  We are concerned with the mathematical study of the Mean Field Games system
(MFGS). In the conventional setup, the MFGS is a system of two coupled
nonlinear parabolic PDEs of the second order in a backward-forward manner,
namely one terminal and one initial conditions are prescribed respectively for
the value function and the population density. In this paper, we show that
uniqueness of solutions to the MFGS can be guaranteed if, among all four
possible terminal and initial conditions, eitheir only two terminal or only two
initial conditions are given. In both cases H\"older stability estimates are
proven. This means that he accuracies of the solutions are estimated in terms
of the given data. Moreover, these estimates readily imply uniqueness of
corresponding problems for the MFGS. The main mathematical apparatus to
establish those results is two new Carleman estimates, which may find
application in other contexts associated with coupled parabolic PDEs.
