A two-sided (simultaneous) optimization technique is presented for the two-player linear quadratic Gaussian multistage game. This direct solution method naturally decomposes the solution into separate deterministic and stochastic terms, which provides insight into the optimal control strategies. We use the optimal governing equations to derive a compact expression for the optimal performance index value. We prove that this two-sided linear solution is indeed optimal out of all possible strategies (linear and nonlinear) by way of a saddle point proof. In addition, we provide some numerical results to show how system noises affect the optimal control strategies and optimal performance index value.