In this paper we study m-discount optimality (m ≥ −1) and Blackwell optimality for a general class of controlled (Markov) diffusion processes. To this end, a key step is to express the expected discounted reward function as a Laurent series, and then search certain control policies that lexicographically maximize the mth coefficient of this series for m = −1,0,1,…. This approach naturally leads to m-discount optimality and it gives Blackwell optimality in the limit as m → ∞.

We consider a variant of the house-selling problem in which the seller has n objects for sale and the buyers come sequentially one by one. The objective of this paper is to find the seller's optimal strategy for choosing his selling price so as to maximize the average reward per selling attempt until the nth house is sold.

This paper studies an admission control M/M/1 queueing system. It shows that the only gain (average) optimal stationary policies with gain and bias which satisfy the optimality equation are of control limit type, that there are at most two and, if there are two, they occur consecutively. Conditions are provided which ensure the existence of two gain optimal control limit policies and are illustrated with an example. The main result is that bias optimality distinguishes these two gain optimal policies and that the larger of the two control limits is the unique bias optimal stationary policy. Consequently it is also Blackwell optimal. This result is established by appealing to the third optimality equation of the Markov decision process and some observations concerning the structure of solutions of the second optimality equation.