Conservation Is Not Allocation: Valuing Clients and Edges in Hierarchical Federated Learning

Hierarchical federated learning routes client updates through edge servers, so contribution accounting must value both clients and edges. A common shortcut sums client values within each edge. For exact Owen values under a fixed partition this sum equals the edge's Shapley value in the quotient game, yet practical methods estimate client values from sampled permutations. We show that the summed ledger is then a Monte Carlo estimator of the quotient game whose efficiency residual is identically zero, so the total is always conserved while individual edges can be misallocated without any visible signal. We propose Federated Composable Contribution Evaluation (FedCCE), which gives each resolution its own estimator. Clients upload per-class feature summaries whose counts, means, and variances compose exactly under union, so the utility of any coalition of clients or edges follows from cached summaries without retraining. FedCCE samples client Owen values under a fixed budget and solves the much smaller edge quotient game exactly, and we prove that the extra utility calls this requires depend on the number of edges but not on the number of clients. Across four image benchmarks, summing budgeted client values identifies the most valuable edge in only 46% of 232 runs with 50 or 100 clients even at the largest budget, whereas FedCCE identifies it in every run and needs roughly 700$\times$ fewer utility evaluations than computing both ledgers exactly at 100 clients.