  We propose and investigate a general class of discrete time and finite state
space mean field game (MFG) problems with potential structure. Our model
incorporates interactions through a congestion term and a price variable. It
also allows hard constraints on the distribution of the agents. We analyze the
connection between the MFG problem and two optimal control problems in duality.
We present two families of numerical methods and detail their implementation:
(i) primal-dual proximal methods (and their extension with nonlinear proximity
operators), (ii) the alternating direction method of multipliers (ADMM) and a
variant called ADM-G. We give some convergence results. Numerical results are
provided for two examples with hard constraints.
