# Bridging Finite and Infinite-Horizon Nash Equilibria in Linear Quadratic Games

## TL;DR

This paper explores the relationship between finite and infinite-horizon Nash equilibria in linear quadratic games by modeling the finite-horizon case as a nonlinear dynamical system, revealing how these equilibria relate and how to transition between them.

## Contribution

It introduces a dynamical systems framework to connect finite and infinite-horizon Nash equilibria in LQ games, including fixed points and periodic orbits, with numerical and simulation evidence.

## Key findings

- Finite-horizon equilibrium corresponds to fixed points of the dynamical system.
- Infinite-horizon equilibria can be obtained via appropriate terminal costs.
- Numerical simulations show convergence to stationary and periodic equilibria.

## Abstract

Finite-horizon linear quadratic (LQ) games admit a unique Nash equilibrium, while infinite-horizon settings may have multiple. We clarify the relationship between these two cases by interpreting the finite-horizon equilibrium as a nonlinear dynamical system. Within this framework, we prove that its fixed points are exactly the infinite-horizon equilibria and that any such equilibrium can be recovered by an appropriate choice of terminal costs. We further show that periodic orbits of the dynamical system, when they arise, correspond to periodic Nash equilibria, and we provide numerical evidence of convergence to such cycles. Finally, simulations reveal three asymptotic regimes: convergence to stationary equilibria, convergence to periodic equilibria, and bounded non-convergent trajectories. These findings offer new insights and tools for tuning finite-horizon LQ games using infinite-horizon.