A MAXIMUM PRINCIPLE APPROACH TO A DETERMINISTIC MEAN FIELD GAME OF CONTROL WITH ABSORPTION

We study a deterministic mean field game on finite and infinite time horizons arising in models of optimal exploitation of exhaustible resources. The main characteristic of our game is an absorption constraint on the players' state process. As a result of the state constraint the optimal time of absorption becomes part of the equilibrium. Using Pontryagin's maximum principle, we prove the existence and uniqueness of equilibria and solve the infinite horizon models in closed form. As players may drop out of the game over time, equilibrium production rates need not be monotone nor smooth.