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 w hich the players receive the same payoff at each outcome, let X represent t he number of pure strategy Nash equilibria occurring. Treating both the cas es 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. Furthermor e, we show that for any fixed positive integer k, the probability that X ex ceeds k approaches one as pure strategy sets increase in size without bound . (C) 1999 Elsevier Science S.A. All rights reserved.