  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, 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.
