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Sequential proportional approval voting

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Flow chart of SPAV calculation

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. It is a special case of Thiele's voting rules, proposed by Danish statistician Thorvald N. Thiele in the early 1900s.[2] It was used in Sweden from 1909 to 1921, when it was replaced by a "party-list" style system, and is still used for some local elections.[3][4]

Description

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Sequential proportional approval voting uses approval voting ballots to elect multiple winners on a round-by-round basis. With approval voting ballots, each voter may support any number of candidates on their ballot as they see fit. For tabulation, each ballot is weighted according to a formula, the candidate with the most support is elected, and the process is repeated until there are no more seats to fill.[5][6]

The aforementioned formula is as follows: where is the number of candidates approved on that ballot who were already elected in the previous rounds, and is the final weight of the ballot. For the first round, is naturally 0, and so each ballot has a weight of 1. A SPAV election with only one seat to fill is identical to an approval voting election. Other weighting formulas may be used while still being referred to as SPAV.

Example

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Illustration of the example election. Candidates with the most votes wins for each round. When a candidate is elected they are removed for the next round.

As a clarifying example, consider an election for a committee with three winners. There are six candidates, representing two main parties: A, B, and C from one party, and X, Y, and Z from another party. About two-thirds of the voters support the first party, and the other third of the voters support the second party. Each voter casts their vote by selecting all the candidates they support. The following table shows the results of the votes. Each row represents a possible candidate support combination and the first column indicates how many ballots were cast with that combination. 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
112
6
4
73
4
1
Total Votes 116 122 126 82 78 77

Because Candidate C has the most support, they are the first winner, , and they cannot win any subsequent rounds. For the second round, any ballot which voted for Candidate C is given a weight of one half. Below is the chart for round 2. A column has been added to indicate the weight of each set of ballots.

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, , because most of the ballots that support A and B also support C and thus already have representation on the council. In round 3, ballots that voted for both candidates X and C have their vote weighted by one third. Any ballot that supports only one of the two winners will be weighted by one half. Ballots that indicate support for neither remain at full weight. Below is a table representing that information.

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, . The final result has two winners from the party that had about two thirds of the votes, and one winner from the party that had about one third of the votes. If approval voting had been used instead, the final committee would have all three candidates from the first party, as they had the highest three vote totals without scaling.

Properties

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SPAV 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][8]

Pareto efficiency Committee monotonicity Support monotonicity with additional voters Support monotonicity without additional voters Consistency inclusion-strategyproofness Computational complexity
Approval voting strong P
Proportional approval voting strong × cand × NP-hard
Sequential proportional approval voting × cand cand × × P

There is a small incentive towards tactical voting where a voter may withhold approval from candidates who are likely to be elected, just like there is with cumulative voting and the single non-transferable vote. SPAV is a much computationally simpler algorithm than harmonic proportional approval voting and other proportional methods, permitting votes to be counted either by hand, rather than requiring a computer to determine the outcome.[9]

When comparing sequential proportional approval to single transferable vote (STV), SPAV is more likely to elect candidates that individually represent the average voter, where STV is more likely to elect a range of candidates that match the distribution of the voters. The larger the number of candidates elected, the smaller the practical difference.[10]

See also

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References

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  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. S2CID 53600917.
  2. ^ E. Phragmén (1899): "Till frågan om en proportionell valmetod." Statsvetenskaplig tidskrifts Vol. 2, No. 2: pp 87-95 [1] Archived 2015-06-18 at the Wayback Machine
  3. ^ Lewis, Edward G. (1950). "Review of Modern Foreign Governments". The American Political Science Review. 44 (1): 209–211. doi:10.2307/1950372. ISSN 0003-0554. JSTOR 1950372. S2CID 152254976.
  4. ^ Humphreys, John H. (2006-01-01). Proportional Representation: A Study in Methods of Election. Archived from the original on 2022-05-11. Retrieved 2022-05-11.
  5. ^ 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.
  6. ^ Steven J. Brams, D. Marc Kilgour (2009): "Satisfaction Approval Voting": p4 [2] Archived 2012-06-28 at the Wayback Machine
  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). arXiv:1611.09928. doi:10.1609/aaai.v31i1.10611. S2CID 17538641. Archived from the original on 2021-06-24. Retrieved 2021-06-24.
  8. ^ Aziz, Haris; Brill, Markus; Conitzer, Vincent; Elkind, Edith; Freeman, Rupert; Walsh, Toby (2014). "Justified Representation in Approval-Based Committee Voting". arXiv:1407.8269 [cs.MA].
  9. ^ Aziz, Haris; Gaspers, Serge; Gudmundsson, Joachim; Mackenzie, Simon; Mattei; Mattei, Nicholas; Walsh, Toby (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.
  10. ^ Faliszewski, Piotr; Skowron, Piotr; Szufa, Stanisław; Talmon, Nimrod (2019-05-08). "Proportional Representation in Elections: STV vs PAV". Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems. AAMAS '19. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 1946–1948. ISBN 978-1-4503-6309-9. Archived from the original on 2022-05-11. Retrieved 2022-05-11.