  This paper studies multidimensional mean field games with common noise and
the related system of McKean-Vlasov forward-backward stochastic differential
equations deriving from the stochastic maximum principle. We first propose some
structural conditions which are related to the submodularity of the underlying
mean field game and are a sort of opposite version of the well known
Lasry-Lions monotonicity. By reformulating the representative player
minimization problem via the stochastic maximum principle, the submodularity
conditions allow to prove comparison principles for the forward-backward
system, which correspond to the monotonicity of the best reply map. Building on
this property, existence of strong solutions is shown via Tarski's fixed point
theorem, both for the mean field game and for the related McKean-Vlasov
forward-backward system. In both cases, the set of solutions enjoys a lattice
structure, with minimal and maximal solutions which can be constructed by
iterating the best reply map or via the fictitious play algorithm.
