Evolutionary Equilibrium with Forward-Looking Players

The stochastic evolutionary game literature is built on three behavioral postulates: ``noisy'' decisionmaking, myopic decisionmaking and random opportunities for choice (inertia). The role of noise is by now well- understood. This paper investigates the significance of the other two postulates. The model is the conventional evolutionary story, where a population of players is randomly matched against each other, and where strategy revision opportunities for each player arrive at random moments. But here players are assumed to have rational expectations about the evolution of play of their opponents, and choose revision policies to maximize the expected present value of the payoff stream. The dynamic programming problem each player solves is studied, and equilibrium is shown to exist. Not surprisingly, myopic play emerges as discount rates become large. More surprising is that, for any discount rate, myopic play also emerges as the arrival rate of strategy revision opportunities shrinks to~0. These results hold for any $K\times K$ symmetric game. For $2\times 2$ coordination games the effects of taking limits in the opposite direction is studied. When the noise term is small, if players are sufficiently patient or if the arrival rate of strategy revision opportunities is sufficiently large, the unique symmetric Markov equilibrium has each player always choosing the payoff- dominant strategy.