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Chapter 10
Similarity of Fuzzy Choice
Functions
In this chapter, we focus on the work of Georgescu [19, 22]. Zadeh  in-
troduced the notion of a similarity relation in order to generalize the notion
of an equivalence relation. Trillas and Valverde  extended the notion of a
similarity relation for an arbitrary t-norm. See also [27, p. 254].
Samuelson  introduced revealed preference theory in order to deﬁne
the rational behavior of consumers in terms of a preference relation. Others
including Uzawa , Arrow , Sen , Suzumura  have developed a
revealed preference theory in a context of choice functions. Preferences of the
individuals can be imprecise, due possibly to human subjectivity or incom-
plete information. Vague preferences can be mathematically modelled by fuzzy
relations, [9-12, 14, 28]. However, the choice can be exact or vague even if the
preference is ambiguous. When the choice is exact, it can be mathematically
described by a crisp choice function. The case of vague preferences and exact
choices considered in  has been presented in Chapter 7. In situations such
as negotiations on electronic markets, the choices are fuzzy.
Various types of fuzzy choice functions have been presented in Chapters 2
and 7. See also [2, 33, 45]. Georgescu [16 , 17] presented a fuzzy generalization
of classic theory on revealed preference which generalized the results of [1, 34,
35].
In this chapter, Georgescu’s degree of similarity of two fuzzy choice func-
tions is presented. It is analogous to the degree of similarity of two fuzzy sets
and induces a similarity relation on the set of all choice functions deﬁned on
a collection of fuzzy choice functions. The degree of similarity of two choice
functions is deﬁned to be the pair (, t), where is a t-norm and t is an element
of the interval [0, 1].
There is an interesting notion involving similarity relations and that is
Poincar´e’s paradox. Poincar´e stated that in the physical world “equal” re-
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