Aggregation Consistency in Multiperiod Arithmetic Attribution

Executive overview

Should linking before or after combining attribution buckets give the same answer? This paper studies that requirement, aggregation consistency, axiomatically and shows that it is far more restrictive than it looks.

Together with bucket anonymity and continuity, it forces every linker into a common-period-coefficient form. Methods that keep only aggregate effects collapse to signed pro-rata and cannot be defined everywhere. Keeping the return path removes the obstruction, and the Aumann–Shapley value of the compounding function is the principled resolution.

Findings, in order

1. Characterization. Aggregation consistency, anonymity and continuity are equivalent to F_k = c(A,R)ᵀx_k, with one coefficient vector shared by all buckets.
2. Reconciliation becomes one condition. Exact reconciliation reduces to the single scalar condition cᵀA = D.
3. Reduced-form uniqueness. If a method may use only aggregate effects G and total D, the only exactly reconciling choice is signed pro-rata, F_k = (D/S)G_k.
4. Totality obstruction. Admissible paths with S = 0 and D ≠ 0 exist, so no everywhere-defined regular reduced-form method can meet the same requirements.
5. Aumann–Shapley resolves it. The cost-sharing value of the compounding function is total, continuous, exactly reconciling, aggregation-consistent and anonymous.
6. Methods classified. A deterministic battery classifies established and experimental linkers and preserves the counterexamples.

Novelty