Detailed balance and phase transition in Nash equilibria of mean-field Ising game

We study a finite-state mean-field game with entropic control cost. In the large population limit, we show that its stationary Nash equilibria satisfy the detailed balance condition and their distributions take the Gibbsian form with the value function playing the role of a Hamiltonian. We apply the entropic control cost to a ferromagnetic mean-field Ising game and investigate its stationary and non-stationary Nash equilibria. We find that the stationary Nash equilibria exhibit a second-order phase transition from disordered to ordered equilibria as temperature decreases. A center-manifold reduction yields non-stationary Nash equilibrium near the critical temperature and reveals the critical slowing down in the relaxation toward the stationary Nash equilibrium.