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In many areas of applied mathematics, engineering, and social and naturalsciences, decentralization of information is a key aspect determining how toapproach a problem. In this review article, we study information structures ina probability theoretic and geometric context. We define informationstructures, place various topologies on them, and study closedness, compactnessand convexity properties on the strategic measures induced by informationstructures and decentralized control/decision policies under varying degree ofrelaxations with regard to access to private or common randomness. Ultimately,we present existence and tight approximation results for optimaldecision/control policies. We discuss various lower bounding techniques,through relaxations and convex programs ranging from classically realizable andclassically non-realizable (such as quantum and non-signaling) relaxations. Foreach of these, we establish closedness and convexity properties and also ahierarchy of correlation structures. As a second main theme, we review andintroduce various topologies on decision/control strategies defined independentof information structures, but for which information structures determinewhether the topologies entail utility in arriving at existence, compactness,convexification or approximation results. These approaches, which we term asthe strategic measures approach and the control topology approach, lead tocomplementary results on existence, approximations and upper and lower boundsin optimal decentralized decision and control.

Geometry of Information Structures, Strategic Measures and associated Control Topologies