Turnpike properties in linear quadratic Gaussian N-player differential games

TL;DR

This paper analyzes the long-term behavior of strategies in N-player linear quadratic Gaussian differential games, establishing exponential convergence and turnpike properties uniformly across players without relying on mean field models.

Contribution

It provides a novel uniform analysis of turnpike properties in finite-horizon and ergodic N-player games, independent of mean field game limits.

Findings

Exponential convergence between finite-horizon and ergodic Riccati solutions.

Turnpike property holds uniformly for all players.

Numerical experiments support theoretical results.

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.