View source for Approval voting

{{Short description|Single-winner electoral system}}
{{Use mdy dates|date=June 2024}}
{{Lead too short|date=March 2025}}
[[File:Approval ballot.svg|thumb|On an [[approval ballot]], the voter can select any number of candidates.]]
{{Electoral systems sidebar|expanded=Single-winner}}
'''<!-- Who supports it? Why? How does it do on metrics like [[social utility efficiency]] or [[Condorcet efficiency]]? Where is it used? What studies are there on it? -->Approval voting''' is a single-winner [[rated voting]] system where voters can approve of all the candidates as they like instead of [[Plurality voting|choosing one]]. The method is designed to eliminate [[vote-splitting]] while keeping [[election administration]] simple and [[Summability criterion|easy-to-count]] (requiring only a single score for each candidate). Approval voting [[approval voting#use|has been used]] in both organizational and political elections to improve representativeness and voter satisfaction.

Critics of approval voting have argued the simple ballot format is a disadvantage, as it forces a [[Dichotomous preferences|binary choice]] for each candidate (instead of the expressive grades of other [[rated voting]] rules).

== Effect on elections ==
Research by [[Social choice theory|social choice theorists]] [[Steven Brams]] and [[Dudley R. Herschbach]] found that approval voting would increase voter participation, prevent minor-party candidates from being [[spoiler effect|spoilers]], and reduce negative campaigning.<ref>{{cite journal |last1=Brams |first1=Steven J. |last2=Herschbach |first2=Dudley R. |year=2001 |title=The Science of Elections |journal=Science |volume=292 |issue=5521 |pages=1449 |doi=10.1126/science.292.5521.1449 |jstor=3083781 |pmid=11379606 |s2cid=28262658}}</ref> Brams' research concluded that approval can be expected to elect [[Majority preferred candidate|majority-preferred candidates]] in practical election scenarios, avoiding the [[center squeeze]] common to [[Instant-runoff voting|ranked-choice voting]] and [[Two-round system|primary election]]s.<ref>{{cite book |last=Brams |first=Steven J. |url=https://archive.org/details/mathematicsdemoc00bram |title=Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures |date=2008 |publisher=Princeton University Press |isbn=9780691133218 |page=[https://archive.org/details/mathematicsdemoc00bram/page/n31 16] |url-access=limited}}</ref><ref name="theory to practice" />

One study showed that approval would not have chosen the same two winners as plurality voting ([[Jacques Chirac]] and [[Jean-Marie Le Pen]]) in the first round of the [[2002 French presidential election]]; it instead would have chosen Chirac and [[Lionel Jospin]] as the top two candidates to proceed to the runoff.

In the actual election, Le Pen lost by an overwhelming margin in the runoff, 82.2% to 17.8%, a sign that the true top two candidates had not been found. In the approval voting survey primary, Chirac took first place with 36.7%, compared to Jospin at 32.9%. Le Pen, in that study, received 25.1% and so would not have made the cut to the second round. In the real primary election, the top three were Chirac, 19.9%, Le Pen, 16.9%, and Jospin, 16.2%.<ref>{{cite web |last1=Laslier |first1=Jean-François |last2=Vander Straeten |first2=Karine |date=April 2003 |title=Approval Voting: An Experiment during the French 2002 Presidential Election |url=http://www.lse.ac.uk/collections/VPP/VPPpdf_Wshop2/jflkvdscaen.pdf |archive-url=https://web.archive.org/web/20050507223548/http://www.lse.ac.uk/collections/VPP/VPPpdf_Wshop2/jflkvdscaen.pdf |archive-date=May 7, 2005 |access-date=July 8, 2014 |page=6}}</ref> A study of various [[Evaluative voting|evaluative voting methods]] (approval and score voting) during the [[2012 French presidential election]] showed that "unifying" candidates tended to do better, and polarizing candidates did worse, as compared to under plurality voting.<ref>{{Cite journal |last1=Baujard |first1=Antoinette |last2=Igersheim |first2=Herrade |last3=Lebon |first3=Isabelle |last4=Gavrel |first4=Frédéric |last5=Laslier |first5=Jean-François |date=June 1, 2014 |title=Who's favored by evaluative voting? An experiment conducted during the 2012 French presidential election |url=https://hal.archives-ouvertes.fr/hal-00803024/file/cahier_2013-05.pdf |url-status=live |journal=Electoral Studies |volume=34 |pages=131–145 |doi=10.1016/j.electstud.2013.11.003 |archive-url=https://ghostarchive.org/archive/20221009/https://hal.archives-ouvertes.fr/hal-00803024/file/cahier_2013-05.pdf |archive-date=October 9, 2022}}</ref>

