  It is well known that the monotonicity condition, either in Lasry-Lions sense
or in displacement sense, is crucial for the global well-posedness of mean
field game master equations, as well as for the uniqueness of mean field
equilibria and solutions to mean field game systems. In the literature, the
monotonicity conditions are always taken in a fixed direction. In this paper we
propose a new type of monotonicity condition in the opposite direction, which
we call the anti-monotonicity condition, and establish the global
well-posedness for mean field game master equations with nonseparable
Hamiltonians. Our anti-monotonicity condition allows our data to violate both
the Lasry-Lions monotonicity and the displacement monotonicity conditions.
