Learning and Mixed-Strategy Equilibria in Evolutionary Games

This paper considers whether Maynard Smith's concept of an evolutionarily stable strategy, or "ESS", can be used to predict long-run strategy frequencies in large populations whose members are randomly paired to play a game, and who adjust their strategies over time according to sensible learning rules. The existing results linking the ESS to stable equilibrium population strategy frequencies when strategies are inherited do not apply to learning, even when each individual always adjusts its strategy in the direction of increased fitness, because the inherited-strategies stability results depend on aggregating across individuals, and this is not possible for learning. The stability of learning must therefore be analyzed for the entire system of individuals' strategy adjustments. The interactions between individuals' adjustments prove to be generically destabilizing at mixed-strategy equilibria, which are saddlepoints of the learning dynamics. Using the inherited-strategies dynamics to describe learning implicitly restricts the system to the stable manifold whose trajectories approach the saddlepoint, masking its instability. Thus, allowing for the interactions between individuals' strategy adjustments extends the widely recognized instability of mixed-strategy equilibria in multi-species inherited-strategies models to single-species (or multi-species) learning models.