The Arrow index of a fuzzy choice function C is a measure of the degree to which C satisfies the Fuzzy Arrow Axiom, a fuzzy version of the classical Arrow Axiom. The main result of this paper shows that A(C) characterizes the degree to which C is full rational. We also obtain a method for computing A(C). The Arrow index allows to rank the fuzzy choice functions with respect to their rationality. Thus, if for solving a decision problem several fuzzy choice functions are proposed, by the Arrow index the most rational one will be chosen.
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This paper is concerned with consistency properties for fuzzy consumers. We introduce the consistency conditions Fα, Fβ, Fδ as fuzzy forms of Sen's properties α, β and δ. Other consistency conditions are also studied: Fα2, Fβ(+), Fγ2 and FPI. The main results are: (1) A fuzzy consumer verifies Fα, Fβ if and only if the congruence axiom WFCA holds; (2) If h is a normal fuzzy consumer then Fδ holds if and only if the associated fuzzy preference relation R is quasi-transitive; (3) A fuzzy consumer is normal if and only if conditions Fα2, Fγ2 hold.
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