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The Frequency of Convergent Games under Best-Response Dynamics

Author

Listed:
  • Samuel C. Wiese

    (University of Oxford
    University of Oxford)

  • Torsten Heinrich

    (University of Oxford
    Chemnitz University of Technology
    University of Oxford)

Abstract

We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium 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 with a given number of pure Nash equilibria 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.

Suggested Citation

  • Samuel C. Wiese & Torsten Heinrich, 2022. "The Frequency of Convergent Games under Best-Response Dynamics," Dynamic Games and Applications, Springer, vol. 12(2), pages 689-700, June.
    • Handle: RePEc:spr:dyngam:v:12:y:2022:i:2:d:10.1007_s13235-021-00401-3
    DOI: 10.1007/s13235-021-00401-3
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    References listed on IDEAS

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    1. Moulin, Herve, 1984. "Dominance solvability and cournot stability," Mathematical Social Sciences, Elsevier, vol. 7(1), pages 83-102, February.
    2. Pei, Ting & Takahashi, Satoru, 2019. "Rationalizable strategies in random games," Games and Economic Behavior, Elsevier, vol. 118(C), pages 110-125.
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    Cited by:

    1. Torsten Heinrich & Yoojin Jang & Luca Mungo & Marco Pangallo & Alex Scott & Bassel Tarbush & Samuel Wiese, 2023. "Best-response dynamics, playing sequences, and convergence to equilibrium in random games," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(3), pages 703-735, September.
    2. Hlafo Alfie Mimun & Matteo Quattropani & Marco Scarsini, 2022. "Best-Response dynamics in two-person random games with correlated payoffs," Papers 2209.12967, arXiv.org, revised Jan 2024.

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