  An overdetermination is introduced in an initial condition for the second
order mean field games system (MFGS). This makes the resulting problem close to
the classical ill-posed Cauchy problems for PDEs. Indeed, in such a problem and
overdetermination in boundary conditions usually takes place. A Lipschitz
stability estimate is obtained. This estimate implies uniqueness. A new
Carleman estimate is derived. The second estimate the author calls
"quasi-Carleman estimate", since it contains two test functions rather than a
single one in conventional Carleman estimates. These two estimates play the key
role. Carleman estimates were not applied to the MFGS prior to the recent work
of Klibanov and Aveboukh in arXiv: 2302.10709, 2023. Applications are
discussed.
