Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games

TL;DR

This paper derives discrete replicator equations from microscopic Wright-Fisher processes incorporating game-theoretic frequency-dependent selection in finite populations, linking stochastic processes to deterministic replicator dynamics.

Contribution

It introduces a microscopic stochastic model with frequency-dependent selection into the Wright-Fisher process, connecting it to discrete replicator equations in finite populations.

Findings

Derived discrete replicator maps from microscopic processes.

Connected master, Fokker-Planck, and Langevin equations.

Established the link between stochastic models and deterministic dynamics.

Abstract

This paper is concerned with exploring the microscopic basis for the discrete versions of the standard replicator equation and the adjusted replicator equation. To this end, we introduce frequency-dependent selection -- as a result of competition fashioned by game-theoretic consideration -- into the Wright--Fisher process, a stochastic birth-death process. The process is further considered to be active in a generation-wise nonoverlapping finite population where individuals play a two-strategy bimatrix population game. Subsequently, connections among the corresponding master equation, the Fokker--Planck equation, and the Langevin equation are exploited to arrive at the deterministic discrete replicator maps in the limit of infinite population size.