Utility representation of an incomplete and nontransitive preference relation

The objective of this paper is to provide continuous utility representation theorems analogous to Debreu's classic utility representation theorem, albeit for preference relations that may fail to be complete and/or transitive. Specifically, we show that every (continuous and) reflexive binary relation on a (compact) metric space can be represented by means of the maxmin, or dually, minmax, of a (compact) set of (compact) sets of continuous...