On finite-horizon approximation of an infinite-horizon feedback Nash equilibrium in discrete-time LQ games

Dynamic games provide a fundamental framework for multi-agent decision-making over time, yet computing feedback Nash equilibria (FNEs) in infinite-horizon discrete-time linear-quadratic (LQ) settings remains computationally challenging. Motivated by the need for tractable and implementable strategies, this paper studies a finite-horizon strategy for approximating a certain infinite-horizon equilibrium. Specifically, at each stage, each player solves a T-stage game and implements only the first-stage control, thereby avoiding the direct solution of coupled infinite-horizon Riccati equations. We first analyze the finite-horizon game and characterize the structure of the associated coupled generalized discrete Riccati difference equations. Based on this analysis, we establish a sufficient condition for uniqueness of the FNE and propose an efficient algorithm that computes it via a sequence of linear equations. We then consider the infinite-horizon game in which players adopt the finite-horizon strategies with heterogeneous prediction horizons and show that, under suitable conditions, the total cost under the finite-horizon strategies converges to  the cost under the limiting infinite-horizon FNE. Moreover, we derive an explicit upper bound on this cost gap in terms of the distance between the corresponding strategy matrices. These results provide theoretical justification and quantitative performance guarantees for finite-horizon strategies in infinite-horizon LQ dynamic games. A nonscalar numerical example illustrates the effectiveness of the proposed framework.