How does the value function of a Markov decision depend on the transition probabilities?

The present work deals with the comparison of (discrete time) Markov decision processes (MDPs), which differ only in their transition probabilities. We show that the optimal value function of an MDP is monotone with respect to appropriately defined stochastic order relations. We also find conditions for continuity with respect to suitable probability metrics. The results are applied to some well-known examples, including inventory control and optimal stopping.