Controlled Markov chains with non-exponential discounting and distribution-dependent costs

This paper deals with a controlled Markov chain in continuous time non-exponential discounting and distribution-dependent cost functional. A definition of closed-loop equilibrium is given its existence uniqueness are established. Due to the time-inconsistency brought by distribution dependence, it proved that locally optimal some appropriate sense. Moreover, shown our problem equivalent mean-field game for infinite-many symmetric players cost.