In this paper, we study the number of pure strategy Nash equilibria in a “random” n -person non-cooperative game in which all players have a countable number of strategies. We provide explicit expressions for the expected number of pure strategy Nash Equilibria, and show that the distribution of the number of pure strategy Nash Equilibria approaches the Poisson distribution with mean 1 as the numbers of strategies of two or more players go to inﬁnity.
Powers, Imelda Yeung, "Limiting Distributions of the Number of Pure Strategy Nash Equilibria in n-Person Games" (1986). Cowles Foundation Discussion Papers. 1042.