  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), and (ii) those that
are policy-dependent (as those introduced by Ho and Chu 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
policy-independent 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.
