On the number of pure strategy Nash equilibria in finite common payoffs games

In a two-person "random" common payoffs game, defined as a finite game in which the players receive the same payoff at each outcome, let X represent the number of pure strategy Nash equilibria occurring. Treating both the cases where players have strictly and weakly ordinal preferences over outcomes, we observe that the expected value of X approaches infinity as the sizes of the pure strategy sets of the players increase without bound. Furthermore, we show that for any fixed positive integer k, the probability that X exceeds k approaches one as pure strategy sets increase in size without bound. (C) 1999 Elsevier Science S.A. All rights reserved.