Group Selection Under the Replicator Dynamic

We consider a society where individuals get divided into different groups. All groups play the same normal form game, and each group is in the basin of attraction of a particular strict Nash equilibrium of the game. Group selection refers to the evolution of the mass of each group under the replicator dynamic. We provide microfoundations to the replicator dynamic using revision protocols, wherein agents migrate between groups based on average payoffs of the groups. Individual selection of strategies within each group happens in two possible ways. Either all agents instantaneously coordinate on a strict Nash equilibrium or all agents change their strategies under the logit dynamic. In either case, the group playing the Pareto efficient Nash equilibrium gets selected. Individual selection under the logit dynamic may slow down the process and introduce non-monotonicity. We then apply the model to the stag hunt game.