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Preface
This book provides a comprehensive study of fuzzy social choice theory. Hu-
man thinking is marked by imprecision and vagueness. These are the very
qualities that fuzzy logic seeks to capture. Thus, social choice theory suggests
itself as a means for modeling the uncertainty and imprecision endemic in
social life. Nonetheless, fuzzy logic has seen little application in the social sci-
ences to include social choice theory. We attempt to partially fill this lacuna.
The main focus of Chapter 1 is the concept of a fuzzy maximal subset of
a set of alternatives X. A fuzzy maximal subset gives the degree to which the
elements of X are maximal with respect to a fuzzy relation on X and a fuzzy
subset of X. This allows for alternative notions of maximality not allowed in
the crisp case. The main result states that a fuzzy maximal subset is not the
zero function if and only if the fuzzy preference relation involved is partially
acyclic.
Chapter 2 deals with fuzzy choice functions. Classical revealed preference
theory postulates a connection between choices and revealed preferences. If it
is assumed that preferences of a set of individuals are not cyclic, then their col-
lective choices are rationalizable. We examine this question in some detail for
the fuzzy case. We determine conditions under which a fuzzy choice function
is rationalizable. We also present results inspired by Georgescu [24]. Desai [16]
characterized the rationality of fuzzy choice functions with reflexive, strongly
connected, and quasi-transitive rationalization in terms of path independence
and the fuzzy Condorcet property. We consider these results as well as those
of Chaudhari and Desai [11] on various types of rationality.
Chapter 3 is concerned with the factorization of a fuzzy preference rela-
tion into the “union” of a strict fuzzy relation and an indifference operator,
where union here means conorm. Such a factorization was motivated by Fono
and Andjiga [12]. The factorization is useful in the examination of Arrowian
type results. We show that there is an inclusion reversing correspondence be-
tween conorms and fuzzy strict preference relations in factorizations of fuzzy
preference relations into their strict preference and indifference components.
In Chapter 4, we consider fuzzy non-Arrowian results. The first author
felt that presenting this material before Arrowian type results would help
emphasize the difference between the fuzzy case and the crisp case. We pro-
vide two types of fuzzy aggregation rules that satisfy certain fuzzy versions of
ix

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