Phragm\'{e}n's Voting Methods and Justified Representation

TL;DR

This paper analyzes Phragmén's load-balancing voting methods, focusing on their properties related to proportional representation, computational complexity, and introduces algorithms for their computation.

Contribution

It provides an axiomatic analysis of Phragmén's methods, demonstrating their proportional and perfect representation properties, and develops algorithms for their practical computation.

Findings

Sequential variant satisfies proportional justified representation.

Optimization variants satisfy perfect representation.

Algorithms for computing Phragmén's methods are developed.

Abstract

In the late 19th century, Swedish mathematician Lars Edvard Phragm\'{e}n proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants - one minimizing the maximal load and one minimizing the variance of loads - and a sequential variant. We study Phragm\'{e}n's methods from an axiomatic point of view, focusing on properties capturing proportional representation. We show that the sequential variant satisfies proportional justified representation, which is a rare property for committee monotonic methods. Moreover, we show that the optimization variants satisfy perfect representation. We also analyze the computational complexity of Phragm\'{e}n's methods and provide mixed-integer programming based algorithms for computing them.