A note on the structure of value spaces in vector-valued Markov decision processes

For a vector-valued Markov decision process with discounted reward criterion, we study the structure of its value spaces defined for all initial states. At first we discuss the relationship between the value spaces, i.e. we verify a linking property for optimality. We next show that if the values of deterministic stationary policies generate a face of the value space, any point of that face can be represented as the value of a randomization of these policies. We also examine whether the value of a randomization of deterministic stationary policies lies on the face generated by the values of these policies.