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Abstract

We consider evolutionary game dynamics in a finite population of size N.
When mutations are rare, the population is monomorphic most of the time. Occasionally
a mutation arises. It can either reach fixation or go extinct. The evolutionary
dynamics of the process under smallmutation rates can be approximated by an embedded
Markov chain on the pure states. Here we analyze how small the mutation rate
should be to make the embedded Markov chain a good approximation by calculating
the difference between the real stationary distribution and the approximated one.
While for a coexistence game, where the best reply to any strategy is the opposite
strategy, it is necessary that the mutation rate μ is less than N−1/2 exp[−N] to ensure
that the approximation is good, for all other games, it is sufficient if themutation rate is
smaller than (N ln N)
−1. Our results also hold for a wide class of imitation processes
under arbitrary selection intensity.