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We present a new notion of solution for mean field games master equations.This notion allows us to work with solutions which are merely continuous. Weprove first results of uniqueness and stability for such solutions. It turnsout that this notion is helpful to characterize the value function of meanfield games of optimal stopping or impulse control and this is the topic of thesecond half of this paper. The notion of solution we introduce is only usefulin the monotone case. We focus in this paper in the finite state space case.

Monotone solutions for mean field games master equations : finite state space and optimal stopping