Fixation times in evolutionary games under weak selection

TL;DR

This paper analytically derives simple formulas for fixation times in evolutionary games under weak selection, showing they depend linearly on payoff differences and are determined by specific payoff matrix components.

Contribution

It provides a novel analytical insight into how mean fixation times depend on payoff matrix parameters under weak selection in stochastic evolutionary dynamics.

Findings

Unconditional mean exit time depends only on constant payoff term v.

Conditional mean exit time depends only on density-dependent term u.

Results apply to two common microscopic evolutionary processes.

Abstract

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, ie strong stochasticity: The payoff difference $\Delta \pi$ is a linear function of the number of $A$ individuals $i$, $\Delta \pi = u i + v$. 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.