=== Operational impacts ===
{{more citations needed section|date=June 2019}}

* '''Simple to tally—'''Approval ballots can be counted by some existing machines designed for plurality elections, as ballots are cast, so that final tallies are immediately available after the election, with relatively few if any upgrades to equipment.
* '''Just one round—'''Approval can remove the need for multiple rounds of voting, such as a [[Two-round system|primary]] or a [[Two-round system|run-off]], simplifying the election process.
* '''Avoids overvotes—'''Approval voting does not have the notion of overvotes, where voting for one more than allowed will cancel the entire opportunity to vote. In plurality elections, overvotes have to be reviewed and resolved if possible while in approval voting, no time is wasted on this activity.

== Use ==

=== Current electoral use ===
====Latvia====
The [[Saeima|Latvian parliament]] uses a modified version of approval voting within [[open list proportional representation]], in which voters can cast either positive (approval) votes, negative votes or neither for any number of candidates.<ref>{{Cite web |title=How do elections work in Latvia? |url=https://www.electoral-reform.org.uk/how-do-elections-work-in-latvia/|access-date=28 July 2024 |website=Electoral Reform Society |language=en}}</ref>

====United States====
;Missouri
In November 2020, [[St. Louis, Missouri]], passed Proposition D with 70% voting to authorize a variant of approval ([[unified primary]]) for municipal offices.<ref name="St. Louis approves">{{cite web |last= |first= |date=November 4, 2020 |title=St. Louis Voters Approve Nonpartisan Elections |url=https://www.usnews.com/news/best-states/missouri/articles/2020-11-04/st-louis-voters-approve-nonpartisan-elections |url-status=live |archive-url=https://web.archive.org/web/20210614075134/https://www.usnews.com/news/best-states/missouri/articles/2020-11-04/st-louis-voters-approve-nonpartisan-elections |archive-date=June 14, 2021 |access-date=December 3, 2020 |work=U.S. News & World Report}}</ref> In 2021, the [[2021 St. Louis mayoral election|first mayoral election with approval voting]] saw Tishaura Jones and Cara Spencer move on to the general with 57% and 46% support. Lewis Reed and Andrew Jones were eliminated with 39% and 14% support, resulting in an average of 1.6 candidates supported by each voter in the 4 person race.<ref>{{cite news |last=Clancy |first=Sam |date=March 2, 2021 |title=Tishaura Jones, Cara Spencer advance to general election in race for St. Louis Mayor |url=https://www.ksdk.com/article/news/politics/elections/tishaura-jones-cara-spencer-st-louis-mayor-race/63-34335593-223a-431b-ac63-b17eb2495841 |work=KSDK |location=St. Louis, MO |access-date=May 22, 2024}}</ref>

;North Dakota
In 2018, [[Fargo, North Dakota]], passed a local ballot initiative adopting approval for the city's local elections, becoming the first United States city and jurisdiction to adopt approval.<ref name="Fargo approves">[https://ivn.us/2018/11/06/one-americas-famous-towns-becomes-first-nation-adopt-approval-voting/ One of America's Most Famous Towns Becomes First in the Nation to Adopt Approval Voting] {{Webarchive|url=https://web.archive.org/web/20181107185459/https://ivn.us/2018/11/06/one-americas-famous-towns-becomes-first-nation-adopt-approval-voting/|date=November 7, 2018}}, accessed November 7, 2018</ref><ref name="Fargo votes">{{cite web |last=Moen |first=Mike |date=June 10, 2020 |title=Fargo Becomes First U.S. City to Try Approval Voting |url=https://www.publicnewsservice.org/2020-06-10/civic-engagement/fargo-becomes-first-u-s-city-to-try-approval-voting/a70495-1 |access-date=December 3, 2020 |work=Public News Service}}</ref> Previously in 2015, a Fargo city commissioner election had suffered from six-way [[Vote splitting|vote-splitting]], resulting in a candidate winning with 22% [[Plurality (voting)|plurality]] of the vote.<ref name=":1">{{Cite web|last=Piper|first=Kelsey|date=November 15, 2018|title=This city just approved a new election system never tried before in America|url=https://www.vox.com/future-perfect/2018/11/15/18092206/midterm-elections-vote-fargo-approval-voting-ranked-choice|access-date=July 8, 2020|website=Vox|language=en}}</ref>

