Theoretically, implicit communication has proven to be a hard nut to crack. From a control stand-point, implicit communication makes problems hard because the same actions that are traditionally used exclusively for control can now communicate as well. From a communications view, there is often another conceptual difficulty: since the source is not specified explicitly, the message can be altered by control actions!

Consequently, even the minimalist toy problem that distills these two difficulties -- the infamous Witsenhausen counterexample -- has remained unsolved for the past four decades. Worse, its discrete counterpart is known to be NP-complete, ruling out the possibility of an algorithmic solution. Since the problem is hard as well as minimalist, it is a bottleneck in understanding implicit communication in particular and decentralized control in general.

The main contribution of this dissertation is two-fold. First, using a sequence of three simplifications of the counterexample, we release this bottleneck by providing the first provably approximately-optimal solutions to the Witsenhausen counterexample. Second, we generalize this sequence of simplifications and propose them as a program for addressing more complicated problems of decentralized control. As an indication of the potential success of this program, we provide approximately-optimal solutions to various problems where implicit communication is possible. Using our refined understanding of implicit communication, we also identify a few practical situations where the phenomena may prove useful.