State Sufficiency and Exact Temporal Composition in Multiperiod Attribution

Executive overview

Attribution systems routinely compress a closed period into a small summary, its portfolio and benchmark returns and linked effects, and discard the raw path. That is only safe if the discarded information could never change a later calculation.

This paper proves that temporal associativity is exactly a sufficiency condition on that stored state, then classifies five widely used methods. Some are safe with a fixed state, some need a small repair, and Aumann–Shapley requires state that grows with the horizon.

Findings, in order

1. Associativity is sufficiency. Leading-block associativity is equivalent to future sufficiency of the standard stored state.
2. Two strengths of safety. Arbitrary associativity needs a stronger two-sided condition, which matters for interior restatements as well as appends.
3. Five methods classified. Cariño and both Frongello directions are safe; Modified Frongello separates the two notions with an exact rational witness; Menchero and Aumann–Shapley fail ordinary sufficiency.
4. Some failures are cheap to repair. Menchero and Modified Frongello admit small exact states that do not grow with the horizon.
5. Aumann–Shapley needs growing state. An exact polynomial state of dimension KT + 1 composes recursively and is generically minimal, so no fixed-size exact state exists.

Novelty