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The Frequency of Convergent Games under Best-Response Dynamics

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  • Samuel C. Wiese
  • Torsten Heinrich

Abstract

Generating payoff matrices of normal-form games at random, we calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of $n$-player, $m$-strategy games. These are perfectly predictable as they must converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games goes to zero as the number of players or the number of strategies goes to infinity. In the $2$-player case, we show that for large games with at least $10$ strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an $n$-partite graph to describe games.

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  • Samuel C. Wiese & Torsten Heinrich, 2020. "The Frequency of Convergent Games under Best-Response Dynamics," Papers 2011.01052, arXiv.org.
  • Handle: RePEc:arx:papers:2011.01052
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    References listed on IDEAS

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    1. Pangallo, Marco & Farmer, J. Doyne & Heinrich, Torsten, "undated". "Best reply structure and equilibrium convergence in generic games," INET Oxford Working Papers 2017-07, Institute for New Economic Thinking at the Oxford Martin School, University of Oxford, revised Mar 2018.
    2. William Stanford, 1996. "The Limit Distribution of Pure Strategy Nash Equilibria in Symmetric Bimatrix Games," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 726-733, August.
    3. Powers, Imelda Yeung, 1990. "Limiting Distributions of the Number of Pure Strategy Nash Equilibria in N-Person Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(3), pages 277-286.
    4. Moulin, Herve, 1984. "Dominance solvability and cournot stability," Mathematical Social Sciences, Elsevier, vol. 7(1), pages 83-102, February.
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