Heterogeneous Mixed Populations of Conformists, Nonconformists, and Imitators

Decision-making individuals are typically either a best-responder who chooses an action that leads to their immediate success, or an imitator , who mimics the actions of the most successful individuals. Conformist best-responders perceive an action to be successful if enough others have done so, whereas nonconformist ones think of an action as successful if few others have done so. Researchers have studied the asymptotic behavior of populations comprising one or two of these types of decision-makers, but not altogether. We consider a population of heterogeneous individuals, each either cooperates or defects , and earns payoffs according to their payoff matrices and the total number of cooperators in the population. The individuals revise their choices asynchronously based on the best-response or imitation update rule. Those who update based on the best-response are a conformist (resp. nonconformist) if their payoff matrix is that of a coordination (resp. anticoordination) game. The state of the system is the distribution of cooperators over conformists and nonconformists with different payoff matrices and the imitators. We obtain necessary and sufficient conditions for equilibrium existence and stability. We find that unless strong conditions are met, only those equilibria where the imitators either all cooperate or all defect can be stable. We then partially characterize the non-singleton minimal positively invariant sets and find conditions for their existence.