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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 couplednonlinear parabolic PDEs of the second order in a backward-forward manner,namely one terminal and one initial conditions are prescribed respectively forthe value function and the population density. In this paper, we show thatuniqueness of solutions to the MFGS can be guaranteed if, among all fourpossible terminal and initial conditions, eitheir only two terminal or only twoinitial conditions are given. In both cases H\"older stability estimates areproven. This means that he accuracies of the solutions are estimated in termsof the given data. Moreover, these estimates readily imply uniqueness ofcorresponding problems for the MFGS. The main mathematical apparatus toestablish those results is two new Carleman estimates, which may findapplication in other contexts associated with coupled parabolic PDEs.

H\"older Stability and Uniqueness for The Mean Field Games System via Carleman Estimates