Structural symmetry in evolutionary games

TL;DR

This paper introduces a new concept of homogeneity in evolutionary games, analyzing how population structure and symmetry influence the equivalence of mutant strategies across different configurations and measures of success.

Contribution

It defines a formal notion of homogeneity for evolutionary processes and provides conditions under which asymmetric games can be simplified to symmetric ones based on population symmetry.

Findings

Homogeneity captures when different mutant configurations have identical evolutionary success measures.

Conditions are derived for process homogeneity in graph-structured populations.

Asymmetric games can be reduced to symmetric games if the population structure has sufficient symmetry.

Abstract

In evolutionary game theory, an important measure of a mutant trait (strategy) is its ability to invade and take over an otherwise-monomorphic population. Typically, one quantifies the success of a mutant strategy via the probability that a randomly occurring mutant will fixate in the population. However, in a structured population, this fixation probability may depend on where the mutant arises. Moreover, the fixation probability is just one quantity by which one can measure the success of a mutant; fixation time, for instance, is another. We define a notion of homogeneity for evolutionary games that captures what it means for two single-mutant states, i.e. two configurations of a single mutant in an otherwise-monomorphic population, to be "evolutionarily equivalent" in the sense that all measures of evolutionary success are the same for both configurations. Using asymmetric games, we argue that the term "homogeneous" should apply to the evolutionary process as a whole rather than to just the population structure. For evolutionary matrix games in graph-structured populations, we give precise conditions under which the resulting process is homogeneous. Finally, we show that asymmetric matrix games can be reduced to symmetric games if the population structure possesses a sufficient degree of symmetry.