The usually overlooked possibility of mixed strategy solutions is approached by applying the method of dual cones to fixed-duration linear quadratic multistage games with bounded controls. It is shown that the value of such games is piecewise quadratic when open-loop strategies are used, and is at best piecewise quadratic when closed-loop strategies are allowed. It is argued that computation of such value functions is generally excessively burdensome, hence supporting the viewpoint that new, approximate, or hybrid methods will be needed to obtain game solutions.