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Best-Response Dynamics, Playing Sequences, And Convergence To Equilibrium In Random Games

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  • Pangallo, Marco
  • Heinrich, Torsten
  • Jang, Yoojin
  • Scott, Alex
  • Tarbush, Bassel
  • Wiese, Samuel
  • Mungo, Luca

Abstract

We show that the playing sequence–the order in which players update their actions–is a crucial determinant of whether the best-response dynamic converges to a Nash equilibrium. Specifically, we analyze the probability that the best-response dynamic converges to a pure Nash equilibrium in random n-player m-action games under three distinct playing sequences: clockwork sequences (players take turns according to a fixed cyclic order), random sequences, and simultaneous updating by all players. We analytically characterize the convergence properties of the clockwork sequence best-response dynamic. Our key asymptotic result is that this dynamic almost never converges to a pure Nash equilibrium when n and m are large. By contrast, the random sequence best-response dynamic converges almost always to a pure Nash equilibrium when one exists and n and m are large. The clockwork best-response dynamic deserves particular attention: we show through simulation that, compared to random or simultaneous updating, its convergence properties are closest to those exhibited by three popular learning rules that have been calibrated to human game-playing in experiments (reinforcement learning, fictitious play, and replicator dynamics).

Suggested Citation

  • Pangallo, Marco & Heinrich, Torsten & Jang, Yoojin & Scott, Alex & Tarbush, Bassel & Wiese, Samuel & Mungo, Luca, 2021. "Best-Response Dynamics, Playing Sequences, And Convergence To Equilibrium In Random Games," INET Oxford Working Papers 2021-02, Institute for New Economic Thinking at the Oxford Martin School, University of Oxford.
  • Handle: RePEc:amz:wpaper:2021-02
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    References listed on IDEAS

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    1. 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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    More about this item

    Keywords

    Best-response dynamics; equilibrium convergence; random games; learning models in games;
    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
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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