  We develop the linear programming approach to mean-field games in a general
setting. This relaxed control approach allows to prove existence results under
weak assumptions, and lends itself well to numerical implementation. We
consider mean-field game problems where the representative agent chooses both
the optimal control and the optimal time to exit the game, where the
instantaneous reward function and the coefficients of the state process may
depend on the distribution of the other agents. Furthermore, we establish the
equivalence between mean-field games equilibria obtained by the linear
programming approach and the ones obtained via the controlled/stopped
martingale approach, another relaxation method used in a few previous papers in
the case when there is only control.
