Mixed Populations of Coordinators, Anticoordinators, and Imitators: Stochastic Stability

Decision-making mechanisms are typically either of imitation, that is, to copy the action of successful fellows, coordination, that is, to take an action if enough fellows have done so, or anticoordination, that is to take an action if few fellows have done so. The resulting decision-making dynamics of a mixture of individuals following these three mechanisms may either eventually reach an equilibrium state where no one tends to switch strategies or enter a nonsingleton positively invariant set, where perpetual switches of strategies will take place. In reality, however, the decisions are subject to mistakes or perturbations. Will any of the equilibrium states “survive” under the perturbed dynamics? That is, will they be visited infinitely often with a nonvanishing frequency under vanishing perturbations. We approach this problem by performing stochastic stability analysis for a population of individuals choosing between cooperation and defection, accordingly earning payoffs based on their payoff matrices, and updating their decisions by following one of the above three mechanisms provided that with a certain probability they may choose the opposite action to what the mechanism puts forward. We find that if an equilibrium consisting of both imitating cooperators and defectors is stochastically stable, so is an extreme equilibrium, where all or none of the imitators cooperate. This highlights the robustness of extreme equilibria and in turn the role of imitation in decision-making dynamics.