  In this work, we study the convergence rate of the $N$-player LQG game with a
Markov chain common noise towards its asymptotic Mean Field Game. By
postulating a Markovian structure via two auxiliary processes for the first and
second moments of the Mean Field Game equilibrium and applying the fixed point
condition in Mean Field Game, we first provide the characterization of the
equilibrium measure in Mean Field Game with a finite-dimensional Riccati system
of ODEs. Additionally, with an explicit coupling of the optimal trajectory of
the $N$-player game driven by $N$ dimensional Brownian motion and Mean Field
Game counterpart driven by one-dimensional Brownian motion, we obtain the
convergence rate $O(N^{-1/2})$ with respect to 2-Wasserstein distance.
