A Maximum Principle approach to deterministic Mean Field Games of Control with Absorption

TL;DR

This paper develops a novel maximum principle approach to deterministic mean field games with absorption constraints, analyzing equilibrium existence, uniqueness, and explicit solutions in resource exploitation models.

Contribution

It introduces a new method for handling state absorption constraints in mean field games, including existence, uniqueness, and closed-form solutions.

Findings

Existence and uniqueness of equilibria are established.

Explicit solutions for infinite horizon models are derived.

Equilibrium production rates can be non-monotone and irregular.

Abstract

We study a class of deterministic mean field games 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. This requires a novel approach when applying Pontyagin'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.