The mechanics of stochastic slowdown in evolutionary games

TL;DR

This paper investigates the stochastic slowdown phenomenon in evolutionary games, revealing how fixation times can be unexpectedly prolonged or shortened depending on selection strength, with detailed analysis of microscopic dynamics and implications for cooperation and defection strategies.

Contribution

It provides a detailed microscopic analysis of stochastic slowdown in evolutionary games, including fixation time behavior and classification of two-player games based on non-monotonic fixation times.

Findings

Slowdown occurs in the prisoner's dilemma and coexistence games.

Fixation times can have multiple extrema as a function of selection strength.

Transient states significantly influence the slowdown effect.

Abstract

We study the stochastic dynamics of evolutionary games, and focus on the so-called `stochastic slowdown' effect, previously observed in (Altrock et. al, 2010) for simple evolutionary dynamics. Slowdown here refers to the fact that a beneficial mutation may take longer to fixate than a neutral one. More precisely, the fixation time conditioned on the mutant taking over can show a maximum at intermediate selection strength. We show that this phenomenon is present in the prisoner's dilemma, and also discuss counterintuitive slowdown and speedup in coexistence games. In order to establish the microscopic origins of these phenomena, we calculate the average sojourn times. This allows us to identify the transient states which contribute most to the slowdown effect, and enables us to provide an understanding of slowdown in the takeover of a small group of cooperators by defectors: Defection spreads quickly initially, but the final steps to takeover can be delayed substantially. The analysis of coexistence games reveals even more intricate behavior. In small populations, the conditional average fixation time can show multiple extrema as a function of the selection strength, e.g., slowdown, speedup, and slowdown again. We classify two-player games with respect to the possibility to observe non-monotonic behavior of the conditional average fixation time as a function of selection strength.