Multi-objective Markov Decision Process: Set-valued Bellman's principle and fixed point perspective

We study infinite-horizon discounted Markov decision processes with multiple cost objectives and finite state and action spaces. We consider the set of value functions generated by randomized Markov policies, where each function describes the expected discounted costs from all initial states under a single policy. We establish a set-valued Bellman equation and characterize this set as the greatest fixed point of the associated Bellman operator. We also show that choosing an attainable cost vector separately for each initial state can produce a function that no single Markov policy realizes. Nevertheless, the attainable costs at individual states satisfy their own Bellman recursion. These results provide a dynamic programming and fixed point framework for multi-objective Markov decision processes.