  The mean field limit of large-population symmetric stochastic differential
games is derived in a general setting, with and without common noise, on a
finite time horizon. Minimal assumptions are imposed on equilibrium strategies,
which may be asymmetric and based on full information. It is shown that
approximate Nash equilibria in the $n$-player games admit certain weak limits
as $n$ tends to infinity, and every limit is a weak solution of the mean field
game (MFG). Conversely, every weak MFG solution can be obtained as the limit of
a sequence of approximate Nash equilibria in the $n$-player games. Thus, the
MFG precisely characterizes the possible limiting equilibrium behavior of the
$n$-player games. Even in the setting without common noise, the empirical state
distributions may admit stochastic limits which cannot be described by the
usual notion of MFG solution.
