Information Structures, Causality, and Nonsequential Stochastic Control I: Design-Independent Properties

Abstract

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 practice, many distributed systems (those subject to deadlock, for instance) are not sequential in this sense. This paper explores the relationship between deadlock freeness, a less restrictive notion of causality, and the properties of a potentially nonsequential generic stochastic control problem formulated within the framework of Witsenhausen’s intrinsic model. A property of the problem’s information structure that is necessary and sufficient to ensure deadlock-freeness is identified and shown to be sufficient to ensure that all of the problem’s control policies possess expected rewards. It is also shown, by example, that there exist stochastic control problems for which all sequential policies are suboptimal. These results subsume Witsenhausen’s “causality” condition (property C), suggest a framework for the optimization of unconstrained nonsequential stochastic control problems, and provide an intuitive design-independent characterization of the cause/effect notion of causality. The results also have game theoretic implications—they suggest, for instance, necessary and sufficient conditions for a finite game to possess an extensive form.

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