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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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