Stochastic Evolutionary Game Dynamics: Foundations, Deterministic Approximation, and Equilibrium Selection

We present a general model of stochastic evolution in games played by large populations of anonymous agents. Agents receive opportunities to revise their strategies by way of independent Poisson processes. A revision protocol describes how the probabilities with which an agent chooses each of his strategies depend on his current payoff opportunities and the current behavior of the population. Over finite time horizons, the population’s behavior is well-approximated by a mean dynamic, an ordinary differential equation defined by the expected motion of the stochastic evolutionary process. Over the infinite time horizon, the population’s behavior is described by the stationary distribution of the stochastic evolutionary process. If limits are taken in the population size, the level of noise in agents’ revision protocols, or both, the stationary distribution may become concentrated on a small set of population states, which are then said to be stochastically stable. Stochastic stability analysis allows one to obtain unique predictions of very long run behavior even when the mean dynamic admits multiple locally stable states. We present a full analysis of the asymptotics of the stationary distribution in two-strategy games under noisy best protocols, and discuss extensions of this analysis to other settings.