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.

Another Look at Partially Observed Optimal Stochastic Control: Existence, Ergodicity, and Approximations without Belief-Reduction

“…to facilitate the infinite horizon analysis. If we have independent static reduction, with measurements Ỹn , by considering the method for static measurements with expanding information structure (this is the information structure that one would obtain for a POMDP under the absolute continuity conditions) in [68,Theorem 5.6] or [54,Theorem 4.7] by expressing all the cost-relevant uncertainty in terms of Ỹn , we have the following version of (2.4)…”

Zero-Sum Games involving Teams against Teams: Existence of Equilibria, and Comparison and Regularity in Information

“…The study of information structures for multi-player setups inevitably necessitates a tedious geometric and topological analysis. Indeed, in the context of team and game problems, non-convexity properties on correlation structures (or strategic measures) and associated non-existence of equilibrium results, (see in particular [1] for nonexistence of equilibria and also [11,25] for convexity properties) and compactness/noncompactness properties [25] (see also [16,32]), require a careful stochastic analysis with many counterexamples and sufficient conditions presented.…”

“…Note that L A (µ) is Borel measurable under the weak convergence topology [25], and we use L A (µ, γ) to denote the strategic measure induced by a particular team strategy γ. We define L CCR (µ) as:…”

“…Closedness and compactness of L C (µ) will be required for some of the results presented here. In general, L C (µ) is not closed under the weak convergence topology [25,Theorem 4.3]. However, we can relate sufficient conditions for the compactness of the space L R (µ) to L C (µ).…”

“…, eliminating the state space from the measure and simplifying notation for convenience. Since the measure on the exogenous variables is fixed by the information structure µ, S i will inherit convexity (as conditional independence will be preserved under convex combinations given that the opponent team policy is fixed) and weak compactness from L C (µ i ) (following from the analysis in the proof of [17, Theorem 3.2]); see [25,Theorem 5.2] which shows that conditional independence is preserved when only one player perturbs the strategic measure (unlike the case where multiple players make simultaneous perturbations in [25,Theorem 4.2]). This applies in this context by viewing each team as a large single-agent.…”