Anomalous finite-size effects in the Battle of the Sexes

TL;DR

This paper investigates how finite population sizes cause extinction of strategies in the Battle of the Sexes game, revealing anomalous effects like prolonged coexistence and flat quasi-stationary distributions.

Contribution

It demonstrates that finite-size fluctuations lead to strategy extinction and characterizes the quasi-stationary distribution and mean extinction times analytically.

Findings

Finite populations cause strategy extinction in the model.

Extinction times increase with system size.

Quasi-stationary distribution is anomalously flat near coexistence.

Abstract

The Battle of the Sexes describes asymmetric conflicts in mating behavior of males and females. Males can be philanderer or faithful, while females are either fast or coy, leading to a cyclic dynamics. The adjusted replicator equation predicts stable coexistence of all four strategies. In this situation, we consider the effects of fluctuations stemming from a finite population size. We show that they unavoidably lead to extinction of two strategies in the population. However, the typical time until extinction occurs strongly prolongs with increasing system size. In the meantime, a quasi-stationary probability distribution forms that is anomalously flat in the vicinity of the coexistence state. This behavior originates in a vanishing linear deterministic drift near the fixed point. We provide numerical data as well as an analytical approach to the mean extinction time and the quasi-stationary probability distribution.