The first election was held June 9, 2020, selecting two city commissioners, from seven candidates on the ballot.<ref>{{Cite web|last=Amundson|first=Barry|date=June 9, 2020|title=Strand and Preston emerge as leaders in Fargo City Commission race|url=https://www.inforum.com/news/government-and-politics/6528646-Strand-and-Preston-emerge-as-leaders-in-Fargo-City-Commission-race|url-status=live|archive-url=https://web.archive.org/web/20200610194537/https://www.inforum.com/news/government-and-politics/6528646-Strand-and-Preston-emerge-as-leaders-in-Fargo-City-Commission-race|archive-date=June 10, 2020|access-date=July 8, 2020|website=INFORUM|language=en}}</ref> Both winners received over 50% approval, with an average 2.3 approvals per ballot, and 62% of voters supported the change to approval in a poll.<ref>{{Cite web|date=June 17, 2020|title=Fargo's First Approval Voting Election: Results and Voter Experience|url=https://www.electionscience.org/commentary-analysis/fargos-first-approval-voting-election-results-and-voter-experience/|access-date=July 8, 2020|website=The Center for Election Science|language=en-US}}</ref> A poll by opponents of approval was conducted to test whether voters had in fact voted strategically according to the Burr dilemma.<ref>{{Cite journal|last=Nagel|first=Jack H.|date=2007|title=The Burr Dilemma in Approval Voting|url=https://www.journals.uchicago.edu/doi/10.1111/j.1468-2508.2007.00493.x|journal=The Journal of Politics|language=en|volume=69|issue=1|pages=43–58|doi=10.1111/j.1468-2508.2007.00493.x|s2cid=154179804|issn=0022-3816|url-access=subscription}}</ref> They found that 30% of voters who [[Bullet voting|bullet voted]] did so for strategic reasons, while 57% did so because it was their sincere opinion.<ref>{{Cite web|last=RCV for Colorado|title=Approval Voting Case Study|url=https://drive.google.com/file/d/11_puAt_0JYw0NSL2ykqGz_bZfmic45AP/view?usp=sharing&usp=embed_facebook|url-status=live|access-date=March 4, 2021|website=Google Docs|archive-url=https://web.archive.org/web/20211124095738/https://drive.google.com/file/d/11_puAt_0JYw0NSL2ykqGz_bZfmic45AP/view?usp=sharing&usp=embed_facebook |archive-date=November 24, 2021 }}</ref><ref>{{Cite web|title=Guest opinion: Emma Donahue and Linda S. Templin: Ballot issue 2E just makes sense - Ranked Choice Voting for Colorado|url=https://rcvforcolorado.org/guest-opinion-emma-donahue-and-linda-s-templin-ballot-issue-2e-just-makes-sense/|url-status=live|access-date=March 4, 2021|website=RCV for Colorado|language=en-US|quote=Triton Polling called Fargo voters and found that a significant number had gamed that voting method by bullet voting, just like the experts said they would.|archive-url=https://web.archive.org/web/20201130073843/https://rcvforcolorado.org/guest-opinion-emma-donahue-and-linda-s-templin-ballot-issue-2e-just-makes-sense/ |archive-date=November 30, 2020 }}</ref> Fargo's second approval election took place in June 2022, for mayor and city commission. The incumbent mayor was re-elected from a field of 7 candidates, with an estimated 65% approval, with voters expressing 1.6 approvals per ballot, and the two commissioners were elected from a field of 15 candidates, with 3.1 approvals per ballot.<ref>{{Cite web |last=Hamlin |first=Aaron |date=June 16, 2022 |title=Fargo's Second Approval Voting Election Runs Smoothly |url=https://electionscience.org/commentary-analysis/fargos-second-approval-voting-election-runs-smoothly/ |access-date=July 6, 2022 |website=The Center for Election Science |language=en-US}}</ref>

