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

2025-08-28

AI Summary

This paper investigates the intrinsic relationship between finite-horizon and infinite-horizon Nash equilibria in linear-quadratic differential games. Addressing the challenge that infinite-horizon equilibrium multiplicity cannot be directly approximated by finite-horizon solutions, we model finite-horizon equilibria as a parametrized nonlinear dynamical system—establishing, for the first time, a dynamical systems bridge between the two regimes: its fixed points correspond exactly to infinite-horizon steady-state equilibria, while its periodic orbits characterize periodic infinite-horizon equilibria. We theoretically prove that arbitrary steady-state or periodic infinite-horizon equilibria can be recovered by appropriately designing the terminal cost. Furthermore, we identify three canonical asymptotic behaviors—convergence, periodic oscillation, and bounded non-convergent dynamics—and validate them numerically. This work provides a novel theoretical framework and computationally tractable tools for the design and regulation of finite-horizon differential games.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Relating finite and infinite-horizon Nash equilibria
• Analyzing dynamical system properties of LQ games
• Identifying convergence patterns in equilibrium trajectories

Innovation

• Finite-horizon equilibrium as nonlinear dynamical system
• Fixed points correspond to infinite-horizon Nash equilibria
• Periodic orbits represent periodic Nash equilibria