The lower convergence tendency of imitators compared to best responders

Imitation is widely observed in nature and often used to model populations of decision-making agents, but it is not yet known under what conditions a network of imitators will reach a state where they are satisfied with their decisions. We show that every network in which agents imitate the best performing strategy in their neighborhood will reach an equilibrium in finite time, provided that all agents are opponent coordinating, i.e., earn a higher payoff if their opponent plays the same strategy as they do. It follows that any non-convergence observed in imitative networks is not necessarily a result of population heterogeneity nor special network topology, but rather must be caused by other factors such as the presence of non-opponent-coordinating agents. To strengthen this result, we show that large classes of imitative networks containing non-opponent-coordinating agents never equilibrate even when the population is homogeneous. Comparing to best-response dynamics where equilibration is guaranteed for every network of homogeneous agents playing 2 × 2 matrix games, our results imply that networks of imitators have a lower equilibration tendency.