  In this paper we examine fully nonlinear mean-field games associated with a
minimization problem. The variational setting is driven by a functional
depending on its argument through its Hessian matrix. We work under fairly
natural conditions and establish improved (sharp) regularity for the solutions
in Sobolev spaces. Then, we prove the existence of minimizers for the
variational problem and the existence of solutions to the mean-field games
system. We also investigate a unidimensional example and unveil new information
on the explicit solutions. Our findings can be generalized to a larger class of
operators, yielding information on a broader range of examples.
