Bayesian Learning in Mean Field Games

We consider a mean-field game model where the cost functions depend on a fixed parameter, called $\mathit{\text{state}}$, which is unknown to players. Players learn about the state from a a stream of private signals they receive throughout the game. We derive a mean field system satisfied by the equilibrium payoff of the game and prove existence of a solution under standard regularity assumptions. Additionally, we establish the uniqueness of the solution when the cost function satisfies the monotonicity assumption of Lasry and Lions at each state.