Abstract

In many areas of engineering and applied mathematics (including stochastic control, networked control, information theory, game theory) decentralization of information is a key aspect determining how to approach a problem. We define information structures, place various topologies on them, and study closedness, compactness and convexity properties on the strategic measures induced by information structures and decentralized control/decision policies under varying degree of relaxations with regard to access to private or common randomness. Ultimately, we present existence and tight approximation results for optimal decision/control policies. We discuss various lower bounding techniques, through relaxations and convex programs ranging from classically realizable and classically non-realizable (such as quantum and non-signaling) relaxations. For each of these, we establish closedness and convexity properties and also a hierarchy of correlation structures.

As a second main theme, we review and introduce various topologies on decision/control strategies defined independent of information structures, but for which information structures determine whether the topologies entail utility in arriving at existence, compactness, convexification or approximation results.

These approaches, which we term as the strategic measures approach and the control topology approach, lead to complementary results on existence, approximations and upper and lower bounds in optimal decentralized stochastic control. Connections with some classical results on control theory will be revisited and examples will be discussed.