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

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