Isomorphism Properties of Optimality and Equilibrium Solutions under Equivalent Information Structure Transformations I: Stochastic Dynamic Teams

Abstract

Static reduction of dynamic stochastic team (or decentralized stochastic control) problems has been an effective method for establishing existence and approximation results for optimal policies. In this Part I of a two-part paper, we address stochastic dynamic teams. Part II addresses stochastic dynamic games. We consider two distinct types of static reductions: (i) those that are policy-independent (as those introduced by Witsenhausen in [35]), and (ii) those that are policy-dependent (as those introduced by Ho and Chu [15], [16] for partially nested dynamic teams). For the first type, we show that there is a bijection between stationary (person-by-person optimal, globally optimal) policies and their policyindependent static reductions, and then we present a variational analysis for convex dynamic teams under policy-independent static reductions. For the second type, although there is a bijection between globally optimal policies of dynamic teams with partially nested information structures and their static reductions, in general there is no bijection between stationary (person-by-person optimal) policies of dynamic teams and their policy-dependent static reductions. We present, however, sufficient conditions under which such a relationship holds and a functional form of the bijection can be constructed. Furthermore, we study the connections between optimality and stationarity concepts when a control-sharing information structure is considered (via an expansion of the information structures) under policy-dependent static reductions. An implication is a convexity characterization of dynamic team problems under policy-dependent static reduction with a control-sharing information structure. Finally, we study multi-stage team problems, where we discuss the connections between optimality concepts through times and players under the introduced static reductions.

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

“…On the other hand, team theory and zero-sum games have many crucial properties in common: They both admit well-defined values (though with possibly non-unique equilibrium policies) and they both possess strong continuity properties in information, as studied recently in [18] (see also [31]). Some general properties of information structures in game theory are studied in detail in [21,2,28,26]. Accordingly, for zerosum games, several positive attributes regarding informational structure-dependent properties of equilibrium solutions arise:…”