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

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 ½Y 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 ≈ .7769 as m increases. For general bimatrix games, the corresponding limit of probabilities is 1 − e⁻¹ ≈ .6321. Thus in this sense, pure strategy Nash equilibria are seen to be significantly more common under the condition of symmetry than otherwise.