Robustness of optimal mixed strategies

Abstract

Mixed strategies, or variable phenotypes, can evolve in fluctuating environments when at the time that a strategy is chosen the consequences of that decision are relatively uncertain. In a previous paper, we have shown several examples of explicit forms of optimal mixed strategies when an environmental distribution and payoff function are given. In many of these examples, the mixed strategy has a continuous distribution. In a recent study, however, Sasaki and Ellner proved that, if the distribution of the environmental parameter is modified in certain ways, the exact ESS distribution becomes discrete rather than continuous. This forces us to take a closer look at the robustness of optimal mixed strategies. In the current paper we prove that such strategies are indeed robust against small perturbations of the environmental distribution and/or the payoff function, in the sense that the optimal strategy distribution for the perturbed system, converges weakly to the optimal strategy distribution for the unperturbed system as the magnitude of the perturbation goes to zero. Furthermore, we show that the fitness difference between the two strategies converges to zero. Thus, although optimal strategies in ‘ideal’ and perturbed systems can be qualitatively different, the difference between the distributions (in a measure theoretic sense) is small.