When a mean field game satisfies certain monotonicity conditions, the
mean field equilibrium is unique and the corresponding value function
satisfies the so called master equation. In general, however, there
can be multiple equilibriums which typically lead to different
values. In this paper we study the set of values over all mean field
equilibriums, which we call the set value of the game. We shall
establish two main properties of the set value: (i) the dynamic
programming principle; (ii) the convergence of the set values of the
corresponding N-player games. We emphasize that the set value is very
sensitive to the choice of the admissible controls. For the dynamic
programming principle, one needs to use closed loop controls (instead
of open loop controls). For the convergence, one has to restrict to
the same type of equilibriums for the N-player game and for the mean
field game. We shall investigate three cases, two in finite state
space models and the other in a diffusion model.