Information structures, causality, and nonsequential stochastic control I: design-independent properties

In control theory, the usual notion of causality---that, at all times, a system's output (action) only depends on its past and present inputs (observations)---presupposes that all inputs and outputs can be ordered, a priori, in time. In reality, many distributed systems (those subject to deadlock, for instance), are not sequential in this sense. In a previous paper (part I) [SIAM J. Control Optim., 30 (1992), pp. 1447--1475], the relationship between a less restrictive notion of causality, deadlock-freeness, and the design-independent properties of a potentially nonsequential generic stochastic control problem formulated within the framework of Witsenhausen's intrinsic model was explored. In the present paper (part II) the properties of individual designs are examined. In particular, a property of a design's information partition that is necessary and sufficient to ensure its deadlock-freeness is identified and shown to be sufficient to ensure its possession of an expected reward. It is also shown, by example, that there exist nontrivial deadlock-free designs that cannot be associated with any deadlock-free information structure. The first result provides an intuitive design-dependent characterization of the cause/effect notion of causality and suggests a framework for the optimization of constrained nonsequential stochastic control problems. The second implies that this characterization is finer than existing design-independent characterizations, including properties C (Witsenhausen) and CI (part I).