Imitation, internal absorption and the reversal of local drift in stochastic evolutionary games

TL;DR

This paper explores how stochastic effects in finite populations influence evolutionary game dynamics, revealing phenomena like drift reversal and internal fixation that differ from deterministic models.

Contribution

It introduces the concept of drift reversal in finite populations and demonstrates that imitation dynamics can cause fixation at internal mixed-strategy points, a novel dynamic.

Findings

Drift reversal occurs in finite populations due to local strategy space effects.

Imitation dynamics can lead to fixation at internal mixed-strategy points.

Internal absorption is a new phenomenon not seen in other microscopic dynamics.

Abstract

Evolutionary game dynamics in finite populations is typically subject to noise, inducing effects which are not present in deterministic systems, including fixation and extinction. In the first part of this paper we investigate the phenomenon of drift reversal in finite populations, taking into account that drift is a local quantity in strategy space. Secondly, we study a simple imitation dynamics, and show that it can lead to fixation at internal mixed-strategy fixed points even in finite populations. Imitation in infinite populations is adequately described by conventional replicator dynamics, and these equations are known to have internal fixed points. Internal absorption in finite populations on the other hand is a novel dynamic phenomenon. Due to an outward drift in finite populations this type of dynamic arrest is not found in other commonly studied microscopic dynamics, not even in those with the same deterministic replicator limit as imitation.