A Deterministic Approximation Approach to the Continuum Logit Dynamic with an Application to Supermodular Games

We consider the logit dynamic in a large population game with a continuum of strategies. The deterministic approximation approach requires us to derive this dynamic as the finite horizon limit of a stochastic process in a game with a finite but large number of strategies and players. We first establish the closeness of this dynamic with a step–wise approximation. We then show that the logit stochastic process is close to the step–wise logit dynamic in a discrete approximation of the original game. Combining the two results, we obtain our deterministic approximation result. We apply the result to large population supermodular games with a continuum of strategies. Over finite but sufficiently long time horizons, the logit stochastic process converges to logit equilibria in a discrete approximation of the supermodular game. By the deterministic approximation approach, so does the logit dynamic in the continuum supermodular game