On Information Structures, Feedback and Causality

A finite number of decisions, indexed by α ∈ A, are to be taken. Each decision amounts to selecting a point in a measurable space (U_α ,F_α ). Each decision is based on some information fed back from the system and characterized by a subfield J_α of the product space (∏ _α U_α ,∏ _α F_α ). The decision function for each α can be any function γ _α measurable from J_α to F_α. A property of the \ J_α \ α ∈ A is defined which assures that the setup has a causal interpretation. This property implies that for any combination of choices of the γ _α, the closed loop equations have a unique solution. The converse implication is false, when card $A > 2$.