From Discrete to Continuous Highest-earning Imitation Dynamics

Abstract

Decision-making by imitating the highest earners has been observed in experimental studies. In two-strategy decision-making problems, this behavior may result in perpetual fluctuations in the population proportions of the two strategies. How these fluctuations evolve for large population sizes remains unclear. This paper addresses this question for a heterogeneous population of players imitating the highest earners. We show that the family of Markov chains describing the discrete population dynamics forms a generalized stochastic approximation process for a good upper semicontinuous differential inclusion--the mean dynamics. Furthermore, we prove that the mean dynamics always equilibrate. Then, by using results from stochastic approximation theory, we show that the amplitudes of fluctuations in the population proportions of the two strategies diminish to zero with probability one, as the population size approaches infinity. Our results suggest that in a well-mixed, large population, imitating the highest earners is unlikely to generate large-scale, perpetual fluctuations.