  We develop the fictitious play algorithm in the context of the linear
programming approach for mean field games of optimal stopping and mean field
games with regular control and absorption. This algorithm allows to approximate
the mean field game population dynamics without computing the value function by
solving linear programming problems associated with the distributions of the
players still in the game and their stopping times/controls. We show the
convergence of the algorithm using the topology of convergence in measure in
the space of subprobability measures, which is needed to deal with the lack of
continuity of the flows of measures. Numerical examples are provided to
illustrate the convergence of the algorithm.
