Quasi-strategies in evolutionary games

Evolutionary Game Theory studies how strategies evolve in populations and predicts the evolutionarily stable state. The theory can be extended to describe stochastic strategies and predicts that the stable stochastic strategy is determined by probabilities that reflect the corresponding mixture of deterministic strategies. However, much less is known about populations of stochastic strategies that evolve at a finite mutation rate, where multiple mutants can segregate in the population at the same time. Here we show that in this limit, the stable state is a distribution of strategies that is the stationary solution of a Fokker-Planck equation. This distribution is equivalent to the quasi-species distribution in population genetics, which we call the "quasi-strategy". We explicitly solve the Fokker-Planck equation for stochastic games with two and three moves, and then show numerically that for iterated stochastic memory-one games at finite mutation rate, the quasi-strategy's mutational robustness allows it to outcompete the "optimal'' generous ZD strategies that cannot be invaded in the small mutation limit. This work suggests that the concept of quasi-strategies may have wide-ranging applications in biological settings in which phenotypic plasticity is observed.