Non-Sequential Decentralized Stochastic Control and Static Reduction

In decentralized stochastic control (or stochastic team theory) and game theory, if there is a pre-defined order in a system in which agents act, the system is called sequential, otherwise it is non-sequential. Much of the literature on stochastic control theory, such as studies on the existence analysis, approximation methods, and on dynamic programming or other analytical or learning theoretic methods, have focused on sequential systems. The static reduction method for sequential control problems (via change of measures or other techniques), in particular, has been shown to be very effective in arriving at existence, structural, approximation and learning theoretic results. Many practical systems, however, are non-sequential where the order of agents acting is random, and dependent on the realization of solution paths and prior actions taken. The study of such systems is particularly challenging as tools applicable for sequential models are not directly applicable. In this paper, we will study static reducibility of non-sequential stochastic control systems, including by change of measure methods. We revisit the notion of Causality (a definition due to Witsenhausen and which has been refined by Andersland and Tekenetzis), and provide an alternative representation using imaginary agents. Via this representation, we show that Causality, under an absolute continuity condition, allows for an equivalent static model whose reduction is policy-independent. This facilitates much of the stochastic analysis available for sequential systems to also be applicable for non-sequential systems. We further show that under more relaxed conditions on the model, such as solvability, such a reduction, when possible at all, is policy-dependent or includes policies as parameters in the cost of the reduced model, and thus has limited utility. We will also present a further reduction method for partially nested causal non-sequential systems.