Pasting of Equilibria and Donsker-type Results for Mean Field Games

TL;DR

This paper explores the relationship between equilibria in discrete and continuous mean field games, establishing existence, pasting properties, and convergence results with rates under certain conditions.

Contribution

It introduces a method to construct multi-period equilibria by pasting single-period solutions and proves convergence of discrete to continuous equilibria with rates under monotonicity.

Findings

Existence of equilibria in single-period mean field games.

Construction of multi-period equilibria via pasting.

Convergence of discrete-time equilibria to continuous-time equilibria with a sharp rate.

Abstract

This paper studies the relation between equilibria in single-period, discrete-time and continuous-time mean field game models. First, for single-period mean field games, we establish the existence of equilibria and then prove the propagation of the Lasry-Lions monotonicity to the optimal equilibrium value, as a function of the realization of the initial condition and its distribution. Secondly, we prove a pasting property for equilibria; that is, we construct equilibria to multi-period discrete-time mean field games by recursively pasting the equilibria of suitably initialized single-period games. Then, we show that any sequence of equilibria of discrete-time mean field games with discretized noise converges (up to a subsequence) to some equilibrium of the continuous-time mean field game as the mesh size of the discretization tends to zero. When the cost functions of the game satisfy the Lasry-Lions monotonicity property, we strengthen this convergence result by providing a sharp convergence rate.