Another Set of Conditions for Strong n (n = −1, 0) Discount Optimality in Markov Decision Processes

Abstract

In this paper we study discrete-time Markov decision processes with average expected costs (AEC) and discount-sensitive criteria in Borel state and action spaces. The costs may have neither upper nor lower bounds. We propose another set of conditions on the system's primitive data, and under which we prove (1) AEC optimality and strong − 1-discount optimality are equivalent; (2) a condition equivalent to strong 0-discount optimal stationary policies; and (3) the existence of strong n (n = −1, 0)-discount optimal stationary policies. Our conditions are weaker than those in the previous literature. In particular, the “stochastic monotonicity condition” in this paper has been first used to study strong n (n = −1, 0)-discount optimality. Moreover, we provide a new approach to prove the existence of strong 0-discount optimal stationary policies. It should be noted that our way is slightly different from those in the previous literature. Finally, we apply our results to an inventory system and a controlled queueing system.