Non-asymptotic Convergence Rates for Mean-Field Games: Weak Formulation and McKean–Vlasov BSDEs

This work is mainly concerned with the so-called limit theory for mean-field games. Adopting the weak formulation paradigm put forward by Carmona and Lacker (Ann Appl Probab 25(3):1189–1231, 2015), we consider a fully non-Markovian setting allowing for drift control and interactions through the joint distribution of players’ states and controls. We provide first a characterisation of mean-field equilibria as arising from solutions to a novel kind of McKean–Vlasov backward stochastic differential equations, for which we provide a well-posedness theory. We incidentally obtain there unusual existence and uniqueness results for mean-field equilibria, which do not require short-time horizon, separability assumptions on the coefficients, nor Lasry and Lions’s monotonicity conditions, but rather smallness—or alternatively regularity—conditions on the terminal reward and a dissipativity condition on the drift. We then take advantage of this characterisation to provide non-asymptotic rates of convergence for the value functions and the Nash-equilibria of the N-player version to their mean-field counterparts, for general open-loop equilibria. An appropriate reformulation of our approach also allows us to treat closed-loop equilibria, and to obtain convergence results for the master equation associated to the problem.