The Exact Boundary of Linear Optimality in Witsenhausen's Counterexample

Witsenhausen’s counterexample has resisted a closed form solution for decades. However, it has remained unknown whether a mathematically tractable representation exists or not. In this paper, we solve this open problem, showing that the optimal policies are either linear or not tame and characterizing the exact regions where the two properties hold. Moreover, in the linear optimal region, we give an essentially unique closed form solution for the optimal policy, and show that no optimal first-stage policy is definable in any $o$-minimal expansion of the ordered real field in the nonlinear optimal region.