Some examples for stable and historic behavior in replicator equations

Abstract

The evolutionary dynamics of zero-sum and non zero-sum games under replicator equations could be drastically different from each other. In zero-sum games, heteroclinic cycles naturally occur whenever the species of the population supersede each other in cyclic fashion (like for the Rock–Paper–Scissors game). In this case, the highly erratic oscillations may cause the divergence of the time averages. In contrast, it is a common belief that all “reasonable” replicator equations of non-zero sum games satisfy “The Folk Theorem of Evolutionary Game Theory” which asserts that $\left(i\right)$ a Nash equilibrium is a rest point; $\left(ii\right)$ a stable rest point is a Nash equilibrium; $\left(iii\right)$ a strictly Nash equilibrium is asymptotically stable; $\left(iv\right)$ any interior convergent orbit evolves to a Nash equilibrium. In this paper, we propose two distinct classes of replicator equations generated by Schur-convex potential functions which exhibit two opposing phenomena: stable/predictable and historic/unpredictable behavior. In the latter case, the time averages of the orbit will slowly oscillate during the evolution of the system and do not converge to any limit. This will eventually cause the divergence of higher-order repeated time averages.