Representing Preference Relations with Nontransitive Indifference by a Single Real-Valued Function

Abstract

Let ≺ be an interval order on a topological space (X, r), and let x ≺* y (x ≺** y) if and only if there exists ξ (there exists η) in X with x ≺ ξ ≺_~ y (respectively, x ≺_~ η ≺ y). Then ≺*_~ and ≺**_~ are complete preorders. In the particular case when ≺ is a semiorder, let x ≺⁰ y if and only if either x ≺* y or x ≺** y. Then ≺ ⁰_~ is a complete preorder, too. We present sufficient conditions for the existence of continuous utility functions representing ≺*_~, ≺**_~ and ≺ ⁰_~ , by using the notion of strong separability of a preference relation, which was introduced by Chateauneuf.