Table Of ContentTRIANGULAR NORM-BASED MEASURES
AND GAMES WITH FUZZY COALmONS
THEORY AND DECISION LffiRARY
General Editors: W. Leinfellner (Vienna) and G. Eberlein (Munich)
Series A: Philosophy and Methodology of the Social Sciences
Series B: Mathematical and Statistical Methods
Series C: Game Theory, Mathematical Programming and Operations Research
Series D: System Theory, Knowledge Engineering and Problem Solving
SERIES C: GAME THEORY, MATHEMATICAL PROGRAMMING
AND OPERATIONS RESEARCH
VOLUME 10
Editor: S. H. Tijs (University of Nijmegen); Editorial Board: H. Keiding (Copenhagen),
J.-F. Mertens (Louvain-Ia-Neuve), H. Moulin (Durham), B. Peleg (Jerusalem), T. E. S.
Raghavan (Chicago), 1. Rosenmiiller (Bielefeld), A. Roth (Pittsburgh), D. Schmeidler (Tel
Aviv), R. Selten (Bonn).
Scope: Particular attention is paid in this series to game theory and operations research, their
formal aspects and their applications to economic, political and social sciences as well as to
socio-biology. It will encourage high standards in the application of game-theoretical
methods to individual and social decision making.
The titles published in this series are listed at the end of this volume.
TRIANGULAR NORM-BASED
MEASURES AND GAMES
WITH FUZZY COALITIONS
by
DAN BUTNARIU
Department of Mathematics and Computer Science,
University of Haifa, Israel
and
ERICH PETER KLEMENT
Institute of Mathematics, Johannes Kepler University, Linz, Austria
~
"
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Butnartu, Dan.
Trtangular norm-based measures and games wtth fuzzy coalttons I by
Dan Butnartu and Ertch Peter Klement.
p. cm. -- (Theory and deciston 11brary. Ser1es C, Game
theory, mathemat1cal programm1ng, and operat1ons research; v. 10)
Includes b1bl1ograph1cal references and 1ndex.
ISBN 978-90-481-4296-5 ISBN 978-94-017-3602-2 (eBook)
DOI 10.1007/978-94-017-3602-2
1. Game theory. 2. Fuzzy sets. I. Klement, E. P. (Er1ch Peter)
II. T1tle. III. Ser1es.
QA269. B88 1993
519.3--dc20 93-24820
ISBN 978-90-481-4296-5
Printed on acid-free paper
All Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To Hadassah and Sieglinde
Contents
Introduction ............ . 1
I Triangular Norm-Based Tribes 7
1 Triangular Norms ....... . 7
2 Triangular Norm-Based Tribes . 27
3 Tribes Based on Fundamental t-Norms . 32
II Triangular Norm-Based Measures 37
4 Disjointness with Respect to t-Norms 37
5 Triangular Norm-Based Measures and their
Representation by Markov Kernels . 41
6 Integral Representation of Too -Measures . 48
7 Integral Representation of Monotone
T -Measures on Generated Tribes . . 54
8 Decomposition of Monotone T-Measures . 57
III Too -Measures 69
9 Jordan Decomposition of Bounded Too -Measures 69
10 Jordan Decomposition of Finite Too -Measures 75
11 Absolute Continuity of Too -Measures ..... 78
12 Vector Too-Measures with Darboux Property 82
13 Nonatomic Too -Measures ........... . 87
14 A Liapounoff Type Theorem for Too -Measures . 95
IV Games with Fuzzy Coalitions 99
15 Functions of Bounded Variation . 99
16 The Space FAC ......... . 107
17 The Space pFNA ........ . 111
18 Diagonal Aumann-Shapley Value on pFNA 115
19 Classical Values Revisited ......... . 123
v
Extensions of the Diagonal Value 127
20 Representation of the Diagonal Value on pFNA
by Frechet Differentials. . . . . . . . . . 127
Vll
21 Dini Derivatives of Games in FBV .... 131
22 Proof of Theorem 20.4 ........... 137
23 Diagonal Value for Gateaux Differentiable
Games ................ 145
24 Proof of Theorem 23.6 . . . . . . . . . . . 152
25 Further Extensions of the Diagonal Value 157
VI Related Topics and Applications 165
26 Games with Crisp Coalitions: Extensions
of the Diagonal Value on pNA .. 165
27 Values, Cores, and Economic Equilibria 168
28 Rate Problem for Services in Bulk: A Game
Theoretical Model . . . . . . . . . . 175
29 Plausibility Measures, Possibility Measures,
and Normalized Valuations on MV Algebras 180
30 Open Problems and Bibliographical Comments 183
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 189
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 197
vIn
Acknowledgment
We are very grateful to Robert J. Aumann, Irine! Dragan, Lloyd S. Shapley, and Lotfi
A. Zadeh for repeatedly giving us the opportunity to discuss various aspects of the
material presented in this work. We greatly benefitted from their deep insights into
different fields and also from their personal support.
