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Utility representation of an incomplete and nontransitive preference relation

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  • Nishimura, Hiroki
  • Ok, Efe A.

Abstract

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.

Suggested Citation

  • Nishimura, Hiroki & Ok, Efe A., 2016. "Utility representation of an incomplete and nontransitive preference relation," Journal of Economic Theory, Elsevier, vol. 166(C), pages 164-185.
    • Handle: RePEc:eee:jetheo:v:166:y:2016:i:c:p:164-185
    DOI: 10.1016/j.jet.2016.07.002
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    Cited by:

    1. Madhav Chandrasekher & Mira Frick & Ryota Iijima & Yves Le Yaouanq, 2022. "Dual‐Self Representations of Ambiguity Preferences," Econometrica, Econometric Society, vol. 90(3), pages 1029-1061, May.
    2. Hiroki Nishimura & Efe A. Ok, 2022. "A class of dissimilarity semimetrics for preference relations," Papers 2203.04418, arXiv.org.
    3. Robert G. Chambers & Tigran Melkonyan & John Quiggin, 2022. "Incomplete preferences, willingness to pay, and willingness to accept," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 74(3), pages 727-761, October.
    4. repec:hal:psewpa:halshs-01998001 is not listed on IDEAS
    5. Cosimo Munari, 2020. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Papers 2009.04151, arXiv.org.
    6. M. Ali Khan & Metin Uyanık, 2021. "Topological connectedness and behavioral assumptions on preferences: a two-way relationship," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 411-460, March.
    7. Cosimo Munari, 2021. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Finance and Stochastics, Springer, vol. 25(1), pages 77-99, January.
    8. Peter Caradonna & Christopher P. Chambers, 2023. "A Note on Invariant Extensions of Preorders," Papers 2303.04522, arXiv.org.
    9. Paolo Leonetti & Giulio Principi, 2022. "Representations of cones and applications to decision theory," Papers 2209.06310, arXiv.org, revised Jan 2023.
    10. Kirby Nielsen & Luca Rigotti, 2022. "Revealed Incomplete Preferences," Papers 2205.08584, arXiv.org, revised Oct 2022.
    11. Weixuan Xia, 2023. "Optimal Consumption--Investment Problems under Time-Varying Incomplete Preferences," Papers 2312.00266, arXiv.org.
    12. Paolo Leonetti, 2022. "Expected multi-utility representations of preferences over lotteries," Papers 2210.04739, arXiv.org, revised Jan 2024.
    13. Wilfried Youmbi, 2024. "Nonparametric Analysis of Random Utility Models Robust to Nontransitive Preferences," Papers 2406.13969, arXiv.org.
    14. Elias Bouacida, 2021. "Identifying Choice Correspondences," Working Papers halshs-01998001, HAL.
    15. Dan Qin, 2021. "A Note on Numerical Representations of Nested System of Strict Partial Orders," Games, MDPI, vol. 12(3), pages 1-9, July.

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    More about this item

    Keywords

    Nontransitive binary relations; Continuous utility; Debreu's theorem; Multi-utility representation;
    All these keywords.

    JEL classification:

    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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