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The distribution of optimal strategies in symmetric zero-sum games

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  • Brandl, Florian

Abstract

Given a skew-symmetric matrix, the corresponding two-player symmetric zero-sum game is defined as follows: one player, the row player, chooses a row and the other player, the column player, chooses a column. The payoff of the row player is given by the corresponding matrix entry, the column player receives the negative of the row player. A randomized strategy is optimal if it guarantees an expected payoff of at least 0 for a player independently of the strategy of the other player. We determine the probability that an optimal strategy randomizes over a given set of actions when the game is drawn from a distribution that satisfies certain regularity conditions. The regularity conditions are quite general and apply to a wide range of natural distributions.

Suggested Citation

  • Brandl, Florian, 2017. "The distribution of optimal strategies in symmetric zero-sum games," Games and Economic Behavior, Elsevier, vol. 104(C), pages 674-680.
    • Handle: RePEc:eee:gamebe:v:104:y:2017:i:c:p:674-680
    DOI: 10.1016/j.geb.2017.06.017
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    References listed on IDEAS

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    1. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    2. McLennan, Andrew & Berg, Johannes, 2005. "Asymptotic expected number of Nash equilibria of two-player normal form games," Games and Economic Behavior, Elsevier, vol. 51(2), pages 264-295, May.
    3. Andrew McLennan, 2005. "The Expected Number of Nash Equilibria of a Normal Form Game," Econometrica, Econometric Society, vol. 73(1), pages 141-174, January.
    4. Laffond, Gilbert & Laslier, Jean-Francois & Le Breton, Michel, 1997. "A Theorem on Symmetric Two-Player Zero-Sum Games," Journal of Economic Theory, Elsevier, vol. 72(2), pages 426-431, February.
    5. David Roberts, 2006. "Nash equilibria of Cauchy-random zero-sum and coordination matrix games," International Journal of Game Theory, Springer;Game Theory Society, vol. 34(2), pages 167-184, August.
    6. Raghavan, T.E.S., 1994. "Zero-sum two-person games," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 20, pages 735-768, Elsevier.
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    More about this item

    Keywords

    Symmetric zero-sum games; Maximin strategies; Random games; Uniqueness of Nash equilibria;
    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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