Characterizable fuzzy preference structures

In this paper, we study the existence, construction and reconstruction of fuzzy preference structures. Starting from the definition of a classical preference structure, we propose a natural definition of a fuzzy preference structure, merely requiring the fuzzification of the set operations involved. Upon evaluating the existence of these structures, we discover that the idea of fuzzy preferences is best captured when fuzzy preference structures are defined using a Łukasiewicz triplet. We then proceed to investigate the role of the completeness condition in these structures. This rather extensive investigation leads to the proposal of a strongest completeness condition, and results in the definition of a one-parameter class of fuzzy preference structures. Invoking earlier results by Fodor and Roubens, the construction of these structures from a reflexive binary fuzzy relation is then easily obtained. The reconstruction of such a structure from its fuzzy large preference relation – inevitable to obtain a full characterization of these structures in analogy to the classical case – is more cumbersome. The main result of this paper is the discovery of a non-trivial characterizing condition that enables us to fully characterize the members of a two-parameter class of fuzzy preference structures in terms of their fuzzy large preference relation. As a remarkable side-result, we discover three limit classes of characterizable fuzzy preference structures, traces of which are found throughout the preference modelling literature.