Fixation probabilities for any configuration of two strategies on regular graphs

TL;DR

This paper derives a comprehensive formula for the probability of cooperation fixation on regular graphs under weak selection, revealing how initial configurations and updating rules influence evolutionary outcomes.

Contribution

It extends fixation probability analysis to any configuration of two strategies on regular graphs, providing a tractable formula for weak selection under death-birth updating.

Findings

Cooperation is never favored under birth-death updating.

A simple formula for weak selection to favor cooperation under death-birth updating.

Strategic placement of cooperators can enhance or suppress cooperation.

Abstract

Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of single, randomly placed mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updating. In contrast, for death-birth (DB) updating, we derive a simple, computationally tractable formula for weak selection to favor cooperation when starting from any configuration containing any number of cooperators and defectors. This formula elucidates two important features: (i) the takeover of cooperation can be enhanced by the strategic placement of cooperators and (ii) adding more cooperators to a configuration can sometimes suppress the evolution of cooperation. These findings give a formal account for how selection acts on all transient states that appear in evolutionary trajectories. They also inform the strategic design of initial states in social networks to maximally promote cooperation. We also derive general results that characterize the interaction of any two strategies, not only cooperation and defection.