ROBUSTNESS OF OPTIMAL MIXED STRATEGIES

Mixed strategies, or variable phenotypes, can evolve in fluctuating en vironments when at the time that a strategy is chosen the consequences of that decision are relatively uncertain. In a previous paper, we ha ve shown several examples of explicit forms of optimal mixed strategie s when an environmental distribution and payoff function are given. In many of these examples, the mixed strategy has a continuous distribut ion. In a recent study, however, Sasaki and Ellner proved that, if the distribution of the environmental parameter is modified in certain wa ys, the exact ESS distribution becomes discrete rather than continuous . This forces us to take a closer look at the robustness of optimal mi xed strategies. In the current paper we prove that such strategies are indeed robust against small perturbations of the environmental distri bution and/or the payoff function, in the sense that the optimal strat egy distribution for the perturbed system, converges weakly to the opt imal strategy distribution for the unperturbed system as the magnitude of the perturbation goes to zero. Furthermore, we show that the fitne ss difference between the two strategies converges to zero. Thus, alth ough optimal strategies in 'ideal' and perturbed systems can be qualit atively different, the difference between the distributions (in a meas ure theoretic sense) is small.