In evolutionary game dynamics, reproductive success increases with the performance in an evolutionary game. If strategy A performs better than strategy B, strategy A will spread in the population. Under stochastic dynamics, a single mutant will sooner or later take over the entire population or go extinct. We analyze the mean exit times (or average fixation times) associated with this process. We show analytically that these times depend on the payoff matrix of the game in an amazingly simple way under weak selection, i.e. strong stochasticity: the payoff difference Δπ is a linear function of the number of A individuals i, Δπ=u i+v. The unconditional mean exit time depends only on the constant term v. Given that a single A mutant takes over the population, the corresponding conditional mean exit time depends only on the density dependent term u. We demonstrate this finding for two commonly applied microscopic evolutionary processes.

A finite array of N globally coupled Stratonovich models exhibits a continuous non-equilibrium phase transition. In the limit of strong coupling, there is a clear separation of timescales of centre of mass and relative coordinates. The latter relax very fast to zero and the array behaves as a single entity described by the centre of mass coordinate. We compute analytically the stationary probability distribution and the moments of the centre of mass coordinate. The scaling behavior of the moments near the critical value of the control parameter ac(N) is determined. We identify a crossover from linear to square root scaling with increasing distance from ac. The crossover point approaches ac in the limit N→∞ which reproduces previous results for infinite arrays. Our results are obtained in both the Fokker–Planck and the Langevin approach and are corroborated by numerical simulations. For a general class of models we show that the transition manifold in the parameter space depends on N and is determined by the scaling behavior near a fixed point of the stochastic flow.