  We formulate the MFG limit for $N$ interacting agents with a common noise as
a single quasi-linear deterministic infinite-dimensional partial differential
second order backward equation. We prove that any its (regular enough) solution
provides an $1/N$-Nash-equilibrium profile for the initial $N$-player game. We
use the method of stochastic characteristics to provide the link with the basic
models of MFG with a major player. We develop two auxiliary theories of
independent interest: sensitivity and regularity analysis for the McKean-Vlasov
SPDEs and the $1/N$-convergence rate for the propagation of chaos property of
interacting diffusions.