We are indebted to Zvi Artstein, Radko Mesiar, Roland Takacs, and Anton Wakol
binger whose solutions to some mathematical problems led to improvements in our
own results.
We greatly appreciate the comments of Alain ChateauneuJ, Ulrich Hohle, Vilem
Novak, Endre Pap, and Siegfried Weber who kindly agreed to read large parts of the
manuscript and whose comments helped us to improve earlier versions of it.
We gratefully acknowledge the following institutions and individuals: the Austrian
Bundesministerium fur Wissenschaft und Forschung, Vienna, the Linzer Hochschul
fonds, the Osterreichische Mathematische Gesellschajt, our colleagues in the Institut
fur Mathematik of the Johannes Kepler Universitiit Linz, Austria (in particular Pe
ter Weip), in the Institut fur Systemtheorie of the Johannes Kepler Universitiit Linz,
Austria (especially Franz Pichler), in the Department of Mathematics and Computer
Science of the University of Haifa, Israel (particularly Yaki Sternfeld, Jonathan Arazi,
Yair Censor, Jonathan Golan, and Izu Vaisman), and in the Department of Mathe
matics of The University of Texas at Arlington (chairman George Fix). Their help
in providing both the scientific environment for our cooperation and the financial
support for our mutual visits to Linz and Haifa, as well as for the visit of the first
author to Arlington, has made this book possible.
We very much appreciate the professional support of Waltraud Eidljorg in typing
several versions of the manuscript, and of Daria Novakova in providing the final
version in U-TEX. Gerard Wearden very much helped us when problems with the
English language occured. All Figures have been drawn using MATHEMATICATM.
Special thanks go to the Editor of this Series, Stephen Tijs, and to Maria M.
Stratta from Kluwer Academic Publishers.
Haifa and Linz, December 1992
Dan Butnariu
Erich Peter Klement
IX
1
Introduction
This book aims to present, in a unified approach, a series of mathematical results con
cerning triangular norm-based measures and a class of cooperative games with Juzzy
coalitions. Our approach intends to emphasize that triangular norm-based measures
are powerful tools in exploring the coalitional behaviour in 'such games. They not
only unify and simplify some technical aspects of the already classical axiomatic the
ory of Aumann-Shapley values, but also provide new perspectives and insights into
these results. Moreover, this machinery allows us to obtain, in the game theoretical
context, new and heuristically meaningful information, which has a significant impact
on balancedness and equilibria analysis in a cooperative environment.
From a formal point of view, triangular norm-based measures are valuations on
subsets of a unit cube [0, 1]X which preserve dual binary operations induced by trian
gular norms on the unit interval [0, 1]. Triangular norms (and their dual conorms) are
algebraic operations on [0,1] which were suggested by MENGER [1942] and which
proved to be useful in the theory of probabilistic metric spaces (see also [WALD
1943]). The idea of a triangular norm-based measure was implicitly used under
various names: vector integrals [DVORETZKY, WALD & WOLFOWITZ 1951], prob
abilities oj Juzzy events [ZADEH 1968], and measures on ideal sets [AUMANN &
SHAPLEY 1974, p. 152].
