# Proportional approval voting

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Proportional approval voting (PAV) is a voting system for multiple-winner elections, in which each voter can vote for as many or as few candidates as the voter chooses. It was developed by the Danish polymath Thorvald N. Thiele, and its sequential approximation was used in Swedish elections up until its replacement by the less computationally laborious party-list system.[1] It was later rediscovered by Forest Simmons in 2001.[2]

PAV works by looking at how "satisfied" each voter is with each potential result or outcome of the of the election. The satisfaction for individual voters of a potential result is calculated based on how many of the successful candidates they voted for. In this particular system, if an individual voted for n successful candidates (and an irrelevant number of unsuccessful ones) then their satisfaction is taken to be

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\ldots +{\frac {1}{n}}}$

Adding up the satisfaction of all the voters with the potential result gives the total satisfaction with that result. The potential result with the highest total satisfaction is chosen as the actual result.

If there was only one winner then proportional approval voting would become simple approval voting. Alternatively, if each voter only voted for all the candidates of a single party then the results would essentially be the same as the D'Hondt method of party-list proportional representation. The method can also be used with Sainte-Laguë divisors.

Without the weighting of satisfaction, i.e. if the numbers of votes for each candidate are simply added up and those with the highest numbers elected, equivalent to satisfaction being n, then this would amount to block approval voting which could have a similar chance of landslide results as block voting.

Proportional approval voting is a computationally complex method of vote counting. If there were c candidates and w winners, then there would be

${\displaystyle {\frac {c!}{w!(c-w)!}}}$

potential results to compare with each vote. If there were 20 candidates for 5 seats then there would be more than 15,000 potential results. Such elections could only reasonably be counted by computer.

A somewhat simpler counting method is sequential proportional approval voting where candidates are elected one-by-one to the winners' circle by approval voting, but in each round the value of the votes of each voter who already has m candidates in the winners' circle is reduced to

${\displaystyle {\frac {1}{m+1}}}$

This was developed by the Danish polymath Thorvald N. Thiele, and used (with adaptations) in Sweden for a short period after 1909.

The system disadvantages minority groups who share some preferences with the majority. In terms of tactical voting, it is therefore highly desirable to withhold approval from candidates who are likely to be elected in any case, as with cumulative voting and the single non-transferable vote.

PAV is strongly monotonic and passes Independence of Irrelevant Ballots (IIB). However, it fails the Universally liked candidate criterion (ULC) and also Perfect Representation In the Limit (PRIL), which is arguably a prerequisite for an approval method to be considered properly proportional.

## Example

2 seats to be filled, four candidates: Andrea (A), Brad (B), Carter (C), and Delilah (D). The ballots are:

• 5: AB
• 17: AC
• 8: D

There are 6 possible results: AB, AC, AD, BC, BD, and CD.

AB AC AD BC BD CD
voters approving 2 successful candidates (satisfaction of 1+1/2) 5 17 0 0 0 0
voters approving 1 successful candidate (satisfaction of 1) 17 5 30 22 13 25
voters approving no successful candidates (satisfaction of 0) 8 8 0 8 17 5
total satisfaction 24.5 30.5 30 22 13 25

Andrea and Carter are elected.

## Similar Systems

Both the Phragmén's Method and Sequential Proportional Approval Voting are very similar systems invented in the early 1900s. Reweighted Range Voting is an extension of this concept to Score Voting, along with Harmonic Voting and its sequential counterpart, Sequential proportional score voting. These systems all derive their reweighting theory as the natural extension of the Jefferson Method to Multi-Member Systems.

Optimised PAV is the version of the method that allows any number of candidates to be elected according to the optimal candidate weights, rather than a fixed number with equal weight. It is not known if this method passes PRIL, but on the assumption that is does, it has been considered, along with COWPEA, as a candidate method for the most accurately proportional approval method.[3] Optimised PAV Lottery elects a fixed number of candidates with equal weight, using the candidate weights as probabilities, and it has parallels with COWPEA Lottery. If it does pass PRIL, then it would pass the four main Holy Grail criteria of PRIL, strong monotonicity, IIB and ULC, along with COWPEA Lottery.

## References

1. Janson, Svante (2012-08-20). "Proportionella valmetoder" (PDF). Typescript, Uppsala. Retrieved 2020-02-28.
2. Simmons, Forest (2001-01-12). "Proportional Representation via Approval Voting". Election-methods mailing list archives. Retrieved 2020-02-28.
3. Pereira, Toby (2023-05-17). "COWPEA (Candidates Optimally Weighted in Proportional Election using Approval voting)". arXiv.