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The Expected Number of Nash Equilibria of a Normal Form Game


  • Andrew McLennan


Fix finite pure strategy sets S1 , … , Sn , and let S= S1 ×⋯× Sn . In our model of a random game the agents' payoffs are statistically independent, with each agent's payoff uniformly distributed on the unit sphere in R -super-S. For given nonempty T1 ⊂ S1 , … , Tn ⊂ Sn we give a computationally implementable formula for the mean number of Nash equilibria in which each agent i's mixed strategy has support T i . The formula is the product of two expressions. The first is the expected number of totally mixed equilibria for the truncated game obtained by eliminating pure strategies outside the sets T i . The second may be construed as the "probability" that such an equilibrium remains an equilibrium when the strategies in the sets Si ∖ Ti become available. Copyright The Econometric Society 2005.

Suggested Citation

  • Andrew McLennan, 2005. "The Expected Number of Nash Equilibria of a Normal Form Game," Econometrica, Econometric Society, vol. 73(1), pages 141-174, January.
  • Handle: RePEc:ecm:emetrp:v:73:y:2005:i:1:p:141-174

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    References listed on IDEAS

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    1. Wheatley, W. Parker, 2003. "Survival And Ownership Of Internet Marketplaces For Agriculture," 2003 Annual meeting, July 27-30, Montreal, Canada 22214, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
    2. Lee, Robin S. & Pakes, Ariel, 2009. "Multiple equilibria and selection by learning in an applied setting," Economics Letters, Elsevier, vol. 104(1), pages 13-16, July.
    3. Bade, Sophie & Haeringer, Guillaume & Renou, Ludovic, 2007. "More strategies, more Nash equilibria," Journal of Economic Theory, Elsevier, vol. 135(1), pages 551-557, July.
    4. repec:eee:gamebe:v:104:y:2017:i:c:p:674-680 is not listed on IDEAS
    5. P. Jean-Jacques Herings & Ronald J. A. P. Peeters, 2003. "Equilibrium Selection In Stochastic Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 5(04), pages 307-326.
    6. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
    7. Park, Andreas & Smith, Lones, 2008. "Caller Number Five and related timing games," Theoretical Economics, Econometric Society, vol. 3(2), June.
    8. P. Herings & Ronald Peeters, 2010. "Homotopy methods to compute equilibria in game theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 119-156, January.
    9. Herings, P. Jean-Jacques & Peeters, Ronald J. A. P., 2004. "Stationary equilibria in stochastic games: structure, selection, and computation," Journal of Economic Theory, Elsevier, vol. 118(1), pages 32-60, September.
    10. Patrick Bajari & Han Hong & Stephen Ryan, 2004. "Identification and Estimation of Discrete Games of Complete Information," NBER Technical Working Papers 0301, National Bureau of Economic Research, Inc.
    11. McLennan, Andrew & Berg, Johannes, 2005. "Asymptotic expected number of Nash equilibria of two-player normal form games," Games and Economic Behavior, Elsevier, vol. 51(2), pages 264-295, May.
    12. Elizabeth Baldwin & Paul Klemperer, 2015. "Understanding Preferences: “Demand Types”, and the Existence of Equilibrium with Indivisibilities," Economics Papers 2015-W10, Economics Group, Nuffield College, University of Oxford.
    13. Takahashi, Satoru, 2008. "The number of pure Nash equilibria in a random game with nondecreasing best responses," Games and Economic Behavior, Elsevier, vol. 63(1), pages 328-340, May.
    14. Manh Hong Duong & The Anh Han, 2016. "On the Expected Number of Equilibria in a Multi-player Multi-strategy Evolutionary Game," Dynamic Games and Applications, Springer, vol. 6(3), pages 324-346, September.
    15. Klaus Kultti & Hannu Salonen & Hannu Vartiainen, 2011. "Distribution of pure Nash equilibria in n-person games with random best replies," Discussion Papers 71, Aboa Centre for Economics.
    16. Arieli, Itai & Babichenko, Yakov, 2016. "Random extensive form games," Journal of Economic Theory, Elsevier, vol. 166(C), pages 517-535.
    17. Ariel Pakes, 2008. "Theory and Empirical Work on Imperfectly Competitive Markets," NBER Working Papers 14117, National Bureau of Economic Research, Inc.

    More about this item

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games


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