Arrow’s Axiom and Full Rationality for Fuzzy Choice Functions

Arrow Index of Fuzzy Choice Function

“…We shall recall some notions and results in [12], [13], [14], [15], [16]. A fuzzy choice space is a pair (X, B) where X is a non-empty set of alternatives and B is a non-empty family of non-zero fuzzy subsets of X.…”

“…In [12], [13], [14], [15] we have considered fuzzy choice functions C for which the domain and the range are made of fuzzy subsets of X. Banerjee fuzzifies only the range of a choice function; we use a fuzzification of both the domain and the range of a choice function. In our case, the available sets of alternatives are fuzzy subsets of X.…”

“…The main theorem of [14] established the equivalence between F AA and the full rationality of fuzzy choice functions. By measuring by the Arrow index the degree of full rationality of a fuzzy choice function, the results of this paper allow to compare them from the point of view of their rationality.…”

“…In [11], [12], [13], [14], [15], [16] we worked with a more general definition of fuzzy choice functions. Banerjee fuzzifies only the range of a choice function, in our approach both the domain and the range of a choice function are made of fuzzy subsets of a universe of alternatives.…”

“…Works [12], [13], [14], [15], [16] are an attempt to build a theory of revealed preference for these fuzzy choice functions. In [15] we have introduced the Fuzzy Arrow Axiom F AA and we have proved that a fuzzy choice function satisfies F AA iff it is full rational.…”