Uncertain Choices: A Comparison of Fuzzy and Probabilistic Approaches

Choices among alternatives in a set can be expressed in three different ways: by means of choice functions, by means of preference relations or using choice probabilities. The connection between the two first formalizations has been widely studied in the literature, both in the crisp or classical context and in the setting of fuzzy relations. However, the connection between probabilistic choice functions and fuzzy choice functions seems to have been forgotten and as far as we know, no literature can be found about it. In this contribution we focus on the comparison of both types of choice functions. We provide a way to obtain the fuzzy choice function from the probabilistic choice function and the other way around. Moreover, we can prove that under Luce's Choice Axiom the fuzzy choice function derived from the probabilistic choice function is G-normal.