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Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems


  • Michael R. Powers

    () (Department of Finance, School of Economics and Management, and Schwarzman Scholars Program, Tsinghua University, Beijing 100084, China)


We identify restrictions on a decision maker’s utility function that are both necessary and sufficient to preserve dominance reasoning in each of two versions of the Two-Envelope Paradox (TEP). For the classical TEP, the utility function must satisfy a certain recurrence inequality. For the St. Petersburg TEP, the utility function must be bounded above asymptotically by a power function, which can be tightened to a constant. By determining the weakest conditions for dominance reasoning to hold, the article settles an open question in the research literature. Remarkably, neither constant-bounded utility nor finite expected utility is necessary for resolving the classical TEP; instead, finite expected utility is both necessary and sufficient for resolving the St. Petersburg TEP.

Suggested Citation

  • Michael R. Powers, 2015. "Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems," Risks, MDPI, Open Access Journal, vol. 3(1), pages 1-9, January.
  • Handle: RePEc:gam:jrisks:v:3:y:2015:i:1:p:26-34:d:44877

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    References listed on IDEAS

    1. Nalebuff, Barry, 1989. "The Other Person's Envelope Is Always Greener," Journal of Economic Perspectives, American Economic Association, vol. 3(1), pages 171-181, Winter.
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    More about this item


    two-envelope paradox; dominance reasoning; von Neumann–Morgenstern utility; heavy-tailed payoffs; boundedness;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G3 - Financial Economics - - Corporate Finance and Governance
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M4 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Accounting
    • K2 - Law and Economics - - Regulation and Business Law


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