  In this note we prove the uniqueness of solutions to a class of Mean Field
Games systems subject to possibly degenerate individual noise. Our results hold
true for arbitrary long time horizons and for general non-separable
Hamiltonians that satisfy a so-called $displacement\ monotonicity$ condition.
This monotonicity condition that we propose for non-separable Hamiltonians is
sharper and more general than the one proposed in our earlier work written
jointly with Gangbo and Zhang. The displacement monotonicity assumptions
imposed on the data provide actually not only uniqueness, but also the
existence and regularity of the solutions. Our analysis uses elementary
arguments and does not rely on the well-posedness of the corresponding master
equations.
