sequential proportional approval voting

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.{{cite book |last=Kilgour |first=D. Marc |editor1=Jean-François Laslier |editor2=M. Remzi Sanver |title=Handbook on Approval Voting |chapter-url=https://books.google.com/books?id=mQBEAAAAQBAJ&pg=PA114 |date=2010 |publisher=Springer |isbn=978-3-642-02839-7 |pages=105–124 |chapter=Approval Balloting for Multi-winner Elections}}Steven J. Brams, D. Marc Kilgour (2009): "Satisfaction Approval Voting": p4 [http://www2.eco.uva.es/presad/SSEAC/documents/Tilburg09-Brams-Kilgour.pdf] {{Webarchive|url=https://web.archive.org/web/20120628092034/http://www2.eco.uva.es/presad/SSEAC/documents/Tilburg09-Brams-Kilgour.pdf|date=2012-06-28}}

The aforementioned formula is as follows: $W=\backslash frac\{1\}\{E+1\}$ where $E$ is the number of candidates approved on that ballot who were already elected in the previous rounds, and $W$ is the final weight of the ballot. For the first round, $E$ 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.

File:SPAV Illustration.png

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.

Because Candidate C has the most support, they are the first winner, $w\_1$, 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.

Despite Candidates A and B having so many votes in the first round, Candidate X is the second winner, $w\_2$, 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 $w\_1$ and $w\_2$ 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 winner remain at full weight. Below is a table representing that information.

Candidate B is the third and final winner, $w\_3$. The final result has two winners from the party that received about two thirds of the votes, and one winner from the party that received about one third of the votes. If ordinary 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 in the first round.

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.{{Cite journal|last1=Sánchez-Fernández|first1=Luis|last2=Elkind|first2=Edith|last3=Lackner|first3=Martin|last4=Fernández|first4=Norberto|last5=Fisteus|first5=Jesús|last6=Val|first6=Pablo Basanta|last7=Skowron|first7=Piotr|date=2017-02-10|title=Proportional Justified Representation|url=https://ojs.aaai.org/index.php/AAAI/article/view/10611|journal=Proceedings of the AAAI Conference on Artificial Intelligence|language=en|volume=31|issue=1|doi=10.1609/aaai.v31i1.10611|s2cid=17538641|access-date=2021-06-24|archive-date=2021-06-24|archive-url=https://web.archive.org/web/20210624210617/https://ojs.aaai.org/index.php/AAAI/article/view/10611|url-status=live|doi-access=free|arxiv=1611.09928}}{{cite arXiv | eprint=1407.8269 | last1=Aziz | first1=Haris | last2=Brill | first2=Markus | last3=Conitzer | first3=Vincent | last4=Elkind | first4=Edith | last5=Freeman | first5=Rupert | last6=Walsh | first6=Toby | title=Justified Representation in Approval-Based Committee Voting | date=2014 | class=cs.MA }}

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.{{cite book |last1=Aziz |first1=Haris |last2=Gaspers |first2=Serge |last3=Gudmundsson |first3=Joachim |last4=Mackenzie |first4=Simon |last5=Mattei |last6=Mattei |first6=Nicholas |last7=Walsh |first7=Toby |title=Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems |chapter=Computational Aspects of Multi-Winner Approval Voting |year=2014 |pages=107–115 |arxiv=1407.3247v1 |isbn=978-1-4503-3413-6}}

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.{{Cite journal |last1=Faliszewski |first1=Piotr |last2=Skowron |first2=Piotr |last3=Szufa |first3=Stanisław |last4=Talmon |first4=Nimrod |date=2019-05-08 |title=Proportional Representation in Elections: STV vs PAV |url=https://dl.acm.org/doi/10.5555/3306127.3331972 |journal=Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems |series=AAMAS '19 |location=Richland, SC |publisher=International Foundation for Autonomous Agents and Multiagent Systems |pages=1946–1948 |isbn=978-1-4503-6309-9 |access-date=2022-05-11 |archive-date=2022-05-11 |archive-url=https://web.archive.org/web/20220511164125/https://dl.acm.org/doi/10.5555/3306127.3331972 |url-status=live }}

• Proportional approval voting
• Satisfaction approval voting
• Reweighted range voting
• Approval voting
• Single transferable vote
• Sainte-Laguë method
• D'Hondt method

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Category:Semi-proportional electoral systems

Category:Cardinal electoral systems

Category:Approval voting