Mean field games with option to buy information

Abstract: We introduce a class of continuous time finite horizon mean field games where the objective function of the representative player depends on a hidden state, in addition to position, control, and the population distribution. While acting on the position dynamics, the agent has the option to pay for seeing the hidden state. We connect the original formulation of our model with a mean field model of optimal control with discretionary stopping, characterize solutions, and give a simple explicitly solvable example. For a class of $N$-player games with compatible information structure, we show that approximate Nash equilibria can be constructed starting from a solution to the limit model.