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

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.

Published in at http://dx.doi.org/10.1214/12-AAP849 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Voter Model Perturbations and Reaction Diffusion Equations
• Rescaled Lotka-Volterra models converge to super-Brownian motion

Cited by in corpus (15)

• Evolutionary dynamics on any population structure
• Evolution of prosocial behavior in multilayer populations
• Evolution of cooperation with asymmetric social interactions
• Fixation probabilities in evolutionary dynamics under weak selection
• Transient amplifiers of selection and reducers of fixation for death-Birth updating on graphs
• A mathematical formalism for natural selection with arbitrary spatial and genetic structure
• Complex Network Modelling with Power-law Activating Patterns and Its Evolutionary Dynamics
• Evolutionary Games on Isothermal Graphs
• The limits of weak selection and large population size in evolutionary game theory
• Evaluating the structure-coefficient theorem of evolutionary game theory
• Evolution of cooperation under a generalized death-birth process
• The conflict between self-interaction and updating passivity in the evolution of cooperation
• Evolutionary games on the lattice: Payoffs affecting birth and death rates
• Fixation properties of multiple cooperator configurations on regular graphs
• Large Fluctuations in Anti-Coordination Games on Scale-Free Graphs