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Concept
Proportional approval voting
Summary
Proportional approval voting (PAV) is a proportional electoral system for selecting committees. It is an extension of the D'Hondt method of apportionment that additionally allows for personal votes (voters vote for candidates, not for a party list). The voters vote via approval ballots where each voter marks those candidates that the voter finds acceptable.
The system was first proposed by Thorvald N. Thiele. It was used in combination with ranked voting in the early 20th century in Sweden, for example between 1909 and 1921 for distributing seats within parties, and in local elections. After 1921 it was replaced by Phragmén's rules. PAV was rediscovered by Forest Simmons in 2001 who gave it the name "proportional approval voting".
PAV selects a committee of a fixed desired size with the highest score, where scores are calculated according to the following formula. Given a committee , for each voter we check how many candidates in the committee the voter approves. Assuming that the voter voted for candidates in , the voter contributes to the score of the value of the -th harmonic number, that is:
The score of is calculated as the sum of the scores garnered from all the voters.
Formally, assume we have a set of candidates , a set of voters , and a committee size . Let denote the set of candidates approved by voter . The PAV score of a committee with size is defined as . PAV select the committee with the maximal score.
Assume 2 seats to be filled, and there are four candidates: Andrea (A), Brad (B), Carter (C), and Delilah (D), and 30 voters. The ballots are:
5 voters voted for A and B
17 voters voted for A and C
8 voters voted for D
There are 6 possible results: AB, AC, AD, BC, BD, and CD.
Andrea and Carter are elected.
Note that Simple Approval shows that Andrea has 22 votes, Carter has 17 votes, Delilah has 8 votes and Brad has 5 votes. In this case, the PAV selection of Andrea and Carter is coincident with the Simple Approval sequence.
Official source
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