Multiwinner approval voting

Multiwinner approval voting[1], also called approval-based committee voting,[2] is a multi-winner electoral system that uses approval ballots. Each voter may select ("approve") any number of candidates, and multiple candidates are elected. The number of elected candidates is usually fixed in advance. For example, it can be the number of seats in a country's parliament, or the required number of members in a committee.

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Multiwinner approval voting is an adaptation of approval voting to multiwinner elections. In a single-winner approval voting system, it is easy to determine the winner: it is the candidate approved by the largest number of voters. In multiwinner approval voting, there are many different ways to decide which candidates will be elected.

Majoritarian approval voting

Block approval voting (unlimited voting)

The straightforward extension of approval balloting to multi-winner elections is called block approval voting and is a type of multiple non-transferable vote, where each voter can select an unlimited number of candidates, and the k candidates with the most approval votes win (where k is the predetermined committee size). This does not provide proportional representation and is subject to the Burr dilemma, among other problems.

Limited block approval voting

Limited block approval voting is a type of multiple non-transferable vote, in which the k candidates with the most votes win, and each voter can select a limited number of candidates, but that limit is more than k. This is subject to the same problems as unlimited block approval voting.

Plurality-at-large and limited voting
Main articles: Plurality block voting and Limited voting

Plurality-at-large or plurality block voting (BV) and limited voting (LV) are both types of the multiple non-transferable vote category of multi-winner systems. Under plurality block voting, each voter has up to as many votes as there are seats to be filled, but no more than one per candidate and under limited voting, each voter has less votes than seats to be filled. Plurality block voting provides majoritarian representation, while limited voting is semi-proportional. Therefore voting reform advocates refrain from calling it approval voting,[3] similarly to how first-past-the-post or single-winner limited voting are not considered approval voting.

Example for comparison

Candidates are running in a 3 member district of the 10000 voters. Voters may not cast a more than one vote for a single candidate.

  • Under block approval voting (unlimited voting) voter may vote for any number of candidates
  • Under limited block approval voting voters may cast 6 votes maximum (twice as many as there are winners)
  • Under (plurality) block voting, voters may cast 3 votes (but do not have to)
  • Under limited (block) voting, voters may cast 2 votes maximum.
  • Under the single non-transferable vote, voters may cast 1 vote.

Party A has about 35% support among the electorate (with one particularly well-like candidate), Party B around 25% (with two well-like candidates) and the remaining voters primarily support independent candidates, but mostly lean towards party B if they have to choose between the two parties. All voters vote sincerely, there is no tactical voting.

Candidate Party Block approval voting Limited block approval voting Plurality block voting Limited (block) voting Single non-transferable vote
Votes % Elected? Votes % Elected? Votes % Elected? Votes % Elected? Votes % Elected?
Candidate A1 Party A 4200 42% 8. 4000 40% 5. 3555 36% 3. Yes 1800 18% 4. 800 8% 4.
Candidate A2 Party A 4500 45% 5. 4100 40% 4. 3700 37% 1. Yes 3500 35% 1. Yes 1900 19% 1. Yes
Candidate A3 Party A 3900 39% 9. 3800 40% 7. 3600 36% 2. Yes 2000 20% 2. Yes 700 7% 9.
Candidate B1 Party B 5200 52% 1. Yes 4900 49% 1. Yes 2600 26% 4. 1750 18% 5. 900 9% 3. Yes
Candidate B2 Party B 5000 50% 2. Yes 4600 46% 3. Yes 2500 25% 5. 1950 20% 3. Yes 1100 11% 2. Yes
Candidate B3 Party B 4700 47% 4. 4200 42% 2. Yes 2400 24% 6. 1425 14% 7. 400 4% 12.
Candidate I1 Independent 4400 44% 6. 3800 38% 7. 2300 23% 8. 1400 14% 8. 800 8% 4.
Candidate I2 Independent 4900 49% 3. Yes 4000 40% 5. 2395 20% 7. 1500 15% 6. 800 8% 4.
Candidate I3 Independent 4400 44% 6. 3700 37% 9. 1900 19% 9. 1300 13% 9. 700 7% 7.
Candidate I4 Independent 3900 39% 9. 3200 32% 10. 1800 15% 10 1200 12% 10. 700 7% 7.
Candidate I5 Independent 2600 26% 11. 2000 20% 11. 650 7% 11. 625 6% 11. 600 6% 10.
Candidate I6 Independent 2300 23% 12. 1700 17% 12. 600 6% 12. 550 6% 12. 500 5% 11.
TOTAL votes cast 50000 44000 28000 19000 10000
TOTAL possible votes 120000 60000 30000 20000 10000
Voters 10000 100% 10000 100% 10000 100% 10000 100% 10000 100%
  • Under the single non-transferable vote (not a type of approval voting), the 3 most popular candidates according to voters first preferences are elected, regardless of party affiliation.
  • Under limited voting, it is most likely that the party with a plurality takes 2 seats (the number of votes each voter has), and the minority party receives the remaining seats.
  • Under (plurality) block voting, the party with plurality support most likely wins all seats.
  • Under limited block approval voting voters, voters of independent candidates may use their extra votes to help candidates other than their top3, which may result in the reversal of the plurality block vote result.
  • Under block approval voting, any party-affiliated or independent candidates particularly popular among the population may be elected, but it is possible that about half of the population can elect no representatives.

