) ∗ RC (x, y) ∗ RC (y, z) ≤ ≤ RC (x, y) ∗ RC (y, z) ∗ [(T (x) ∗ C(T )(y) ∗ RC (x, y)) → C(T )(x)] = = RC (x, y) ∗ RC (y, z) ∗ [RC (x, y) → C(T )(x)] ≤ ≤ RC (x, y) ∗ [RC (x, y) → C(T )(x)] = RC (x, y) ∗ C(T )(x) ≤ ≤ C(T )(x) = C(T )(x) ∗ T (z) ≤ RC (x, z). • C(T )(z) = 1 Thus W F CA(C) ∗ RC (x, y) ∗ RC (y, z) ≤ RC (x, y)∗RC (y, z)∗[(T (y)∗C(T )(z)∗RC (y, z)) → C(T )(y)]∗W F CA(C) = RC (x, y) ∗ RC (y, z) ∗ [RC (y, z) → C(T )(y)] ∗ W F CA(C) = RC (x, y) ∗ RC (y, z) ∗ C(T )(y) ∗ W F CA(C) ≤ RC (x, y) ∗ C(T )(y) ∗ [(T (x) ∗ C(T )(y) ∗ RC (x, y)) → C(T )(x)] = = RC (x, y) ∗ C(T )(y) ∗ [(RC (x, y) ∗ C(T )(y)) → C(T )(x)] = RC (x, y) ∗ C(T )(y) ∗ C(T )(x) ≤ C(T )(x) = C(T )(x) ∗ T (z) ≤ RC (x, z). Therefore the inequality (a) is veriﬁed for all the three cases. From (a) it follows immediately that W F CA(C) ≤ (RC (x, y) ∗ RC (y, z)) → RC (x, z) for any x, y, z ∈ X, therefore W F CA(C) ≤

[(RC (x, y) ∗ RC (y, z)) → RC (x, z)] = T rans(RC ).

x,y,z∈X

The following two results hold for the G¨ odel t–norm ∧.

214

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

ˆ ∧ T rans Proposition 9.30. If C is a fuzzy choice function then E(C, C) (RC ) ≤ W F CA(C). Proof. Let S ∈ B and x, y ∈ X. We shall prove that ˆ ∧ T rans(RC ) ∧ RC (x, y) ∧ C(S)(y) ∧ S(x) ≤ C(S)(x). (a) E(C, C) Let z ∈ X. Knowing that C(S)(y) ∧ S(z) ≤ RC (y, z) and applying Proposition 4.29: ˆ ∧ T rans(RC ) ∧ RC (x, y) ∧ C(S)(y) ∧ S(x) ∧ S(z) ≤ E(C, C) ˆ ∧ T rans(RC ) ∧ RC (x, y) ∧ RC (y, z) ∧ S(x) ≤ RC (x, z) E(C, C) From this one gets: ˆ ∧ T rans(RC ) ∧ RC (x, y) ∧ C(S)(y) ∧ S(x) ≤ S(z) → RC (x, z) E(C, C) for each z ∈ X, therefore ˆ E(C, C)∧T rans(RC )∧RC (x, y)∧C(S)(y)∧S(x) ≤

[S(z) → RC (x, z)].

z∈X

We deduce that ˆ ∧ T rans(RC ) ∧ RC (x, y) ∧ C(S)(y) ∧ S(x) ≤ S(x) ∧ E(C, C)

[S(z) →

z∈X

ˆ RC (x, z)] = C(S)(x). By applying Lemma 9.7 (i) we obtain ˆ ∧ T rans(RC ) ∧ RC (x, y) ∧ C(S)(y) ∧ S(x) ≤ E(C, C) ˆ ∧ C(S)(x) ˆ E(C, C) ≤ C(S)(x) and (a) is proved. According to (a), for any S ∈ B and x, y ∈ X we have ˆ ∧ T rans(RC ) ≤ (S(x) ∧ C(S)(y) ∧ RC (x, y)) → C(S)(x) E(C, C) from where ˆ ∧ T rans(RC ) ≤ E(C, C)

[(S(x) ∧ C(S)(y) ∧ RC (x, y)) →

x,y∈X S∈B

C(S)(x)] = W F CA(C) The following theorem is the ﬁrst main result of this section. Theorem 9.31. If C is a fuzzy choice function on (X, B) then ˆ ∧ T rans(RC ) = W F CA(C) = SF CA(C) = E(C, C) ˆ ∧ T rans(R ¯ C ). = E(C, C)

9.3 Revealed Preference and Congruence Indicators

215

ˆ ≤ E(RC , R ¯ C ), hence, by PropoProof. According to Theorem 9.10, E(C, C) sition 4.29: ˆ ∧ T rans(RC ) ≤ E(RC , R ¯ C ) ∧ T rans(RC ) ≤ T rans(R ¯ C ); E(C, C) ˆ ∧ T rans(R ¯ C ) ≤ E(RC , R ¯ C ) ∧ T rans(R ¯ C ) ≤ T rans(RC ); E(C, C) From these two inequalities it follows immediately ˆ ∧ T rans(RC ) = E(C, C) ˆ ∧ T rans(R ¯ C ). E(C, C) Let S ∈ B and x, y ∈ X. Since RC (x, y) ≤ WC (x, y) we have (S(x) ∧ C(S)(y) ∧ WC (x, y)) → C(S)(x) ≤ (S(x) ∧ C(S)(y) ∧ RC (x, y)) → C(S)(x). From this, SF CA(C) ≤ W F CA(C). For the converse inequality W F CA(C) ≤ SF CA(C) we have to show that for any S ∈ B and x, y ∈ X (a) W F CA(C) ≤ (S(x) ∧ C(S)(y) ∧ WC (x, y)) → C(S)(x). Inequality (a) is equivalent with (b) W F CA(C) ∧ S(x) ∧ C(S)(y) ∧ WC (x, y) ≤ C(S)(x). According to Proposition 9.29, W F CA(C) ≤ T rans(RC ) and according to Lemma 4.30, T rans(RC ) ∧ WC (x, y) ≤ RC (x, y). Then W F CA(C) ∧ S(x) ∧ C(S)(y) ∧ WC (x, y) ≤ T rans(RC ) ∧ S(x) ∧ C(S)(y) ∧ WC (x, y) ≤ ≤ S(x) ∧ C(S)(y) ∧ RC (x, y) from where one obtains W F CA(C) ∧ S(x) ∧ C(S)(y) ∧ WC (x, y) ≤ S(x) ∧ C(S)(y) ∧ RC (x, y) ∧ W F CA(C) ≤ ≤ S(x) ∧ C(S)(y) ∧ RC (x, y) ∧ [(S(x) ∧ C(S)(y) ∧ RC (x, y)) → C(S)(x)] = = S(x) ∧ C(S)(y) ∧ RC (x, y) ∧ C(S)(x) ≤ C(S)(x). With this (b) is proved, therefore W F CA(C) ≤ SF CA(C). It follows W F CA(C) = SF CA(C). Let S ∈ B and x ∈ X. Since C(S) is a normal fuzzy subset of X there exists z ∈ X such that C(S)(z) = 1, therefore S(z) = 1. Then

216

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

ˆ C(S)(x) = S(x) ∧

[S(u) → RC (x, u)] ≤ S(x) ∧ [S(z) → RC (x, z)] =

u∈X

= S(x) ∧ [1 → RC (x, z)] = S(x) ∧ RC (x, z). From this it follows ˆ W F CA(C) ∧ C(S)(x) ≤ W F CA(C) ∧ S(x) ∧ RC (x, z) = S(x) ∧ C(S)(z) ∧ RC (x, z) ∧ [(T (u) ∧ C(T )(v) ∧ RC (u, v)) → u,v∈X T ∈B

C(T )(u)] ≤

≤ S(x) ∧ C(S)(z) ∧ RC (x, z) ∧ [(S(x) ∧ C(S)(z) ∧ RC (x, z)) → C(S)(x)] = = S(x) ∧ C(S)(z) ∧ RC (x, z) ∧ C(S)(x) ≤ C(S)(x). ˆ ˆ One obtains W F CA(C) ≤ C(S)(x) → C(S)(x). Since C(S)(x) ≤ C(S)(x), ˆ then C(S)(x) → C(S)(x) = 1 therefore ˆ ˆ W F CA(C) ≤ [C(S)(x) ↔ C(S)(x)] = E(C, C). S∈B x,y∈X

ˆ ∧ T rans(RC ). By Proposition 9.29, it follows that W F CA(C) ≤ E(C, C) ˆ ∧ T rans(RC ) ≤ W F CA(C) is ensured The converse inequality E(C, C) ˆ ∧ T rans(RC ). With this by Proposition 9.30, therefore W F CA(C) = E(C, C) the theorem is proved. The previous theorem establishes a tight connection between the indicators of weak and strong fuzzy congruence axioms, the indicator of G–normality and ¯ C . The comthe degree of transitivity of the revealed preferences RC and R mon value between W F CA(C) and SF CA(C) equals the minimum between ¯ C ), therefore W F CA(C) and SF CA(C) express how N ormG (C) and T rans(R transitive G–rational the choice function C is. The indicators W F CA(C) and ¯C SF CA(C) are a reﬁned manner of computing the rationality of C. Since R has a much simpler form than RC , from the computational point of view, the last expression is preferred for obtaining these two indicators. The following results are true for the case when ∗ is the Lukasiewicz t– norm and in the presence of hypotheses H1 and H2.

9.3 Revealed Preference and Congruence Indicators

217

Lemma 9.32. Let x, y ∈ X, Q1 and Q2 be two fuzzy preference relations on ˜ ∗ R(x, y) ≤ X. Then E(Q1 , Q2 ) ∗ Q1 (x, y) ≤ Q2 (x, y). In particular, E(R, R) ˜ y). R(x, Proposition 9.33. Let C : B → F(X) be a fuzzy choice function on the fuzzy ˜ ≤ W F CA(C). choice space (X, B). Then E(R, R) Proof. Let C : B → F(X) be a fuzzy choice function. We intend to prove that ˜ ≤ R(x, y) ∗ C(S)(y) ∗ S(x) → C(S)(x). E(R, R) ˜ ∗ R(x, y) ∗ C(S)(y) ∗ By Lemma 4.3 (1), this is equivalent with E(R, R) S(x) ≤ C(S)(x). By Lemma 9.32, ˜ ∗ R(x, y) ∗ C(S)(y) ∗ S(x) ≤ R(x, ˜ y) ∗ C(S)(y) ∗ S(x). E(R, R) ˜ y) = ¬P˜ (y, x) and C(S)(x) = ¬¬C(S)(x), we intend to prove Since R(x, ˜ y) ∗ C(S)(y) ∗ S(x) = ¬P˜ (y, x) ∗ C(S)(y) ∗ S(x) ≤ ¬¬C(S)(x). R(x, By Lemma 4.3 (1), this is equivalent with ¬P˜ (y, x) ∗ C(S)(y) ∗ S(x) ∗ ¬C(S)(x) = 0. By Lemma 4.3 (1) and Lemma 4.6 (9) this is equivalent with C(S)(y) ∗ S(x) ∗ ¬C(S)(x) ≤ P˜ (y, x). If we explicate P˜ (y, x) we get C(S)(y) ∗ S(x) ∗ ¬C(S)(x) ≤

[C(T )(y) ∗ ¬C(T )(x) ∗ T (x)],

T ∈B

which is true.

Proposition 9.34. Let C : B → F(X) be a fuzzy choice function on the fuzzy ˜ choice space (X, B). Then W F CA(C) ≤ E(R, R). Proof. Let C : B → F(X) be a fuzzy choice function. ˜ = ˜ y)], we have to prove the folSince E(R, R) [R(x, y) ↔ R(x, S∈B x,y∈S

˜ y) for each S ∈ B lowing two inequalities: (a) W F CA(C) ≤ R(x, y) → R(x, ˜ y) → R(x, y) for each S ∈ B and and x, y ∈ S and (b) W F CA(C) ≤ R(x, x, y ∈ S.

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9 Similarity and Rationality Indicators for Fuzzy Choice Functions

By Lemma 4.3 (1), this gets down to proving the following two inequalities: ˜ y) and (b’) W F CA(C) ∗ R(x, ˜ y) ≤ R(x, y). (a’) W F CA(C) ∗ R(x, y) ≤ R(x, We prove ﬁrst (a’). Since R(x, y) = ¬P (y, x) and by Lemma 4.3 (1), (a’) is equivalent with W F CA(C) ∗ R(x, y) ∗ P˜ (y, x) = 0. We explicate the left–hand side of the above inequality. W F CA(C) ∗ R(x, y) ∗ P˜ (y, x) = W F CA(C) ∗ R(x, y) ∗ (C(S)(y) ∗ ¬C(S)(x) ∗ S(x)) = W F CA(C) ∗

S∈B

[R(x, y) ∗ C(S)(y) ∗ ¬C(S)(x) ∗ S(x)] =

S∈B

[W F CA(C) ∗ R(x, y) ∗ C(S)(y) ∗ ¬C(S)(x) ∗ S(x)].

