The effect of the opting-out strategy on conditions for selection to favor the evolution of cooperation in a finite population.

Abstract

We consider a Prisoner s Dilemma (PD) that is repeated with positive probability 1-ρ only between cooperators as a result of an opting-out strategy adopted by all individuals. The population is made of N pairs of individuals and is updated at every time step by a birth-death event according to a Moran model. Assuming an intensity of selection of order 1/N and taking 2N2 birth-death events as unit of time, a diffusion approximation exhibiting two time scales, a fast one for pair frequencies and a slow one for cooperation (C) and defection (D) frequencies, is ascertained in the limit of a large population size. This diffusion approximation is applied to an additive PD game, cooperation by an individual incurring a cost c to the individual but providing a benefit b to the opponent. This is used to obtain the probability of ultimate fixation of a single C introduced as a single mutant in an all D population under selection, which can be compared to the probability under neutrality, 1/(2N), as well as the corresponding probability for a single D introduced as a single mutant in an all C population under selection. This gives conditions for cooperation to be favored by selection. We show that these conditions are satisfied when the benefit-to-cost ratio, b/c, exceeds some increasing function of ρ that is approximately given by (1+ρ)/(1-ρ). This condition is more stringent, however, than the condition for tit-for-tat (TFT) to be favored against always-defect (AllD) in the absence of opting-out.