  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 (EW) 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 $2\times2$
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 EW 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.
