  We address a class of McKean-Vlasov (MKV) control problems with common noise,
called polynomial conditional MKV, and extending the known class of linear
quadratic stochastic MKV control problems. We show how this polynomial class
can be reduced by suitable Markov embedding to finite-dimensional stochastic
control problems, and provide a discussion and comparison of three
probabilistic numerical methods for solving the reduced control problem:
quantization, regression by control randomization, and regress later methods.
Our numerical results are illustrated on various examples from portfolio
selection and liquidation under drift uncertainty, and a model of interbank
systemic risk with partial observation.
