Phragm\'en Rules for Degressive and Regressive Proportionality

TL;DR

This paper introduces a new family of approval-based voting rules that interpolate between degressive and regressive proportionality, analyzing their properties in Euclidean issue spaces.

Contribution

It generalizes Phragmén's Sequential Rule to encompass both degressive and regressive proportionality principles in approval voting.

Findings

The new rules effectively balance minority and majority preferences.

Comparison shows the rules' behavior varies smoothly between degressive and regressive extremes.

Analysis in Euclidean space provides insights into their practical applicability.

Abstract

We study two concepts of proportionality in the model of approval-based committee elections. In degressive proportionality small minorities of voters are favored in comparison with the standard linear proportionality. Regressive proportionality, on the other hand, requires that larger subdivisions of voters are privileged. We introduce a new family of rules that broadly generalize Phragm\'en's Sequential Rule spanning the spectrum between degressive and regressive proportionality. We analyze and compare the two principles of proportionality assuming the voters and the candidates can be represented as points in an Euclidean issue space.