Decision Hierarchy Invariance in Investment Performance Attribution

Executive overview

A report may reconcile at every level while changing the assignment of responsibility when elementary decisions are regrouped and their allocations are recomputed. This paper separates adding an existing allocation ledger from recalculating a Shapley allocation after groups become the players.

An exact Harsanyi-dividend certificate measures the difference. For a fixed finite game, invariance across all partitions is equivalent to invariance across all bipartitions and to a unique decomposition into an additive game and a normalized complement-symmetric game. Signed higher-order interactions can cancel, so stable allocations do not establish that nonlinear interactions are absent.

Findings, in order

1. Two meanings of aggregation. Adding fixed atomic allocations differs from recomputing allocations in a quotient game.
2. An exact grouping certificate. A rational linear operator on Harsanyi dividends identifies the allocation difference and gives a sharp coordinatewise bound.
3. Bipartitions suffice. For the fully specified fixed game, testing all complementary two-block partitions certifies every partition.
4. Signed interactions can cancel. The invariant subspace has dimension 2ⁿ⁻¹ + n − 1 for n ≥ 1. For four players, each triple dividend equals minus one half of the four-way dividend, with singleton and pair dividends unrestricted.
5. A declared hierarchy is a model choice. Owen and multilevel values respect specified coalition structures; they do not make every regrouping equivalent to the original atomic allocation.
6. Economic interpretation remains conditional. Synthetic bond repricing and multiasset controls test a stipulated model, not observed causal ownership, investment skill or empirical superiority.

Novelty

Related research

• 02Hierarchical Reconciliation with Narrative Preservation

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• 03Aggregation Consistency in Multiperiod Arithmetic Attribution

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• 09What Return Attribution Cannot Tell Us About Decision Risk

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