The logit dynamic in supermodular games with a continuum of strategies: A deterministic approximation approach

We consider large population supermodular games with pairwise interaction and a continuous strategy set. Our objective is to establish convergence of the logit dynamic in such games to logit equilibria. For this purpose, we apply the deterministic approximation approach, which interprets a deterministic dynamic as an approximation of a stochastic process. 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 then apply this result to supermodular games. 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.