Evolutionary matrix-game dynamics under imitation in heterogeneous populations

Abstract

Perpetual fluctuations between decisions are reported as the dominant asymptotic outcome of imitative behaviours, 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 non-singleton minimal positively invariant set, a fluctuation set, in the long run. In addition to finding all equilibria, we characterize the fluctuation sets, provide necessary conditions and sufficient conditions for their existence, and approximate their basins of attraction. Our results theoretically explain earlier reported simulation results and shed new light on the boundedly rational nature of imitation behaviours.