  We propose a distributionally robust return-risk model for Markov decision
processes (MDPs) under risk and reward ambiguity. The proposed model optimizes
the weighted average of mean and percentile performances, and it covers the
distributionally robust MDPs and the distributionally robust chance-constrained
MDPs (both under reward ambiguity) as special cases. By considering that the
unknown reward distribution lies in a Wasserstein ambiguity set, we derive the
tractable reformulation for our model. In particular, we show that that the
return-risk model can also account for risk from uncertain transition kernel
when one only seeks deterministic policies, and that a distributionally robust
MDP under the percentile criterion can be reformulated as its nominal
counterpart at an adjusted risk level. A scalable first-order algorithm is
designed to solve large-scale problems, and we demonstrate the advantages of
our proposed model and algorithm through numerical experiments.
