Large Fluctuations and Fixation in Evolutionary Games

TL;DR

This paper develops a semi-classical WKB approach to analyze large fluctuations and fixation probabilities in evolutionary games, providing accurate results beyond weak selection limits and outperforming traditional Fokker-Planck methods.

Contribution

It introduces a generalized WKB method for stochastic evolutionary dynamics with multiple absorbing states, enabling precise analysis of fixation phenomena under finite selection.

Findings

Accurately computed fixation probabilities and mean fixation times.

Validated analytical results with extensive numerical simulations.

Demonstrated superiority over Fokker-Planck approximation at finite selection intensity.

Abstract

We study large fluctuations in evolutionary games belonging to the coordination and anti-coordination classes. The dynamics of these games, modeling cooperation dilemmas, is characterized by a coexistence fixed point separating two absorbing states. We are particularly interested in the problem of fixation that refers to the possibility that a few mutants take over the entire population. Here, the fixation phenomenon is induced by large fluctuations and is investigated by a semi-classical WKB (Wentzel-Kramers-Brillouin) theory generalized to treat stochastic systems possessing multiple absorbing states. Importantly, this method allows us to analyze the combined influence of selection and random fluctuations on the evolutionary dynamics \textit{beyond} the weak selection limit often considered in previous works. We accurately compute, including pre-exponential factors, the probability distribution function in the long-lived coexistence state and the mean fixation time necessary for a few mutants to take over the entire population in anti-coordination games, and also the fixation probability in the coordination class. Our analytical results compare excellently with extensive numerical simulations. Furthermore, we demonstrate that our treatment is superior to the Fokker-Planck approximation when the selection intensity is finite.