  The goal of the paper is to develop the theory of finite state mean field
games with major and minor players when the state space of the game is finite.
We introduce the finite player games and derive a mean field game formulation
in the limit when the number of minor players tends to infinity. In this limit,
we prove that the value functions of the optimization problems are viscosity
solutions of PIDEs of the HJB type, and we construct the best responses for
both types of players. From there, we prove existence of Nash equilibria under
reasonable assumptions. Finally we prove that a form of propagation of chaos
holds in the present context and use this result to prove existence of
approximate Nash equilibria for the finite player games from the solutions of
the mean field games. this vindicate our formulation of the mean field game
problem.
