Utility representation of an incomplete and nontransitive preference relation

Hiroki Nishimura, Efe A. Ok

    Research output: Contribution to journalArticlepeer-review


    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 utility functions. This notion of “maxmin multi-utility representation,” generalizes the recently proposed notions of “multi-utility representation” for preorders and “justifiable preferences” for complete and quasitransitive relations. As such, our main representation theorems lead to some new characterizations of these special cases as well.

    Original languageEnglish (US)
    Pages (from-to)164-185
    Number of pages22
    JournalJournal of Economic Theory
    StatePublished - Nov 1 2016


    • Continuous utility
    • Debreu's theorem
    • Multi-utility representation
    • Nontransitive binary relations

    ASJC Scopus subject areas

    • Economics and Econometrics


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