  We identify structural assumptions which provide solvability of the Nash
system arising from a linear-quadratic closed-loop game, with stable properties
with respect to the number of players. In a setting of interactions governed by
a sparse graph, both short-time and long-time existence of a classical solution
for the Nash system set in infinitely many dimensions are addressed, as well as
convergence to the solution to the respective ergodic problem as the time
horizon goes to infinity; in addition, equilibria for the infinite-dimensional
game are shown to provide $\epsilon$-Nash closed-loop equilibria for the
$N$-player game. In a setting of generalized mean-field type (where the number
of interactions is large but not necessarily symmetric), directly from the
$N$-player Nash system estimates on the value functions are deduced on an
arbitrary large time horizon, which should pave the way for a convergence
result as $N$ goes to infinity.
