The effect of variability in payoffs on average abundance in two-player linear games under symmetric mutation

Classical studies in evolutionary game theory assume constant payoffs. Randomly fluctuating environments in real populations make this assumption idealistic. In this paper, we study randomized two-player linear games in a finite population in a succession of birth-death events according to a Moran process and in the presence of symmetric mutation. Introducing identity measures under neutrality that depend on the mutation rate and calculating these in the limit of a large population size by using the coalescent process, we study the first-order effect of the means, variances and covariances of the payoffs on average abundance in the stationary state under mutation and selection. This shows how the average abundance of a strategy is driven not only by its mean payoffs but also by the variances and covariances of its payoffs. In Prisoner's Dilemmas with additive cost and benefit for cooperation, where constant payoffs always favor the abundance of defection, stochastic fluctuations in the payoffs can change the strategy that is more abundant on average in the stationary state. The average abundance of cooperation is increased if the variance of any payoff to cooperation against cooperation or defection, or their covariance, is decreased, or if the variance of any payoff to defection against cooperation or defection, or their covariance, is increased. This is also the case for a Prisoner's Dilemma with independent payoffs that is repeated a random number of times. As for the mutation rate, it comes into play in the coefficients of the variances and covariances that determine average abundance. Increasing the mutation rate can enhance or lessen the condition for a strategy to be more abundant on average than another.