Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs

TL;DR

This paper provides rigorous, explicit approximations for fixation probabilities in evolutionary games on regular graphs, confirming the sharpness of benefit-to-cost rules for cooperation evolution under specific updating mechanisms.

Contribution

It introduces a first-order approximation framework for voter model perturbations on finite graphs, validating and sharpening existing benefit-to-cost rules for cooperation.

Findings

Explicit first-order fixation probability approximations for regular graphs.

Validation and sharpening of Ohtsuki et al.'s benefit-to-cost rules.

Rigorous proof of the rules' validity and sharpness.

Abstract

We study two of the simple rules on finite graphs under the death-birth updating and the imitation updating discovered by Ohtsuki, Hauert, Lieberman and Nowak [Nature 441 (2006) 502-505]. Each rule specifies a payoff-ratio cutoff point for the magnitude of fixation probabilities of the underlying evolutionary game between cooperators and defectors. We view the Markov chains associated with the two updating mechanisms as voter model perturbations. Then we present a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs subject to small perturbation in terms of the voter model fixation probabilities. In the context of regular graphs, we obtain algebraically explicit first-order approximations for the fixation probabilities of cooperators distributed as certain uniform distributions. These approximations lead to a rigorous proof that both of the rules of Ohtsuki et al. are valid and are sharp.