Markov decision processes with restricted observations: Finite horizon case

In this article we consider a Markov decision process subject to the constraints that result from some observability restrictions. We assume that the state of the Markov process under consideration is unobservable. The states are grouped so that the group that a state belongs to is observable. So, we want to find an optimal decision rule depending on the observable groups instead of the states. This means that the same decision applies to all the states in the same group. We prove that a deterministic optimal policy exists for the finite horizon. An algorithm is developed to compute policies minimizing the total expected discounted cost over a finite horizon.