On finite-horizon approximation of a feedback Nash equilibrium in LQ games

TL;DR

This paper proposes a finite-horizon approximation method for feedback Nash equilibria in infinite-horizon discrete-time linear-quadratic games, providing theoretical guarantees and an efficient computation algorithm.

Contribution

It introduces a finite-horizon strategy approach for infinite-horizon LQ games, characterizes the Riccati equations, and establishes convergence and performance bounds.

Findings

Finite-horizon strategies approximate infinite-horizon equilibria effectively.

An efficient algorithm computes the finite-horizon feedback Nash equilibrium.

The cost gap between finite and infinite-horizon strategies is explicitly bounded.

Abstract

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.