The replicator dynamics for multilevel selection in evolutionary games
We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner’s Dilemma or the Hawk–Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner’s Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors at the between-group level. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, finding a threshold intensity of between-group selection required to sustain density steady states and the survival of cooperation. When all-cooperator groups are most fit, the average and most abundant group compositions at steady state range from featuring all-defector groups when individual-level selection dominates to featuring all-cooperator groups when group-level selection dominates. When the most fit groups have a mix of cooperators and defectors, then the average and most abundant group compositions always feature a smaller fraction of 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 be sustained at all in the case where between-group competition favors an even mix of cooperators and defectors.
KeywordsMultilevel selection Evolutionary game theory Replicator dynamics
I would like to thank Carl Veller for initial discussions and advice on the problem of multilevel selection in evolutionary games. I am grateful to Carl Veller, Simon Levin, Joshua Plotkin, Chai Molina, and an anonymous referee for helpful comments on the manuscript and to Peter Constantin, Robin Pemantle, Fernando Rossine, Dylan Morris, George Constable, Chadi Saad-Roy and Gergely Boza for helpful discussions.
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