Extrapolating Weak Selection in Evolutionary Games

This work is inspired by a 2013 paper from Arne Traulsens lab at the Max Plank Institute for Evolutionary Biology [10]. They studied the small mutation limit of evolutionary games. It has been shown that for 2 x 2 games the ranking of the strategies does not change as strength of selection is increased [11]. The point of the 2013 paper is that when there are three or more strategies the ordering can change as selection is increased. Wu et al [10] did numerical computations for fixed N. Here, we will instead let the strength of selection {beta} = c/N and let N [->] {infty} to obtain formulas for the invadability probabilities{phi} ij that determine the rankings. These formulas, which are integrals on [0, 1], are intractable calculus problems but can be easily evaluated numerically. Here, we concentrate on simple formulas for the ranking order when c is small or c is large. The results we have obtained for five concrete examples lead us to doubt the accuracy of insights that are derived in the small mutation limit for games with three strategies.