  In decentralized stochastic control (or stochastic team theory) and game
theory, if there is a pre-defined order in a system in which the 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 and thus has limited utility. We will also present a further
reduction method for partially nested causal non-sequential systems.