S∈B

We consider an arbitrary member of this union. By Lemma 4.3 (2), (3): W F CA(C) ∗ R(x, y) ∗ C(S)(y) ∗ ¬C(S)(x) ∗ S(x) = [R(s, t) ∗ C(T )(t) ∗ T (s) → C(T )(s)]∗ T ∈B s,t∈T

∗R(x, y) ∗ C(S)(y) ∗ ¬C(S)(x) ∗ S(x) ≤ [R(x, y)∗C(S)(y)∗S(x) → C(S)(x)]∗R(x, y)∗C(S)(y)∗S(x)∗¬C(S)(x) ≤

R(x, y) ∗ C(S)(y) ∗ S(x) ∗ C(S)(x) ∗ ¬C(S)(x) = 0. Therefore W F CA(C)∗R(x, y)∗ P˜ (y, x) = 0. Now we prove (b’). By Proposition 6.5 and Lemma 4.3 (3) ˜ y) ≤ W F CA(C) ∗ R(x, y) ≤ R(x, y). W F CA(C) ∗ R(x, Theorem 9.35. Let C : B → F(X) be a fuzzy choice function. Then ˜ W F CA(C) = E(R, R). Proof. By Propositions 9.33 and 9.34. Theorem 9.36. Let C be a fuzzy choice function on (X, B). Then the following properties hold: (i) W F CA(C)2 ≤ SF CA(C) ≤ W F CA(C); ˜ (ii) W F CA(C) = W AF RP (C) = E(R, R); (iii) W AF RP (C)5 ≤ SAF RP (C) ≤ W AF RP (C).

9.3 Revealed Preference and Congruence Indicators

219

Proof. (i) The inequality SF CA(C) ≤ W F CA(C) is obvious. We prove now the inequality W F CA(C)2 ≤ SF CA(C). This is equivalent with proving that W F CA(C)2 ∗ W (x, y) ∗ C(S)(y) ∗ S(x) ≤ C(S)(x). First we shall prove that W F CA(C) ≤ E(R, W ). Since R(x, y) ≤ W (x, y), E(R, W ) = [R(x, y) ↔ W (x, y)] = [W (x, y) → R(x, y)]. x,y∈X

x,y∈X

By Proposition 4.30 and Proposition 9.29 we obtain that W F CA(C) ≤ T rans(R)(x, y) ≤ W (x, y) → R(x, y), that implies that W F CA(C) ≤ E(R, W ). By Proposition 9.32, Lemma 4.3 (2) and the deﬁnition of W F CA(C), W F CA(C)2 ∗ W (x, y) ∗ C(S)(y) ∗ S(x) ≤ W F CA(C) ∗ E(R, W ) ∗ W (x, y) ∗ C(S)(y) ∗ S(x) ≤ W F CA(C) ∗ R(x, y) ∗ C(S)(y) ∗ S(x) ≤ [R(x, y) ∗ C(S)(y) ∗ S(x) → C(S)(x)] ∗ R(x, y) ∗ C(S)(y) ∗ S(x) ≤ C(S)(x). (ii) It suﬃces to prove W F CA(C) = W AF RP (C). First we prove the inequality W F CA(C) ≤ W AF RP (C); this means to prove that W F CA(C) ≤ P˜ (x, y) → ¬R(y, x) ⇔ W F CA(C) ∗ P˜ (x, y) ∗ R(y, x) = 0. Suppose by absurdum there exist x, y ∈ X such that W F CA(C)∗ P˜ (x, y)∗ R(y, x) > 0. Then [P˜ (x, y) → ¬R(y, x)] ∗ R(y, x) ∗ C(S0 )(x) ∗ S0 (y) ∗ ¬C(S0 )(y) = [ (C(S)(x) ∗ S(y) ∗ ¬C(S)(y)) → ¬R(y, x)] ∗ R(y, x) ∗ C(S0 )(x) ∗ S0 (y) ∗ S∈B

¬C(S0 )(y) = C(S0 )(x) ∗ S0 (y) ∗ C(S0 )(y) ∗ ¬R(y, x) ∗ R(y, x) = 0. We have obtained W AF RP (C)∗R(y, x)∗C(S0 )(x)∗S0 (y)∗¬C(S0 )(y) = 0, which is a contradiction. (iii) The inequality SAF RP (C) ≤ W AF RP (C) is obvious. We prove now W AF RP (C)5 ≤ SAF RP (C). By (ii) and Proposition 9.29 we obtain ˜ 3 ≤ T rans(R). ˜ W AF RP (C)4 ≤ T rans(R) ∗ E(R, R) ˜ ∗ T (R)(x, ˜ ˜ y). We obtain by comBy Proposition 4.30, T rans(R) y) ≤ R(x, putation ˜ ˜ ∗ T (R)(x, ˜ ˜ y) ⇒ W AF RP (C)4 ∗ T (R)(x, y) ≤ T rans(R) y) ≤ R(x, ˜ ˜ W AF RP (C)5 ∗ T (R)(x, y) = W AF RP (C) ∗ W AF RP (C)4 ∗ T (R)(x, y) ≤

220

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

˜ ∗ T (R)(x, ˜ ˜ y) ≤ ≤ W AF RP (C) ∗ T rans(R) y) ≤ W AF RP (C) ∗ R(x, ¬R(y, x), by the deﬁnition of W AF RP (C). Finally, we have obtained ˜ W AF RP (C)5 ∗ T (R)(x, y) ≤ ¬R(y, x), consequently W AF RP (C)5 ≤ ˜ T (R)(x, y) → ¬R(y, x). Theorem 9.36 proved for the Lukasiewicz t–norm establishes some connections between the indicators of revealed preference and the indicators of congruence. In particular, Theorem 9.36 (ii) oﬀers a way of computing the indicators W F CA(C) and SF CA(C).

9.4 The Arrow Index of a Fuzzy Choice Function In this section we shall introduce the Arrow index A(C) of a fuzzy choice function C. A(C) measures the degree to which C veriﬁes F AA. We prove an extensionability property of the Arrow index with respect to the similarity relation of two fuzzy choice functions. Deﬁnition 9.37. Let C be a fuzzy choice function on (X, B). The Arrow index A(C) of C is deﬁned by A(C) = [(I(S1 , S2 )∧S1 (x)∧C(S2 )(x)) → E(S1 ∩C(S2 ), C(S1 )]. S1 ,S2 ∈B x∈X

We remark that A(C) = 1 iﬀ C veriﬁes the Fuzzy Arrow Axiom. Intuitively, the real number A(C) represents the degree of truth of the statement “The fuzzy choice function C veriﬁes Fuzzy Arrow Axiom”. According to Theorem 6.38, the Fuzzy Arrow Axiom characterizes the full rationality of a fuzzy choice function C and the congruence axioms W F CA and SF CA as well. Then the Arrow index A(C) can be considered a measure of the full rationality of C. The following theorem shows how the similarity relation E(., .) behaves with respect to the Arrow index.

9.4 The Arrow Index of a Fuzzy Choice Function

221

Theorem 9.38. If C, C are two fuzzy choice functions on (X, B) then A(C)∧ E(C, C ) ≤ A(C ). Proof. Let S1 , S2 ∈ B and x ∈ X. We shall prove that

A(C) ∧ E(C, C ) ∧ I(S1 , S2 ) ∧ S1 (x) ∧ C (S2 )(x) ≤ ≤ E(S1 ∩ C (S2 ), C (S1 ))

(9.1)

It suﬃces to prove that for any y ∈ X the following two inequalities hold

A(C) ∧ E(C, C ) ∧ I(S1 , S2 ) ∧ S1 (x) ∧ C (S2 )(x) ≤ ≤ (S1 (y) ∧ C (S2 )(y)) → C (S1 )(y)

(9.2)

and

A(C) ∧ E(C, C ) ∧ I(S1 , S2 ) ∧ S1 (x) ∧ C (S2 )(x) ≤ ≤ C (S1 )(y) → (S1 (y) ∧ C (S2 )(y))

(9.3)

By Lemma 4.3 (1), the inequality (9.2) is equivalent to

A(C) ∧ E(C, C ) ∧ I(S1 , S2 ) ∧ S1 (x) ∧ C (S2 )(x) ∧ S1 (y) ∧ C (S2 )(y) ≤ ≤ C (S1 )(y)(9.4) According to Lemma 4.5 (1) C (S1 )(y) → (S1 (y) ∧ C (S2 )(y)) = (C (S1 )(y) → S1 (y)) ∧ (C (S1 )(y) → C (S2 )(y)) = C (S1 )(y) → C (S2 )(y) hence the inequality (9.3) is equivalent to

A(C) ∧ E(C, C ) ∧ I(S1 , S2 ) ∧ S1 (x) ∧ C (S2 )(x) ∧ C (S1 )(y) ≤ ≤ C (S2 )(y)

(9.5)

222

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

From the deﬁnition of A(C) we infer that

A(C) ≤ ≤ (I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x)) → E(S1 ∩ C(S2 ), C(S1 ))

(9.6)

Applying (9.6) and Lemma 9.7 and Lemma 3.12 (vii) de extensionabilitate we get A(C) ∧ E(C, C ) ∧ I(S1 , S2 ) ∧ S1 (x) ∧ C (S2 )(x) ∧ S1 (y) ∧ C (S2 )(y) ≤ ≤ A(C) ∧ S1 (x) ∧ C(S2 )(x) ∧ I(S1 , S2 ) ∧ S1 (y) ∧ C (S2 )(y) ∧ E(C, C ) ≤ ≤ I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ [(I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x)) → E(S1 ∩ C(S2 ), C(S1 )] ∧ S1 (y) ∧ C (S2 )(y) ∧ E(C, C ) = = I(S1 , S2 )∧S1 (x)∧C(S2 )(x)∧E(S1 ∩C(S2 ), C(S1 ))∧S1 (y)∧C (S2 )(y)∧ E(C, C ) ≤ ≤ S1 (y) ∧ C(S2 )(y) ∧ E(S1 ∩ C(S2 ), C(S1 )) ∧ E(C, C ) ≤ C(S1 )(y) ∧ E(C, C ) ≤ C (S1 )(y). Then (9.4) has been proved. Applying again (9.6) and Lemma 9.7 and Lemma 3.12 (vii) we obtain A(C) ∧ E(C, C ) ∧ I(S1 , S2 ) ∧ S1 (x) ∧ C (S2 )(x) ∧ C (S1 )(y) ≤ I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ [(I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x)) → E(S1 ∩ C(S2 ), C(S1 )] ∧ E(C, C ) ∧ C (S1 )(y) = = I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ∧ E(S1 ∩ C(S2 ), C(S1 )) ∧ E(C, C ) ∧ C (S1 )(y) ≤ ≤ C(S1 )(y) ∧ E(S1 ∩ C(S2 ), C(S1 )) ∧ E(C, C ) ≤ S1 (y) ∧ C(S2 )(y) ∧ E(C, C ) ≤ C(S2 )(y) ∧ E(C, C ) ≤ C (S2 )(y). Hence (9.5) was proved. Thus (9.2) and (9.3) follow, hence (9.1) holds for all S1 , S2 ∈ B and x ∈ X. By Lemma 4.3 (1) we get A(C)∧E(C, C ) ≤ (I(S1 , S2 )∧S1 (x)∧C (S2 )(x)) → E(S1 ∩C (S2 ), C (S1 )) for all S1 , S2 ∈ B and x ∈ X. Therefore

9.5 Consistency Indicators

A(C) ∧ E(C, C ) ≤

223

[(I(S1 , S2 ) ∧ S1 (x) ∧ C (S2 )(x)) → E(S1 ∩

S1 ,S2 ∈B x∈X

C (S2 ), C (S1 )]

i.e. A(C) ∧ E(C, C ) ≤ A(C ). The proof is ﬁnished.

9.5 Consistency Indicators In Chapter 6 the consistency properties F α and F β , fuzzy versions of Sen’s consistency conditions α and β have been studied. This section concerns the consistency indicators F α(C) and F β(C) associated with a fuzzy choice function C. F α(C) represents the degree in which C veriﬁes the consistency condition F α and F β(C) the degree in which C veriﬁes F β. A representative result of Chapter 6 shows that a fuzzy choice function C veriﬁes the congruence axiom W F CA if and only if it veriﬁes F α and F β. In this section we will prove the equality W F CA(C) = F α(C) ∧ F β(C), that is a strong reﬁnement of the mentioned theorem. In order to prove it some preliminary propositions with intrinsic interest are needed. We ﬁx a continuous t–norm ∗ and a fuzzy choice function C deﬁned on a fuzzy choice space (X, B). Deﬁnition 9.39. The consistency indicators F α(C) and F β(C) are deﬁned by

F α(C) =

[I(S, T ) ∗ S(x) ∗ C(T )(x) → C(S)(x)]

S,T ∈B x∈X

F β(C) =

[I(S, T ) ∗ C(S)(x) ∗ C(T )(y) → (C(S)(x) ↔ C(T )(y))]

S,T ∈B x∈X

Lemma 9.40. The following assertions hold:

(1) F α(C) = 1 iﬀ C veriﬁes F α; (2) F β(C) = 1 iﬀ C veriﬁes F β.

