Witsenhausen's counterexample: A view from optimal transport theory

We formulate Witsenhausen's decentralized stochastic control setup as an optimization problem involving the quadratic Wasserstein distance and the minimum mean-square error. Classical results are recovered as immediate consequences of transport-theoretic properties. New results and bounds on the optimal cost are also obtained. In particular, we show that the optimal controller is a strictly increasing function with a real analytic left inverse.