Semi-infinite weighted Markov decision processes with perturbation

Abstract

In this paper, Weighted reward Perturbed Markov Decision Processes with finite state and countable action spaces (semi-infinite WMDP for short) are considered. The ”weighted reward” refers to appropriately normalized convex combination of the discounted and the long-run average reward criteria. This criterion allows the controller to trade-off short-term rewards versus long-run rewards. In every application where both the discounted and the long-run average criteria have been proposed in the past, there is clearly a rationale for considering the weighted criterion. Of course, as with all Markov decision models, the standard weighted criterion model assumes that all the transition probabilities are known precisely. Since, in most applications this would not be the case, we consider the perturbed version of the weighted reward model. In the case of perturbations, we prove that for many models a nearly optimal strategy can be found in the class of relatively “simple ultimately deterministic” strategies. These are strategies which behave just like deterministic stationary strategies, after a certain point of time.