  In this paper we propose two new monotonicity conditions that could serve as
sufficient conditions for uniqueness of Nash equilibria in mean field games. In
this study we aim for $unconditional\ uniqueness$ that is independent of the
length of the time horizon, the regularity of the starting distribution of the
agents, or the regularization effect of a non-degenerate idiosyncratic noise.
Through a rich class of simple examples we show that these new conditions are
not only in dichotomy with each other, but also with the two widely studied
monotonicity conditions in the literature, the Lasry-Lions monotonicity and
displacement monotonicity conditions.
