Witsenhausen's Standard Form Revisited: A General Dynamic Program for Decentralized Stochastic Control and Existence Results

Abstract

For sequential stochastic control problems with standard Borel measurement and control action spaces, we introduce a general dynamic programming formulation, establish its well-posedness, and provide new existence results for optimal policies. Our dynamic program builds in part on Witsenhausen's standard form, but with a different formulation for the state, action, and transition dynamics. Using recent results on measurability properties of strategic measures in decentralized control, we obtain a controlled Markov model with standard Borel state and state dependent action sets. This allows for a well-posed formulation for the controlled Markov model for a general class of sequential decentralized stochastic control in that it leads to well-defined dynamic programming recursions through universal measurability properties of the value functions for each time stage. In addition, new existence results are obtained for optimal team policies in decentralized stochastic control. These state that for a static team with independent measurements, it suffices for the cost function to be continuous (only) in the actions for the existence of an optimal policy under mild compactness conditions. These also apply to dynamic teams which admit static reductions with independent measurements.