  This paper answers a question raised by Doyle on the relevance of the
Witsenhausen counterexample as a toy decentralized control problem. The
question has two sides, the first of which focuses on the lack of an external
channel in the counterexample. Using existing results, we argue that the core
difficulty in the counterexample is retained even in the presence of such a
channel. The second side questions the LQG formulation of the counterexample.
We consider alternative formulations and show that the understanding developed
for the LQG case guides the investigation for these other cases as well.
Specifically, we consider 1) a variation on the original counterexample with
general, but bounded, noise distributions, and 2) an adversarial extension with
bounded disturbance and quadratic costs. For each of these formulations, we
show that quantization-based nonlinear strategies outperform linear strategies
by an arbitrarily large factor. Further, these nonlinear strategies also
perform within a constant factor of the optimal, uniformly over all possible
parameter choices (for fixed noise distributions in the Bayesian case).
  Fortuitously, the assumption of bounded noise results in a significant
simplification of proofs as compared to those for the LQG formulation.
Therefore, the results in this paper are also of pedagogical interest.
