  We present a novel framework for mean field games with finite state space and
common noise, where the common noise is given through shocks that occur at
random times. We first analyze the game for up to $n$ shocks, in which case we
are able to characterize mean field equilibria through a system of
parameterized and coupled forward-backward equations. We establish existence
and uniqueness of solutions to this system for small time horizons. In
addition, we show that mean field equilibria for the $n$-shock setting
constitute approximate equilibria for the corresponding mean field game with
infinitely many common shocks. Our results are illustrated in a corruption
detection model with random audits.
