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On the expected number of equilibria in a multi-player multi-strategy evolutionary game

In this paper, we analyze the mean number $E(n,d)$ of internal equilibria in a general $d$-player $n$-strategy evolutionary game where the agents' payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next we characterize the asymptotic behavior of $E(2,d)$, estimating its lower and upper bounds as $d$ increases. Two important consequences are obtained from this analysis. On the one hand, we show that in both cases the probability of seeing the maximal possible number of equilibria tends to zero when $d$ or $n$ respectively goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger $n$ and $d$, numerical results are provided and discussed.