  We establish existence of Markov chains of mean-field type with unbounded
jump intensities by means of a fixed point argument using the Total Variation
distance. We further show existence of nearly-optimal controls and, using a
Markov chain backward SDE approach, we suggest conditions for existence of an
optimal control and a saddle-point for respectively a control problem and a
zero-sum differential game associated with payoff functionals of mean-field
type, under dynamics driven by such Markov chains of mean-field type.
