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Pure Nash Equilibria and Best-Response Dynamics in Random Games

Author

Listed:
  • Ben Amiet

    (School of Mathematics, Monash University, Melbourne, Victoria 3800, Australia)

  • Andrea Collevecchio

    (School of Mathematics, Monash University, Melbourne, Victoria 3800, Australia)

  • Marco Scarsini

    (Department of Economics and Finance, Luiss, 00196 Rome, Italy)

  • Ziwen Zhong

    (School of Mathematics, Monash University, Melbourne, Victoria 3800, Australia)

Abstract

In finite games, mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies and the payoffs are independent and identically distributed with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of pure Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a pure Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that various phase transitions depend only on a single parameter of the model, that is, the probability of having ties.

Suggested Citation

  • Ben Amiet & Andrea Collevecchio & Marco Scarsini & Ziwen Zhong, 2021. "Pure Nash Equilibria and Best-Response Dynamics in Random Games," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1552-1572, November.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:4:p:1552-1572
    DOI: 10.1287/moor.2020.1102
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    References listed on IDEAS

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    Cited by:

    1. J'anos Flesch & Arkadi Predtetchinski & Ville Suomala, 2021. "Random perfect information games," Papers 2104.10528, arXiv.org.
    2. Torsten Heinrich & Yoojin Jang & Luca Mungo & Marco Pangallo & Alex Scott & Bassel Tarbush & Samuel Wiese, 2021. "Best-response dynamics, playing sequences, and convergence to equilibrium in random games," Papers 2101.04222, arXiv.org, revised Nov 2022.
    3. Ben Amiet & Andrea Collevecchio & Kais Hamza, 2020. "When "Better" is better than "Best"," Papers 2011.00239, arXiv.org.
    4. 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 2021-23, Institute for New Economic Thinking at the Oxford Martin School, University of Oxford.

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