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A note on discontinuity and approximate equilibria in games with infinitely many players

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  • Rachmilevitch, Shiran

Abstract

Peleg (1969) showed that it is possible for a game with countably many players and finitely many pure strategies to have no Nash equilibrium. In his example not only Nash, but even perfect ϵ-equilibrium fails to exist. However, the example is based on tail utility functions, and these have infinitely many discontinuity points. I demonstrate non-existence of perfect ϵ-equilibrium under a milder form of discontinuity: I construct a game with countably many players, finitely many pure strategies and no perfect ϵ-equilibrium, in which one player has a utility function with a single discontinuity point, and the utility of every other player is not only continuous, but depends on finitely many coordinates.

Suggested Citation

  • Rachmilevitch, Shiran, 2020. "A note on discontinuity and approximate equilibria in games with infinitely many players," Economics Letters, Elsevier, vol. 193(C).
    • Handle: RePEc:eee:ecolet:v:193:y:2020:i:c:s0165176520301816
    DOI: 10.1016/j.econlet.2020.109267
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    References listed on IDEAS

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    1. Hannu Salonen, 2010. "On the existence of Nash equilibria in large games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(3), pages 351-357, July.
    2. Rachmilevitch, Shiran, 2016. "Approximate equilibria in strongly symmetric games," Journal of Mathematical Economics, Elsevier, vol. 66(C), pages 52-57.
    3. Cingiz, Kutay & Flesch, János & Herings, P. Jean-Jacques & Predtetchinski, Arkadi, 2016. "Doing it now, later, or never," Games and Economic Behavior, Elsevier, vol. 97(C), pages 174-185.
    4. Basu Kaushik, 1994. "Group Rationality, Utilitarianism, and Escher's Waterfall," Games and Economic Behavior, Elsevier, vol. 7(1), pages 1-9, July.
    5. Voorneveld, Mark, 2010. "The possibility of impossible stairways: Tail events and countable player sets," Games and Economic Behavior, Elsevier, vol. 68(1), pages 403-410, January.
    6. Radner, Roy, 1980. "Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives," Journal of Economic Theory, Elsevier, vol. 22(2), pages 136-154, April.
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    More about this item

    Keywords

    Approximate equilibrium; Discontinuous games; Infinite games; Equilibrium non-existence; Tail events;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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