  Static reduction of information structures (ISs) is a method that is commonly
adopted in stochastic control, team theory, and game theory. One approach
entails change of measure arguments, which has been crucial for stochastic
analysis and has been an effective method for establishing existence and
approximation results for optimal policies. Another approach entails
utilization of invertibility properties of measurements, with further
generalizations of equivalent IS reductions being possible. In this paper, we
demonstrate the limitations of such approaches for a wide class of stochastic
dynamic games and teams, and present a systematic classification of static
reductions for which both positive and negative results on equivalence
properties of equilibrium solutions can be obtained: (i) those that are
policy-independent, (ii) those that are policy-dependent, and (iii) a third
type that we will refer to as static measurements with control-sharing
reduction (where the measurements are static although control actions are
shared according to the partially nested IS). For the first type, we show that
there is a bijection between Nash equilibrium (NE) policies under the original
IS and their policy-independent static reductions, and establish sufficient
conditions under which stationary solutions are also isomorphic between these
ISs. For the second type, however, we show that there is generally no
isomorphism between NE (or stationary) solutions under the original IS and
their policy-dependent static reductions. Sufficient conditions (on the cost
functions and policies) are obtained to establish such an isomorphism
relationship between Nash equilibria of dynamic non-zero-sum games and their
policy-dependent static reductions. For zero-sum games and teams, these
sufficient conditions can be further relaxed.
