  We consider a multi-player stochastic differential game with linear
McKean-Vlasov dynamics and quadratic cost functional depending on the variance
and mean of the state and control actions of the players in open-loop form.
Finite and infinite horizon problems with possibly some random coefficients as
well as common noise are addressed. We propose a simple direct approach based
on weak martingale optimality principle together with a fixed point argument in
the space of controls for solving this game problem. The Nash equilibria are
characterized in terms of systems of Riccati ordinary differential equations
and linear mean-field backward stochastic differential equations: existence and
uniqueness conditions are provided for such systems. Finally, we illustrate our
results on a toy example.
