Statistics of the number of equilibria in random social dilemma evolutionary games with mutation

In this paper, we study analytically the statistics of the number of equilibria in pair-
wise social dilemma evolutionary games with mutation where a game's payoff entries are random
variables. Using the replicator-mutator equations, we provide explicit formulas for the probability
distributions of the number of equilibria as well as other statistical quantities. This analysis is
highly relevant assuming that one might know the nature of a social dilemma game at hand (e.g.,
cooperation vs coordination vs anti-coordination), but measuring the exact values of its payoff
entries is difficcult. Our delicate analysis shows clearly the in
uence of the mutation probability on these probability distributions, providing insights into how varying this important factor impacts
the overall behavioural or biological diversity of the underlying evolutionary systems. Keywords:
evolutionary game theory, social dilemmas, replicator-mutator equation.