Finite-population evolution with rare mutations in asymmetric games

TL;DR

This paper models evolution in asymmetric games with multiple finite populations, showing that rare mutations simplify the analysis to a Markov chain over pure states, enabling easier calculation of stationary distributions and insights into evolutionary dynamics.

Contribution

It introduces a method to analyze asymmetric multi-population evolution with rare mutations, deriving fixation probabilities and stationary distributions using population genetics tools.

Findings

Stationary distribution can be computed via an ergodic Markov chain over monomorphic states.

Fixation probabilities are frequency-independent due to asymmetry.

Application to various game models demonstrates the approach's effectiveness.

Abstract

We model evolution according to an asymmetric game as occurring in multiple finite populations, one for each role in the game, and study the effect of subjecting individuals to stochastic strategy mutations. We show that, when these mutations occur sufficiently infrequently, the dynamics over all population states simplify to an ergodic Markov chain over just the pure population states (where each population is monomorphic). This makes calculation of the stationary distribution computationally feasible. The transition probabilities of this embedded Markov chain involve fixation probabilities of mutants in single populations. The asymmetry of the underlying game leads to fixation probabilities that are derived from frequency-independent selection, in contrast to the analogous single-population symmetric-game case (Fudenberg and Imhof 2006). This frequency independence is useful in that it allows us to employ results from the population genetics literature to calculate the stationary distribution of the evolutionary process, giving sharper, and sometimes even analytic, results. We demonstrate the utility of this approach by applying it to a battle-of-the-sexes game, a Crawford-Sobel signalling game, and the beer-quiche game of Cho and Kreps (1987).