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

In infinite-horizon discrete-time linear-quadratic (LQ) dynamic games, computing feedback Nash equilibria (FNEs) remains computationally challenging.

Motivated by this, we study a finite-horizon strategy for approximating one of the infinite-horizon FNEs.

The finite-horizon strategy is as follows.

Each player $i$ has an individual prediction horizon $T^i$.

In the infinite-horizon game, at each stage, each player $i$ computes its control in the following way: player $i$ envisions an auxiliary $T^i$-stage game in which the same set of players play, computes the unique FNE of the auxiliary game using a standard method, and implements only the first-stage control.

Our main result is, under suitable conditions, the total cost under these finite-horizon strategies converges to that under one of the infinite-horizon FNEs when all players' prediction horizons tend to infinity.

Moreover, we derive an explicit cubic-polynomial upper bound on this cost gap with respect to the distance between the corresponding strategy matrices.

This strategy is tractable and implementable, as it avoids the direct solution of the coupled algebraic Riccati equations (CARE) of infinite-horizon LQ games.