Feedback and Open-Loop Nash Equilibria for LQ Infinite-Horizon Discrete-Time Dynamic Games

TL;DR

This paper analyzes Nash equilibrium strategies in infinite-horizon linear-quadratic discrete-time dynamic games, providing structural insights, linking optimal control solutions to equilibrium strategies, and illustrating findings with a numerical example.

Contribution

It offers new structural insights into feedback Nash equilibria, connects optimal control solutions to open-loop strategies, and characterizes OL-NE strategies in infinite-horizon LQ games.

Findings

Structural properties of F-NE solutions revealed

Connection established between DP and Pontryagin's principle

Characterization of OL-NE strategies achieved

Abstract

We consider dynamic games defined over an infinite horizon, characterized by linear, discrete-time dynamics and quadratic cost functionals. Considering such linear-quadratic (LQ) dynamic games, we focus on their solutions in terms Nash equilibrium strategies. Both Feedback (F-NE) and Open-Loop (OL-NE) Nash equilibrium solutions are considered. The contributions of the paper are threefold. First, our detailed study reveals some interesting structural insights in relation to F-NE solutions. Second, as a stepping stone towards our consideration of OL-NE strategies, we consider a specific infinite-horizon discrete-time (single-player) optimal control problem, wherein the dynamics are influenced by a known exogenous input and draw connections between its solution obtained via Dynamic Programming and Pontryagin's Minimum Principle. Finally, we exploit the latter result to provide a characterization of OL-NE strategies of the class of infinite-horizon dynamic games. The results and key observations made throughout the paper are illustrated via a numerical example.