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

️Yu-Ting Chen
️Mon Apr 01 2013

April 2013 Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs

Yu-Ting Chen

Ann. Appl. Probab. 23(2): 637-664 (April 2013). DOI: 10.1214/12-AAP849

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.

Citation

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Yu-Ting Chen. "Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs." Ann. Appl. Probab. 23 (2) 637 - 664, April 2013. https://doi.org/10.1214/12-AAP849

Information

Published: April 2013

First available in Project Euclid: 12 February 2013

Digital Object Identifier: 10.1214/12-AAP849

Subjects:

Primary: 60K35 , 91A22

Secondary: 60J10 , 60J28

Keywords: Coalescing random walks , evolution of cooperation , Evolutionary game theory , interacting particle systems , perturbations of Markov chains , voter model , voter model perturbations