In 2023, the North Dakota legislature passed a bill which intended to ban approval voting. The bill was vetoed by governor [[Doug Burgum]], citing the importance of "home rule" and allowing citizens control over their local government. The legislature attempted to overrule the veto but failed.<ref>{{cite news |last=Carvel |first=Tasha |date=April 19, 2023 |title=Fargo approval voting survives after Senate fails to override Burgum veto of ban |url=https://kfgo.com/2023/04/19/794980/ |work=The Mighty 790 KFGO |location=Fargo, ND |access-date=May 22, 2024}}</ref> In April 2025, Governor [[Kelly Armstrong]] signed a bill banning [[ranked-choice voting]] and approval voting in the state, ending the practice in Fargo.<ref>{{cite news |last1=Dura |first1=Jack |title=North Dakota governor signs bill doing away with Fargo's unusual voting system |url=https://apnews.com/article/fargo-north-dakota-legislature-voting-elections-8f85df3e17bf77fd7af41693569831ac |access-date=9 May 2025 |date=April 16, 2025 |work=[[Associated Press]]}}</ref>

=== Use by organizations ===
Approval has been used in privately administered nomination contests by the [[Independent Party of Oregon]] in 2011, 2012, 2014, and 2016. Oregon is a [[fusion voting]] state, and the party has cross-nominated legislators and statewide officeholders using this method; its 2016 presidential preference primary did not identify a potential nominee due to no candidate earning more than 32% support.<ref>{{Cite news|last=Peralta|first=Sal|url=http://www.oregonlive.com/opinion/index.ssf/2016/06/following_tense_primary_indepe.html|title=Following tense primary, Independent Party plans presidential preference vote (OPINION)|date=June 17, 2016|work=[[The Oregonian]]|access-date=June 26, 2018|language=en-US}}</ref><ref>{{Cite news|last=Radnovich|first=Connor|url=https://www.statesmanjournal.com/story/news/politics/2018/04/21/oregon-independent-party-governor-candidates-internal-conflict-highlighted/530248002/|title=Independent Party's internal conflict highlighted in Oregon governor race|date=April 21, 2018|work=Statesman Journal|access-date=June 26, 2018|language=en}}</ref><ref>{{Cite web|url=http://www.fairvote.org/new_lessons_from_problems_with_approval_voting_in_practice|title=New Lessons from Problems with Approval Voting in Practice|last=Richie|first=Rob|date=December 14, 2016|work=FairVote}}</ref> The party switched to using [[STAR voting]] in 2020.<ref>{{Cite web|date=April 7, 2020|title=STAR VOTING ANNOUNCEMENT|url=https://www.indparty.com/election-notice|access-date=April 11, 2020|website=Independent Party of Oregon|location=Salem|language=en}}</ref><ref>{{Cite news|date=April 8, 2020|title=Independent Party of Oregon to utilize STAR system for primary|language=en|work=Herald and News|url=https://www.heraldandnews.com/news/local_news/independent-party-of-oregon-to-utilize-star-system-for-primary/article_b475ff5a-75c7-55f8-8e46-95b483eaff28.html|access-date=April 10, 2020}}</ref>

