  In this article we consider finite Mean Field Games (MFGs), i.e. with finite
time and finite states. We adopt the framework introduced in Gomes Mohr and
Souza in 2010, and study two seemly unexplored subjects. In the first one, we
analyze the convergence of the fictitious play learning procedure, inspired by
the results in continuous MFGs. In the second one, we consider the relation of
some finite MFGs and continuous first order MFGs. Namely, given a continuous
first order MFG problem and a sequence of refined space/time grids, we
construct a sequence finite MFGs whose solutions admit limits points and every
such limit point solves the continuous first order MFG problem.
