From mean field games to the best reply strategy in a stochastic framework

TL;DR

This paper introduces a simplified MPC-based method for approximating Nash equilibria in N-player stochastic differential games, connecting mean field game strategies with best reply strategies to improve tractability.

Contribution

It demonstrates that MPC can approximate best reply strategies in stochastic games and reduces complex PDE systems to a single PDE, enhancing computational simplicity.

Findings

MPC approximates Nash equilibrium controls effectively.

The mean field limit simplifies the PDE system from two to one.

The approach is validated through applications to existing literature.

Abstract

This paper builds on the work of Degond, Herty and Liu by considering N-player stochastic differential games. The control corresponding to a Nash equilibrium of such a game is approximated through model predictive control (MPC) techniques. In the case of a linear quadratic running-cost, considered here, the MPC method is shown to approximate the solution to the control problem by the best reply strategy (BRS) for the running cost. We then compare the MPC approach when taking the mean field limit with the popular mean field game (MFG) strategy. We find that our MPC approach reduces the two coupled PDEs to a single PDE, greatly increasing the simplicity and tractability of the original problem. We give two examples of applications of this approach to previous literature and conclude with future perspectives for this research.