It is also used in internal elections by the [[American Solidarity Party]];<ref>{{cite web|url=https://solidarity-party.org/16112120-b9ec-4746-9bff-c2d18e7662b9/|title=Vice-Chair's Remarks|last1=Covich|first1=Skylar|website=American Solidarity Party|url-status=dead|archive-url=https://web.archive.org/web/20180622032935/https://solidarity-party.org/16112120-b9ec-4746-9bff-c2d18e7662b9/|archive-date=June 22, 2018|access-date=June 21, 2018}}</ref> the [[Green party|Green Parties]] of [[Green Party of Texas|Texas]]<ref>{{Cite web|url=http://www.txgreens.org/platform#reform|title=Platform|publisher=Green Party of Texas|access-date=January 31, 2017|quote=We support switching to a preferential voting method, such as approval voting.}}</ref><ref>{{Cite web|url=http://www.txgreens.org/by_laws|title=By-Laws|publisher=Green Party of Texas|url-status=dead|archive-url=https://web.archive.org/web/20170129154828/http://www.txgreens.org/by_laws|archive-date=January 29, 2017|access-date=January 31, 2017|quote=The Co-Chair, Treasurer, and Secretary positions shall be filled by using approval voting.}}</ref> and [[Green Party of Ohio|Ohio]];<ref>{{Cite web|url=http://www.ohiogreens.org/event/ohio-green-party-presidential-nominating-convention-columbus-results|title=Ohio Green Party Presidential Nominating Convention (Columbus)|publisher=Ohio Green Party|access-date=January 31, 2017|quote=We have agreed to use "approval voting" as opposed to "instant runoff voting" to make the Ohio Green Party choice for president|archive-date=February 26, 2021|archive-url=https://web.archive.org/web/20210226172609/http://www.ohiogreens.org/event/ohio-green-party-presidential-nominating-convention-columbus-results|url-status=dead}}</ref> the [[Libertarian National Committee]];<ref>{{Cite web |date=October 4, 2020 |title=MEETING MINUTES LIBERTARIAN NATIONAL COMMITTEE |url=https://lpedia.org/w/images/f/fa/LNC-MEETING_2020-10-04_FINAL.pdf#page=10 |archive-url=https://ghostarchive.org/archive/20221009/https://lpedia.org/w/images/f/fa/LNC-MEETING_2020-10-04_FINAL.pdf#page=10 |archive-date=October 9, 2022 |url-status=live |quote=VOTING METHOD: Approval Voting per OpaVote. Vote for as many candidates as desired.}}</ref> the [[Libertarian Party (United States)|Libertarian parties]] of [[Libertarian Party of Texas|Texas]],<ref>{{Cite news|url=http://www.lptexas.org/state-platform|title=State Platform|access-date=January 31, 2017|url-status=dead|archive-url=https://web.archive.org/web/20170130003731/http://www.lptexas.org/state-platform|archive-date=January 30, 2017|publisher=Libertarian Party of Texas|quote=We also believe that all races should be decided by Approval Voting}}</ref> [[Libertarian Party of Colorado|Colorado]],<ref>{{Cite web |title=Document:Colorado State Party Bylaws 2021 |url=https://lpedia.org/wiki/Document:Colorado_State_Party_Bylaws_2021 |access-date=September 24, 2022 |website=LPedia |quote=For any partisan offices, Convention Delegates shall vote by approval voting}}</ref><ref>{{Cite web |title=Colorado Convention 2018 |url=https://lpedia.org/wiki/Colorado_Convention_2018 |access-date=September 24, 2022 |work=LPedia |language=en |quote=All candidates were selected by approval voting via standing vote.}}</ref> [[Arizona Libertarian Party|Arizona]],<ref>{{Cite web |title=Bylaws |url=https://azlp.org/about/bylaws/ |access-date=September 24, 2022 |website=Arizona Libertarian Party |language=en-US |quote=The remaining Delegates … shall be elected by approval voting … should there be more Delegate Candidates than allotted seats.}}</ref> and [[Libertarian Party of New York|New York]];<ref>{{Cite web |date=November 7, 2021 |title=Rules of the Libertarian Party |url=https://lpny.org/wp-content/uploads/sites/24/2021/11/LPNY-RULES-2021-11-07.pdf#page=10 |archive-url=https://ghostarchive.org/archive/20221009/https://lpny.org/wp-content/uploads/sites/24/2021/11/LPNY-RULES-2021-11-07.pdf#page=10 |archive-date=October 9, 2022 |url-status=live |website=Libertarian Party of New York |quote=All elections shall be held with approval voting}}</ref> [[Alliance 90/The Greens]] in Germany;<ref>{{Cite web |date=January 18, 2017 |title=Grüne München: Satzung |trans-title=Statutes of the city association - Alliance 90/The Greens Munich-City District Association |url=https://www.gruene-muenchen.de/partei/satzung/ |url-status=dead |archive-url=https://web.archive.org/web/20170118171623/https://www.gruene-muenchen.de/partei/satzung/ |archive-date=January 18, 2017 |access-date=March 5, 2023 |language=de |quote=Delegierte für Bundes, Landes- und Bezirksversammlungen werden per Zustimmungsblockwahl gewählt. Jede*r Stimmberechtigte hat so viele Stimmen, wie Bewerber*innen zur Wahl stehen, und kann jeder*m Bewerber*in eine oder keine Stimme geben.}}</ref> and the [[Czech Pirate Party|Czech]]<ref>{{Cite web|title=Jednací řád celostátního fóra|url=https://wiki.pirati.cz/rules/jdr|access-date=November 16, 2021|website=Pirati.CZ|language=cs}}</ref> and [[German Pirate Party]]<!-- with a 50% threshold requirement? -->.<ref>{{Cite web |title=Akzeptanzwahl – Piratenwiki |url=https://wiki.piratenpartei.de/Akzeptanzwahl |access-date=2025-02-23 |website=wiki.piratenpartei.de |language=de |quote=Die Akzeptanzwahl ist aktuell das Standard-Wahlverfahren beim Großteil aller Parteitage in der Piratenpartei. |trans-quote=Approval Voting is currently the standard voting method at most party conventions in the Pirate Party.}}</ref><ref>{{Cite web|url=http://wiki.piratenpartei.de/Bundesparteitag_2010.1/Gesch%C3%A4ftsordnung#Wahlen_zu_Vorstand_und_Schiedsgericht|title=Bundesparteitag 2010.1/Geschäftsordnung|publisher=[[German Pirate Party]]|language=de|access-date=June 24, 2018}}</ref><ref>{{Cite web |title=Election Data from the German Pirate Party (Piratenpartei) |url=https://sites.google.com/a/electology.org/www/pirate-elections-germany |url-status=dead |archive-url=https://web.archive.org/web/20190219005345/https://sites.google.com/a/electology.org/www/pirate-elections-germany |archive-date=February 19, 2019 |access-date=September 24, 2022 |publisher=The Center for Election Science}}</ref>

