## The Annals of Applied Probability

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

Yu-Ting Chen

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 12 February 2013

https://projecteuclid.org/euclid.aoap/1360682025

Digital Object Identifier
doi:10.1214/12-AAP849

Mathematical Reviews number (MathSciNet)
MR3059271

Zentralblatt MATH identifier
1267.91019

#### Citation

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. https://projecteuclid.org/euclid.aoap/1360682025

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