Extrapolating weak selection in evolutionary games.

Abstract

This work is inspired by a 2013 paper from Arne Traulsen's lab at the Max Plank Institute for Evolutionary Biology (Wu et al. in PLoS Comput Biol 9:e1003381, 2013). They studied evolutionary games when the mutation rate is so small that each mutation goes to fixation before the next one occurs. It has been shown that for [Formula: see text] games the ranking of the strategies does not change as strength of selection is increased (Wu et al. in Phys Rev 82:046106, 2010). The point of the 2013 paper is that when there are three or more strategies the ordering can change as selection is increased. Wu etal. (2013) did numerical computations for a fixed population size N. Here, we will instead let the strength of selection [Formula: see text] where c is fixed and let [Formula: see text] to obtain formulas for the invadability probabilities [Formula: see text] that determine the rankings. These formulas, which are integrals on [0,1], are intractable calculus problems, but can be easily evaluated numerically. Here, we use them to derive simple formulas for the ranking order when c is small or c is large.