  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.
