Mutation-selection equilibrium in games with multiple strategies

Abstract

In evolutionary games the fitness of individuals is not constant but depends on the relative abundance of the various strategies in the population. Here we study general games among n strategies in populations of large but finite size. We explore stochastic evolutionary dynamics under weak selection, but for any mutation rate. We analyze the frequency dependent Moran process in well-mixed populations, but almost identical results are found for the Wright-Fisher and Pairwise Comparison processes. Surprisingly simple conditions specify whether a strategy is more abundant on average than 1/n, or than another strategy, in the mutation-selection equilibrium. We find one condition that holds for low mutation rate and another condition that holds for high mutation rate. A linear combination of these two conditions holds for any mutation rate. Our results allow a complete characterization of nxn games in the limit of weak selection.

Fig. 1

Strategy abundance (mean frequency) in the game given by the payoff matrix (35). Colored lines show the critical conditions under which one of the three strategies exceeds an abundance of 1/3. For small mutation rates, S₁ is favored over S₃, but for large mutation rate, S₃ is favored over S₁. All three strategies have equal abundance at the intersection of all boundaries.

Fig. 2

Simulation results for strategy abundances as a function of the rescaled mutation rate μ = N u in the game of payoff matrix (35), at λ = 4.6. The population size is N = 30 and the selection strength is δ = 0.003, which means N δ = 0.09. The solid lines are the theoretical curves given by (20), and the dotted line marks the average abundance 1/3. The intersections of the lines are located at the critical values given by (3) and (4). The highest possible value of the mutation rate at this system size is μ = 30, which corresponds to mutation probability u = 1, where all densities are equal.

Fig. 3

Strategy abundance in the interaction between AllC, AllD, and TFT in the probability simplex S₃. Dark areas are inaccessible to the evolutionary dynamics. Red lines show thresholds where a strategy abundance crosses 1/3, the thresholds are given in terms of b/c. Blue lines depict thresholds where two strategy abundances are identical. (a) For small mutation rates, the abundance of AllC is never above 1/3 and it is never greater than the abundance of TFT. (b) For high mutation rates, the abundance of AllC is above 1/3 in the yellow shaded area, but again it never exceeds the abundance of TFT.