Evolutionary Matrix-Game Dynamics Under Imitation in Heterogeneous Populations

TL;DR

This paper analyzes how imitation-driven decision-making in heterogeneous populations leads to equilibrium or fluctuation dynamics, revealing the conditions for stability and the impact of game types on convergence.

Contribution

It characterizes fluctuation sets in imitative dynamics, provides conditions for their existence, and explains the role of game types in population stability.

Findings

Fluctuation sets are almost surely reached in long-term dynamics.

Exclusive coordination or prisoner's dilemma populations always reach equilibrium.

Heterogeneous populations playing anticoordination games often do not converge.

Abstract

Decision-making individuals often imitate their highest-earning fellows rather than optimize their own utilities, due to bounded rationality and incomplete information. 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, provide necessary and sufficient conditions for their existence, and approximate their basins of attraction. 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. Moreover, we show that except for the two extreme equilibria where all individuals play the same strategy, almost all other equilibria are unstable as long as the population is heterogeneous. Our results theoretically explain earlier reported simulation results and shed new light on the boundedly rational nature of imitation behaviors.