Mathematical aspects of degressive proportionality

TL;DR

This paper examines the mathematical properties of apportionment functions used for seat allocation in the European Parliament, identifying conditions that lead to similar solutions despite theoretical flexibility.

Contribution

It provides necessary and sufficient conditions for apportionment functions and analyzes how constraints influence the similarity of solutions.

Findings

Identified conditions for apportionment functions

Demonstrated constraints lead to similar seat allocations

Presented specific families of apportionment functions

Abstract

We analyze properties of apportionment functions in context of the problem of allocating seats in the European Parliament. Necessary and sufficient conditions for apportionment functions are investigated. Some exemplary families of apportionment functions are specified and the corresponding partitions of the seats in the European Parliament among the Member States of the European Union are presented. Although the choice of the allocation functions is theoretically unlimited, we show that the constraints are so strong that the acceptable functions lead to rather similar solutions.