On the distribution of the number of internal equilibria in random evolutionary games

In this paper, we study the distribution of the number of internal equilibria of a multi-player two-strategy random evolutionary game. Using techniques from the random polynomial theory, we obtain a closed formula for the probability that the game has a certain number of internal equilibria. In addition, by employing Descartes’ rule of signs and combinatorial methods, we provide useful estimates for this probability. Finally, we also compare our analytical results with those obtained from samplings.