Proportional approval voting

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Proportional approval voting (PAV) is a theoretical 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 Forest Simmons in 2001.

If there was only one winner then proportional approval voting would become simple approval voting.

1 Counting votes
2 Example
3 Proportionality
4 Tactical voting
5 Complexity
6 Variants
7 See also

[edit] Counting votes

The first thing that happens in PAV is that a list of all potential outcomes of an election is compiled by a computer. For instance, if there were 3 candidates A, B, and C, then the potential outcomes of that election if two candidates were to be elected would be AB, CA, and BC (of course, to insure proportionality more candidates would usually be elected). Then the system compares each of the voters ballots to each outcome. Each voter is given a "satisfaction" score with each potential outcome based on how many candidates that they voted for that are in the given outcome. They are given one satisfaction point for having one candidate in the outcome, and an extra half of a point for having another, and so on, according to this formula:

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

(This is in essence the D'Hondt quota, but the webster quota could also be used to give more advantages to smaller parties)

Then the satisfaction of all voters is added up, and that gives the potential outcome its satisfaction score. The potential result with the highest total satisfaction is chosen as the actual result.

[edit] Example

Here is an example using the candidates mentioned above. Let's suppose voters cast their ballots in the following way:

10 voters: A, B
20 voters: B, C
20 voters: A
30 voters: B
10 voters: C

The satisfaction with the election of just A and B is calculated like so:

15=10*1.5 is given from the first set of ballot, as both candidates will be elected in this outcome.
20=20*1 is given from the second set, as only one candidate is elected out of that outcome.
20=20*1 is given from the third, as only their only candidate would be elected in that outcome.
30=30*1 is given from the fourth, as only their only candidate would be elected in that outcome.
0=10*0 is given from the fifth, as none of their candidate would be elected in such an outcome.

Therefore, the satisfaction with A and B getting elected is taken as 15 + 20 + 20 + 30 + 0, or 85.

The election of B and C would get a total satisfaction score of 70 by the same method, and the election of A and C would get a total satsifaction score of 60.

So A and B are elected. This system would obviously need a computer to process, but other than making the voters understand the basic math it would create very few technical problems.

[edit] Proportionality

It is easy to see why this system is proportional, as a large block of people voting can easily get a single candidate elected, but their voting power is reduced as they get more and more seats filled. By the same token, a small bloc would be unlikely to get a large amount of people in, but they would be able to get their first choice in. Unlike some PR systems, a party cannot alter its outcome by running more candidates. Having fewer candidates than the number of seats does not make that candidate more likely to become elected. Adding a new candidate to the field should have little effect in the overall outcome as long as it doesn't get elected.

If almost all voters 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.

[edit] Tactical voting

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.

[edit] Complexity

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

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

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.

[edit] Variants

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

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

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

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. If you limited the people to as many votes as there were seats instead of allowing them to cast for as many as they wanted, it would simply become the proportional form of plurarity voting.

See main article Sequential proportional approval voting