Policy-Dependent and Policy-Independent Static Reduction of Stochastic Dynamic Teams and Games and Fragility of Equivalence Properties

In stochastic control, information structure arguments have been crucial for stochastic analysis. Such an approach is often called static reduction in dynamic team theory (or decentralized stochastic control) and has been an effective method for establishing existence and approximation results for optimal policies. In this paper, we classify such static reductions into three categories: (i) those that are policy-independent (introduced by Witsenhausen in [17]), (ii) those that are policy-dependent (introduced by Ho and Chu [7], [8] for partially nested dynamic teams), and (iii) static measurement with control-sharing reduction (where the measurements become static although control actions are shared according to the partially nested information structure). For these reductions, while there exist bijection relationships between glob-ally optimal solutions of dynamic teams and their reductions, in general there is no bijection for person-by-person optimal policies. We also establish a similar result (but not identical) concerning stationary solutions. We present sufficient conditions under which bijection relationships hold. Under static measurement with control-sharing reduction, connections between optimality concepts can be established under relaxed conditions. An implication is a convexity characterization of dynamic teams under static measurement with control-sharing reduction. Some counterparts for stochastic games are also discussed.