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

This paper builds on the work of Degond, Herty and Liu in [ 16 ] by considering \begin{document}$ N $\end{document} -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.