Evolutionary game dynamics in a Wright-Fisher process. (English) Zbl 1110.92028
Summary: Evolutionary game dynamics in finite populations can be described by a frequency dependent, stochastic Wright-Fisher process. We consider a symmetric game between two strategies, \(A\) and \(B\). There are discrete generations. In each generation, individuals produce offspring proportional to their payoff. The next generation is sampled randomly from this pool of offspring. The total population size is constant. The resulting Markov process has two absorbing states corresponding to homogeneous populations of all \(A\) or all \(B\). We quantify frequency dependent selection by comparing the absorption probabilities to the corresponding probabilities under random drift. We derive conditions for selection to favor one strategy or the other by using the concept of total positivity. In the limit of weak selection, we obtain the \(1/3\) law: if \(A\) and \(B\) are strict Nash equilibria then selection favors replacement of \(B\) by \(A\), if the unstable equilibrium occurs at a frequency of \(A\) which is less than \(1/3\).
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