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Chapter 10
Similarity of Fuzzy Choice
Functions
In this chapter, we focus on the work of Georgescu [19, 22]. Zadeh [47] in-
troduced the notion of a similarity relation in order to generalize the notion
of an equivalence relation. Trillas and Valverde [40] extended the notion of a
similarity relation for an arbitrary t-norm. See also [27, p. 254].
Samuelson [34] introduced revealed preference theory in order to define
the rational behavior of consumers in terms of a preference relation. Others
including Uzawa [42], Arrow [1], Sen [35], Suzumura [38] 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 [3] 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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