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Pure Nash Equilibria and Best-Response Dynamics in Random Games

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Listed:
  • Ben Amiet
  • Andrea Collevecchio
  • Marco Scarsini
  • Ziwen Zhong

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 i.i.d. 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 Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that a multitude of phase transitions depend only on a single parameter of the model, that is, the probability of having ties.

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  • Ben Amiet & Andrea Collevecchio & Marco Scarsini & Ziwen Zhong, 2019. "Pure Nash Equilibria and Best-Response Dynamics in Random Games," Papers 1905.10758, arXiv.org, revised Jun 2020.
    • Handle: RePEc:arx:papers:1905.10758
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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.
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    Cited by:

    1. Ben Amiet & Andrea Collevecchio & Kais Hamza, 2020. "When "Better" is better than "Best"," Papers 2011.00239, arXiv.org.
    2. J'anos Flesch & Arkadi Predtetchinski & Ville Suomala, 2021. "Random perfect information games," Papers 2104.10528, arXiv.org.
    3. 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.
    4. 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.

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