  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, 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.
