Geometry of information structures, strategic measures and associated stochastic control topologies

Abstract

In many areas of applied mathematics, decentralization of information is a ubiquitous attribute affecting how to approach a stochastic optimization, decision and estimation, or control problem. In this review article, we present a general formulation of information structures under a probability theoretic and geometric formulation. 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 degrees 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 theoretic and non-signaling) relaxations. For each of these, we establish closedness and convexity properties and also a hierarchy of correlation structures. As a further 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 decision, estimation, and control problems.