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We develop an exhaustive study of Markov decision process (MDP) under meanfield interaction both on states and actions in the presence of common noise,and when optimization is performed over open-loop controls on infinite horizon.Such model, called CMKV-MDP for conditional McKean-Vlasov MDP, arises and isobtained here rigorously with a rate of convergence as the asymptotic problemof N-cooperative agents controlled by a social planner/influencer that observesthe environment noises but not necessarily the individual states of the agents.We highlight the crucial role of relaxed controls and randomization hypothesisfor this class of models with respect to classical MDP theory. We prove thecorrespondence between CMKV-MDP and a general lifted MDP on the space ofprobability measures, and establish the dynamic programming Bellman fixed pointequation satisfied by the value function, as well as the existence of-optimalrandomized feedback controls. The arguments of proof involve an originalmeasurable optimal coupling for the Wasserstein distance. This provides aprocedure for learning strategies in a large population of interactingcollaborative agents. MSC Classification: 90C40, 49L20.

Mean-field Markov decision processes with common noise and open-loop controls