## Abstract

We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n × n payoff matrix. Players can use mixed strategies, which are given by the vector (p_{1}, ..., p_{n}). Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S_{n}. We study evolutionary dynamics in a well-mixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutation-selection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the Hawk-Dove game. We prove general statements about the relationship between games with pure and with mixed strategies.

Original language | English (US) |
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Pages (from-to) | 50-57 |

Number of pages | 8 |

Journal | Journal of Theoretical Biology |

Volume | 261 |

Issue number | 1 |

DOIs | |

State | Published - Nov 7 2009 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Modeling and Simulation
- Biochemistry, Genetics and Molecular Biology(all)
- Immunology and Microbiology(all)
- Agricultural and Biological Sciences(all)
- Applied Mathematics

## Keywords

- Evolutionary dynamics
- Evolutionary game theory
- Finite population size
- Hawk-Dove game
- Stochastic process