Games with Juzzy coalitions are described in this book by special functions on
subsets of a unit cube [0, 1]x. Particular classes of such games considered previously
were called ideal set Junctions [AUMANN & SHAPLEY 1974, p. 144], Juzzy games
[AUBIN 1974 a,b; 1982, p. 320], generalized games [AUBIN 1981] or, simply,
games [RUCKLE 1982, 1987]. They were studied because of their relevance to
the interaction between values, cores and equilibria, an interaction which became
self-evident in [AUMANN & SHAPLEY 1974, Chapters IV - VI]. The connection
between cooperative games and certain classes of triangular norm-based measures
was implicitly noted by DVORETZKY, WALD & WOLFOWITZ [1950], SHAPLEY
[1967] and AUMANN & SHAPLEY [1974, p. 152]. The present book is a first
attempt to exploit this relationship.
The proper domain for a game with fuzzy coalitions is a triangular norm-based
tribe which, in our interpretation, is a collection of Juzzy coalitions, the formation
and dismantling of which obey some exogeneous rules modeled in terms of algebraic
operations induced by triangular norms.
The term Juzzy coalition, first introduced in [AUBIN 1974a, b], denotes a collection
of economic agents, i.e., players, who transfer fractions of [their] representability
[BILLOT 1990] to a collective decision maker, the fuzzy coalition. The fraction
of representability a player transfers to a specific coalition is called the degree oj
membership (of the player in that coalition), and it may be any real number in the
interval [0,1].
Formally, fuzzy coalitions are modelled by fuzzy sets [ZADEH 1965]' i.e., functions
2
A : X ---+ [0,1], where X is the set of players and A(x) represents the degree of
membership of the player x in the fuzzy coalition A. However, the fuzzy coalitions
are closely linked to a fixed underlying triangular norm-based tribe !Y, i.e., we shall
use the term fuzzy coalition only for fuzzy sets which are contained in!Y. This
automatically imposes a measurability condition upon the function A : X ---+ [0,1]
(see Theorem 3.2), where fuzzy coalitions are proper generalizations of ideal sets in
the sense of AUMANN & SHAPLEY [1974, p. 142]. It turns out that tribes of ideal
sets are exactly what we shall call generated tribes.
The concept of a fuzzy coalition, i.e., a collective decisional entity where members
may have gradual degrees of membership, may be somewhat surprising for scientists
used to working with the concept of coalition as a collection of players who are
indistinguishable from each other except for their marginal contribution to the total
worth of the coalition, as it emerged from the work of NEUMANN & MORGENSTERN
[1944] and most of the literature based on it. Such crisp collections of players will
be called crisp coalitions, and they are, in fact, particular fuzzy coalitions.
Of course, in many real cooperative games only crisp coalitions are possible. How
ever, in other cases the restriction to crisp coalitions only is an excessive id~alization.
For instance, consider a cooperative game in which each coalition is a group of share
holders of a company. The worth of a coalition is determined by the total investment
(endowment) of its members in this company. Assume that two members of a coali
tion invest the same amount (and, consequently, have the same share in and the same
marginal contribution to this coalition) and assume that this amount represents 90 %
of the individual worth of the first player and only 5 % of the second player's indi
vidual worth. In such a case, it is hard to imagine that the two players will have
equal interest in the performance of the coalition because of their different degrees
of personal involvement. The use of fuzzy coalitions makes it easy to represent the
different involvement of the two players.
Some authors have linked the introduction of fuzzy coalitions to the convexification
of a set of crisp coalitions. In our general approach, the convexity of a tribe of
coalitions plays little role: a tribe is a convex set of functions if and only if it is
generated, but the consistency of the solution concepts we deal with does not depend
on whether the tribe is generated or not (see Theorem 18.4).
From a game theoretical point of view, the main concept investigated in this work
is that of Aumann-Shapley value. We use it in the more specific framework of games
with fuzzy coalitions, at the same time preserving the original plausibility conditions:
efficiency, symmetry, positivity and additivity. Aumann-Shapley values for games
with fuzzy coalitions have the same natural interpretation as the original concept
[AUMANN & SHAPLEY 1974, p. 15].
The principal question is whether diagonal Aumann-Shapley values for games
with fuzzy coalitions exist and, if so, how they can be represented in an analytical
way. We start our discussion at the point which is marked by the contributions of
AUMANN & SHAPLEY [1974], MERTENS [1980, 1988] and RUCKLE [1982, 1987],
