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**Optimal mixed strategies in stochastic environments.**
*(English)*
Zbl 0823.92015

We will discuss the case of non-structured populations where the environment is characterized by a one-dimensional stochastic parameter that varies over the generations according to an ergodic process. It is shown how the optimal strategy in such situations can, in principle, be found. This also leads to a criterion to decide whether pure or mixed strategies are optimal for any given case. Our results make it possible to calculate optimal strategies in stochastic environments explicitly for complicated situations. These include, for instance, a number of cases with continuous strategy sets.

As an illustration, we then derive the optimal strategy for the situation in which the payoff function (specifying the relationship between trait values, environment, and expected contributions to the population size in the next generation) has a Gaussian form and the environmental parameter has a normal distribution. After that, more general forms of environmental distributions as well as payoff functions are considered, leading to explicit solutions for specific cases, using results from the statistical literature, and numerical solutions otherwise.

Initially we assume that individuals have no information about the current value of the environmental parameter, but they do “know” its distribution over the generations. This amounts to assuming that the environment is stable over a large number of generations. Afterwards, we generalize the model for situations where individuals can obtain indications of the environmental conditions using some external cue.

As we will show, environmental variability usually leads to mixed rather than pure optimal strategies. Furthermore, there is a minimum environmental variance below which pure strategies are optimal, which depends on the form of the payoff function and, in case there is information about the environment, on the accuracy of such information.

We investigate the difference between long-term reproductive success of different strategies in variable environments. Our results indicate that there is a strong selection pressure for mixed strategies to evolve unless it is possible for individuals to acquire highly accurate information about the environmental conditions.

Finally, we examine the robustness of the optimal solution for slight perturbations of the form of the payoff function and/or the distributions of the environmental parameter. We show that, for the cases considered in this paper, the long-term reproductive success of the optimal strategy derived without perturbation is arbitrary close to that of the real optimal strategy when there are perturbations.

As an illustration, we then derive the optimal strategy for the situation in which the payoff function (specifying the relationship between trait values, environment, and expected contributions to the population size in the next generation) has a Gaussian form and the environmental parameter has a normal distribution. After that, more general forms of environmental distributions as well as payoff functions are considered, leading to explicit solutions for specific cases, using results from the statistical literature, and numerical solutions otherwise.

Initially we assume that individuals have no information about the current value of the environmental parameter, but they do “know” its distribution over the generations. This amounts to assuming that the environment is stable over a large number of generations. Afterwards, we generalize the model for situations where individuals can obtain indications of the environmental conditions using some external cue.

As we will show, environmental variability usually leads to mixed rather than pure optimal strategies. Furthermore, there is a minimum environmental variance below which pure strategies are optimal, which depends on the form of the payoff function and, in case there is information about the environment, on the accuracy of such information.

We investigate the difference between long-term reproductive success of different strategies in variable environments. Our results indicate that there is a strong selection pressure for mixed strategies to evolve unless it is possible for individuals to acquire highly accurate information about the environmental conditions.

Finally, we examine the robustness of the optimal solution for slight perturbations of the form of the payoff function and/or the distributions of the environmental parameter. We show that, for the cases considered in this paper, the long-term reproductive success of the optimal strategy derived without perturbation is arbitrary close to that of the real optimal strategy when there are perturbations.

### MSC:

92D15 | Problems related to evolution |

91A40 | Other game-theoretic models |

91A60 | Probabilistic games; gambling |