Table Of ContentStudies in Economic Theory
Editors
CharalambosD.Aliprantis
PurdueUniversity
DepartmentofEconomics
WestLafayette,IN47907-1310
USA
NicholasC.Yannelis
UniversityofIllinois
DepartmentofEconomics
Champaign,IL 61820
USA
Springer-Verlag Berlin Heidelberg GmbH
Titles in the Series
M.A.Khanand N.C.Yannelis(Eds.)
Equilibrium Theoryin InfiniteDimensional
Spaces
C.D.Aliprantis,KC.Border
and W.A.].Luxemburg(Eds.)
PositiveOperators,RieszSpaces,andEconomics
D.G.Saari
GeometryofVoting
C.D.Aliprantisand K C.Border
Infinite DimensionalAnalysis
].-P.Aubin
DynamicEconomicTheory
M.Kurz(Ed.)
EndogenousEconomicFluctuations
].-F.Laslier
TournamentSolutionsand MajorityVoting
A.Alkan,C.D.Aliprantis and N.C.Yannelis
(Eds.)
Theoryand Applications
].C.Moore
MathematicalMethodsforEconomicTheoryI
].C.Moore
MathematicalMethodsforEconomic Theory2
M.Majumdar,T.Mitraand K Nishimura
Optimizationand Chaos
K K Sieberg
CriminalDilemmas
M.Florenzanoand C.LeVan
FiniteDimensionalConvexityandOptimization
K Vind
Independence,Additivity,Uncertainty
T.Casonand C.Noussair(Eds.)
AdvancesinExperimentalMarkets
Fuad Aleskerov
Bernard Monjardet
Utility
Maximization,
Choice
and Preference
With 55Figures
and27Tables
, Springer
ProfessorFuadAleskerov
InstituteofControlSeiences
65ProfsoyuznayaStr.
117997Moskow
Russia
ProfessorBernardMonjardet
MSE- UniversiredeParis I
CERMSEM
106-112bddel'Höpital
75647Paris Cedex13
France
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Aleskerov,Fuad:
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Berlin;Heidelberg,NewYork;Barcelona;HongKong;London;Milan:
Paris;Tokyo:Springer,2002
(StudiesinEconomicTheory;16)
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Acknowledgments
We appreciate the patience and he1pof our families, first of all of our
wives, Inna A1eskerov and Cornelia Monjardet, and our chi1dren, Sara
and Tair Aleskerov and Sylvain Monjardet.
Several colleagues and friends made many helpful comments: Rafig
Agaev, Pavel Chebotarev, Tümer Kapan, Viacheslav Yakuba and the
Aleskerovs'students in thecourse 'Advancedbinaryrelations' in Bogaz
i<;i University.
Viacheslav Yakuba prepared all figures. Mrs. NataliaAndryushina put
a lot of work into preparing the text in camera-ready form. Mr. Vadim
Shkol'nikov helped to solve all computer problems. Weare particularly
gratefu1 to these friends.
We are grateful to Professor C. Aliprantis and Dr. W. Mueller for sup
port this project and for their help with our work.
Fuad Aleskerov's work on these research topics was initiated in 1976
by discussions with his dear friend and teacher Mark Aizerman. His
friendly supervision (not only in science) is unforgettable.
Recently his work has been partly supported by Bogazici University
(Istanbul), the Institute of Control Seiences (Moscow)and the Russian
Foundation for Basic Research (grant #02-01-00301). An invitation
from Universite ParisI allowedhim to workon themanuscript in a very
comfortable environment. All this support is gratefully acknowledged.
Aleskerov would also like to thank several colleagues and friends who
have supported hirn over the years: Kenneth Arrow, Salvador Barberä,
Hasan Ersel, Herve Moulin, Charles Plott, Iveri Prangishvili, Amartya
Sen.
Bernard Monjardet's work on this book was supported by CERMSEM
(Universite Paris I). Among his CERMSEM colleagues he would partic
u1arlylike to thank Cuong LeVan, A1ainChateauneuf, Patrick Gourdel
and Eric Soutiffor material or scientific support. He would like also to
mention that his interest in these topics was aroused a long time ago by
G. T. Guilbaud's fascinating lectures and kept alive by contacts with
many people, in particular, Mark Aizerman, Jean-Paul Doignon, Peter
Fishburn and Philippe Vincke.
Naturally, all shortcomings are our own responsibility.
