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.