Pref.Tools: ABC Voting

Compute Proportional Approval Voting (PAV) online

Proportional Approval Voting (PAV) is a multi-winner voting method based on approval votes that allows to elect a committee that proportionally represents the voters' interests.

Portrait of Thorvald N. ThieleThis voting method was first proposed in 1895 by Danish mathematician Thorvald N. Thiele. It was developed to address the needs of Swedish elections, which sought a more representative and proportional process for selecting parliament members, and to decide on party lists.

Proportional Approval Voting is a voting method for multi-winner elections, where the aim is to elect a committee that represents the voters. It is tailored to ensure a proportional outcome, where every voter has approximately equal influence on the outcome.

There are several variants of Proportional Approval Voting. The main variant is based on maximizing a particular objective function. Because this is a difficult optimization problem, computers are needed to perform the computations. There are also variants of Proportional Approval Voting that are based on simpler rules, most importantly Sequential Proportional Approval Voting (seq-PAV) which elects committee members one at a time. This sequential variant has an easy interpretation: It first elects the candidate with the most approvals. Then it reduces the weight of all the voters who approved this candidate to 1/2. Then it selects the next candidate, who has the most (weighted) approvals. This process is repeated until the committee is full. After a voter has a second approved candidate elected, their weight is reduced to 1/3, then to 1/4, and so on. There is also a "reverse sequential" variant which starts with every candidate elected and then repeatedly kicks out the weakest candidate.

All these variants work with approval ballots, also called unordered ballots, where voters can vote for an arbitrary number of candidates, without ordering them in order of preferences. However, there are also variants of PAV that use ordered ballots, though we do not discuss them on this website.

Screenshot of Proportional Approval Voting online calculator
Click to go to the online ABC voting rule calculator with support for PAV and its variants.

This website features a free and easy-to-use online calculator that that allows users to compute the results of PAV and its variants. By inputting election data, users can observe the outcomes generated by each rule and compare them. The computations are performed on the user's device.

Besides Proportional Approval Voting, the calculator supports many other approval-based committee voting rules, including standard approval voting, Phragmen's voting rules, and the Method of Equal Shares. A total of 27 voting rules are available for immediate computation. The calculator is based on the popular open-source abcvoting python package.

Book cover of 'Multi-Winner Voting with Approval Preferences' For more information about Proportional Approval Voting, you can refer to the 2023 book Multi-Winner Voting with Approval Preferences by Martin Lackner and Piotr Skowron, which is published by Springer and which is available for free for PDF download from the publisher. You can also refer to the 2016 survey paper "Phragmén's and Thiele's election methods" by Svante Janson, which discusses historical background, all the various variants of Thiele's rules, and their properties.