  We study a Hessian-dependent functional driven by a fully nonlinear operator.
The associated Euler-Lagrange equation is a fully nonlinear mean-field game
with free boundaries. Our findings include the existence of solutions to the
mean-field game, together with H\"older continuity of the value function and
improved integrability of the density. In addition, we prove the reduced free
boundary is a set of finite perimeter. To conclude our analysis, we prove a
$\Gamma$-convergence result for the functional.
