We consider some Mean Field Games with large discount factor in the cost functional and cheap controls. We show that in the limit the mass distribution solves a nonlocal nonlinear continuity equation and the (rescaled) feedback producing the mean field equilibrium converges to the gradient of the running cost, so that the limit dynamics follows the steepest descent of the running cost associated to the limit mass distribution.
For suitable choices of the underlying control system and running costs our results establish a rigorous connection among Mean-Field Games and various agent-based models such as the aggregation equation and some models of crowd dynamics and flocking. This is joint work with Pierre Cardaliaguet, University Paris-Dauphine.
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