Nash equilibria in random games with right fat-tailed distributions

We study the distribution of the number of mixed strategy Nash equilibria in two-player games where each player’s payoffs are independently drawn from an identical distribution. When the payoff distributions are sufficiently right fat-tailed, we characterize the Nash equilibria by best reply cycles of pure strategies, and we show that the expected number of Nash equilibria is approximately $$\sqrt{\pi mn/\left( m+n\right) }$$ π m n / m + n in a random $$m\times n$$ m × n asymmetric game and approximately n/2 in a random $$n\times n$$ n × n symmetric game. We also provide new lower bounds for the expected number of Nash equilibria in a random game with any type of payoff distribution.