Proportional approval voting

Proportional approval voting (PAV) is a proportional electoral system for multiwinner elections. It is a multiwinner approval method that extends the highest averages method of apportionment commonly used to calculate apportionments for party-list proportional representation.^[1] However, PAV allows voters to support only the candidates they approve of, rather than being forced to approve or reject all candidates on a given party list.^[2]

In PAV, voters cast approval ballots marking all candidates they approve of; each voter's ballot is then treated as if all candidates on the ballot were on their own "party list." Seats are then apportioned between candidates in a way that ensures all coalitions are represented proportionally.

History

PAV is a special case of Thiele's voting rule, proposed by Thorvald N. Thiele.^[3]^[4] It was used in combination with ranked voting in the Swedish elections from 1909 to 1921 for distributing seats within parties and in local elections.^[4] PAV was rediscovered by Forest Simmons in 2001,^[5] who gave it the name "proportional approval voting."

Method

Like its close cousin, satisfaction approval voting, PAV can be thought of as selecting a committee by testing all possible committees, then choosing the committee with the most votes. In satisfaction approval voting, each voter's ballot is split equally between all $r$ candidates they approve of, giving each one $1/r$ votes. If voters are perfectly strategic, and only support as many candidates as their party is entitled to, SAV creates a proportional result.

PAV makes one modification to remove this need for strategy: it only splits a voter's ballot after they have elected a candidate. As a result, voters can freely approve of losing candidates without diluting their ballot. Voters contribute a whole vote to the first candidate they support who is elected; half a vote to the second candidate; and so on.

Thus, if a ballot approves of $r$ candidates who are elected, that ballot contributes the $r$ -th harmonic number to that committee's vote total. In other words:^[5]^[6]

\mathrm {H} (r)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{r}}

The score for a given committee is calculated as the sum of the scores garnered from all the voters. We then choose the committee with the highest score.

Formally, assume we have a set of candidates $C=\{c_{1},c_{2},\ldots ,c_{m}\}$ , a set of voters $N=\{1,2,\ldots ,n\}$ , and a committee size $k$ . Let $A_{i}$ denote the set of candidates approved by voter $i\in N$ . The PAV score of a committee $W\subseteq C$ with size $|W|=k$ is defined as $\textstyle \mathrm {sc} _{\mathrm {PAV} }(W)=\sum _{i=1}^{n}\mathrm {H} (|A_{i}\cap W|)$ . PAV selects the committee $W$ with the maximal score.

Example 1

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.

	AB	AC	AD	BC	BD	CD
score from the 5 voters voting for AB	$5\cdot \left(1+{\tfrac {1}{2}}\right)=7.5$	$5\cdot 1=5$	$5\cdot 1=5$	$5\cdot 1=5$	$5\cdot 1=5$	$5\cdot 0=0$
score from the 17 voters voting for AC	$17\cdot 1=17$	$17\cdot \left(1+{\tfrac {1}{2}}\right)=25.5$	$17\cdot 1=17$	$17\cdot 1=17$	$17\cdot 0=0$	$17\cdot 1=17$
score from the 8 voters voting for D	$8\cdot 0=0$	$8\cdot 0=0$

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