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An undecidable statement regarding zero-sum games

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  • Fey, Mark

Abstract

In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist “paradoxical” two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory.

Suggested Citation

  • Fey, Mark, 2024. "An undecidable statement regarding zero-sum games," Games and Economic Behavior, Elsevier, vol. 145(C), pages 19-26.
    • Handle: RePEc:eee:gamebe:v:145:y:2024:i:c:p:19-26
    DOI: 10.1016/j.geb.2024.02.004
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    References listed on IDEAS

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    1. Baye, Michael R. & Kovenock, Dan & de Vries, Casper G., 2012. "The Herodotus paradox," Games and Economic Behavior, Elsevier, vol. 74(1), pages 399-406.
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    Cited by:

    1. Chris Fields & James F. Glazebrook, 2024. "Nash Equilibria and Undecidability in Generic Physical Interactions—A Free Energy Perspective," Games, MDPI, vol. 15(5), pages 1-22, August.

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    More about this item

    Keywords

    Zero-sum game; Undecidable; Herodotus paradox; Set theory;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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