Optimality in a min-max sense for constrained difference equations in the presence of disturbances is studied as a two-player zero-sum game. Sufficient conditions that permit the evaluation and to upper bound the cost of solutions to such systems are presented. Cost evaluation results are also presented for the case in which a control input aims to minimize a cost functional while the objective of disturbance is to maximize it. Sufficient conditions in the form of Hamilton-Jacobi-Isaacs equations are provided to certify closed-loop saddle point optimality. The results are illustrated in an example featuring a linearized and discretized model of an inverted pendulum.