A Study On Rationality Conditions Of Fuzzy Choice Functions

In the research on choice functions, the investigation of rationality conditions is of extremely important significance no matter whether the involved choice function is crisp or not. In this thesis, we systematically discuss the connections between some extensively used rationality conditions of fuzzy choice function in the literature, make an exploration on the issue of how to characterize rationality of a fuzzy choice function. In addition, we study some links between fuzzy choice functions and fuzzy preference relations. The major research aspects and results are as follows.Firstly, we summarize the relationships between some rationality conditions for crisp choice functions which are proposed by Arrow, Richter, Sen et al. We also present some new results, and thus complete the existing conclusions in the theory of crisp choice function.Secondly, based on the results in crisp case, we putforward some rationality conditions for fuzzy choice functions and establish links between these conditions. As a consequence, some work done by Arrow, Richter, Sen et al. is generalized.We note that many conclusions true in crisp case are no longer valid in the generalization when choice functions are fuzzified. However, if we impose some restrictions on fuzzy choice functions, then some satisfactory results are still available. For this purpose, we deeply investigate relationships between rationality conditions under the assumption that every involved choice set is normal. The major result is that a choice function is transitive rational is equivalent to that the choice function can be rationalized or the choice function satisfies WFCA (SFCA) or the choice function satisfies Fα and Fβ condition. Consequently, most results concerning crisp choice functions are accordingly extended. Meanwhile, we point out that WAFRP or SAFRP can not be employed to characterize the transitive rationality of fuzzy choice functions like in crisp case.Finally, we make an investigation into the interrelationship between fuzzy choice function and fuzzy preference relation. We draw the conclusion that, if the involved fuzzy preference relation is reflexive and the fuzzy choice function is normal, then they are uniquely determined byeach other. This result generalizes the result obtained by Arrow who shows that crisp choice function and ordering can be uniquely determined by each other.To sum up, rationality conditions of fuzzy choice functions with the domain being crisp are systematically investigated, and characterization conditions of a transitive rational fuzzy choice function are provided in the thesis. The research lays a foundation for how to characterize the rationality of a choice function in the general case.