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9 Similarity and Rationality Indicators for Fuzzy Choice Functions

The real number F α(C) (resp. F β(C)) is called the F α–consistency indicator (resp. the F β–consistency indicator) of C. F α(C) expresses the degree to which C veriﬁes the condition F α and F β(C) the degree to which C veriﬁes F β. The results of this section are proved under the hypotheses H1 and H2 and by assuming that ∗ is the G¨ odel t–norm. Proposition 9.41. If C is a fuzzy choice function on (X, B) then F α(C) ≤ ¯ C ). E(RC , R ¯C ) = ¯ C ⊆ RC is always true, E(RC , R Proof. Since the inclusion R ¯ C (x, y)) = y) ↔ R

(RC (x,

x,y∈X

¯ C (x, y)). (RC (x, y) → R

x,y∈X

¯ C (x, y). Let S ∈ B. By Lemma 4.3 We prove that F α(C) ≤ RC (x, y) → R (2) F α(C) ∧ I([x, y], S) ∧ [x, y](x) ∧ C(S)(x) = = [((I(S, T ) ∧ S(u) ∧ C(T )(u)) → C(S)(u)] ∧ I([x, y], S) ∧ S,T ∈B u∈X

[x, y](x) ∧ C(S)(x) ≤ ≤ ((I([x, y], S)∧[x, y](x)∧C(S)(x)) → C([x, y])(x))∧I([x, y], S)∧[x, y](x)∧ C(S)(x) = I([x, y], S) ∧ [x, y](x) ∧ C(S)(x) ∧ C([x, y])(x) ≤ C([x, y])(x). We remark that I([x, y], S) = ([x, y](u) → S(u)) = u∈X

= ([x, y](x) → S(x)) ∧ ([x, y](y) → S(y)) = = (1 → S(x)) ∧ (1 → S(y)) = S(x) ∧ S(y).

Then the above inequality becomes: F α(C) ∧ I([x, y], S) ∧ [x, y](x) ∧ C(S)(x) = = F α(C) ∧ S(x) ∧ S(y) ∧ C(S)(x) = = F α(C) ∧ C(S)(x) ∧ S(y) ≤ C([x, y])(x). since C(S)(x) ≤ S(x). We have obtained F α(C) ∧ C(S)(x) ∧ S(y) ≤ C([x, y])(x) for each S ∈ B

9.5 Consistency Indicators

F α(C) ∧ RC (x, y) = F α(C) ∧

(C(S)(x) ∧ S(y)) =

S∈B

¯ C (x, y). C(S)(x) ∧ S(y)) ≤ C([x, y])(x) = R

225

(F α(C) ∧

S∈B

¯ C (x, y) and by Lemma 4.3 (1), Finally, F α(C) ∧ RC (x, y) ≤ R ¯ C (x, y). F α(C) ≤ RC (x, y) → R Proposition 9.42. If C is a fuzzy choice function on the fuzzy choice space ˆ (X, B) then W F CA(C) ≤ E(C, C). Proof. Let S ∈ B and x ∈ X. ˆ It suﬃces to prove that W F CA(C) ≤ C(S)(x) → C(S)(x). By Lemma 4.3 (1), this is equivalent to ˆ W F CA(C) ∧ C(S)(x) ≤ C(S)(x). By hypothesis H1, C(S) is a normal fuzzy subset of X, therefore there exists v ∈ X such that C(S)(v) = 1 and S(v) = 1. ˆ By applying the deﬁnitions of W F CA(C) and C(S) and Lemma 4.3 (2), (6), (11) we obtain ˆ W F CA(C) ∧ C(S)(x) = = [(RC (s, t) ∧ C(T )(t) ∧ T (s)) → C(T )(s)] ∧ S(x) ∧ (S(u) → T ∈B s,t∈X

RC (x, u)) ≤

u∈X

≤ [(RC (x, v) ∧ C(S)(v) ∧ S(x)) → C(S)(x)] ∧ S(x) ∧ [S(v) → RC (x, v)] = = [(RC (x, v) ∧ S(x)) → C(S)(x)] ∧ S(x) ∧ RC (x, v) = = S(x) ∧ [S(x) ∧ (RC (x, v) → C(S)(x))] ∧ RC (x, v) = = S(x) ∧ [RC (x, v) → C(S)(x)] ∧ RC (x, v) = = S(x) ∧ RC (x, v) ∧ C(S)(x) ≤ C(S)(x).

Proposition 9.43. If C is a fuzzy choice function on the fuzzy choice space ˆ ≤ F α(C). (X, B), then E(C, C) Proof. By applying the deﬁnition of F α(C), we have to prove

226

9 Similarity and Rationality Indicators for Fuzzy Choice Functions

ˆ ≤ E(C, C)

[(I(S, T ) ∧ S(x) ∧ C(T )(x)) → C(S)(x)].

S,T ∈B x∈X

It suﬃces to prove that for any S, T ∈ B and x ∈ X, ˆ ∧ I(S, T ) ∧ S(x) ∧ C(T )(x) ≤ C(S)(x). E(C, C) ˆ = ˆ E(C, C) ρ(C(S)(x), C(S)(x)) =

S∈B x∈X

ˆ (C(S)(x) → C(S)(x)) since

S∈B x∈X

ˆ ˆ ρ(C(S)(x), C(S)(x)) = C(S)(x) → C(S)(x). By applying Lemma 4.3 (12) and Lemma 4.5 (7) we get ˆ )(x) = I(S, T )∧S(x)∧ C(T (S(u) → T (u))∧S(x)∧T (x)∧ (T (u) → RC (x, u)) ≤ ≤ S(x) ∧

u∈X

u∈X

[(S(u) → T (u)) ∧ (T (u) → RC (x, u))] ≤

u∈X

≤ S(x) ∧

ˆ (S(u) → RC (x, u)) ≤ C(S)(x)

u∈X

ˆ )(x) ≤ C(S)(x). ˆ We have proved that I(S, T ) ∧ S(x) ∧ C(T By applying Lemma 4.3 (2) we compute ˆ ˆ ˆ )(x) ≤ E(C, C)∧I(S, T )∧S(x)∧C(T )(x) ≤ E(C, C)∧I(S, T )∧S(x)∧ C(T ˆ ∧ I(S, T ) ∧ S(x) ∧ C(T ˆ )(x) ≤ E(C, C) ˆ ∧ C(S)(x) ˆ ≤ E(C, C) = ˆ )(y) → C(S )(y)] ∧ C(S)(x) ˆ = [C(S ≤ S ∈B y∈X

ˆ ˆ ≤ [C(S)(x) → C(S)(x)] ∧ C(S)(x) = ˆ = C(S)(x) ∧ C(S)(x) = C(S)(x).

Proposition 9.44. If C is a fuzzy choice function on the fuzzy choice space (X, B), then W F CA(C) ≤ F α(C). Proof. By Propositions 9.42 and 8.43. Proposition 9.45. If C is a fuzzy choice function on the fuzzy choice space (X, B) then W F CA(C) ≤ F α(C) ∧ F β(C).

9.5 Consistency Indicators

227

Proof. We have proved in Proposition 9.44 that W F CA(C) ≤ F α(C). We still have to prove that W F CA(C) ≤ F β(C). Suppose by absurdum W F CA(C) ≤ F β(C). Then there exist S, T ∈ B, x, y ∈ X, such that W F CA(C) ≤ (I(S, T ) ∧ C(S)(x) ∧ C(S)(y)) → [C(T )(y) ↔ C(T )(x)], i.e. W F CA(C) ∧ I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ C(T )(x) ↔ C(T )(y) = [C(T )(x) → C(T )(y)] ∧ [C(T )(x) → C(T )(y)]. Therefore W F CA(C) ∧ I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ C(T )(x) → C(T )(y) or W F CA(C) ∧ I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ C(T )(y) → C(T )(x). Suppose W F CA(C) ∧ I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ≤ C(T )(x) → C(T )(y). It follows (a) W F CA(C) ∧ I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ∧ C(T )(x) ≤ C(T )(y). One remarks W F CA(C) ∧ I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ∧ C(T )(x) ≤ ≤ W F CA(C) ∧ C(S)(y) ∧ S(x) ∧ C(T )(x) ≤ C(S)(y) ∧ S(x) ∧ C(T )(x) ∧ S(y) because C(S)(y) ∧ S(y) = C(S)(y). Since C(S)(y) ∧ S(x) ≤ RC (y, x) one obtains (b) W F CA(C) ∧ I(S, T ) ∧ C(S)(x) ∧ C(S)(y) ∧ C(T )(x) ≤ W F CA(C) ∧ RC (y, x) ∧ C(T )(x) ∧ S(y). By (a) and (b) one infers (c) W F CA(C) ∧ RC (y, x) ∧ C(T )(x) ∧ S(y) ≤ C(T )(y). By applying Lemma 4.3 (2) we obtain: W F CA(C) ∧ [RC (y, x) ∧ C(T )(x) ∧ S(y)] = = [RC (x, y)∧C(S)(y)∧S(x) → C(S)(x)]∧(RC (y, x)∧C(T )(x)∧ x,y∈X S∈B

S(y)) ≤

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9 Similarity and Rationality Indicators for Fuzzy Choice Functions

[RC (y, x) ∧ C(T )(x) ∧ S(y) → C(T )(y)] ∧ (RC (y, x) ∧ C(T )(x) ∧ S(y)) = RC (y, x) ∧ C(T )(x) ∧ S(y) ∧ C(T )(y) ≤ C(T )(y). We obtain W F CA(C) ∧ RC (y, x) ∧ C(T )(x) ∧ S(y) ≤ C(T )(y) that contradicts (c). Proposition 9.46. If C is a fuzzy choice function on the fuzzy choice space (X, B) then F α(C) ∧ F β(C) ≤ W F CA(C). Proof. Let S ∈ B and x, y ∈ X. F α(C) ∧ I([x, y], S) ∧ [x, y](y) ∧ C(S)(y) ≤ (I([x, y], S) ∧ [x, y](y) ∧ C(S)(y)) → C([x, y])(y)) ∧ I([x, y], S) ∧ [x, y](y) ∧ C(S)(y) = I([x, y], S) ∧ [x, y](y) ∧ C(S)(y) ∧ C([x, y])(y) ≤ C([x, y])(y). Since I([x, y], S) = S(x) ∧ S(y) we have F α(C) ∧ S(x) ∧ C(S)(y) ∧ RC (x, y) = F α(C) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ RC (x, y) = F α(C) ∧ I([x, y], S) ∧ [x, y](y) ∧ C(S)(y) ∧ S(x) ∧ S(y) ∧ RC (x, y) ≤ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ RC (x, y). Explicating RC (x, y), we obtain (c) F α(C) ∧ S(x) ∧ C(S)(y) ∧ RC (x, y) ≤ F α(C) ∧ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧

(C(H)(x) ∧ H(y)) =

H∈B

(F α(C) ∧ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ C(H)(x) ∧ H(y)).

H∈B

We prove that (d) F α(C) ∧ C(H)(x) ∧ H(y) ≤ C([x, y])(x). Indeed, according to the deﬁnition of F α(C) F α(C) ∧ C(H)(x) ∧ H(x) = F α(C) ∧ C(H)(x) ∧ H(y) ∧ H(x) = F α(C) ∧ I([x, y], S) ∧ [x, y](x) ∧ C(H)(x) ≤ [I([x, y], H) ∧ [x, y](x) ∧ C(H)(x) → C([x, y])(x)] ∧ I([x, y], H) ∧ [x, y](x) ∧ C(H)(x)] =

9.5 Consistency Indicators

229

I([x, y], H) ∧ [x, y](x) ∧ C(H)(x) ∧ C([x, y])(x) ≤ C([x, y])(x).

