  Mean field games allow to describe tractable models of dynamic games with a
continuum of players, explicit interaction and heterogeneous states. Thus,
these models are of great interest for socio-economic applications. A
particular class of these models are games with finite state and action space,
for which recently in Neumann (2020a) a semi-explicit representation of all
stationary equilibria has been obtained. In this paper we investigate whether
these stationary equilibria are stable against model perturbations. We prove
that the set of all games with only essential equilibria is residual and obtain
two characterization results for essential stationary equilibria.
