Hyper-relations, choice functions, and orderings of opportunity sets

Abstract

We prove a coincidence of the class of multi-preference hyper-relations and the class of decent hyper-relations (DHR), that is the class of binary relations on opportunity sets satisfying monotonicity, no-dummy, stability with respect to contraction and extension, and the union property. We study subclasses of DHR. In order to pursue our analysis, we establish a canonical bijection between DHR and the class of no-dummy heritage choice functions. From this we obtain that the no-dummy heritage choice functions have multi-criteria rationalizations with reflexive binary relations. We also prove that the restriction of this bijection to two subclasses of DHR, namely the transitive decent hyper-relations, and the ample hyper-relations, is a bijection between these subclasses and the classes of closed no-dummy choice functions and no-dummy path-independent choice functions (Plott functions), respectively.