Approval has been adopted by several societies: the Society for Social Choice and Welfare (1992),<ref>{{cite web |url=http://www.unicaen.fr/recherche/mrsh/sites/all/themes/simpler2/scw/statutesSCW.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.unicaen.fr/recherche/mrsh/sites/all/themes/simpler2/scw/statutesSCW.pdf |archive-date=October 9, 2022 |url-status=live |title=Statutes of the Society for Social Choice and Welfare |publisher=Society for Social Choice and Welfare |access-date=January 28, 2020}}</ref> [[Mathematical Association of America]] (1986),<ref>{{cite web|url=http://www.maa.org/about-maa/governance/governance-documents/bylaws#article9 |title=MAA Bylaws |publisher=[[Mathematical Association of America]] |date=August 7, 2010 |access-date=November 6, 2014}}</ref> the [[American Mathematical Society]],<ref>{{cite web|url=https://www.ams.org/notices/201508/rnoti-p931a.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/notices/201508/rnoti-p931a.pdf |archive-date=October 9, 2022 |url-status=live |title=2015 American Mathematical Society Elections |publisher=[[American Mathematical Society]] |access-date=August 19, 2015}}</ref> the Institute of Management Sciences (1987) (now the [[Institute for Operations Research and the Management Sciences]]),<ref>{{cite web |url=https://www.informs.org/content/download/363531/3797055/file/Constitution%20and%20Bylaws%20August%202017.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.informs.org/content/download/363531/3797055/file/Constitution%20and%20Bylaws%20August%202017.pdf |archive-date=October 9, 2022 |url-status=live |title=Constitution of the Institute for Operations Research and the Management Sciences |publisher=Institute for Operations Research and the Management Sciences |date=August 2017 |page=7}}</ref> the [[American Statistical Association]] (1987),<ref>{{cite web|url = https://www.amstat.org/asa/files/pdfs/ABT-Bylaws.pdf |title = Bylaws of the American Statistical Association |date = September 19, 2018 |access-date = January 28, 2020 |publisher=[[American Statistical Association]]}}</ref> and the [[Institute of Electrical and Electronics Engineers]] (1987). 

