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 Y/2 approaches the Poisson distribution with mean 1/2 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} ≈ 0.7769 as m increases. For general bimatrix games, the corresponding limit of probabilities is 1 − e^{− 1} ≈ 0.6321. Thus in this sense, pure strategy Nash equilibria are seen to be significantly more common under the condition of symmetry than otherwise.