The effect of mutation on equilibrium properties of deterministic and random evolutionary games

The replicator-mutator equation is a set of differential equations describing the evolution of frequencies of different strategies in a population that takes into account both selection and mutation mechanisms. It is a fundamental mathematical framework for the modelling, analysis and simulation of complex biological, economical and social systems and has been utilized in the study of, just to name a few, population genetics, autocatalytic reaction networks, language evolution and the evolution of cooperation. In this extended abstract, we report our recent works on the statistics of the equilibria of the replicator-mutator equation. For two-player social dilemmas (namely the Prisoner’s Dilemma, Snow Drift, Stag Hunt and Harmony), we characterize (stable) equilibrium points and analytically calculate the probability of having a certain number of equilibria when the payoff entries are uniformly distributed. For multi-player random games whose pay-offs are independently distributed according to a normal distribution, by employing techniques from random polynomial theory, we compute the expected or average number of internal equilibria. In addition, we perform extensive simulations by sampling and averaging over a large number of possible payoff matrices to compare with and illustrate analytical results. Our results provide new insights into the behavioural diversity of dynamical systems, including biological, social and artificial life ones.