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A sandwich theorem for generic n × n two person games

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  • Sun, Ching-jen

Abstract

We study the structure of Nash equilibria in generic n×n games. A game is said to have a sandwich structure in Nash equilibria if there is a mixed strategy Nash equilibrium “inside” every collection of pure strategy Nash equilibria. A sufficient condition, which solely relies on the ordinal information of the game, is given for a generic n×n game to have a sandwich structure in Nash equilibria. We provide a lower bound on the number of Nash equilibria and determine the stability of each equilibrium in games with a sandwich structure in Nash equilibria. Moreover, when the number of pure strategy Nash equilibria is equal to the number of pure strategies available to each player, the exact structure of Nash equilibria can be determined.

Suggested Citation

  • Sun, Ching-jen, 2020. "A sandwich theorem for generic n × n two person games," Games and Economic Behavior, Elsevier, vol. 120(C), pages 86-95.
    • Handle: RePEc:eee:gamebe:v:120:y:2020:i:c:p:86-95
    DOI: 10.1016/j.geb.2019.12.004
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    References listed on IDEAS

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    1. Faruk Gül & David Pearce & Ennio Stacchetti, 1993. "A Bound on the Proportion of Pure Strategy Equilibria in Generic Games," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 548-552, August.
    2. McLennan, Andrew & Park, In-Uck, 1999. "Generic 4 x 4 Two Person Games Have at Most 15 Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 26(1), pages 111-130, January.
    3. Keiding, Hans, 1997. "On the Maximal Number of Nash Equilibria in ann x nBimatrix Game," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 148-160, October.
    4. Demichelis, Stefano & Germano, Fabrizio, 2002. "On (un)knots and dynamics in games," Games and Economic Behavior, Elsevier, vol. 41(1), pages 46-60, October.
    5. McLennan, Andrew, 1997. "The Maximal Generic Number of Pure Nash Equilibria," Journal of Economic Theory, Elsevier, vol. 72(2), pages 408-410, February.
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    Cited by:

    1. Pahl, Lucas, 2023. "Polytope-form games and index/degree theories for extensive-form games," Games and Economic Behavior, Elsevier, vol. 141(C), pages 444-471.

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    More about this item

    Keywords

    Nash equilibrium; Lefschetz-Hopf theorem; Index; Stability;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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