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The goal of the paper is to develop the theory of finite state mean fieldgames 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 formulationin 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 viscositysolutions of PIDEs of the HJB type, and we construct the best responses forboth types of players. From there, we prove existence of Nash equilibria underreasonable assumptions. Finally we prove that a form of propagation of chaosholds in the present context and use this result to prove existence ofapproximate Nash equilibria for the finite player games from the solutions ofthe mean field games. this vindicate our formulation of the mean field gameproblem.

Finite State Mean Field Games with Major and Minor Players