The Replicator Dynamics for Multilevel Selection in Evolutionary Games

We consider a stochastic model for evolution of cooperation in the Prisoner's Dilemma played in group-structured populations. Selection operates at two levels: individuals compete with individuals in their group, while groups compete with other groups. In the limit of infinite population size, we derive a non-local PDE describing the probability distribution of groups in the population. We characterize the long-time behavior of our system, with an emphasis on understanding the most frequent group compositions at steady state.

When average payoff of groups is maximized by all-cooperator groups, steady state composition ranges from all-defector groups when individual-level selection dominates to all-cooperator groups when group-level selection dominates. When group payoff is maximized by a mix of cooperators and defectors, then the steady state features a fewer cooperators than required for the optimal mix, even in the limit where group-level selection is infinitely stronger than individual-level selection. In such cases, the conflict between the two levels of selection cannot be decoupled, and cooperation cannot survive when between-group competition favors perfect coexistence of cooperators and defectors.