Arrow’s Axiom and Full Rationality for Fuzzy Choice Functions

A classical result for crisp choice functions shows the equivalence between Arrow axiom and the property of full rationality. In this paper we study a fuzzy form of Arrow axiom formulated in terms of the subsethood degree and of the degree of equality (of fuzzy sets). We prove that a fuzzy choice function satisfies Fuzzy Arrow Axiom if and only if it is (fuzzy) full rational. We also show that these conditions are also equivalent with weak and strong fuzzy congruence axioms WFCA and SFCA. It is studied the Arrow index, a new concept that indicates the degree to which a fuzzy choice function satisfies the Fuzzy Arrow Axiom. The author wishes to express her gratitude to the two anonymous reviewers for their valuable comments and discussions that contributed to a higher quality of the paper. Some of their comments can be found in the final version of the paper. An abstract of this paper was presented at the First Spain Italy Netherlands Meeting on Game Theory SING, Maastricht, The Netherlands, June 2005.