Necessity of remaining conditions in Lee–Bareinboim (2020, Thm. 2) for immateriality

Prove necessity of the remaining assumptions required by Theorem 2 of Lee and Bareinboim (2020)—in particular, the LB‑factorizability conditions beyond condition (I) and any further theorem‑specific constraints—for establishing immateriality in insoluble multi‑decision scoped graphs, by showing that materiality cannot be excluded whenever any of these conditions is violated.

Background

This work proves necessity of one key assumption (LB‑factorizability condition I) used by Lee and Bareinboim’s criterion, demonstrating that materiality can arise whenever that condition is violated. The authors highlight that for the other conditions involved in the criterion, necessity remains unsettled.

Resolving the necessity of these remaining conditions—e.g., LB‑factorizability conditions II–III and the additional assumptions in Theorem 2—would either complete the proof of the criterion’s completeness or show that it is incomplete, thereby clarifying the limits of current graphical methods for insoluble decision problems.