  In this manuscript we derive a new nonlinear transport equation written on
the space of probability measures that allows to study a class of deterministic
mean field games and master equations, where the interaction of the agents
happens only at the terminal time. The point of view via this transport
equation has two important consequences. First, this equation reveals a new
monotonicity condition that is sufficient both for the uniqueness of MFG Nash
equilibria and for the global in time well-posedness of master equations.
Interestingly, this condition is in general in dichotomy with both the
Lasry--Lions and displacement monotonicity conditions, studied so far in the
literature. Second, in the absence of monotonicity, the conservative form of
the transport equation can be used to define weak entropy solutions to the
master equation. We construct several concrete examples to demonstrate that MFG
Nash equilibria, whether or not they actually exist, may not be selected by the
entropy solutions of the master equation.
