Sequential proportional approval voting

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Sequential proportional approval voting (SPAV) or reweighted approval voting (RAV)[1] is an electoral system that extends the concept of approval voting to a multiple winner election. It is a simplified version of proportional approval voting. Proposed by Danish statistician Thorvald N. Thiele in the early 1900s,[2] it was used (with adaptations for party lists) in Sweden for a short period after 1909.[3]


This system converts Approval Voting into a multi-round rule,[4] selecting a candidate in each round and then reweighing the approvals for the subsequent rounds. The first candidate elected is the Approval winner (w1). The value of all ballots that approve of w1 are reduced in value from 1 to 1/2 and the approval scores recalculated. Next, the unelected candidate who has the highest approval score is elected (w2). Then the value of ballots that approve of both w1 and w2 are reduced in value to 1/3, and the value of all ballots that approve of either w1 or w2 but not both are reduced in value to 1/2.[5]

At each stage, the unelected candidate with the highest approval score is elected. Then the value of each voter’s ballot is set at 1/(1+m) where m is the number of candidates approved on that ballot who were already elected, until the required number of candidates is elected. This reweighting is based on the D'Hondt method (Jefferson method).

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.

It is however a much computationally simpler algorithm than proportional approval voting, permitting votes to be counted either by hand or by computer, rather than requiring a computer to determine the outcome of all but the simplest elections.[6]


For this example, there is an election for a committee with 3 winners. There are six candidates from two main parties: A, B, and C from one party, and X, Y, and Z from another party. About 2/3 of the voters support the first party, and the other roughly 1/3 of the voters support the second party. Each voter casts their vote by selecting the candidates they support. The following table shows the results of the votes. Each row starts by saying how many voters voted in that way and marks each candidate that group of voters supported. The bottom row shows the number of votes each candidate received.

Votes from 200 voters
# of votes Candidate A Candidate B Candidate C Candidate X Candidate Y Candidate Z
Total Votes 116 122 126 82 78 77

Because Candidate C has the most support, they are the first winner, w1, and their vote is not counted in later rounds. For the second round, anyone who voted for Candidate C has their vote counted as only 1/2. Below is the chart for round 2. A second column on the left has been added to indicate the weight of each ballot.

Second Round Results
# of votes Weight of Vote Candidate A Candidate B Candidate C Candidate X Candidate Y Candidate Z
112 1/2
6 1/2
4 1/2
73 1
4 1/2
1 1
Weighted Votes 58 61 78 76 75

Despite Candidates A and B having so many votes in the first round, Candidate X is the second winner, w2, because not as many of the votes for Candidate X were halved. In round 3, anyone who voted for either Candidates C or X has their vote count 1/2, and anyone who voted for both has their vote count 1/3. If anyone had voted for neither, their vote would remain at 1. Below is that table.

Third Round Results
# of votes Weight of Vote Candidate A Candidate B Candidate C Candidate X Candidate Y Candidate Z
112 1/2
6 1/2
4 1/3
73 1/2
4 1/3
1 1/2
Weighted Votes 57 1/3 60 1/3 38 1/3 37 5/6

Candidate B is the third and final winner, w3. The final result has 2/3 winners from the party that had about 2/3 of the votes, and 1/3 winner from the party that had about 1/3 of the votes. If approval voting had been used instead, the final committee would be all three candidates from the first party, as they had the highest three vote totals without scaling.


Sequential-PAV satisfies the fairness property called justified representation whenever the committee size is at most 5, but might violate it when the committee size is at least 6.[7]

See also[edit]


  1. ^ Brams, Steven; Brill, Markus (2018). "The Excess Method: A Multiwinner Approval Voting Procedure to Allocate Wasted Votes". SSRN Electronic Journal. doi:10.2139/ssrn.3274796. ISSN 1556-5068.
  2. ^ E. Phragmén (1899): "Till frågan om en proportionell valmetod." Statsvetenskaplig tidskrifts Vol. 2, No. 2: pp 87-95 [1]
  3. ^ Aziz, H., Brill, M., Conitzer, V., et al. (2014): "Justified Representation in Approval-Based Committee Voting", arXiv:1407.8269 p5 [2]
  4. ^ Kilgour, D. Marc (2010). "Approval Balloting for Multi-winner Elections". In Jean-François Laslier; M. Remzi Sanver (eds.). Handbook on Approval Voting. Springer. pp. 105–124. ISBN 978-3-642-02839-7.
  5. ^ Steven J. Brams, D. Marc Kilgour (2009): "Satisfaction Approval Voting": p4 [3]
  6. ^ Aziz, Haris; Serge Gaspers, Joachim Gudmundsson, Simon Mackenzie, Nicholas Mattei, Toby Walsh (2014). "Computational Aspects of Multi-Winner Approval Voting". Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems. pp. 107–115. arXiv:1407.3247v1. ISBN 978-1-4503-3413-6.CS1 maint: multiple names: authors list (link)
  7. ^ Sánchez-Fernández, Luis; Elkind, Edith; Lackner, Martin; Fernández, Norberto; Fisteus, Jesús; Val, Pablo Basanta; Skowron, Piotr (2017-02-10). "Proportional Justified Representation". Proceedings of the AAAI Conference on Artificial Intelligence. 31 (1). ISSN 2374-3468.