Quantitative propagation of chaos for mean field Markov decision process with common noise

We investigate propagation of chaos for mean field Markov Decision Process with common noise (CMKV-MDP), and when the optimization is performed over randomized open-loop controls on infinite horizon. We first state a rate of convergence of order MNγ, where MN is the mean rate of convergence in Wasserstein distance of the empirical measure, and γ ∈ (0,1] is an explicit constant, in the limit of the value functions of N-agent control problem with asymmetric open-loop controls, towards the value function of CMKV-MDP. Furthermore, we show how to explicitly construct (ε+O(MNγ))-optimal policies for the N-agent model from ε-optimal policies for the CMKV-MDP. Our approach relies on sharp comparison between the Bellman operators in the N-agent problem and the CMKV-MDP, and fine coupling of empirical measures.