Thresholds quantifying proportionality criteria for election methods

TL;DR

This paper introduces thresholds for various election methods based on different proportionality criteria, reformulating known criteria and analyzing multiple methods including classical and lesser-known ones.

Contribution

It provides a unified framework for thresholds related to proportionality criteria across diverse election methods, including reformulations and detailed analyses.

Findings

Thresholds calculated for many election methods

Reformulation of DPC, PSC, JR, PJR, EJR criteria

Analysis of classical and lesser-known methods

Abstract

We define several different thresholds for election methods by considering different scenarios, corresponding to different proportionality criteria that have been proposed by various authors. In particular, we reformulate the criteria known as DPC, PSC, JR, PJR, EJR in our setting. We consider multi-winner election methods of different types, using ballots with unordered lists of candidates or ordered lists, and for comparison also methods using only party lists. The thresholds are calculated for many different election methods. The considered methods include classical ones such as BV, SNTV and STV (with some results going back to e.g. Droop and Dodgson in the 19th century); we also study in detail several perhaps lesser known methods by Phragm\'en and Thiele. There are also many cases left as open problems.