Sequential proportional approval voting

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Sequential Proportional Approval Voting (SPAV) is a candidate based system of proportional representation. This system of voting was invented by Dane Thorvald N. Thiele in the early 1900’s. Candidates are listed individually on the ballot paper and voters may approve as few or as many of the candidates as they wish.

The total approval votes for the candidates are determined and the candidate with the most votes (the highest approval score) is declared elected. The value of all ballots that approve the candidate elected first are then reduced in value from 1 to 0.5 and the approval scores recalculated.

The unelected candidate who now has the highest approval score is elected. The value of all ballots that approve two elected candidates are now set at 1/3 and the value of all ballots that approve one candidate are now set at 0.5.

At each stage of the election the candidate with the highest approval score is elected and the value of the ballots set at

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

where m is the number of candidates approved on that ballot that have been elected at previous stages of the election. For example a ballot that approves 0 already elected candidates has a value of 1, a ballot that approves 1 already elected candidate has a value of 0.5, a ballot that approves 4 already elected candidates has a value of 0.2, etc

In Sequential Proportional Approval Voting the D’Hondt Highest Average formula is applied to the individual ballot papers whereas in conventional Party-list proportional representation it is applied to the vote totals for each list.

The following 100 approval ballots are cast in an election for 3 seats where 2 factions each present 3 candidates. Only one voter fails to vote on factional lines.

53: A1 A2 A3

1: A1 A2

1: A1 B1 B2

43: B1 B2 B3

1: B1 B2

1: B1

The table below shows the first stage Approval scores for the election:

Candidate	Approval score
A1	55
A2	54
A3	53
B1	46
B2	45
B3	43

Under multi-member Approval voting (the use of Approval voting to fill more than one seat in which the N most approved candidates are elected) all the candidates of faction A are elected since they are the 3 most approved candidates and faction B has no representation.

Multi-member Approval voting is used to elect the Board of Trustees of the Wikimedia Foundation.

Using SPAV the result of the election would be as follows:

Candidate	Stage 1 Approval Score	Stage 2 Approval Score	Stage 3 Approval Score
A1	55	Elected stage 1	Elected stage 1
A2	54	27	27
A3	53	26.5	26.5
B1	46	45.5	Elected stage 2
B2	45	44.5	22.33
B3	43	43	21.5

Candidates A1, B1 and A2 are elected.

Stage 1

At the first stage of the election candidate A1 is elected as the most approved candidate. The value of all ballot papers that approve A1 is then reduced to 1/(1+1) = 0.5.

Stage 2

At the second stage of the election candidate B1 has the greatest Approval score and is therefore the winner of the second seat. The value of all ballot papers that approve only candidate B1 is now reduced to 0.5. The value of the single ballot paper that approves both A1 and B1 is reduced to 1/(1+2)= 0.33.

Stage 3

At the final stage of the election candidate A2 has the greatest Approval score and is the winner of the final seat.

The values of the votes at the various stages of the count are shown in the table below:

Candidates Approved	Votes	Stage 1 value	Stage 2 value	Stage 3 value
A1 A2 A3	53	1.00	0.50	0.50
A1 A2	1	1.00	0.50	0.50
A1 B1 B2	1	1.00	0.50	0.33
B1 B2 B3	43	1.00	1.00	0.50
B1 B2	1	1.00	1.00	0.50
B1	1	1.00	1.00	0.50

The winners of this election under multi-member Approval Voting are A1,A2 and A3. Under Sequential proportional approval voting candidates A1,B1 and A2 are elected and the B faction obtains representation.