On the axioms of revealed preference in fuzzy consumer theory

The revealed preference is a central subject in classical consumer theory. Authors like Samuelson, Arrow, Richter, Sen, Uzawa and others have proposed an axiomatic setting of revealed preference theory.Consequently revealed preference axioms WARP and SARP and congruence axioms WCA and SCA have been considered.An important theorem of Sen establishes the equivalence between these axioms provided the family of budgets includes all non-empty finite sets of bundles.Fuzzy consumer theory (=fuzzy choice functions) is a topic that appears in a lot of papers. Particularly, Banerjee studies in fuzzy context axioms of revealed preference and congruence extending some results of Arrow and Sen.In this paper we modify the Banerjee definition of a fuzzy choice function (=fuzzy consumer) and we study some fuzzy versions of the axioms of revealed preference and congruence. Banerjee fuzzifies only the range of a consumer; we use a fuzzification of both the domain and the range of a consumer. The axioms WAFRP, SAFRP, WFCA, SFCA generalize to fuzzy consumer theory the well-known axioms WARP, SARP, WCA, SCA. Our main result establishes some connections between WAFRP, SAFRP, WFCA, SFCA extending a significant part of Sen theorem.Generally, we work in a fuzzy set theory based on a continuous t-norm, but some results are obtained for Gödel t-norm and others are obtained for Lukasiewicz t-norm.