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The Limit Distribution of Pure Strategy Nash Equilibria in Symmetric Bimatrix Games

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  • William Stanford

    (Department of Economics (M/C 144), University of Illinois at Chicago, 601 South Morgan Street, Room 2103, Chicago, Illinois 60607-7121)

Abstract

In a “random” symmetric bimatrix game, let X and Y represent the numbers of symmetric and asymmetric pure strategy Nash equilibria occurring, respectively. We find the probability distributions of both X and Y depending on m , the number of pure strategies for each of the two players. We show the distribution of X approaches the Poisson distribution with mean one and the distribution of ½ Y approaches the Poisson distribution with mean ½ as m increases. We determine the joint distribution of X and Y and the limit distribution of X + Y . From this we see the probability of at least one pure strategy Nash equilibrium approaches 1 - e -1.5 (approx) .7769 as m increases. For general bimatrix games, the corresponding limit of probabilities is 1 - e -1 (approx) .6321. Thus in this sense, pure strategy Nash equilibria are seen to be significantly more common under the condition of symmetry than otherwise.

Suggested Citation

  • William Stanford, 1996. "The Limit Distribution of Pure Strategy Nash Equilibria in Symmetric Bimatrix Games," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 726-733, August.
  • Handle: RePEc:inm:ormoor:v:21:y:1996:i:3:p:726-733
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    File URL: http://dx.doi.org/10.1287/moor.21.3.726
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    Cited by:

    1. 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.

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    Keywords

    bimatrix game; Nash equilibria;

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