Witsenhausen's counterexample, shown in the figure below, is a deceptively simple toy problem in decentralized stochastic control. It was formulated by Hans Witsenhausen in 1968.[1] It is a counterexample to a natural conjecture that one can generalize a key result of centralized linear–quadratic–Gaussian control systems—that in a system with linear dynamics, Gaussian disturbance, and quadratic cost, affine (linear) control laws are optimal—to decentralized systems. Witsenhausen constructed a two-stage linear quadratic Gaussian system where two decisions are made by decision makers with decentralized information and showed that for this system, there exist nonlinear control laws that outperform all linear laws. The problem of finding the optimal control law remains unsolved.[2]
