Policy iteration method for discounted infinite horizon mean field games: the semi-Lagrangian approach

We study the policy iteration method for solving discounted infinite-horizon mean field games. At the continuous level, a policy iteration algorithm can be used to establish the existence and uniqueness of solutions for mean field games with a large discount factor lambda. At a discrete level, it can be used to compute a solution of the problem. To implement the method, we employ a semi-Lagrangian method, where the Hamilton-Jacobi-Bellman equation is first discretized in time using the dynamic programming principle and then in space by projecting onto a grid. To support our theoretical findings, we present numerical examples in both one and two dimensions.