On the discrete-time origins of the replicator dynamics: from convergence to instability and chaos

We consider three distinct discrete-time models of learning and evolution in games: a biological model based on intra-species selective pressure, the dynamics induced by pairwise proportional imitation, and the exponential/multiplicative weights algorithm for online learning. Even though these models share the same continuous-time limit-the replicator dynamics-we show that second-order effects play a crucial role and may lead to drastically different behaviors in each model, even in very simple, symmetric 2x2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} games. Specifically, we study the resulting discrete-time dynamics in a class of parametrized congestion games, and we show that (i) in the biological model of intra-species competition, the dynamics remain convergent for any parameter value; (ii) the dynamics of pairwise proportional imitation exhibit an entire range of behaviors for larger time steps and different equilibrium configurations (stability, instability, and even Li-Yorke chaos); while (iii) in the exponential/multiplicative weights algorithm, increasing the time step (almost) inevitably leads to chaos (again, in the formal, Li-Yorke sense). This divergence of behaviors comes in stark contrast to the globally convergent behavior of the replicator dynamics, and serves to delineate the extent to which the replicator dynamics provide a useful predictor for the long-run behavior of their discrete-time origins.