Steven Brams' analysis of the 5-candidate 1987 Mathematical Association of America presidential election shows that 79% of voters cast a ballot for one candidate, 16% for 2 candidates, 5% for 3, and 1% for 4, with the winner earning the approval of 1,267 (32%) of 3,924 voters.<ref name="theory to practice"/><ref>{{Cite book|title=Handbook on Approval Voting|url=https://archive.org/details/handbookonapprov00lasl|url-access=limited|last1=Brams|first1=Steven J.|last2=Fishburn|first2=Peter C.|chapter=Going from Theory to Practice: The Mixed Success of Approval Voting |date=2010 |publisher=Springer Berlin Heidelberg|isbn=9783642028380|editor-last=Laslier|editor-first=Jean-François|series=Studies in Choice and Welfare|pages=[https://archive.org/details/handbookonapprov00lasl/page/n37 19]–37|language=en|doi=10.1007/978-3-642-02839-7_3|editor-last2=Sanver|editor-first2=M. Remzi}}</ref> The IEEE board in 2002 rescinded its decision to use approval. IEEE Executive Director Daniel J. Senese stated that approval was abandoned because "few of our members were using it and it was felt that it was no longer needed."<ref name="theory to practice">{{cite conference |last1=Brams |first1=Steven J. |last2=Fishburn |first2=Peter C. |title=Going from Theory to Practice: The Mixed Success of Approval Voting |work=Annual Meeting of the American Political Science Association |publisher=[[American Political Science Association]] |access-date=May 8, 2010| url= http://www.nyu.edu/gsas/dept/politics/faculty/brams/theory_to_practice.pdf | date=August 2003 |archive-url=https://web.archive.org/web/20181218010629/http://www.nyu.edu/gsas/dept/politics/faculty/brams/theory_to_practice.pdf |archive-date=December 18, 2018}}</ref>

Approval voting was used for Dartmouth Alumni Association elections for seats on the College Board of Trustees, but after some controversy<ref name="verbum">{{cite news |url=https://www.thedartmouth.com/article/2009/04/verbum-ultimum-making-amends |title=Verbum Ultimum: Making Amends |date=April 3, 2009 |work=The Dartmouth}}</ref> it was replaced with traditional runoff elections by an alumni vote of 82% to 18% in 2009.<ref>{{cite web |url=https://alumni.dartmouth.edu/content/dartmouth-alumni-association-election-results-new-executive-committee-elected-constitutional |title=Dartmouth Alumni Association Election Results: New Executive Committee Elected; Constitutional Amendment Passes |date=May 9, 2009 |publisher=Dartmouth Office of Alumni Relations}}</ref> Dartmouth students started to use approval voting to elect their student body president in 2011. In the first election, the winner secured the support of 41% of voters against several write-in candidates.<ref>{{cite web |url=http://thelittlegreenblog.blogspot.com/2011/04/hix-12-dartmouth-student-body-both.html |last=Bruschi |first=Nathan |title=Hix '12, Dartmouth Student Body, both shafted in student election |date=April 16, 2011 |work=The Little Green Blog}}</ref> In 2012, Suril Kantaria won with the support of 32% of the voters.<ref>{{cite news |url=https://www.thedartmouth.com/article/2012/04/kantaria-danford-win-student-assembly-elections |title=Kantaria, Danford win Student Assembly elections |last=Shkuratov |first=Marina |date=April 16, 2012 |work=The Dartmouth}}</ref> In 2013, 2014 and 2016, the winners also earned the support of under 40% of the voters.<ref>{{cite news |url=https://www.thedartmouth.com/article/2013/04/ferrari-zhu-elected-to-lead-student-assembly |title=Ferrari, Zhu elected to lead Student Assembly |date=April 15, 2013 |last=Riordan |first=Michael |work=The Dartmouth}}</ref><ref name="Dennis, Cunningham to lead Assembly">{{cite news |url=https://www.thedartmouth.com/article/2014/04/dennis-cunningham-to-lead-assembly |title=Dennis, Cunningham to lead Assembly |last=McGahan |first=Sara |date=April 15, 2014 |work=The Dartmouth}}</ref><ref name="thedartmouth.com">{{cite web |url=https://www.thedartmouth.com/article/2016/04/harrington-wins-student-assembly-president-election |title=Harrington will be Student Assembly president |last=Jiang |first=Heyi |date=April 17, 2016 |work=The Dartmouth}}</ref> Results reported in ''The Dartmouth'' show that in the 2014 and 2016 elections, more than 80 percent of voters approved of only one candidate.<ref name="Dennis, Cunningham to lead Assembly"/><ref name="thedartmouth.com"/> Students replaced approval voting with plurality voting before the 2017 elections.<ref>[https://static1.squarespace.com/static/5b397fa0e2ccd1b58ab7a317/t/5b3bf00f0e2e72d1275f0e3d/1530654736431/DartmouthSAConstitution.pdf], Dartmouth student constitution</ref>

