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Lipschitz Bernoulli Utility Functions

Author

Listed:
  • Efe A. Ok

    (Department of Economics and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012)

  • Nik Weaver

    (Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130)

Abstract

We obtain several variants of the classic von Neumann–Morgenstern expected utility theorem with and without the completeness axiom in which the derived Bernoulli utility functions are Lipschitz. The prize space in these results is an arbitrary separable metric space, and the utility functions are allowed to be unbounded. The main ingredient of our results is a novel (behavioral) axiom on the underlying preference relations, which is satisfied by virtually all stochastic orders. The proof of the main representation theorem is built on the fact that the dual of the Kantorovich–Rubinstein space is (isometrically isomorphic to) the Banach space of Lipschitz functions that vanish at a fixed point. An application to the theory of nonexpected utility is also provided.

Suggested Citation

  • Efe A. Ok & Nik Weaver, 2023. "Lipschitz Bernoulli Utility Functions," Mathematics of Operations Research, INFORMS, vol. 48(2), pages 728-747, May.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:2:p:728-747
    DOI: 10.1287/moor.2022.1270
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    References listed on IDEAS

    as
    1. Kazuhiro Hara & Efe A. Ok & Gil Riella, 2019. "Coalitional Expected Multi‐Utility Theory," Econometrica, Econometric Society, vol. 87(3), pages 933-980, May.
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    3. Dubra, Juan & Maccheroni, Fabio & Ok, Efe A., 2004. "Expected utility theory without the completeness axiom," Journal of Economic Theory, Elsevier, vol. 115(1), pages 118-133, March.
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    5. Grandmont, Jean-Michel, 1972. "Continuity properties of a von Neumann-Morgenstern utility," Journal of Economic Theory, Elsevier, vol. 4(1), pages 45-57, February.
    6. Simone Cerreia‐Vioglio & David Dillenberger & Pietro Ortoleva, 2015. "Cautious Expected Utility and the Certainty Effect," Econometrica, Econometric Society, vol. 83, pages 693-728, March.
    7. Evren, Özgür, 2014. "Scalarization methods and expected multi-utility representations," Journal of Economic Theory, Elsevier, vol. 151(C), pages 30-63.
    8. Eddie Dekel & Barton L Lipman & Aldo Rustichini & Todd Sarver, 2007. "Representing Preferences with a Unique Subjective State Space: A Corrigendum -super-1," Econometrica, Econometric Society, vol. 75(2), pages 591-600, March.
    9. Todd Sarver, 2008. "Anticipating Regret: Why Fewer Options May Be Better," Econometrica, Econometric Society, vol. 76(2), pages 263-305, March.
    10. , & ,, 2015. "Hidden actions and preferences for timing of resolution of uncertainty," Theoretical Economics, Econometric Society, vol. 10(2), May.
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    Cited by:

    1. Joseph Root & Evan Sadler, 2026. "A Theory of Network Games Part 1: Utility Representations," Papers 2602.16071, arXiv.org, revised Feb 2026.
    2. Stelios Arvanitis, 2025. "Universal Choice Spaces and Expected Utility: A Banach-type Functorial Fixed Point," Working Paper 1534, Economics Department, Queen's University.
    3. Leonetti, Paolo, 2025. "Expected multi-utility representations of preferences over lotteries," Journal of Mathematical Economics, Elsevier, vol. 121(C).

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