By applying the deﬁnition of F β(C) and (c) F β(C) ∧ F α(C) ∧ S(x) ∧ C(S)(y) ∧ RC (x, y) ≤ [F α(C) ∧ F β(C) ∧ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(H)(x) ∧ H(y)]. H∈B

Considering one member of this union and applying (d) F α(C) ∧ F β(C) ∧ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(H)(x) ∧ H(y) ≤ ≤ F α(C) ∧ F β(C) ∧ C([x, y])(y) ∧ S(x) ∧ S(y) ∧ C(S)(y) ∧ C([x, y])(x) = = F α(C) ∧ F β(C) ∧ I([x, y], S) ∧ C([x, y])(x) ∧ C([x, y])(y) ∧ C(S)(y) ≤ [(I([x, y], S) ∧ C([x, y])(x) ∧ C([x, y])(y)) → (C(S)(x) ↔ C(S)(y))] ∧ (I([x, y], S) ∧ C([x, y])(x) ∧ C([x, y])(y) ∧ C(S)(y) ≤ ≤ (C(S)(x) ↔ C(S)(y)) ∧ C(S)(y). Consequently, F β(C) ∧ F α(C) ∧ S(x) ∧ C(S)(y) ∧ RC (x, y) ≤ ≤ (C(S)(x) ↔ C(S)(y)) ∧ C(S)(y) ≤ C(S)(x). By Lemma 4.3 (1) F β(C) ∧ F α(C) ≤ (S(x) ∧ C(S)(y) ∧ RC (x, y)) → C(S)(x), therefore F β(C) ∧ F α(C) ≤

[(S(x) ∧ C(S)(y) ∧ RC (x, y)) → C(S)(x)] =

S∈B x,y∈X

W F CA(C).

Theorem 9.47. If C is a fuzzy choice function on the fuzzy choice space (X, B) then F α(C) ∧ F β(C) = W F CA(C). Proof. By Propositions 9.45 and 9.46. Remark 9.48. From Theorem 9.47 it follows that a fuzzy choice function C veriﬁes W F CA iﬀ it veriﬁes conditions F α and F β (Theorem 6.13). Accordingly, Remark 9.48 says that the statements “C veriﬁes W F CA” and “C veriﬁes F α and F β” are simultaneously true, and Theorem 9.47 strengthens this result, showing that their degrees of truth are equal.

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9 Similarity and Rationality Indicators for Fuzzy Choice Functions

Example 9.49. In this example a particular class of fuzzy choice function is studied. Let X = {x, y} and B = {A, B} where A, B ∈ F(X) are given by A = αχ{x} + χ{y} ; B = χ{x} + βχ{y} (0 ≤ α, β ≤ 1). Consider the function C : B → F(X) deﬁned by: C(A) = γχ{x} + χ{y} ; C(B) = χ{x} + δχ{y} (0 ≤ γ ≤ α, 0 ≤ δ ≤ β). For any α, β, γ, δ as above, C is a fuzzy choice function on (X, B). First we compute the fuzzy preference relation RC : RC (x, x) = RC (y, y) = 1; RC (x, y) = (C(A)(x) ∧ A(y)) ∨ (C(B)(x) ∧ BC (y)) = β ∨ γ; RC (y, x) = (C(A)(y) ∧ A(x)) ∨ (C(B)(y) ∧ B(x)) = α ∨ δ. Thus  1

β∨γ



. α∨δ 1 We also obtain I(A, A) = I(B, B) = 1, I(A, B) = β, I(B, A) = α. RC = 

We intend to compute the congruence indicator W F CA(C) and the consistency indicators F α(C) and F β(C). We obtain: W F CA(C) = [(α ∧ β) → γ] ∧ [(α ∧ β) → δ] = (α ∧ β) → (γ ∧ δ); F α(C) = (α ∧ β) → (γ ∧ δ); F β(C) = [(β ∧ γ) → δ] ∧ [(α ∧ δ) → γ]. In particular, the fuzzy choice function C veriﬁes W F CA iﬀ α ∧ β = γ ∧ δ; C veriﬁes F β iﬀ β ∧ γ ≤ δ and α ∧ δ ≤ γ. Next we verify the equality W F CA(C) = F α(C) ∧ F β(C) proved in Theorem 9.46 Since F α(C) = W F CA(C), it suﬃces to prove F α(C) ≤ F β(C), that means to check whether the following inequality holds: (α ∧ β) → (γ ∧ δ) ≤ [(β ∧ γ) → δ] ∧ [(α ∧ δ) → γ]. This gets down to proving: (α ∧ β) → (γ ∧ δ) ≤ (β ∧ γ) → δ and

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231

(α ∧ β) → (γ ∧ δ) ≤ (α ∧ δ) → γ. We will prove only the ﬁrst inequality. By Lemma 4.3 (1) we have to prove: [(α ∧ β) → (γ ∧ δ)] ∧ (β ∧ γ) ≤ δ. By computing the left–hand right member of this inequality and by Lemma 4.3 (2), we get: [(α ∧ β) → (γ ∧ δ)] ∧ (β ∧ γ) = = γ ∧ β ∧ (β → [α → (γ ∧ δ)]) ≤ ≤ α ∧ β ∧ [α → (γ ∧ δ)] = β ∧ α ∧ (γ ∧ δ) ≤ δ.

10 Applications

In making a choice, a set of alternatives and a set of criteria are usually needed. According to [119], the alternatives and the criteria are deﬁned as follows: “Alternatives are usually mutually exclusive activities, objects, projects, or models of behaviour among which a choice is possible”. “Criteria are measures, rules and standards that guide decision making. Since decision making is conducted by selecting or formulating diﬀerent attributes, objectives or goals, all three categories can be referred as criteria. That is, criteria are all those attributes, objectives or goals which have been judged relevant in a given decision situation by a particular decision maker (individual or group)”. In the real world the human vagueness and imprecision prevail and fuzzy logic is introduced as a tool for modelling the acts of choice. In this chapter we shall present three possible applications of fuzzy revealed preference theory. They represent models of decision making based on the ranking of alternatives according to fuzzy choices. An agent’s decision is based on the ranking of alternatives according to diﬀerent criteria. This ranking is obtained by using fuzzy choice problems and the instrument by which it is established is the degree of dominance associated with a fuzzy choice function.

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 233–263 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

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10 Applications

10.1 Fuzzy Choices Support for Agent–based Automated Negotiations In recent years, the rapid changing business environment demands the business organizations have more frequent and closer collaboration with their business partners. However, such business negotiation process has suﬀered from high information fragmentation, time and cost overrun, and dispute in negotiation for discussing multiple issues for reaching mutual beneﬁcial agreements for a business goal [67], [28]. This brings many diﬃculties for the decisionmakers. Software agents, as a set of computational programs or problem solving entities, oﬀer new means and tools, which are characterized as autonomous, cooperative, and learning in concurrent and collaborative working environment for accomplishing a speciﬁc task [114], [78]. Among them, one of the most essential characters of the multi-agent systems is that several agents on behalf of diﬀerent parties are able to interact and communicate with each other and negotiate on multiple issues towards to automatically cope with the problems in the environment [105], [38]. In an online marketplace, potential buyer agents can exchange information with seller agents based on their own preferences. Especially, the electronic marketplace allows the buyer agents to search for the best available oﬀers and this is the same for the seller agent to seek for the most suitable orders. In this section, we focus on agent negotiation approach; we do not consider negotiation protocol and negotiation languages, aiming to minimize the involvement of human beings and to bring a negotiation system. Thus, all parties involved in negotiation refer to the software agents, which intend to form an automated negotiation. According to [115], the multi-attribute negotiation process can be simpliﬁed into three steps: (1) attribute evaluation, (2) utility determination, and (3) attribute planning. Among them, attribute evaluation is the ﬁrst essential step whereby the value of the attributes is evaluated, based

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on the preferences of the participants. Commonly, there are two categories of the attributes: quantitative (e.g. cost and time) and qualitative (e.g. quality, safety, and environment). In this example, we will focus on quantitative attributes. Here a new approach is proposed and presented by employing agents in the electronic marketplace to carry out automated negotiation on behalf of diﬀerent parties and participants in the business transaction on multiple issues.

10.1.1 Related Work Paper [110] discusses a negotiation situation on an electronic market with one seller and multiple buyers. The negotiation takes place according to multiple criteria that regard attributes of the products or services negotiated. Next we will give a short description of their model. A person wants to sell a product on an electronic market and he has to choose one buyer among several buyers. The negotiation issues here refer to the multiple criteria that consist of the attributes of the product (e. g. price, delivery times, etc.). The negotiator (=seller) receives oﬀers from the potential buyers and consequently, he deﬁnes his preferences, represented by a fuzzy preference relation on the set of alternatives (=buyers). By analyzing this fuzzy preference relation by means of α-cut levels, the oﬀers of the buyers will be ranked. The simplest and most common solution for choosing the best buyer is to evaluate each oﬀer by associating a weight to every negotiable issue, to calculate a weighted sum for each oﬀer and to choose the oﬀer with the highest value. This solution assumes that all oﬀers are comparable. In the model proposed in [110] the ranking of the oﬀers is partial, based on pairwise comparisons, compared to other approaches where the rank order was linear. This approach is closer to real life situations. Next we give the main steps of the approach in [110].

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10 Applications

Suppose there are m criteria C1 , . . . , Cm and n oﬀers x1 , . . . , xn of n buyers. The preference of the seller with respect to the values speciﬁed by the buyers are represented in a matrix P = (pij )n×m . The real number pij ∈ [0, 1] shows the degree of preference of the seller with respect to the value oﬀered by xi on criterion Cj . The row Pi = (pi1 , . . . , pim ) is the vector of the seller’s preferences with respect to the i-th buyer. A pairwise comparison between vectors Pi and Pj , i, j = 1, . . . , n is required in order to establish a ranking of the oﬀers. For such vectors Pi and Pj the real number Inc(Pi , Pj ) shows the degree of inclusion of Pi in Pj :

(1) Inc(Pi , Pj ) =

1 m

m

min(1, 1 − Pik + Pjk ).

k=1

This degree of inclusion is summarized in the matrix D = (Inc(Pi , Pj ))n×n that is a fuzzy preference relation on the set of alternatives X = {x1 , . . . , xn }. Notice that P is reﬂexive but not transitive. By computing its transitive closure Q one obtains a fuzzy preorder. The properties of matrix Q allow for a better interpretation of the oﬀers [110]. Starting from Q, we consider the α-cuts at various levels, obtaining crisp preorders Qα , α ∈ (0, 1] [40]. Each preorder Qα leads to an equivalence relation Eα : (x, y) ∈ Eα iﬀ (x, y) ∈ Qα ∧ (y, x) ∈ Qα Denote by [x]α the equivalence class of x ∈ X with respect to Eα . Deﬁne the order relation ≤α on the quotient set X/Eα of the equivalence classes: [x]α ≤α [y]α iﬀ (x, y) ∈ Qα . With ≤α one can establish a hierarchy of oﬀers for each criterion.

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237

10.1.2 Fuzzy Choices in Automated Negotiations This subsection studies a modiﬁed form of the problem described in the previous subsection. If there the ranking of alternatives is based on a fuzzy preference relation, here this ranking will result from a fuzzy choice function. We will arrive at a choice problem, and for the ranking of oﬀers for each criterion we will apply the degree of dominance introduced previously. We start from a similar situation as above and we indicate a way to obtain the matrix P of fuzzy preferences. A person wants to sell a product described by some attributes and he registers it on an electronic market. He has to choose one buyer among several buyers for his product, and his choice will be made according to the oﬀers that he gets. The oﬀers are made with respect to the criteria given by the attributes of the product. Therefore we are in face of a multiple-criteria negotiation problem with one seller and multiple buyers. Our treatment of the problem diﬀers from [110]. In [110] the ranking of alternatives is derived from a somehow given fuzzy preference relation and not from the act of choice. In our case the ranking of alternatives is based on a choice function associated with a preference relation P and on some (crisp) available sets of alternatives deﬁned by the thresholds e1 , . . . , en . The negotiation issues (=criteria) are denoted by C1 , . . . , Cm . The seller proposes the values b1 , . . . , bm for his product. The i-th potential buyer of the product responds with the oﬀer ai1 , . . . , aim . In order to have a fuzzy choice problem, we can assume that 0 < aij ≤ 1, respectively 0 < bj ≤ 1 for any i = 1, . . . , n, j = 1, . . . , m. Otherwise, this can be obtained by dividing all values aij , bj by a convenient power of 10. In this way the values aij , bj preserve their initial values, hence the preferences remain the same. These values are summarized in the following table:

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10 Applications

C1 C2 . . . Cm Seller’s oﬀer

b1 b 2 . . . b m

First buyer’s oﬀer a11 a12 . . . a1m Second buyer’s oﬀer a21 a22 . . . a2m ... n-th buyer’s oﬀer an1 an2 . . . anm

The values proposed by the seller generally diﬀer from the values proposed by the buyers. The distance |bj − aij | measures the closeness of the values that belong to the buyers and to the seller with respect to criterion j. If |bj − aij | < |bj − akj | then obviously the seller will prefer the oﬀer aij to the oﬀer akj . The number pij = 1−|bj −aij | will represent the degree to which the seller prefers his value bj to the value aij suggested by the buyer i on criterion j. If the number pij tends to 1, then aij reaches bj therefore the seller and the buyer i reach consensus with respect to negotiation issue j. When pij reaches 0 then there will be a dissension between them. The values pij are represented in a matrix P with n rows and m columns. Consider now similarly as in the previous model [110] the matrix D = (Inc(Pi , Pj ))n×n , where Pi is the i-th row and Inc(Pi , P j) is given by (1). According to [110], matrix D on the set of the buyers comprises the inclusion degrees of the rows Pi . Denote by Q = (qij )n×n the transitive closure of D. In diﬀerent stages of negotiation, it is possible that the buyers’ oﬀers are exaggerated for the seller. Therefore, we need to introduce the thresholds e1 , . . . , en ∈ (0, 1). If |bj − aij | ≤ ej then we say that the values aij are admissible. By introducing these thresholds, the buyers whose oﬀers diﬀer to a large extent from the seller’s proposal are eliminated. Deﬁne the crisp subsets S1 , . . . , Sm of X:

