Evolutionary game dynamics with three strategies in finite populations

TL;DR

This paper introduces a stochastic model for three-strategy evolutionary game dynamics in finite populations, analyzing fixation probabilities and demonstrating conditions under which strategies like TFT can invade populations.

Contribution

It presents a new Moran process-based model for three strategies, incorporating global and local fixation probabilities to better understand evolutionary outcomes.

Findings

Global fixation probability determines strategy success regardless of initial ratios.

Local fixation probability influences strategy success only in certain initial conditions.

A single TFT individual can invade and dominate the population under specific circumstances.

Abstract

We propose a model for evolutionary game dynamics with three strategies $A$, $B$ and $C$ in the framework of Moran process in finite populations. The model can be described as a stochastic process which can be numerically computed from a system of linear equations. Furthermore, to capture the feature of the evolutionary process, we define two essential variables, the {\em global} and the {\em local} fixation probability. If the {\em global} fixation probability of strategy $A$ exceeds the neutral fixation probability, the selection favors $A$ replacing $B$ or $C$ no matter what the initial ratio of $B$ to $C$ is. Similarly, if the {\em local} fixation probability of $A$ exceeds the neutral one, the selection favors $A$ replacing $B$ or $C$ only in some appropriate initial ratios of $B$ to $C$. Besides, using our model, the famous game with AllC, AllD and TFT is analyzed. Meanwhile, we find that a single individual TFT could invade the entire population under proper conditions.