Turnpike properties in linear quadratic Gaussian N-player differential games

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.