House-monotone multi-level apportionment has logarithmic quota discrepancy

2026-08-03

AI Summary

This study addresses the compatibility between quota constraints and seat monotonicity in multi-level integer apportionment: specifically, whether an apportionment rule can simultaneously satisfy quota compliance and seat monotonicity even when quota bounds are imposed only at the root node. Focusing on complete binary comb-like hierarchies, the work establishes for the first time the nonexistence of such rules, revealing a fundamental conflict between static feasibility and dynamically monotone allocation paths. By employing tools including midpoint embeddings, prefix deviation analysis, and the van der Corput sequence, the authors derive a worst-case lower bound of Θ(log D) on quota deviation and provide an explicit upper bound of log D / (3 log 2) + 1, thereby refuting a long-standing conjecture regarding the existence of quota- and monotonicity-compliant apportionment rules in this setting.

Abstract

Multi-level apportionment allocates integer seats through a hierarchy of groups. Schmidt-Kraepelin, Suksompong, and Wijaya proved that, at every fixed house size, lower and upper quota can be satisfied simultaneously; they also constructed house-monotone rules satisfying either quota separately. They left open whether one rule can satisfy lower quota, upper quota, and house monotonicity together, even when quota is required only relative to the root. We give a negative answer. For a full binary comb with $D$ equally entitled leaves, every house-monotone allocation sequence induces a sequence of seat recipients. Quota for the nested comb groups would force every grid-aligned prefix discrepancy to be below one. A midpoint embedding then bounds the full interval discrepancy by this quantity plus $1/2$, contradicting Schmidt's logarithmic lower bound. Conversely, a binary van der Corput seat schedule has comb-prefix error at most $\log D/(3\log 2)+1$. Thus the optimal worst-case error on the comb is $Θ(\log D)$, and for sufficiently large finite $D$ no house-monotone quota rule exists. The proof isolates a static--dynamic gap: each house size admits a quota-feasible allocation, but the feasible allocations cannot be embedded into one monotone path. In quantization language, the result characterizes the order of the embedded-quantization penalty for progressive one-hot rounding on the comb.

Problem

• multi-level apportionment
• house monotonicity
• quota
• discrepancy
• allocation

Innovation

• house-monotone apportionment
• quota discrepancy
• multi-level allocation
• van der Corput sequence
• static-dynamic gap