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239

(2) Sj = {xi ∈ X||bj − aij | ≤ ej }. Take B = {S1 , . . . , Sm } and (X, B) is the choice space. In this manner we have obtained a context similar to [14], with the diﬀerence that here B contains only S1 , . . . , Sm . On the choice space (X, B) deﬁne the fuzzy choice function C by

(3) C(Sj )(xi ) =

Q(xi , y)

y∈Sj

We remark that by identifying a crisp subset of X with its characteristic function, the fuzzy choice function deﬁned in (3) is a particular case of the class of fuzzy choice functions introduced by Deﬁnition 5.30 (i). In this context the degree of dominance DSj (xi ) gets the form:

(4) DSj (xi ) =

[C(Sj )(y) → C(Sj )(xi )]

y∈X

The degree of dominance DSj (xi ) reﬂects the dominance of alternative xi with respect to criterion j. Ordering the set {DSj (x1 ), . . . , DSj (xn )} one obtains a ranking of alternatives x1 , . . . , xn with respect to criterion j. Extending the approach, we can compute the aggregated degree of dominance of the alternative xi with respect to criterion j:

(5) D(xi ) =

1 n

n

DSj (xi ), i = 1, . . . , n.

j=1

The fuzzy choice function deﬁned by us is based on the preferences pij derived from the buyers’ oﬀers and on the values bj proposed by the seller; obviously the seller’s choices are potential. The fuzzy choice function gives the seller information about the ranking of alternatives by means of potential choices. This information helps the seller to negotiate with those buyers situated in a superior position in this hierarchy.

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10 Applications

10.1.3 Numerical Illustration In this section we shall illustrate the theoretical analysis from above with one example, pointing out the two approaches presented in Subsections 10.1.1 and 10.1.2 . Consider the case of a seller that registers a product on an electronic market and describes it according to three criteria, for example Price, Delivery time and Warranty. Five buyers are interested in buying this product. The following table summarizes the reservation prices proposed by the seller and the reservation prices of the buyers: C1 C2 C3 Seller’s oﬀer

0.5 0.6 0.7

First buyer’s oﬀer 0.47 0.57 0.41 Second buyer’s oﬀer 0.45 0.55 0.55 Third buyer’s oﬀer 0.39 0.56 0.68 Fourth buyer’s oﬀer 0.48 0.58 0.67 Fifth buyer’s oﬀer 0.46 0.53 0.66

The values of the thresholds are e1 = 0.05, e2 = 0.06 and e3 = 0.07. Next we present the main results. The matrix P describes the seller’s preferences with respect to the three criteria. 

 0.97 0.97 0.71

   0.95   P =  0.89    0.98  0.96

  0.95 0.85    0.96 0.98 .   0.98 0.97   0.93 0.96

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241

For example the value p11 = 0.97 represents the degree to which the seller prefers his reservation price to the ﬁrst buyer’s reservation price with respect to the ﬁrst criterion (in this case the degree of preference is very high). If we compare the reservation prices of the ﬁve potential buyers with respect to the ﬁrst criterion of the product and look at the ﬁrst column of P , the seller will prefer to a high degree to keep his reservation price to the oﬀers of the buyers. We compute next the degrees of inclusion Inc(Pi , Pj ) according to (1) and we obtain the matrix D: 

 1 0.98 0.97 1 0.98

   0.95   D =  0.91    0.90  0.91

  1 0.98 1 0.99    0.95 1 0.99 0.98 .   0.94 0.96 1 0.97   0.96 0.97 1 1

The transitive closure of D is matrix Q: 

 1 0.98 0.98 1 0.98

   0.95   Q =  0.95    0.95  0.95

  1 0.98 1 0.99    0.96 1 0.99 0.98 .   0.96 0.97 1 0.97   0.96 0.97 1 1

The crisp sets S1 , S2 and S3 corresponding to the three criteria that characterize the product are given by (2): S1 = {x1 , x2 , x4 , x5 } S2 = {x1 , x2 , x3 , x4 } S3 = {x1 , x2 , x3 }

The corresponding fuzzy choice functions are:

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10 Applications

C(S1 ) = 0.98χ{x1 } + 0.95χ{x2 } + 0.95χ{x4 } + 0.95χ{x5 } C(S2 ) = 0.98χ{x1 } + 0.95χ{x2 } + 0.95χ{x3 } + 0.95χ{x4 } C(S3 ) = 0.98χ{x1 } + 0.95χ{x2 } + 0.95χ{x3 } . The corresponding degrees of dominance are represented in the following table: DSi (xj ) x1 x2

x3

x4

x5

S1

1 0.95 0 0.95 0.95

S2

1 0.95 0.95 0.95 0

S3

1 0.95 0.95 0

0

The aggregated degrees of dominance are according to (5): D(x1 ) = 1, D(x2 ) = 0.95, D(x3 ) = 0.63, D(x4 ) = 0.63, D(x5 ) = 0.31. Therefore the order of the oﬀers is D(x5 ) < D(x3 ) = D(x3 ) < D(x2 ) < D(x1 ). Figure 10.1 illustrates the hierarchy of alternatives according to each criteria and overall.

10.1.4 Van de Walle et al.’s Approach We apply for the same input data Van de Walle et al.’s approach aiming to compare the two methods. As in [110], we consider the α–cuts at the levels {0.95, 0.96, 0.97, 0.98, 0.99, 1} and consequently the matrix Q at these levels:

10.1 Fuzzy Choices Support for Agent–based Automated Negotiations

243

x1

x2 x1

x1

x1

x3 , x4 x2 , x4 , x5

x 2 , x3

x2 , x3 , x4

x5

S1

S3

S2

Overall

Fig. 10.1. Hierarchy of axioms





Q0.96

11   0 1   = 0 1   0 1  01

111









11111 11         0 1 1 1 1 0 1 1 1 1         1 1 1 , Q0.97 =  0 0 1 1 1 , Q0.98 =  0 0         0 0 1 1 1 0 0 1 1 1     111 00111 00     10010 10010         0 1 0 1 1 0 1 0 1 0         Q0.99 =  0 0 1 1 0 , Q1 =  0 0 1 0 0 ,         0 0 0 1 0 0 0 0 1 0     00011 00011

111

  1 1 1   1 1 1 ,   0 1 0  011

If we analyze the fuzzy preference matrix at the cut-levels, we notice that at α = 0.96, oﬀer 1 is separated from the rest of the oﬀers, therefore it will be preferred. At α = 0.97, oﬀers 1 and 2 are both separated from oﬀers 3, 4 and 5 respectively. Still oﬀer 1 is preferred to oﬀer 2. At α = 0.98, all oﬀers

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are diﬀerent. Oﬀer 1 is preferred to oﬀer 2, oﬀer 2 is preferred to oﬀer 3, oﬀer 3 is preferred to oﬀer 5 and oﬀer 5 is preferred to oﬀer 4. At α = 0.99 oﬀer 4 is the least preferred, while oﬀers 1, 2, 3, and 5 are equally preferred by the buyer. Finally, at α = 1 oﬀers 2 and 3 are not mutually comparable and neither with the rest, oﬀer 4 is the least preferred to oﬀers 1 and 5 which cannot be compared either.

Comparison of the Two Approaches Previously we have developed a simple multi-issue negotiation model with one seller and multiple buyers. The vagueness of the preferences and choices that characterizes the transactions on electronic markets and that is speciﬁc to the process of decisionmaking is very diﬃcult to model. So far there exist some agent-based emarketplaces that try to represent this vagueness. Comprehensive classiﬁcations and descriptions of existing agent-based e-marketplaces with crisp and/or fuzzy preferences of sellers and buyers are made in [67], [71]. In the negotiation stages between the buyers’ oﬀers and the seller’s proposals big diﬀerences might exist. By introducing the thresholds e1 , . . . , en , these diﬀerences will be eliminated. By the presence of the thresholds our model diﬀers from [110]. Another diﬀerence consists in the way the alternatives are ranked. In [110] the ranking is based on a preference relation and here the ranking is based on the degree of dominance that directly expresses the act of choice.

Conclusion In this example, we have presented a new approach to process vague preferences and evaluate information that can better support decision-making in fuzzy (vague) environments speciﬁcally in electronic marketplaces. With this model, we have shown how the decision maker (the seller) can precisely choose

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an alternative through the rank oﬀers according to the degree of dominance of fuzzy choice functions. Such algorithms could be developed for software agents in the multi-agent systems to semi-autonomously act to provide decision support. However, our study still has some limitations. Firstly, our approach did not discuss the qualitative data, which usually are important aspects in the negotiation process; this would be a topic for future research. Secondly, an interesting problem would be to analyze the seller’s criteria with the same method. This would help the seller to focus on the most important criteria.

10.2 Reducing Adverse Selection by Fuzzy Choices Adverse selection was originally used in insurance markets to describe a situation when the buyer of an insurance policy usually knows more about her individual risk than the insurance company does. Adverse selection was later studied for other types of markets where information asymmetry might exist. The most well-known situation of adverse selection has been analyzed by Akerlof [5] for the market of used cars (see also [62], [20], [42]). We shortly present this model. Two types of car are sold on a market at the same price: good-quality cars and bad-quality cars. The sellers perfectly know the quality of the cars while the buyers do not know what type of cars they buy. Consequently only the bad-quality cars will dominate the market and they will drive away the good-quality cars. This situation has negative consequences, in the sense that the market cannot allocate the products eﬃciently and that leads to market failures. The analysis of markets with asymmetric information represents one of the most challenging research topics in microeconomic theory. Asymmetry of information is predominant in market interactions. We can exemplify this situation by various instances. A seller is better informed on the

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quality of the product she sells than the prospective buyer. A job applicant has better information on her working abilities than the potential employer. The buyer of an insurance policy usually knows more about her individual risk than the insurance company does. All these situations are characterized by a common feature: one party is better informed than the other. This phenomenon is called adverse selection. In many economic situations, the adverse selection might obstruct transactions that are beneﬁcial for both sides of the market. In this example we propose a model of decision making of one buyer and several sellers based on the ranking of alternatives according to fuzzy choices. Here the criteria of choice are derived from the partial information existing in the model. These criteria are represented by fuzzy available sets of alternatives. Diﬀerent from other possible solutions, our model tries to correct the adverse selection that appears as the result of asymmetric information between buyers and sellers by interpreting the vectors that contain the partial information (=qualitative) as criteria for the buyers in decision making and by combining it with the certain information existing in the model. In this way the decision making process of buyers in the presence of partial information will be improved.

10.2.1 The Harmfulness of Adverse Selection Regardless of the types of market (labor, insurance and so on), a potential consequence of information asymmetry is adverse selection. We discuss brieﬂy this situation by comparing products a and b. On the marketplace of second hand goods (see e.g. the market of second-hand cars [5]), it is usually the case that only the sellers know the real quality of the products they sell and the buyers know the average quality of the products; consequently, the buyers are willing to pay the average price P . Therefore, only the sellers that have the worst qualitative products (b: lemons=the worst quality car in Akerlof

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terminology) can accept the average price P (P > Pb ). The sellers with inferior products would be the eagerest to sell their products. Then the sellers who have better quality products (a) will not be willing to sell their merchandise below the average price P (P < Pa ) and they will quit the market gradually. This leads to a decrease in the average price. Then the total quality of the marketplace decreases, the average price gets lower, and the eﬀect of “adverse selection” occurs. This is due to the fact that the sellers with high-quality products will hesitate to enter the market and accordingly, only the sellers with low-quality products can enter and dominate the market. In this way, the low-quality products “destroy” the market. Relying on the price competence, the low-quality products keep the high-quality products out of the market and also destroy the conﬁdence of the consumers in the market. Furthermore, such phenomenon also leads to total decrease of the overall quality of the products; at the macroeconomic level, social welfare decreases.

10.2.2 Conventional Solutions for Reducing the Eﬀects of Adverse Selection The fundamental method to improve or avoid the social phenomenon of adverse selection is that the buyers (whatever the employers or the customers) should have the abilities to identify the quality of the merchandise by using information. The following is a review of conventional possible approaches to avoid adverse selection. The core target is to ﬁgure out the hidden attributes of an object, which show the cost performance. In general, the high-quality merchant should know the way to convey information and enable consumers to ﬁnd information. (1) Sellers providing quality guarantees or making advertisement to the buyers as “signals” can prove the goods have high quality and worth the price. Media is also a good way to keep asymmetry information away. Spence [99] formulated ﬁrst this research problem on “signaling” on the job market.

