Optimal Control and Potential Games in the Mean Field

TL;DR

This paper explores a comprehensive mean field control framework with non-Markovian dynamics, establishing a connection to potential mean field games and illustrating with diverse examples including flocking and synchronization models.

Contribution

It introduces a general theory linking mean field optimal control problems with potential mean field games, including non-Markovian dynamics, common noise, and jumps.

Findings

Minimizers of the control problem are Nash equilibria of the associated mean field game.

The mean field games are necessarily potential, facilitating analysis and solution.

Examples include control with price interactions, flocking, and Kuramoto models.

Abstract

We study a mean field optimal control problem with general non-Markovian dynamics, including both common noise and jumps. We show that its minimizers are Nash equilibria of an associated mean field game of controls. These types of games are necessarily potential, and the Nash equilibria derived as the minimizers of the control problem are closely connected to McKean-Vlasov equations of Langevin type. To illustrate the general theory, we present several examples, including a mean field game of controls with interactions through a price variable, and mean field Cucker-Smale Flocking and Kuramoto models. We also establish the invariance property of the value function, a key ingredient used in our proofs.