Emergence of cooperation in finite populations under biased selection

We study the effects of biased selection on evolutionary games in finite populations. Biased selection is induced by including a number of invariant agents, who do not evolve, into an otherwise competing and evolving population. The invariant agents react differently when they encounter different types of variant agents, helping the cooperators to attain a higher payoff than the defectors by a difference Δ. The probability of a single cooperator invading and taking over a population of all defectors is evaluated based on the Moran process. When M invariant agents are introduced into the evolving population of size N, their presence helps promote the replacement of defectors by cooperators, in comparison with the case of unbiased selection with M=0. While the prisoner’s dilemma cannot sustain a cooperative population without the invariant agents, it is possible for cooperation to emerge with biased selection. We show that for a given N (M), there is a threshold of M (N) above (below) which an initial population of all defectors evolve into one of all cooperators. The dependence of the threshold on the game’s payoffs, intensity of selection, and the values of M and Δ is studied. Beyond the prisoner’s dilemma, biased selection also promotes cooperation in a bigger part of payoff parameter space corresponding to other types of games.