Contents
1 Introduction 1
1.1 Background and overview 1
1.2 Content of the chapters . 8
2 Preference, Utility, and Choice: Classic Models 15
2.1 Introduction... .. .... .. . 15
2.2 Binary relations and preferences 16
2.3 Binary relations and choice . . 26
2.4 Utility and choice . . . . . . . . . 30
2.5 Expansion-contraction axioms . . 34
2.6 Relations between the classes of choice functions 43
2.7 Concluding remarks and guide to the literature . 46
3 Utility Maximization Within a Constant Threshold or
a Threshold Depending on One Alternative 51
3.1 Introduction.. .......... ..... ........ 51
3.2 Non-negative threshold functions: the interval choicecase 52
3.3 Interval orders, semiorders, and interval choice 54
3.4 Properties of interval orders and semiorders . . . . . .. 67
3.5 Interval choice and maximal antichains . . . . . . . . . . 84
3.6 Choice functions rationalizable by interval orders and
semiorders . . . . . . . . . . . . . . . . . . . 90
3.7 Arbitrary threshold functions and biorders . . . 95
3.8 Choice functions rationalizable by biorders. . . . 104
3.9 Concluding remarks and guide to the literature . 107
4 Utility Maximization Within a Threshold Depending
on Both Compared Alternatives 113
4.1 Introduction... .. . . . . . 113
4.2 Representation theorems . . . . . . 113
4.3 Additive threshold function . . . . 117
4.4 Multiplicative threshold function . 119
4.5 Properties of choice functions . . . 136
4.6 Concluding remarks and guide to the literature . 137
viii
5 Utility Maximization Within
a Context-dependent Threshold 139
5.1 Introduction .. .... . . .. . 139
5.2 Analysis ofchoice models . . . . . 140
5.3 Properties of choice functions . . . 145
5.4 Inclusive and strongly inclusive maximization . 154
5.5 Over-value choice . . . . . . . . . . . . . . . . . 159
5.6 Binary rationalizability of choice . . . . . . . . 164
5.7 Utility maximization within an additive threshold func-
tion 171
5.8 Properties of simple semiorders . . . . . . . . . . . . . . 182
5.9 Utility maximization within a non-decreasing threshold
function . . . . . . . . . . . . . . . . . . . . . . . 185
5.10 Concluding remarks and guide to the literature . 188
6 Conclusion 193
List of Notations 201
References 207
Index 221
1. Introduction
1.1 Background and overview
A classically rational individual is a maximizer: he chooses the best
alternative according to some fixed utility function.
This idea that a rational individual facing a choice between various
alternatives chooses those which maximize his utility function goes
back at least to the eighteenth century theory with Bentham [35], and
this utility maximization paradigm is always central in the fields of
social sciences concerned with the questions of individual choice. But
what is utility? In the nineteenth century economists such as Jevons,
Menger or Walras (see, e.g., [39]) wanted to construct a cardinal util
ity function, where the utility of the combination of two alternatives
is the sum of their utilities. At the beginning of the twentieth century
this measurable concept of utility was strongly criticized by Pareto
on the basis of an ordinal concept. In [121, 122], Pareto builds the
following coherent mathematical framework to support this concept.
There exists a set A of possible (mutually exclusive) alternatives for
the individual: for instance, the set ofpossible commodity bundles for
the consumer. It is assumed that the set A can be partitioned into
indifference classes (the Edgeworth 'indifference curves'), themselves
completely ordered. Ifalternatives x and y are inthesamedass, the in
dividualisindifferent between them. Ifthey are intwodifferent classes,
he strictly prefers one of these alternatives to the other, for instance,
x to y,and then he alsostrictly prefers any alternative indifferent to x
to any alternative indifferent to y. Nowone can assign numbers to the
'indifference' classes, with the only restriction that they must respect
the preferences between them: if a dass U is preferred to a dass V, it
must receive a greater number. If the alternative x is in the indiffer
ence dass numbered k, one can say that its utility! is k. Now, when
the individual has to choosefrom asubsetX offeasible alternatives, he
chooses the alternatives with maximum utility. This individual choice
model is called the utility maximization (choice) model.
The utility maximization model is equivalent to a model using the
1Note that there is an infinite number of possible systems of such utilities since any
monotone transformation ofthe assigned numbers provides a new valid system.
