Local Optimality of Almost Piecewise-Linear Quantizers for Witsenhausen's Problem

We pose Witsenhausen's problem as a leader-follower game of incomplete information. The follower makes a noisy observation of the leader's action (who moves first) and chooses an action minimizing her expected deviation from the leader's action. Knowing this, the leader who observes the realization of the state chooses an action that minimizes her distance to the state of the world and the ex-ante expected deviation from the follower's action. We study the perfect Bayesian equilibria of the game and identify a class of "near-piecewise-linear equilibria" when leader cares much more about being close to the follower than the state, and the state is highly volatile. As a major consequence of this result, we prove the existence of a set of local minima for Witsenhausen's problem in form of slopey quantizers, which are at most a constant factor away from the optimal cost.