Proportional approval voting

Methods such as Proportional approval voting,[4][5] Sequential proportional approval voting, and the Method of Equal Shares[6] [7] have the advantage of guaranteeing proportional representation in case all supporters of a party approve all candidates of that party. In the general case, proportional representation is replaced by a more general requirement called justified representation.

In these methods, the voters fill out a standard approval-type ballot, but the ballots are counted in a specific way that produces proportional representation. The exact procedure depends on which method is being used.

Party-approval voting

Party-approval voting (also called approval-based apportionment)[8] is a method in which each voter can approve one or more parties, rather than approving individual candidates. It is a combination of multiwinner approval voting with party-list voting.

Other methods

Other ways of extending Approval voting to multiple winner elections are satisfaction approval voting[9] excess method,[10] and minimax approval.[11] These methods use approval ballots but count them in different ways.

Usage

Multiwinner approval voting, while less common than standard approval voting, is used in several places.

Block approval voting
  • Korean villages used block approval voting for competitive elections following the surrender of Japan, according to observations made by journalist Anna Louise Strong in 1946: "In one village there were twelve candidates, of whom five were to be chosen for the Village Committee. Each voter was given twelve cards, bearing the names of the candidates. He then cast his chosen ones into the white box and the rejected ones into the black."[12]
  • Several Swiss cantons elect their government using such methods and so do French cities with population below 1000.[13]
  • In 1963, the proportional representation system in East Germany was replaced by a procedure in which the candidates had to receive more than 50% of the votes. Had more candidates than seats in this constituency won the majority, the order of the list would determine who would join the Volkskammer.

References
  1. Aziz, Haris; Gaspers, Serge; Gudmundsson, Joachim; Mackenzie, Simon; Mattei, Nicholas; Walsh, Toby (2014-07-11). "Computational Aspects of Multi-Winner Approval Voting". arXiv:1407.3247 [cs.GT].
  2. Aziz, Haris; Brill, Markus; Conitzer, Vincent; Elkind, Edith; Freeman, Rupert; Walsh, Toby (2017). "Justified representation in approval-based committee voting". Social Choice and Welfare. 48 (2): 461–485. doi:10.1007/s00355-016-1019-3. S2CID 8564247.
  3. "Voting Systems Confused with Approval Voting". The Center for Election Science. Retrieved March 12, 2020.
  4. http://www2.math.uu.se/~svante/papers/sjV9.pdf
  5. Brill, Markus; Laslier, Jean-François; Skowron, Piotr (2016). "Multiwinner Approval Rules as Apportionment Methods". arXiv:1611.08691 [cs.GT].
  6. Peters, Dominik; Skowron, Piotr (2020). "Proportionality and the Limits of Welfarism". Proceedings of the 21st ACM Conference on Economics and Computation. EC'20. pp. 793–794. arXiv:1911.11747. doi:10.1145/3391403.3399465. ISBN 9781450379755. S2CID 208291203.
  7. Pierczyński, Grzegorz; Peters, Dominik; Skowron, Piotr (2020). "Proportional Participatory Budgeting with Additive Utilities". Proceedings of the 2021 Conference on Neural Information Processing Systems. NeurIPS'21. arXiv:2008.13276.
  8. Brill, Markus; Gölz, Paul; Peters, Dominik; Schmidt-Kraepelin, Ulrike; Wilker, Kai (2020-04-03). "Approval-Based Apportionment". Proceedings of the AAAI Conference on Artificial Intelligence. 34 (2): 1854–1861. arXiv:1911.08365. doi:10.1609/aaai.v34i02.5553. ISSN 2374-3468. S2CID 208158445.
  9. Plaza, Enric. Technologies for political representation and accountability (PDF). CiteSeerX 10.1.1.74.3284. Retrieved 2011-06-17.
  10. https://as.nyu.edu/content/dam/nyu-as/faculty/documents/Excess%20Method%20(final).pdf
  11. LeGrand, Rob; Markakis, Evangelos; Mehta, Aranyak (2007). Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems - AAMAS '07 (PDF). p. 1. doi:10.1145/1329125.1329365. ISBN 9788190426275. S2CID 13870664. Retrieved 2011-06-17.
  12. Strong, Anna. "In North Korea: First Eye-Witness Report". Marxists Internet Archive. Retrieved May 14, 2019.
  13. Vander Straeten, Karine; Lachat, Romain; Laslier, Jean-François (2018). Stephenson, Laura B.; Aldrich, John H.; Blais, André (eds.). Chapter 9: Strategic voting in multi-winner elections with approval balloting: An application to the 2011 regional government election in Zurich. pp. 178–202. {{cite encyclopedia}}: |work= ignored (help)
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