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In this note, we study a class of deterministic finite-horizon linear-quadratic difference games with coupled-affine inequality constraints involving both state and control variables. We show that the necessary conditions for the existence of generalized open-loop Nash equilibria in this game class lead to two strongly coupled discrete-time linear complementarity systems. Subsequently, we derive the sufficient conditions by establishing an equivalence between the solutions of these systems and the convexity of players' objective functions. These existence conditions are then reformulated as a solution to a linear complementarity problem, providing a numerical method for computing these equilibria. We illustrate our results using a network flow game with constraints.

In this letter, we study a class of linear-quadratic mean-field-type difference games with coupled affine inequality constraints. We show that the mean-filed-type equilibrium can be characterized by the existence of a multiplier process which satisfies some implicit complementarity conditions. Further, we show that the equilibrium strategies can be computed by reformulating these conditions as a single large-scale linear complementarity problem. We illustrate our results with an energy storage problem arising in the management of microgrids.