Table Of ContentLecture Notes in Economics
and Mathematical Systems 373
Editorial Board:
H. Albach, M. Beckmann (Managing Editor)
P. Dhrymes, G. Fandel, G. Feichinger, W. Hildenbrand
W. Krelle (Managing Editor)
H. P. Kiinzi, K. Ritter, U. Schittko, P. Schonfeld, R. Seiten, W. Trockel
Managing Editors:
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut flir Gesellschafts- und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, W-5300 Bonn, FRG
Antoine Billot
Economic Theory
of Fuzzy Equilibria
An Axiomatic Analysis
Springer-VerJag Bcrlin HeideJberg GmbH
Author
Dr. Antoine Billot
Department of Economics
University of Paris 2 (Pantheon-Assas)
92, rue d' Assas
F-75006 Paris
ISBN 978-3-540-54982-6 ISBN 978-3-662-01050-1 (eBook)
DOI 10.1007/978-3-662-01050-1
This work is subject to copyright. AII rights are reserved, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, re-use of
illustrations, recitation, broadcasting, reproduction on microfilms or in any other way,
and storage in data banks. Duplication of this publicat ion 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-Verlag.
Violations are Iiable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg in 1992
Typesetting: Camera ready by author
42/3140-543210 -Printed on acid-free paper
In memoriam of Guillaume Artur du Plessis
Acknowledgements
My thanks go to Maurice Desplas, Didier Dubois, Victor Oinsburgh, Bertrand Munier,
Pierre Salmon, Bernard Walliser, Hans-Jiirgen Zimmermann and Daniel Vitry, who read large
parts of the manuscript based on my Ph.D. dissertation; the book owes much to their detailed
suggestions. Of course, all errors remain my responsability.
I should also like to thank Marc Chenais and Cyrille Piatecki for their helpful comments
and David Sarfas for his translation.
Finally, my greatest debt is to Oaude Ponsard. His comments have been more than
perceptive: he saved me from several errors and made numerous suggestions which have resulted
in a considerably wider perspective for this book. He was my Ph.D. Director and I hope to have
been one of his friends.
The writing and translation of this book were supported by the Centre d'Economie
Financiere et Bancaire of the University of Paris 2, Pantheon-Assas. I should like to thank their
Directors, Michele de Mourgues and Claude Vedel.
Preface
Fuzzy set theory, which started not much more than 20 years ago as a generalization of classical
set theory, has in the meantime evolved into an area which scientifically, as well as from the
point of view of applications, is recognized as a very valuable contribution to the existing
knowledge. To an increasing degree, however, fuzzy set theory is also used in a descriptive,
factual sense or as a decision making technology. Most of these applications of fuzzy set theory
are in the areas of fuzzy control, multi-criteria analysis, descriptive decision theory and expert
systems design. In economics, the application of fuzzy set theory is still very rare. Apart from
Professor Ponsard and his group, who have obviously recognized the potential offuzzy set theory
in economics much better than others, only very few economists are using this new tool in order
to model economic systems in a more realistic way than often possible by traditional approaches,
and to gain more insight into structural interdependences of economic systems. I consider it,
therefore, particularly valuable that Dr. Billot, in his book, makes a remarkable contribution in
this direction.
There seems to be one major difference between Dr. Billot's contribution and the other
publications in which fuzzy set theory is applied to either game theory, group decision making
or economic theory: Some authors use fuzzy set theory in order to gain a mathematically more
attractive or efficient model for their problems of concern, others use a specific problem area
just in order to demonstrate the capabilities of fuzzy set theory without a real justification of
whether this "fuzzification" of a traditional theory makes sense or is justified by better results.
For Dr. Billot, not surprisingly similar to Professor Ponsard, the economic thinking seems to be
the predominant factor and fuzzy set theory is only a tool to improve an economic model or
x
theory where it seems possible or appropriate. Hence, his book should not primarily be considered
as a contribution in fuzzy set theory, but as one in economic theory, which merges microeconomic
considerations with macroeconomic observations.
On an axiomatic basis, the author analyses flrst how the assumption of speciflc fuzzy
preferences bear on equilibria of classical and newly defmed economic systems. He proceeds
in three steps: In the flrst chapter of his book, he presents speciflc kinds of fuzzy individual
preferences and in a second chapter, he investigates their repercussions on aggregation proce
dures in the sense of group decision making or welfare economics, in the third chapter he focuses
his attention on fuzzy cooperative and non-cooperative games. In both fourth and flfth chapters,
he applies the results of the flrst three chapters to the analysis of equilibria in a speciflc market
model.
In introducing his view of fuzzy preferences, Dr. Billot does not follow the so-called
"Anglo-saxon, cardinalist school", but he proceeds along the lines of the European or French
school in concentrating on orders and allowing for different degrees of preferences. He also
distinguishes between relative, absolute, local and global preferences and postulates the "do
minated irrationality principle". He can thus prove that given a certain preference structure
(X ,1((.,.», there exists a fuzzy utility function ifX is a totally preordered, countable set. To arrive
at that result he introduces a number of notions and deflnitions which partly deviate from those
used in classical fuzzy set theory. This concerns particularly the notions of reflexivity and
transitivity and also that of a fuzzy preorder which for Dr. Billot is a fuzzy relation which is
reflexive (f.s.) And f-transitive. New is also the introduction of a "fuzzy incoherent preference"
which is a binary fuzzy relation which is reflexive (f.s.) but not transitive. For the following
parts of his book, the author bases to a large extent of his proof of the existence of a continuous
utility function, given that there exists a fuzzy preorder and continuous fuzzy preferences as
well as a connected set of decision alternatives ( it is also shown in the flrst part that Boolean
preorders are a special case of above ).
