On the distribution of pure strategy equilibria in finite games with vector payoffs

Abstract Two players, each with vector payoffs, no notion of substitution rates between coordinates, and no recourse to mixed strategies, play a finite matrix game. Thus preferences over outcomes are incomplete and a pure equilibrium (PE) is a pure strategy pair in which the choice of each player is maximal in his pure strategy set (no worse than anything else) given the other player's strategy. The coordinates of each payoff vector are real-valued random variables. For two-dimensional payoffs, we calculate the mean and variance of the number of PE points in the game and show that for any positive integer k , the probability of at least k PE points approaches one as pure strategy sets increase in size, and as a corollary, that this extends to n -dimensional payoffs for n ≥ 3. Again for two-dimensional payoffs, in the limit as pure strategy sets expand, the distribution is normal with mean and identical variance increasing without bound. For symmetric bimatrix games with n -dimensional payoffs, the exact distribution of symmetric PE points is binomial with parameters depending on n and the size of pure strategy sets, and a similar central limit result applies. Thus, in the cases we study, the need to consider mixed strategy equilibria vanishes in the limit for large games when payoffs are vector valued.