On Multi-Level Apportionment

TL;DR

The paper generalizes classical apportionment to multi-level hierarchies by modeling entitlements on a rooted tree and enforcing upper and lower quotas relative to ancestors. It proves that allocations exist satisfying both quotas, and shows that level-by-level implementations of Adams' (upper quota) and Jefferson's (lower quota) methods preserve their respective quota properties with house monotonicity. The quota method is extended to multi-level settings but cannot guarantee both quotas simultaneously in general, prompting the Upper-Compliant Quota variant and highlighting open questions about joint guarantees. An extensive empirical study on binary and sparse 4-ary trees illustrates performance trade-offs among methods and provides practical guidance, complemented by available code.

Abstract

Apportionment refers to the well-studied problem of allocating legislative seats among parties or groups with different entitlements. We present a multi-level generalization of apportionment where the groups form a hierarchical structure, which gives rise to stronger versions of the upper and lower quota notions. We show that running Adams' method level-by-level satisfies upper quota, while running Jefferson's method or the quota method level-by-level guarantees lower quota. Moreover, we prove that both quota notions can always be fulfilled simultaneously.

Figures (4)

• Figure 1: Example tree structures of single-level apportionment (left) and multi-level apportionment (right).
• Figure 2: Illustration of reducing an arbitrary tree to a full binary tree in the proof of \ref{['lemma:reduction']}. Shaded nodes are the introduced intermediate nodes.
• Figure 3: Example tree structure for which \ref{['algo:quota1']} fails upper quota.
• Figure 4: Example tree structure where \ref{['algo:quota2']} fails lower quota.

Theorems & Definitions (24)

• Definition 2.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• proof : Proof of \ref{['lemma:reduction']}
• proof : Proof of \ref{['lemma:both-quotas-full-binary']}
• Lemma 4.1
• proof
• Theorem 4.2
• proof