Deterministic evolutionary game dynamics in finite populations

TL;DR

This paper introduces a new microscopic birth-death process for evolutionary game dynamics that becomes fully deterministic under strong selection and recovers the Moran process under weak selection, applicable to finite populations.

Contribution

A novel birth-death process that provides deterministic limits in finite populations and extends existing models under weak selection.

Findings

Explicit fixation probabilities and times derived

Deterministic dynamics depend on initial conditions in cyclic games

Process generalizes classical evolutionary dynamics

Abstract

Evolutionary game dynamics describes the spreading of successful strategies in a population of reproducing individuals. Typically, the microscopic definition of strategy spreading is stochastic, such that the dynamics becomes deterministic only in infinitely large populations. Here, we introduce a new microscopic birth--death process that has a fully deterministic strong selection limit in well--mixed populations of any size. Additionally, under weak selection, from this new process the frequency dependent Moran process is recovered. This makes it a natural extension of the usual evolutionary dynamics under weak selection. We find simple expressions for the fixation probabilities and average fixation times of the new process in evolutionary games with two players and two strategies. For cyclic games with two players and three strategies, we show that the resulting deterministic dynamics crucially depends on the initial condition in a non--trivial way.