Fixation probabilities for the Moran process in evolutionary games with two strategies: graph shapes and large population asymptotics

TL;DR

This paper refines the classification of evolutionary outcomes in the Moran process with two strategies by linking them to graph shapes and analyzing their behavior in large populations, providing new asymptotic formulas.

Contribution

It introduces a graph shape characterization for fixation probabilities and derives asymptotic formulas for large populations, refining previous classifications.

Findings

Each evolutionary scenario corresponds to a specific graph shape.

Some scenarios are impossible in large populations due to asymptotic behavior.

New formulas approximate fixation probabilities as population size grows.

Abstract

This paper is based on the complete classification of evolutionary scenarios for the Moran process with two strategies given by Taylor et al. (B. Math. Biol. 66(6): 1621--1644, 2004). Their classification is based on whether each strategy is a Nash equilibrium and whether the fixation probability for a single individual of each strategy is larger or smaller than its value for neutral evolution. We improve on this analysis by showing that each evolutionary scenario is characterized by a definite graph shape for the fixation probability function. A second class of results deals with the behavior of the fixation probability when the population size tends to infinity. We develop asymptotic formulae that approximate the fixation probability in this limit and conclude that some of the evolutionary scenarios cannot exist when the population size is large.