Fuzzy Choice Functions. A Revealed Preference Approach

281 Pages·2007·1.256 MB·english

by  Irina Georgescu

Irina Georgescu
Fuzzy Choice
Functions
A Revealed Preference Approach
ABC
Dr.IrinaGeorgescu
P.O.Box1-432
014700
Bucharest,Romania
E-mail:[email protected]
LibraryofCongressControlNumber:2006939263
ISSNprintedition:1434-9922
ISSNelectronicedition:1860-0808
ISBN-10 3-540-68997-4SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-68997-3SpringerBerlinHeidelbergNewYork
(cid:1)c Springer-VerlagBerlinHeidelberg2007
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
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 Fα and Fβ...................................119
6.3 Quasi-transitivity and Condition Fδ .......................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
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
consumerintermsofsomepreferencerelationassociatedwithademandfunc-
tion. 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 sec-
ond axiom assures that the demand function can be reconstructed from a
revealed preference relation.
Tomake arational choice isamore generalproblemthatgoes beyondthe
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
plausiblehypothesisisthat,havingadesideratum,eachofthemisrationalin
the act of choice.
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
choicefunctionC thereareassociatedmorepreferencerelations,eachofthem
leading to a different notion of rationality.
In [8] The Weak Axiom of Revealed Preference WARP and The Strong
Axiom of Revealed Preference SARP are formulated. WARP and SARP
are abstract versions of The Weak Axiom of Consumer Behavior and of The
StrongAxiomofConsumerBehavior.TheresultsobtainedbyUzawa,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 suf-
fices 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 WCA
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 WCA, SCA,
the revealed preference axioms WARP, 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.
Theclassictheoryofrevealedpreferenceisfoundedontwomainconcepts:
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 pref-
erences and vague choices. The vague preferences will be modelled by fuzzy
preference relations: they show the degree to which an alternative x is pre-
ferred 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]andavastliteraturehasbeendedicatedtothissubject(foranexhaustive
overviewseethemonograph[40]aswellas[86],[69]).Someauthorsbuildtheir
resultsonthethesisthatsocialchoiceisgovernedbyfuzzypreferences(hence
modelledthroughfuzzybinaryrelations)buttheactofchoiceisexact(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.
Thenanynon-emptysubsetS ofX isanavailablesetofalternatives.Wehave
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
makingproblemscanfindnaturalsolutionsinfuzzyrevealedpreferencetheory
(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; ifS ∈ 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
choicefunctioncontainsfuzzysubsetsofX.Ofcoursetheclassoffuzzychoice
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 re-
vealedpreferencetheoryfromconsumers([87],[88],[63],[111])toanaxiomatic
approach ([8], [84], [85], [91], [92], [93], [100], [101], [102], [108], [109], etc.) is
traced.
1 Introduction 5
Chapter3referstonotionsandclassicresultsofrevealedpreferencetheory
(crisp choice functions, G–rationality, M–rationality, revealed preference and
congruence axioms, consistency conditions etc.).
Chapter4discussescontinuoust–norms,fuzzysetsandfuzzyrelationsand
theirproperties.InthelastpartofthechapterafuzzyversionoftheSzpilrajn
theorem is proved.
Chapter5hasthreesections.Thefirstsectiondealswithexactchoicefunc-
tionsgeneratedbyfuzzypreferencerelations.SomepropertiesoftheOrlovsky
function [79] are presented in detail. The second section introduces the fuzzy
choice functions, the preference relations associated with them and formu-
lates the axioms of revealed preference WAFRP, SAFRP and congruence
WFCA, SFCA. WAFRP and SAFRP (respectively WFCA and SFCA)
are fuzzy versions of the classical axioms WARP and SARP (respectively
WCAandSCA).Following[61],[100]wealsoconsidertheaxiomsWAFRP◦
and HAFRP (a fuzzy version of HARP). The fuzzy versions of classical ax-
ioms 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
lineintroducedbyUzawa,ArrowandSen.Theclassoffuzzychoicefunctions
studied in this chapter is subject to hypotheses (H1) and (H2); they come
naturally from the assumption that the domain of choice functions includes
allfinitesetsofalternatives.InSection6.1westudytherelationsbetweenthe
axiomsWAFRP,SAFRP,WFCA,SFCAandfourotherrationalitycondi-
tions. The main result (Theorem 6.7) establishes equivalences or implications
between these properties; some are true for an arbitrary continuous t-norm