F. Aleskerov et al., Utility Maximization, Choice and Preference
© Springer-Verlag Berlin Heidelberg 2002
2 1. Introduction
formalism of binary relations. It is assumed that the individual has a
preference relation P on the set of alternatives. Then, when he must
choose between two alternatives x and y, he chooses x if he (strictly)
prefers x to y (denoted by xPy). More generally, when a subject has
to choose a subset X of feasible alternatives, he chooses the alterna
tives in X 'undominated' with respect to his preference relation P,
i.e., those alternatives y such that there does not exist another al
ternative x in X which is more preferable than y. These alternatives
are called the maximal alternatives of P in X, and one can say that
the individual chooses the alternatives maximizing his preference rela
tion. This general model ofchoice based on the maximization ofsome
preference relation, irrespective of its properties, has been called the
pair-dominant choice or the (classical) rational choice model. But in
order to be equivalent to the utility maximization model, the prefer
ence relation P cannot be arbitrary. It must be a weak order, 2 which
means, in particular, that it must satisfy two properties. Firstly, it
must be transitive, i.e., if our rational subject prefers x to y and y
and y to z, he must prefer x to z.Secondly, the associated indifference
relation I - x is indifferent to y if x is not preferred to y and y is not
preferred to x - must be transitive as weIl: xly and yIz must imply
xlz. In other words, to say that the choice of an agent follows the
utility maximization model is equivalent to assuming that his prefer
ence relation satisfies these two transitivity conditions - a hypothesis
which cannot be tested directly.
Another milestone in the utility theory was made in 1938by Samuel
son [148J when he proposed to base this theory on the observations
of the effective choices of an agent rather than on 'a priori' assump
tions about his preferences. Then in the context of the consumers'
choice problem in a competitive economy, he introduced a famous con
sistency condition later called Samuelson's Weak Axiom of Revealed
Preferences. This axiom implies that the consumer's choices can be
described in terms ofutility maximization or, equivalently, in terms of
a preference relation maximization, this relation being a weak order.
The Samuelson's revealed preference theory was developed by many
economists, and from the 1950s in a more abstract context (see, e.g.,
[50, 95]). There is a universal set A of possible (mutually exclusive)
2At least iftheset Asarisfiessorneconditions, for instance,ifit is finite, an assurnption
which is always made in this book.
1. Introduction 3
alternatives and a family offeasible subsets of A representing different
sets of alternatives whieh can be submitted to an agent. Formally, a
choice function C associates to each feasible set X a subset C(X).
Properties of such functions were studied, and characteristic condi
tions (the so-called rationality axioms) were established whieh guar
antee that the choiees defined by a choiee function can be described
in terms of utility maximization (such choiee functions are called fully
or utility rationalizable).
The seminal work in whieh the three above descriptions of rational
choice (in terms of utility maximization, in terms of maximization of a
weak order, and in terms of rationality axioms) met was that of Arrow
[30]. This line of research was continued by several authors, with
partieularly significant contributions from Sen [155]-[160]. In these
works the question of how classieally rational choiee can be described
received an exhaustive answer. Surveys ofthe theory ofrational choiee
can be found in [157, 166].
Today, the utility (or weak order) maximization paradigm is still a
foundation of the classieal demand theory in mieroeconomies. How
ever, economists understand the deficiences of this model (see, for
instance, [108]).
Let us discuss threedifferent critics' attitudes to this model. First, it
was strongly debated by Simon in the 1950on the basis of his bounded
rationality approach [163]. He claimed that individuals do not neces
sarily look for the best alternative(s) in a feasible set, but rather they
accept alternatives whieh they consider satisfactory, i.e., they have a
satisfying rather than an optimizing behavior. Sen, one of the lead
ing economists who formulated the paradigm of rational choiee in its
modern form, has also been one of its major crities [158, 159, 160].
He constructed some brain experiments in whieh rational behavior
turns out not to be based on utility maximization. In their clas
sie experiments, the psychologists Kanheman and Tversky [101, 168]
showed that very often individuals do not follow this paradigm, and
they proposed explanations and alternative approaches. Other discus
sions on the applicability of this paradigm, and suggestions for mod
ifying it and the corresponding references can be found, for instance,
in [33,84, 85, 87, 157].
In fact, as early as in the 1930s Georgescu-Roegen [88] and then
Armstrong [28] claimed that abasie assumption of the classie utility
model, narnely, the transitivity of the indifference relation, is highly