The second chapter focuses on the aggregation of fuzzy preferences and on the social
choice, i.e. the usual problem of the theory of group decision making. Dr. Billot flrst describes
traditional theory (evaluation cycles of "Condorcet" ) and Arrow's impossibility theorem. In
the framework of Arrow's axiomatic system, he now deflnes requirements for a group decision
function which is supposed to be analogue to an individual preference function. His axioms are
essentially that the group preference constitutes a fuzzy preorder, that the group preferences are
a function of the individual preferences, the axiom of unanimity, the independence of alternatives
and the request that the group decision function be not dictatorial. Dr. Billot reformulates the
XI
requirements for a social welfare function in his axioms Pl through P6, introduces a "planner"
as an artifact, focuses on specific lexicographic aggregation rules and eventually shows in his
theorems 4 and 5, respectively, that -given fuzzy preference relation for the agents -a translation
from the Arrovian dictator into a more democratical structure is possible. I find the new structures
introduced by Dr. Billot, even though they are sometimes surprising, intuitively appealing and
productive.
Two further major topics are of concern to Dr. Billot : Equilibrium points of fuzzy games
and fuzzy economic equilibria. In the third chapter of the book, he fU'St focuses on non
cooperative games and then he analyses cooperative games. He presents some generalizations
to the theorem of Kakutani and an application to Nash equilibria. Then, the agents of the game
are equipped with non-dichotomous behaviours which eventually yield ordinal strategic utility
functions. It should be noted that the termf-convex defmed and used by the author does not refer
to the membership function, as in fuzzy set theory, but to the support of a fuzzy set. Of particular
importance is the introduction by the author of the notion of a peripheral core which is defmed
to be a specific fuzzy set. He can then show that this core under certain circumstances is non
empty. In effect, the author transforms the situations without equilibria by introducing fuzzy
behaviours of the agents and by augmenting the number of equilibrium solutions.
The author then proceeds to investigate the compatibility of this fuzzy behaviour in an
economy with production. He focuses his attention on the factor labour and eliminates for the
sake of transparency other possible activities in an economy. Essentially, he tries to show that
the introduction of fuzzy preferences does not impeach the existence of a general equilibrium.
He also tries to show that this behavioural fuzziness can generate some non-standard equilibria
for the Walrasian approach. After the examination of different approaches existing already in
the literature, he presents a model of this simplified economy which he calls a GLE (for
Good-Labour-Economy ). He then presents two results concerning the existence of equilibria,
one concerning an GLE in the sense of Ponsard in which the goals of the agents and their
constraints are fuzzy and the other concerning an GLE in which only preferences are fuzzy in
the sense of the fU'St chapter of this book. The principal result of this chapter is the verification
of the consistency of fuzzy choice behaviour and the existence of a general eqUilibrium.
In the last chapter, he focuses on dis-equilibria in the labour market including different
kinds of unemployment. The demonstration of the possible existence of an unvolontary
unemployment which should be compatible with a Walrasian general equilibrium depends on
the notion of free disposal and on that fuzzy preference relations. The author shows that there
exists some Walrasian general equilibria with disequilibria of tasks by using the usual defmition
XII
of a free disposal equilibrium. In characterizing the unemployed, the author focuses on those
agents whose preference relations are typically that of unvolontary Walrasian unemployed who
persist in offering labour even if the equilibrium wage is and stays zero. Those unemployed are
locally coherent and globally indifferent between a zero-offer and a non-zero-offer of tasks, but
the absolute preferences of the non-zero offer is bigger than the absolute preference for the
zero-offer. By contrast to the traditional microeconomic theory in which behaviour of the agents
is more or less unified for the analysis, the author studies diverse agents and examines the
pertinence of the usual equilibria when he introduces a new behavioural mode within the
theoretical models of traditional economic theory. He defines, for the economy under consi
deration, an equilibrium for unemployment and shows that these equilibria exist under certain
conditions.
Altogether, this book presents a very intriguing analysis of classical problems with new
tools which lead to interesting results. The author can be congratulated on his results and it can
be hoped that many economic theorists start from Dr. Billot's work and advance economic theory
along the lines indicated.
Aachen, January 1991 H.-f. Zimmermann
Contents
General Introduction 1
Chapter 1 : Individual Fuzzy Relation of Preference 5
S.l. The fuzzy binary relation of preference 6
S.2. The paradox of indifference 15
S.3. Fuzzy utility function on a countable set 19
S.4. Fuzzy utility function on a convex set 23
Chapter 2 : Aggregation ofF uzzy Preferences 31
S.l. Arrovian dictator and fuzzy preferences 32
S.2. Fuzzy Coalitions and democracy 40
Chapter 3 : Fuzzy Games 57
S.l. Fixed points and Nash-equilibrium 59
S.2. Prudent behaviour and equilibria 73
S.3. Cooperative fuzzy games 82
Chapter 4 : Fuzzy General Equilibrium 99
S.l. Tasks and quasi wage-rates 100
S.2. The existence of a fuzzy general equilibrium 111
Chapter 5: Underemployment Fuzzy Equilibrium 121
S.l. Existence of an underemployment equilibrium 122
S.2. Involuntary unemployment and dominated irrationality 133
General Conclusion 143
Annexes 145
Bibliography 151
