Asynchronous Decision-Making Dynamics Under Imitation Update Rule in Heterogeneous Populations

Decision-making individuals are often likely to imitate their highest-earning fellows rather than optimizing their own utilities, due to bounded rationality and incomplete information on how their utilities depend on their own and others' decisions. Perpetual fluctuations between decisions have been reported as the dominant asymptotic outcome of imitative behaviors, yet little attempt has been made to characterize them, particularly in heterogeneous populations. We study a finite well-mixed heterogeneous population of individuals choosing between the two strategies, cooperation and defection, and earning based on their payoff matrices that can be unique to each individual. At each time step, an arbitrary individual becomes active to update her decision by imitating the highest earner in the population. We show that almost surely the dynamics reach either an equilibrium state or a minimal positively invariant set, a fluctuation set, in the long run. In addition to finding all equilibria, for the first time, we characterize the fluctuation sets and provide necessary and sufficient conditions for their existence. We also find that exclusive populations of individuals playing coordination or prisoner's dilemma games always equilibrate, implying that cycles and non-convergence in imitative populations are due to individuals playing anticoordination games. Our results theoretically explain earlier reported simulation results, shed new lights on the boundedly rational nature of imitation behaviors, and pave the way for future research on controlling such populations.