On Multi-Level Apportionment

TL;DR

This paper introduces a hierarchical extension of apportionment methods, demonstrating how different classical methods satisfy quota principles at multiple levels and can be combined to meet both upper and lower quota constraints.

Contribution

It generalizes apportionment to hierarchical group structures and proves that classical methods can be adapted to satisfy quota conditions at each level.

Findings

Adams' method level-by-level satisfies upper quota.

Jefferson's and quota methods satisfy lower quota.

Both quota notions can be simultaneously fulfilled.

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.