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10 Applications

At the same time the uninformed buyers could “screen” for quality of the products. (2) By risk assessment, products are evaluated both through superﬁcial and hidden appearances. (3) The governmental policies should be directed to quality control through authentication organizations to identify the inferior products. (4) Rational consumers can also “vote by foot” to reveal their preference. 10.2.3 A Multiple Criteria Decision Making Model In this subsection we propose a multiple criteria decision making model based on the ranking of alternatives according to fuzzy choices. Here the criteria are derived from the partial information existing in the model. Our model tries to correct the adverse selection that appears as the result of asymmetric information between buyers and sellers by interpreting the vectors that contain the partial information (=quality) as criteria for the buyers in decision making. A buyer is interested in purchasing m types of products P1 , . . . , Pm . To acquire them n potential sellers present their oﬀers x1 , . . . , xn . On these products certain information such as the cost, the delivery time etc. exists, but also partial information such as the quality of the products exists. The partial information on the quality of the products is summarized in the following table: P1 P2 . . . Pm First seller’s information a11 a12 . . . a1m Second seller’s information a21 a22 . . . a2m ... n-th seller’s information an1 an2 . . . anm

The number aij ∈ (0, 1], i = 1, . . . , n, j = 1, . . . , m describes the quality of the product Pj in oﬀer xi . From the n oﬀers (=alternatives) only one has to be

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chosen. For a better decision, the buyer hires m experts, one for each product. The experts can say an opinion only on the certain information regarding the product. The company will take into account the partial (qualitative) information of the product, each vector aj , j = 1, . . . , m corresponding to a product. The vectors aj play the role of constraints that should be considered in the act of choice. Since the information is partial - expressed by numbers in the unit interval - there will be vague criteria, therefore fuzzy sets. When deciding on the oﬀer xi , the buyer will consider the aggregated opinion of the experts and the information in vectors aj . This is the reason why we might consider these vectors as criteria in the process of decision making. The result of the expertise of the k-th expert is given in a matrix k k Qk = (qij ) of dimension n × n. In interpretation, qij means that the k-th

expert considers the oﬀer xi at least as good as oﬀer xj as far as the product Pk is concerned. k We impose the natural condition of reﬂexivity on matrix Qk : qii = 1 for

i = 1, . . . , n. This way the opinion of each expert is modelled as a fuzzy reﬂexive preference relation. The information collected from all experts is aggregated in the matrix m k 1 Q = (qij )n×n : qij = m qij , i, j = 1 . . . , n. k=1

The information given by matrix Q regards the m products. Notice that 0 < qij ≤ 1 for all i, j = 1, . . . , n, hence Q is a fuzzy preference relation. The number qij shows the degree to which the expertise overall decided that alternative xi is at least as good as alternative xj . Obviously Q is reﬂexive. So far we have a preference fuzzy relation given by Q and m vectors aj = (a1j , . . . , anj ), j = 1, . . . , m. As discussed above, the vectors a1 , . . . , am can be considered criteria that the buyer should consider in choosing the best oﬀer. Now we are in the position to formulate a problem with fuzzy choices. For simplicity we will use the G¨ odel t-norm.

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10 Applications

Let us denote by X = {x1 , . . . , xn } the set of alternatives (=oﬀers) and by S1 , . . . , Sm the fuzzy subsets of X: Sj (xi ) = aij for all i = 1, . . . , n, j = 1, . . . , m. Denote β = {S1 , . . . , Sm }. (X, β) represents the fuzzy choice space. Next we investigate this choice problem. To the fuzzy preference relation Q on X we assign a fuzzy choice function C : β → F(X) deﬁned by: C(S)(x) = S(x) ∧ [S(y) → Q(x, y)], for any S ∈ β and x ∈ X y∈X

Recall that for this fuzzy choice function, C(S)(x) is the degree of truth of the statement “x is one of the Q-greatest alternatives verifying criterion S”. In theory it is possible to ﬁnd S ∈ β such that C(S)(xi ) = 0 for i = 1, . . . , n. In this case C is no longer a fuzzy choice function. As we cannot repeat the expertise until C becomes a fuzzy choice function, we have to make the assumption that for any S ∈ B, C(S)(xi ) > 0, for any i = 1, . . . , n. This assumption is very realistic in practice: for each of the m products there will always exist some information regarding its quality from each seller. Once the fuzzy choice function has been constructed, two issues are raised: 1) To ﬁnd to what extent the mathematical form of the choice (given by C) matches the expertise; 2) To ﬁnd a procedure to establish the hierarchy of the alternatives according to each of the m criteria and overall. We discuss now these issues. 1) The fuzzy choice function C has been generated by the fuzzy preference relation Q which was decided by the experts. In its turn, the fuzzy choice function C generates a fuzzy revealed preference relation R: R(xi , xj ) = [C(S)(xi ) ∧ S(xj )] for any alternatives xi , xj ∈ X. S∈B

The fuzzy revealed preference relation R reﬂects the preferences of the buyer displayed by the act of choosing and Q reﬂects the preferences of the buyer according to the expertise. Since in theory R is not always reﬂexive, we will replace the elements of the main diagonal with 1. In practice this

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251

assumption is normal. The fuzzy choice function C reﬂects faithfully the expertise if Q and R are suﬃciently close to each other. The closeness between Q and R can be measured with the grade of similarity. There areseveral modalities to deﬁne it [113], among which we recall  1 if Q=R=∅      min(Q(x, y), R(x, y)) M (Q, R) = x,y∈X  otherwise    max(Q(x, y), R(x, y))   x,y∈X

In interpretation, if the degree of similarity is 0.8, we say that “the choice is made with the grade of similarity of 0.8”. 2) To establish the hierarchy of the alternatives with respect to each criterion we will apply the degree of dominance for fuzzy choice functions introduced in [48]. Recall that the degree of dominance ranks the alternatives with respect to the act of choice and not to the preference relation. For a global hierarchy of the oﬀers we also apply the aggregated degree m 1 of dominance [48]: D(x) = m DSj (x). By ordering the elements of the set j=1

{D(x)|x ∈ X} one obtains an overall hierarchy of alternatives. In the ﬁnal act of making a choice, the buyer will decide which oﬀer to choose.

10.2.4 Numerical Illustration In this subsection we shall illustrate the above algorithm with a numerical example. Let us take the particular case n = 5 alternatives, m = 3 criteria. The set of alternatives is X = {x1 , . . . , x5 }. The activity of the three experts is materialized in the following preference matrices:

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

 1 0.7 0.8 0.3 0.5

   0.5   Q1 =  0.7    0.7  0.3



 1 0.3 0.5 0.7 0.8

     0.7 1 1 0.7 0.6 0.7      0.3 1 0.7 0.8 , Q2 =  0.6 0.5      0.5 0.7 0.8 0.4 1 0.6    0.4 0.7 0.8 1 0.7 0.6   1 0.7 0.7 0.6 0.8      0.8 1 0.4 0.5 0.6      Q3 =  0.3 1 1 0.2 0.6       0.5 0.6 0.7 1 0.8    0.5 0.8 0.9 0.8 1

  0.3 0.8 0.7    1 0.7 0.3 ,   0.8 1 0.3   0.5 0.5 1

The three criteria are given by the following fuzzy subsets of S, where χ{X} is the characteristic function of X. S1 = 0.3χ{x1 } + 0.6χ{x2 } + 0.8χ{x3 } + 0.5χ{x4 } + 0.6χ{x5 } ; S2 = 0.4χ{x1 } + 0.7χ{x2 } + 0.8χ{x3 } + 0.9χ{x4 } + 0.2χ{x5 } ; S3 = 0.7χ{x1 } + 0.6χ{x2 } + 0.5χ{x3 } + 0.1χ{x4 } + 0.4χ{x5 } . We follow the following steps.

Step 1

The matrix of the fuzzy preferences Q is obtained from the matrices Q1 , Q2 and Q3 : 

 1 0.56 0.66 0.53 0.7

   0.66   Q =  0.53    0.56  0.5

  1 0.46 0.63 0.66    0.6 1 0.53 0.56 .   0.7 0.63 1 0.56   0.6 0.7 0.7 1

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253

Step 2

The fuzzy choice function resulting from Q is computed by: 5 C(Sj )(xi ) = Sj (xi ) ∧ (Sj (xu ) → Q(xi , xu )), i = 1, . . . , 5, j = 1, . . . , 3. u=1

For example, C(S1 )(x1 ) = S1 (x1 ) ∧ [S1 (x1 ) → Q(x1 , x1 )] ∧ [S1 (x2 ) → Q(x1 , x2 )] ∧ [S1 (x3 ) → Q(x1 , x3 )] ∧ [S1 (x4 ) → Q(x1 , x4 )] ∧ [S1 (x5 ) → Q(x1 , x5 )] = 0.3 ∧ [0.3 → 1] ∧ [0.6 → 1] ∧ [0.8 → 0.6] ∧ [0.5 → 0.53] ∧ [0.6 → 0.7] = 0.3. After all computations, we obtain the table: C(Sj )(xi ) x1

x2

x3

x4 x5

S1

0.3 0.46 0.56 0.5 0.6

S2

0.4 0.46 0.53 0.63 0.2

S3

0.56 0.46 0.5 0.1 0.4

Step 3

The elements of the fuzzy revealed preference matrix R associated to C are calculated by the formula: R(xi , xj ) = [C(S1 )(xi )∧S1 (xj )]∨[C(S2 )(xi )∧S2 (xj )]∨[C(S3 )(xi )∧S3 (xj )], i, j = 1, . . . , 5. The matrix R is :



 0.56    0.46   R =  0.5    0.4  0.4

0.56 0.56 0.4 0.4

  0.46 0.46 0.46 0.46    0.56 0.56 0.53 0.56 .   0.63 0.63 0.63 0.5   0.6 0.6 0.5 0.6

We replace R by its reﬂexive closure R :

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

 1 0.56 0.56 0.4 0.4

   0.46   R =  0.5    0.4  0.4

  1 0.46 0.46 0.46    0.56 1 0.53 0.56 .   0.63 0.63 1 0.5   0.6 0.6 0.5 1

Step 4 Now we ﬁnd out how similar matrices Q and R are. For this, ﬁrst we compute: 

 1 0.56 0.56 0.4 0.4

   0.46   Q ∧ R =  0.5    0.4  0.4



 1 0.56 0.66 0.53 0.7

       0.66 1 0.46 0.63 0.66  1 0.46 0.46 0.46        0.56 1 0.53 0.56 , Q ∨ R =  0.53 0.6 1 0.53 0.56 .        0.56 0.7 0.63 1 0.56  0.63 0.63 1 0.5     0.6 0.6 0.5 1 0.5 0.6 0.7 0.7 1 5

The grade of similarity of Q and R is M (Q, R ) =

i,j=1 5

(Q ∧ R )(xi , xj ) =

(Q ∨ R )(xi , xj )

i,j=1 15.17 17.03

= 0.89.

Step 5

The degrees of dominance of alternatives in X with respect to criteria S1 , S2 and S3 are represented in the table:

10.2 Reducing Adverse Selection by Fuzzy Choices

DSj (xi ) x1 x2

255

x3 x4 x5

S1

0.3 0.46 0.56 0.5 0.6

S2

0.4 0.46 0.56 0.9 0.2

S3

0.7 0.46 0.5 0.1 0.4

The aggregated degrees of dominance of the alternatives are

D(x1 ) = 0.46, D(x2 ) = 0.46, D(x3 ) = 0.53, D(x4 ) = 0.5, D(x5 ) = 0.4.

Therefore the order of the alternatives is: D(x5 ) < D(x1 ) = D(x2 ) < D(x4 ) < D(x3 ).

10.2.5 Discussion Now we will make an analysis of the results from the previous section in the context of our example. We deal with a situation where a company wants to buy 3 types of products. The oﬀers come from 5 producers. The set of alternatives (=oﬀers) is given by S = {x1 , . . . , x5 }. The partial information on the product i (e.g. its quality) is represented in the vector ai : a1 = (0.3, 0.6, 0.8, 0.5, 0.6); a2 = (0.4, 0.7, 0.8, 0.9, 0.2); a3 = (0.7, 0.6, 0.5, 0.1, 0.4). It means that a11 = 0.3 represents the degree to which information on the quality of the ﬁrst product given by the ﬁrst producer exists. The certain information on the products (e.g. cost, delivery times) is also taken into consideration, by the participation of 3 experts in the decision 1 making. For example, q12 = 0.56 means that the ﬁrst expert considers that

x1 is preferred to x2 , etc. The overall expertise is reﬂected in the aggregated

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matrix Q of fuzzy preferences. The element q12 = 0.56 can be interpreted as oﬀer x1 is preferred to oﬀer x2 with the degree of intensity 0.56. From the fuzzy choice functions obtained above the fuzzy revealed preference R is derived. Oﬀer 1 is revealed preferred to oﬀer 2 to the extent 0.56 and to oﬀer 3 to the extent 0.56, etc. One ﬁrst conclusion shows that the result of the expertise coincides with the preferences of the company based on fuzzy choices with the grade of similarity 0.89. Another conclusion is that according to criterion S1 , the company might choose oﬀer 5, according to criterion S2 oﬀer 4 and according to criterion S3 oﬀer 1. If the company considers all criteria, it might choose oﬀer 3.

