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Pure Nash Equilibria and Best-Response Dynamics in Random Games

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  • Ben Amiet
  • Andrea Collevecchio
  • Marco Scarsini

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 actions, 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.

Suggested Citation

  • Ben Amiet & Andrea Collevecchio & Marco Scarsini, 2019. "Pure Nash Equilibria and Best-Response Dynamics in Random Games," Papers 1905.10758, arXiv.org, revised Aug 2019.
  • Handle: RePEc:arx:papers:1905.10758
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    File URL: http://arxiv.org/pdf/1905.10758
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    References listed on IDEAS

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    3. Rinott, Yosef & Scarsini, Marco, 2000. "On the Number of Pure Strategy Nash Equilibria in Random Games," Games and Economic Behavior, Elsevier, vol. 33(2), pages 274-293, November.
    4. 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.
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    6. Roughgarden,Tim, 2016. "Twenty Lectures on Algorithmic Game Theory," Cambridge Books, Cambridge University Press, number 9781316624791.
    7. Stanford, William, 1999. "On the number of pure strategy Nash equilibria in finite common payoffs games," Economics Letters, Elsevier, vol. 62(1), pages 29-34, January.
    8. Roughgarden,Tim, 2016. "Twenty Lectures on Algorithmic Game Theory," Cambridge Books, Cambridge University Press, number 9781107172661.
    9. Marco Pangallo & Torsten Heinrich & J Doyne Farmer, 2017. "Best reply structure and equilibrium convergence in generic games," Papers 1704.05276, arXiv.org, revised Sep 2018.
    10. Stanford, William, 1997. "On the distribution of pure strategy equilibria in finite games with vector payoffs," Mathematical Social Sciences, Elsevier, vol. 33(2), pages 115-127, April.
    11. Ozan Candogan & Ishai Menache & Asuman Ozdaglar & Pablo A. Parrilo, 2011. "Flows and Decompositions of Games: Harmonic and Potential Games," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 474-503, August.
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