Precise geometric criterion for complementarity vs substitution at a belief

Establish a precise geometric criterion that determines, for any finite-action Bayesian decision problem and two information channels i and j at a belief b, whether the second-order interaction ΔVoI(j | i, b) is positive (complements) or negative (substitutes), expressed in terms of which decision boundaries—i.e., boundaries between the piecewise-linear decision regions induced by the value function V—are crossed by the posterior distributions generated by channels i and j.

Background

The paper proves an interior complementarity theorem ensuring ΔVoI(j | i, b) ≥ 0 when all posteriors from channel i remain within the current decision region, and a converse theorem showing that ΔVoI(j | i, b) < 0 requires channel i to generate posteriors that cross a decision boundary. However, boundary crossing is necessary but not sufficient for substitution, leaving a gap between these conditions.

A worked example suggests a boundary-based rule of thumb: channels that cross different decision boundaries tend to complement each other, whereas channels that cross the same boundary tend to substitute. Formalizing this intuition into a precise, general geometric criterion would close the gap and unify the necessary and sufficient conditions.