Turnpike properties in linear quadratic Gaussian N-player differential games

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 24 September 2025
Accepted: 6 April 2026

Abstract

We consider the long-time behavior of equilibrium strategies and state trajectories in a linear quadratic N-player game with Gaussian initial data. By comparing the finite-horizon game with its ergodic counterpart, we establish exponential convergence estimates between the solutions of the finite-horizon generalized Riccati system and the associated algebraic system arising in the ergodic setting. Building on these results, we prove the convergence of the time-averaged value function and derive a turnpike property for the equilibrium pairs of each player. Importantly, our approach avoids reliance on the mean field game limiting model, allowing for a fully uniform analysis with respect to the number of players N. As a result, we further establish a uniform turnpike property for the equilibrium pairs between the finite-horizon and ergodic games with N players. Numerical experiments are also provided to illustrate and support the theoretical results.