  The mean field games (MFG) theory has broad application in mathematical
modeling of social phenomena. The Mean Field Games System (MFGS) is the key to
the MFG theory. This is a system of two nonlinear parabolic partial
differential equations with two opposite directions of time 0<t<T. The topic of
Coefficient Inverse Problem (CIPs) for the MFGS is a newly emerging one. A CIP
for the MFGS is studied. The input data are Dirichlet and Neumann boundary
conditions either on a part of the lateral boundary (incomplete data) or on the
whole lateral boundary (complete data). In addition to the initial conditions
at t=0, terminal conditions at t=T are given. The terminal conditions mean the
final overdetermination. The necessity of assigning all these input data is
explained. Holder and Lipschitz stability estimates are obtained for the cases
of incomplete and complete data respectively. These estimates imply uniqueness
of the CIP.
