Proportionality Degree of Multiwinner Rules

TL;DR

This paper evaluates how well various approval-based multiwinner election rules produce committees that proportionally represent voters, using quantitative measures to compare their proportionality guarantees and tradeoffs with efficiency.

Contribution

It introduces a quantitative framework to measure and compare the proportionality of different multiwinner election rules, revealing a hierarchy of their guarantees.

Findings

PAV has stronger proportionality guarantees than its sequential version.

Sequential PAV outperforms Phragmén's Sequential Rule in proportionality.

Tradeoff analysis shows a balance between proportionality and utilitarian efficiency.

Abstract

We study multiwinner elections with approval-based preferences. An instance of a multiwinner election consists of a set of alternatives, a population of voters---each voter approves a subset of alternatives, and the desired committee size $k$; the goal is to select a committee (a~subset) of $k$ alternatives according to the preferences of the voters. We investigate a number of election rules and ask whether the committees that they return represent the voters proportionally. In contrast to the classic literature, we employ quantitative techniques that allow to measure the extent to which the considered rules are proportional. This allows us to arrange the rules in a clear hierarchy. For example, we find that Proportional Approval Voting (PAV) has better proportionality guarantees than its sequential counterpart, and that Phragm\'{e}n's Sequential Rule is worse than Sequential PAV. Yet, the loss of proportionality for the two sequential rules is moderate and in some contexts can be outweighed by their other appealing properties. Finally, we measure the tradeoff between proportionality and utilitarian efficiency for a broad subclass of committee election rules.