Nonuniqueness versus Uniqueness of Optimal Policies in Convex Discounted Markov Decision Processes

Abstract

From the classical point of view, it is important to determine if in a Markov decision process (MDP), besides their existence, the uniqueness of the optimal policies is guaranteed. It is well known that uniqueness does not always hold in optimization problems (for instance, in linear programming). On the other hand, in such problems it is possible for a slight perturbation of the functional cost to restore the uniqueness. In this paper, it is proved that the value functions of an MDP and its cost perturbed version stay close, under adequate conditions, which in some sense is a priority. We are interested in the stability of Markov decision processes with respect to the perturbations of the cost-as-you-go function.