In this talk we present some recent results about quantitative convergence of Nash equilibria (both open and closed loop) of a general class of N-player stochastic differential games, when agents interact through the empirical measure supported on both states and controls. Our analysis is based on a careful comparison between open and closed loop Nash equilibria, and the mean field Nash equilibrium. A particular challenge is to understand the properties of an additional fixed point map and to obtain dimension-free estimates as N increases to infinity. Our quantitative analysis does not impose implicit assumptions on the fixed point map nor uses the master equation, but it relies on displacement monotonicity techniques. If time permits we will discuss how to make use of our approach on the N-player Nash system to show the well-posedness of the underlying master equation as well. The talk will be based on joint works with Joe Jackson (University of Chicago).

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