10.2.6 Conclusions In this section, a new theoretical approach to correcting information asymmetry has been presented. This multiple criteria decision making model is explained in the context of product marketplaces. The model will help buyers make better purchase decisions in the presence of qualitative product information. Yet incentives for sellers to be dishonest might remain, but the buyers’ decisions will be improved. We have found that by interpreting the vectors that contain the partial information (represented by qualitative attributes) as available fuzzy sets of alternatives one obtains as a mathematical model a fuzzy choice problem. By this, from the preferences Q(xi , xj ) resulted from the expertise one goes to the revealed preferences R(xi , xj ) associated with the choice function. R(xi , xj ) reﬂects directly the act of choice and comprises both the exact and the vague information. Moreover, by using the degree of similarity one can compare the preferences Q(xi , xj ) with the revealed preferences R(xi , xj ). To a greater degree of similarity corresponds a better correction of the adverse selection. This way the similarity measures the way the adverse selection has been reduced.

10.3 Application 3

257

In the example there has been considered that the experts have processed only the certain information. An open problem is what happens when the experts process both the exact and a part of the vague information. This study assumed that there is no conﬂict on the experts’ judgments. Another open problem is what happens in case of a conﬂict on the experts’ judgments and in what way the buyers will be aﬀected.

10.3 Application 3 A producer manufactures m types of products P1 , . . . , Pm . Every year he organizes an auction to sell his products. n companies x1 , . . . , xn are interested in participating in this auction. The bidding concerns the right to sell the products. The sales obtained in year T are given in the following table: P1 P2 . . . Pm x1 a11 a12 . . . a1m x2 a21 a22 . . . a2m ... xn an1 an2 . . . anm

where aij denotes the number of units of product Pj sold by company xi in year T . For the year T + 1 the producer would like to increase the number of sales with the n companies. The companies give an estimation of the sales for year T + 1 contained in a matrix (cij ) with n rows and m columns; cij denotes the number of units of product Pj that the company xi estimates to sell in year T + 1. In making his choice, the producer will choose those companies that have an eﬃcient sales market, good marketing, and a good image. The decision analysis will require two aspects: (a) the sales aij for year T ;

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(b) the estimated sales cij for year T + 1. The sales for year T can be considered results of the act of choice, or more clearly, values of a choice function, and the preferences will be given by the revealed preference relation associated with these choice functions. With the resulting preference relation and the estimated sale for the year T + 1, a fuzzy choice function can be deﬁned. This choice function will be used to rank the companies with respect to each type of product. Dividing the values aij and cij respectively by a power of 10 conveniently chosen we may assume that 0 ≤ aij , cij ≤ 1 for each i = 1, . . . , n and j = 1, . . . , m. In establishing the mathematical model the following steps are needed: –(A) To build a fuzzy choice function from the sales of year T . The set of alternatives is X = {x1 , . . . , xn }. For each j = 1, . . . , m denote by Sj the subset of X whose elements are those companies that have had “good” sales for product Pj in year T . Only the companies whose sales are greater than a threshold ej are considered. If H = {S1 , . . . , Sm } then (X, H) is a fuzzy choice space (we will identify Sj with its characteristic function). The sales (aij ) of year T lead to a choice function C : H → F(X) deﬁned by: (1) C (Sj )(xi ) = aij for each j = 1, . . . , m and xi ∈ Sj . This context is similar to [14]. There H contains all non-empty ﬁnite subsets of X. –(B) The choice function C gives a fuzzy revealed preference relation R on X deﬁned:

(2) R(xi , xj ) =

{C (Sk )(xi )|xi , xj ∈ Sk } = {aik |xi , xj ∈ Sk }

for any xi , xj ∈ X. R(xi , xj ) represents the degree to which alternative xi is preferred to alternative xj as a consequence of current sales.

10.3 Application 3

259

Since in most cases R is not reﬂexive, we replace it by its reﬂexive closure R . –(C) From the fuzzy revealed preference matrix R and the matrix cij of estimated sales one can deﬁne a fuzzy choice function C, whose values will estimate the potential sales for the year T + 1. Starting from C one will rank the alternatives for each type of product. The set of alternatives is X = {x1 , . . . , xn }. For each j = 1, . . . , m, Aj will denote the fuzzy subset of X given by (3) Aj (xi ) = cij for any i = 1, . . . , n. Take A = {A1 , . . . , Am }. One obtains the fuzzy choice space (X, A). The choice function C : A → F(X) is deﬁned by: (4) C(Aj )(xi ) = Aj (xi ) ∧ = cij ∧

n

n

[Aj (xk ) → R (xi , xk )]

k=1

[ckj → R (xi , xk )]

k=1

for any i = 1, . . . , n and j = 1, . . . , m.

Applying the degree of dominance for the fuzzy choice function C one will obtain a ranking of the companies with respect to each product. This ranking gives the information that the mathematical model described above oﬀers to the producer with respect to the sales activity for the following year. We present next the algorithm of this problem. The input data are: m= the number of types of products n=the number of companies aij =the matrix of sales for year T cij =the matrix of estimated sales for year T + 1 (e1 , . . . , em )=the threshold vector Assume 0 ≤ aij ≤ 1, 0 ≤ cij ≤ 1 for any i = 1, . . . , n and j = 1, . . . , m. From the mathematical model we can derive the following steps:

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10 Applications

Step 1 Determine the subsets S1 , . . . , Sm of X = {x1 , . . . , xn } by Sk = {xi ∈ X|aik ≥ ek }, k = 1, . . . , m. Step 2 Compute the matrix of revealed preferences R = (R(xi , xj )) by R(xi , xj ) = aik . xi ,xj ∈Sk

Replace R with its reﬂexive closure R . Step 3 Determine the fuzzy sets A1 , . . . , Am Aj = cij χ{x1 } + . . . + cnj χ{xn } for j = 1, . . . , m Step 4 Obtain the choice function C applying (4) Step 5 Determine the degrees of dominance DAj (xi ), i = 1, . . . , n and j = 1, . . . , m. Step 6 Rank the set of alternatives with respect to each product Pj by ranking the set {DAj (x1 ), . . . , DAj (xn )}. For a better understanding of this model we present a numerical illustration. Consider the initial data m = 3 products and n = 5 companies willing to sell these products. The sales for year T are given in the following table: P1 P2 P3 x1 0.3 0.6 0.7 x2 0.8 0.1 0.5 x3 0.7 0.6 0.1 x4 0.1 0.8 0.7 x5 0.8 0.1 0.7

The estimated sales for year T + 1 are given in the following table:

10.3 Application 3

261

P1 P2 P3 x1 0.5 0.7 0.7 x2 0.8 0.3 0.6 x3 0.8 0.7 0.2 x4 0.2 0.8 0.8 x5 0.8 0.2 0.8

The thresholds are e1 = e2 = e3 = 0.2. We follow now the steps described above. Step 1 The subsets S1 , S2 , S3 of X are: S1 = {x1 , x2 , x3 , x5 }, S2 = {x1 , x3 , x4 }, S3 = {x1 , x2 , x4 , x5 }. Step 2 We compute the matrix of revealed preferences R. Then we replace it by its reﬂexive closure R .  0.7 0.7 0.6 0.7    0.8 0.8 0.8 0.5   R =  0.7 0.7 0.7 0.6    0.8 0.8 0.8 0.8  0.8 0.8 0.8 0.7 For example, R(x1 , x2 ) =

 0.7



 1 0.7 0.6 0.7 0.7

       0.8 1 0.8 0.5 0.8  0.8        0.7 . R =  0.7 0.7 1 0.6 0.7 .        0.8 0.8 0.8 1 0.7  0.7     0.8 0.8 0.8 0.7 1 0.8 a1k = a11 ∨ a13 = 0.3 ∨ 0.7 = 0.7.

x1 ,x2 ∈Sk

Step 3 The fuzzy sets A1 , A2 , A3 are:

A1 = 0.5χ{x1 } + 0.8χ{x2 } + 0.8χ{x3 } + 0.2χ{x4 } + 0.8χ{x5 } ; A2 = 0.7χ{x1 } + 0.3χ{x2 } + 0.7χ{x3 } + 0.8χ{x4 } + 0.2χ{x5 } ; A3 = 0.7χ{x1 } + 0.6χ{x2 } + 0.2χ{x3 } + 0.8χ{x4 } + 0.8χ{x5 } . Step 4 The corresponding fuzzy choice functions are:

C(A1 ) = 0.5χ{x1 } + 0.8χ{x2 } + 0.7χ{x3 } + 0.2χ{x4 } + 0.8χ{x5 } C(A2 ) = 0.6χ{x1 } + 0.3χ{x2 } + 0.6χ{x3 } + 0.8χ{x4 } + 0.2χ{x5 }

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C(A3 ) = 0.7χ{x1 } + 0.5χ{x2 } + 0.2χ{x3 } + 0.7χ{x4 } + 0.7χ{x5 } . Step 5 The corresponding degrees of dominance are represented in the table: DAj (xi ) x1 x2 x3 x4 x5 A1

0.5 0.8 0.7 0.2 0.8

A2

0.6 0.3 0.6 0.8 0.2

A3

0.7 0.5 0.2 0.8 0.7

The producer needs to decide one contractor according to the dominant values for each criterion. According to criterion A1 , DA1 (x4 ) < DA1 (x1 ) < DA1 (x3 ) < DA1 (x2 ) = DA1 (x5 ), therefore companies 2 and 5 will be chosen. According to criterion A2 , DA2 (x5 ) < DA2 (x2 ) < DA2 (x1 ) = DA2 (x3 ) < DA2 (x4 ), therefore company 4 will be chosen. According to criterion A3 , DA3 (x3 ) < DA3 (x2 ) < DA3 (x1 ) = DA3 (x5 ) < DA3 (x4 ), therefore company 4 will be chosen. In the situation described above the sales of year T are regarded as the result of the act of choice. As such we deal with a ﬁrst problem of fuzzy choices (in the sense of Banerjee) where the choice function C is deﬁned by the values of the sales of year T . The fuzzy revealed preference relation R associated with C expresses the preferences of the buyers in year T . R gives only partial information on the manner in which the sales of year T + 1 will occur. Estimative values of the sales of year T + 1 will be taken into account. For instance, these estimative values can be obtained by statistical samplings. By combining the estimations of year T + 1 with the preferences of year T one

10.3 Application 3

263

obtains a new fuzzy choice problem (in the general sense of this thesis) that models the sales of year T + 1. The multicriterial hierarchy of alternatives obtained from the second fuzzy choice problem will be useful to the company in organizing the sales of year T + 1.

