On the discrete-time origins of 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 two by two 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 for different equilibrium configurations exhibit an entire range of behaviors for large step size (stability, instability, and even Li-Yorke chaos); while (iii) for the exponential / multiplicative weights (EW) algorithm increasing step size will (almost) inevitably lead 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.