Coding strategies for a class of decentralized control problems with limited communication

We propose a large class of decentralized control problems with non-classical information structure for which a coding strategy is optimal. This class is a generalized version of the hats problem with statistically dependent hat colors where implicit communication via action is allowed. We propose a sufficient condition on the joint distribution of the hat colors which guarantees the optimality of a binary sum coding strategy. We explore the connection between this version of the hats problem and hypercube graph theory, and use that to show that verifying our proposed sufficient condition is computationally tractable.