11 Concluding Remarks

According to [3] pp. 1–2, the general (crisp) choice theory has three main sources: “The ﬁrst is that many problems of applied mathematics and control theory boil down to choosing the “best” options, in some sense, from a given set. For example, plans, control and strategies are the options in optimal control models, mathematical programming,etc., the set options to be compared being deﬁned by the constraints”. “The second reason for interest in general choice theory is that many economic and social models examine questions of individual choice.This is the case of consumers demand models,in the models of competitive market, and in more general economic models like those of Walras, Arrow–Debreu, and so on. Some models of psychological phenomena are similar”. “The third reason for interest in choice theory is due not to models of economic activity, but to models of political (democratic, in particular) processes– models that consider diﬀerent voting procedures and investigate their outcomes”. Each one of these domains generates, at the level of the general theory, speciﬁc results. In particular, the revealed preference theory for choice functions was inspired – to a great extent – from the consumer revealed preference

Irina Georgescu: Fuzzy Choice Functions, StudFuzz 214, 265–270 (2007) www.springerlink.com

c Springer-Verlag Berlin Heidelberg 2007

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theory (Samuelson, Houthakker, Ville). Therefore, the present book will follow the line of the results of consumer revealed preference theory. Shortly, the contributions of this book consist in the following: (1) A theoretical framework for fuzzy revealed preference theory. Our approach diﬀers from that of Banerjee by considering the available sets as fuzzy subsets of a universe X of alternatives. This led to a new notion, the degree of availability S(x) of an alternative X with respect to S. In interpretation, the available sets will model vague attributes or criteria: if x is an alternative and S is a vague criterion, then S(x) will mean the degree to which x veriﬁes the criterion S. In this way, the domain, as well as the range of a fuzzy choice function consists of fuzzy subsets of the domain of alternatives. This general deﬁnition of fuzzy choice functions allows us a natural formulation, in fuzzy setting, of the deﬁnitions and the results of the classical revealed theory, as well as some concepts and propositions, speciﬁc to the fuzzy case. In order to develop the fuzzy version of the direction Uzawa–Arrow–Sen we work under the natural hypotheses H1 and H2. (2) The treatment, in this general framework, of the main themes of revealed preference theory (rationality, revealed preference and congruence axioms, consistency conditions). The deﬁnition of the two concepts of rationality (G–rationality and M – rationality) connects the fuzzy choice functions and the fuzzy preference relations deﬁned on the set of alternatives. The axioms of revealed preference and the axioms of congruence are conditions that express the rationality of fuzzy choice functions. Beside them, the consistency properties ensure a rational behaviour of the choices with respect to the expansion and the contraction of the available sets, as well as with respect to some fuzzy set operations. The statement of various conditions and results, as well their proofs, is not a simple translation of the situations from the case of crisp choice functions. One can extend only an implication of Richter’s theorem: every totally rational fuzzy choice function is congruous (cf. Theorem 7.16). In the attempt

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267

to generalize the converse implication one obtains the notion of semirationality and a surprising result holds: every fuzzy choice function is semirational. Sometimes, it happens that a crisp property has many diﬀerent fuzzy versions. For instance, to the axiom W ARP correspond the axioms W AF RP and W AF RP ◦ . W AF RP is a suitable condition if hypotheses H1 and H2 hold (see Theorem 6.7), while W AF RP is better in the general case (see Theorem 7.3). In obtaining the proofs of fuzzy extensions of some classical results in revealed preference theory, there exists two kinds of limitations. The ﬁrst limitation is related to the continuous t–norm that deﬁnes the context. In the case of fuzzy form of the Arrow–Sen theorem, some equivalences or implications are true for an arbitrary continuous t–norm, others are true only for the Lukasiewicz or G¨ odel t-norm (see Theorem 6.7 and Example 6.8). The G¨odel t–norm is more eﬀective for revealed preference axioms (conditions (i)–(iv) are equivalent) and the Lukasiewicz t–norm is more eﬀective for congruence axioms(conditions (iii)–(vii) are equivalent). Reﬂecting on Theorem 6.7 and Example 6.8, it can be said that the product t–norm has a reduced signiﬁcance in fuzzy revealed preference theory. The two types of limitation appear in deﬁning some properties of fuzzy revealed preference relations. For instance, the totality and the strong totality of a fuzzy preference relation are two properties that correspond to the crisp totality. The fuzzy Arrow–Sen Theorem and the consistency conditions (Sections 6.2 and 6.3) uses the strong totality of fuzzy preference relations, while for the extension of Richter theorem (Section 7.9) the totality is more appropriate. (3) The deﬁnition of the degree of dominance as a tool in obtaining the hierarchy of alternatives. We have introduced the degree of dominance of an alternative, as a method of ranking the alternatives according to diﬀerent criteria. The criteria can be taken as the available sets of alternatives. The degree of dominance of an alternative x with respect to an available set S reﬂects the position of x’s

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towards the other alternatives. This notion expresses the dominance of an alternative with regard to the act of choice, not to a preference relation. With the degree of dominance one can build a hierarchy of alternatives for each available set S. If one deﬁnes a concept of aggregated degree of dominance (that uniﬁes the degrees of dominance with regard to various available sets) one obtains an overall hierarchy of alternatives. (4) The deﬁnition of new concepts, as similarity and the indicators of fuzzy choice functions are introduced for obtaining a deeper insight on fuzzy revealed preference theory. The analysis of the fuzzy phenomena requires reasoning that operate with truth-values of the statements, expressed by numbers in the real interval [0, 1]. Thus, instead of checking whether a fuzzy choice function C satisﬁes a property P or not, it is more appropriate to have a “measure” of the degree to which C satisﬁes P . In our case, the property P could be the rationality of a fuzzy choice function, a revealed preference axiom, etc. Therefore, in Chapter 9, we have deﬁned the rationality indicators RatG (C), RatM (C), the normality indicators N ormG (C), N ormM (C), the revealed preference indicators W AF RP (C), SAF RP (C), the congruence indicators W F CA(C), SF CA(C), the Arrow index A(C) and the consistency indicators F α(C), F β(C), as truth–values of the corresponding properties studied in the previous chapters. These indicators represent a kind of second order fuzziness in the analysis of revealed preference and the numerical connections established between them improve the relations between the corresponding properties. For instance, the equality W F CA(C)=min(F α(C), F β(C)) generalizes the theorem that asserts the equivalence between W F CA and the conjunction of the consistency conditions F α and F β. A second important notion in this chapter is the degree of similarity of two fuzzy choice functions. It is used for deﬁning or for computing the afore mentioned indicators. By means of these indicators we can compare two fuzzy choice functions with respect to each axiom or property. For example, if C1 , C2 are two fuzzy

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269

choice functions such that W F CA(C1 ) ≥ W F CA(C2 ), then we can say that C2 satisﬁes the axiom W F CA to a greater extent than C1 . On this basis, any family of fuzzy choice functions can be ranked with respect to a condition. (5) The analysis of the three applications of Chapter 8 leads to the following conclusions: -all these applications describe concrete economic situations where partial information or human subjectivity appears. -the mathematical modeling is done by formulating some fuzzy choice problems where criteria are represented by fuzzy available sets of alternatives. -the degree of dominance is the mathematical instrument on which the algorithms of multicriterial hierarchy are based. There are cases when for a decision–making problem we have several fuzzy choice problems as mathematical models (resulting, for example, from several expertises or from the analysis of diﬀerent data sets). Accordingly, the treatment of this situation requires two levels: Level a. One will determine which choice problem is the most adequate model for the multicriterial decision–making problem; Level b. Once the most adequate fuzzy choice problem being chosen, one will select the optimal alternative. At level a, one will choose the fuzzy choice function with the greatest rationality indicator. At level b, by applying the degree of dominance one will obtain a hierarchy of alternatives for each criterion; the agent will make a decision considering the information obtained like this. The aggregation of the individual fuzzy preferences into a collective fuzzy relation and the crisp choice rules based on fuzzy preferences have been treated by numerous authors (see [15], [16], [41], [81], [97], etc. ). In this book this issue has not been approached. In future, the following important problems are open: (I) the way the individual fuzzy preferences aggregate into a fuzzy choice function

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(II) the way the individual fuzzy choice functions aggregate into a collective fuzzy choice function. Papers [75], [76], [77] contain a voting theory and group decision making approach from a tripartite point of view: probabilistic, fuzzy and rough set theory. It would be desirable that the fuzzy choice functions theory from this book be connected to the problematic of these papers. On the other hand, in [37], there are studied fuzzy rough sets, a concept that intends to unify the fuzzy and rough sets aspects of imprecise phenomena. An open problem is whether fuzzy rough set theory can be an adequate framework for the development of a theory of revealed preference.

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109. Uzawa H (1959) Preference and rational choice in the theory of consumption. In Arrow K J, Karlin S, Suppes P (eds) Mathematical Methods in the Social Sciences. Stanford, Stanford University Press 110. Van de Walle, Heitsch S, Faratin P (2001) Coping with one-to-many multicriteria negotiations in electronic markets. Proceedings of the Second International Workshop on Negotiations in Electronic Markets: beyond Price Discovery, Munich, Germany 747–751 111. Ville J (1946) Sur les conditions d’existence d’une oph´elimit´e totale et d’un indice de prix. Annales de l’Universit´e de Lyon 9:32–39 112. Wang X (2004) A note on congruence conditions of fuzzy choice functions. Fuzzy Sets and Systems 145:355–358 113. Wang X, De Baets B, Kerre E E (1995) A comparative study of similarity measures. Fuzzy Sets and Systems 73:259–268 114. Wooldridge M, Jennings N R (1995) Intelligent agents: theory and practice. The Knowledge Engineering Review 10:115–152 115. Xue X L, Li D, Shen P, Wang Y W (2005) An agent–based framework for supply chain coordination in construction. Automation in Construction 14:413– 430 116. Ying M (1999) Perturbations of fuzzy reasoning. IEEE Transactions on Fuzzy Systems 7:625–629 117. Zadeh L A (1965) Fuzzy sets. Information and Control 8:338–353 118. Zadeh L A (1971) Similarity relations and fuzzy orderings. Information Sciences 3:177–200 119. Zeleny M (1982) Multiple Criteria Decision Making. McGraw-Hill, New York 120. Zimmermann H J (1984) Fuzzy Set Theory and Its Applications. Kluwer

Index

(∗, δ)-equality

66, 189

irreﬂexive reﬂexive

adverse selection alternative

246–248

9, 26, 169, 233

dominant

42, 137

Arrow index

220–223

Arrow-Sen theorem

attribute

total

43, 44, 107, 113

asymmetric information

regular

available set

245

30

28

transitive

28

biresiduum

56

budget equation budget set

87

11

11

9

C-connected sequence

41, 97

characteristic function

26

choice Banerjee choice function Banerjee’s dominance binary relation acyclic

26

28

31

function fuzzy

32 87

preference-based problem

28

asymmetric part of complete

87, 93, 172

169

31, 36

antisymmetric

inverse

28

9, 31

availability degree

fuzzy

35

symmetric part of

79

Arrow Axiom

28

symmetric

172, 173, 175

pairwise dominant

28

set 30

77

33

32

space

32

commodity bundle

52, 63

complement

26

26

congruence

158

10

280

Index

axioms of

consistency conditions Fα Fα

fuzzy

120

120

F β

120

F β(+) Fδ

G-normal

133

45

γ

45

γ2

100

M-normal

102 100 158, 163

totally rational 46

choice problem choice space relation

46 10, 11

100, 160 88

88

82

Banerjee-transitive

82

9, 31, 169, 233 complete

decision making

63

irreﬂexive

233

63

quasi-transitive

degree of equality

reﬂexive

59

of dominance

169, 173, 175, 237

regular

127

63 63

of reﬂexivity

66

strongly complete

of similarity

189, 192

strongly total

of strong completeness of strong totality of symmetry of transitivity

demand function 27

symmetric

set

67 51

63

63

63

63

transitive

66

11

67

total

67

degree of membership

diagonal

101

53, 63

acyclic

consumer theory criterion

G-rational

strongly complete rational

45

δ

102

semirational

46

β(+)

139

M-rational

45

87

96

full rational

132

α2 β

congruous

127

F γ2

49

choice function

131

Fβ

28

45

120

F α2

α

equivalence relation

172, 176, 184

63

50

normal

59

Fuzzy Arrow Axiom F AA Fuzzy Chernoﬀ Axiom F CA

137, 220 143

Index Fuzzy Dual Chernoﬀ Axiom F DCA 143

partial order strict

G-normal

39

G-normality

98

G-rationality

path independence

46, 135

preference relation

26, 33

53

regular

G-rationalization

34, 100

grade of similarity

256

preorder

35, 44, 124 28

price vector Hansson’s Axiom of Fuzzy Revealed Preference HAF RP

proﬁle

10

12

97

Hansson’s Axiom of Revealed Preference HARP

28

71, 158

fuzzy

98

Q-greatest element Q-maximal element

41

indicators

residuated lattice 211–220

residuum

of consistency

223–231

revealed preference

200–210

of revealed preference indiﬀerence relation intersection M-normal

fuzzy 211–220

14, 16

91

84

Richter theorem

44, 71, 109, 158, 160

Richter–Hansson–Suzumura theory 145

98 98

M-rationalization

score function

34, 100

multiple criteria decision making negation

54

RPWD

39

M-rationality

33, 99

Reward for Pairwise Weak Dominance

30, 61

26

M-normality

33, 99

55

of congruence

of rationality

281

248

56

Orlovsky function

88, 89

Orlovsky score

78

26

crisp

51

fuzzy

51

similarity relation

76

generalized

set

78

singleton

40, 124

social welfare function strict preference

Pairwise Strict Dominance PSD Pairwise Weak Dominance PWD

65, 158, 191, 220

84 84

12

30, 61

Strong Axiom of Consumer Behaviour SACB

18

282

Index

Strong Axiom of Fuzzy Revealed Preference SAF RP

union

96, 97

Uzawa–Arrow–Sen theory

Strong Axiom of Revealed Preference SARP

Uzawa-Arrow-Sen theory

42

Strong Fuzzy Congruence Axiom SF CA 96

voting theory

59

Szpilrajn theorem

71

Weak Axiom of Consumer Behaviour 17

Weak Axiom of Fuzzy Revealed Preference W AF RP

95, 97

Weak Axiom of Revealed Preference

53

continuous

58, 63

W ARP

40

Weak Congruence Axiom W CA

54

Lukasiewicz product

91, 107

12

W ACB

subsethood degree

G¨ odel

145

40

Strong Congruence Axiom SCA

t-norm

26

Weak Fuzzy Congruence Axiom W F CA

54

96

54

transitive closure

42

29, 64

weak order

28

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