  Optimal design of distributed decision policies can be a difficult task,
illustrated by the famous Witsenhausen counterexample. In this paper we
characterize the optimal control designs for the vector-valued setting assuming
that it results in an internal state that can be described by a continuous
random variable which has a probability density function. More specifically, we
provide a genie-aided outer bound that relies on our previous results for
empirical coordination problems. This solution turns out to be not optimal in
general, since it consists of a time-sharing strategy between two linear
schemes of specific power. It follows that the optimal decision strategy for
the original scalar Witsenhausen problem must lead to an internal state that
cannot be described by a continuous random variable which has a probability
density function.
