The Annals of Applied Probability

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

Yu-Ting Chen

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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.

Article information

Ann. Appl. Probab., Volume 23, Number 2 (2013), 637-664.

First available in Project Euclid: 12 February 2013

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 91A22: Evolutionary games
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J28: Applications of continuous-time Markov processes on discrete state spaces

Evolutionary game theory evolution of cooperation interacting particle systems voter model perturbations voter model coalescing random walks perturbations of Markov chains


Chen, Yu-Ting. Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs. Ann. Appl. Probab. 23 (2013), no. 2, 637--664. doi:10.1214/12-AAP849.

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