  We consider N-player and mean field games in continuous time over a finite
horizon, where the position of each agent belongs to {-1,1}. If there is
uniqueness of mean field game solutions, e.g. under monotonicity assumptions,
then the master equation possesses a smooth solution which can be used to prove
convergence of the value functions and of the feedback Nash equilibria of the
N-player game, as well as a propagation of chaos property for the associated
optimal trajectories. We study here an example with anti-monotonous costs, and
show that the mean field game has exactly three solutions. We prove that the
value functions converge to the entropy solution of the master equation, which
in this case can be written as a scalar conservation law in one space
dimension, and that the optimal trajectories admit a limit: they select one
mean field game soution, so there is propagation of chaos. Moreover, viewing
the mean field game system as the necessary conditions for optimality of a
deterministic control problem, we show that the N-player game selects the
optimizer of this problem.