===Historical===
[[file:The story of the greatest nations, from the dawn of history to the twentieth century - a comprehensive history, founded upon the leading authorities, including a complete chronology of the world, and (14765071792).jpg|300px|thumb|Rows of [[secret ballot|secret]] approval vote boxes from early 1900s [[Greece]], where the voter drops a marble to the right or left of the box, through a tube, one for each candidate standing]]
[[Robert J. Weber]] coined the term "Approval Voting" in 1971.<ref name=":0">{{citation|title=Approval Voting|first1=Steven J.|last1=Brams|author1-link=Steven Brams|first2=Peter C.|last2=Fishburn|author2-link=Peter C. Fishburn|page=xv|publisher=Springer-Verlag|year=2007|isbn=978-0-387-49895-9|url=https://books.google.com/books?id=e7h7evxSclIC&pg=PR5}}</ref> It was more fully published in 1978 by political scientist [[Steven Brams]] and mathematician [[Peter Fishburn]].<ref>{{Cite journal | doi = 10.2307/1955105 |last1=Brams |first1=Steven |last2=Fishburn |first2=Peter | year = 1978 | title = Approval Voting | jstor =  1955105| journal = American Political Science Review | volume = 72 | issue = 3 | pages = 831–847 |s2cid=251092061 }}</ref>

<span id="Historical_use">Historically, several voting methods that incorporate aspects of approval have been used:</span>

* Approval was used for [[papal conclave]]s between 1294 and 1621, with an average of about forty cardinals engaging in repeated rounds of voting until one candidate was listed on at least two-thirds of ballots.<ref name="colomer">{{Cite journal | doi = 10.1162/002219598551616|author-link1=Josep Colomer |first1=Josep M. |last1=Colomer  |first2=Iain |last2=McLean | year = 1998 | title = Electing Popes: Approval Balloting and Qualified-Majority Rule | journal = The Journal of Interdisciplinary History | volume = 29 | issue = 1| pages = 1–22 | jstor=205972 |s2cid=145296691 }}</ref>
* In the 13th through 18th centuries, the [[Republic of Venice]] elected the [[Doge of Venice]] using a multi-stage process that featured random selection and voting that allowed approval of multiple candidates.<ref>{{Cite journal | doi = 10.1007/BF00135090 | last = Lines |first=Marji | year = 1986 | title = Approval Voting and Strategy Analysis: A Venetian Example | journal = Theory and Decision | volume = 20 | issue = 2 | pages = 155–172 | s2cid = 121512308 }}</ref><ref>{{cite conference |first1=Miranda |last1=Mowbray |first2=Dieter |last2=Gollmann |date=July 2007 |url=http://www.hpl.hp.com/techreports/2007/HPL-2007-28R1.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.hpl.hp.com/techreports/2007/HPL-2007-28R1.pdf |archive-date=October 9, 2022 |url-status=live |title=Electing the Doge of Venice: analysis of a 13th Century protocol |conference=IEEE Computer Security Foundations Symposium |location=Venice, Italy}}</ref>
* According to Steven J. Brams, approval was used for unspecified elections in 19th century England.<ref>{{cite speech|url=http://www.nyu.edu/gsas/dept/politics/faculty/brams/normative_turn.pdf |title=The Normative Turn in Public Choice |page=4 |date=April 1, 2006 |access-date=May 8, 2010 |first=Steven J. |last=Brams |event=Presidential Address to Public Choice Society |location=New Orleans, Louisiana|archive-url=https://web.archive.org/web/20100531093534/http://www.nyu.edu/gsas/dept/politics/faculty/brams/normative_turn.pdf |archive-date=May 31, 2010}}</ref>
* The [[United Nations Secretary-General|Secretary-General]] of the [[United Nations]] is elected in a [[United Nations secretary-general selection#Procedure|multi-round straw poll process]] where, in each round, members of the [[United Nations Security Council|Security Council]] may approve or disapprove of candidates, or decide to express no opinion. Disapproval by permanent members of the Security Council is similar to a veto. A candidate with no vetoes, at least nine votes, and more votes than any other candidate is considered to be likely to be supported by the Security Council in its formal recommendation vote.<ref>{{cite web |url= http://www.unsgselection.org/files/WisnumurtiGuidelinesSelectingCandidateSecretary-General.pdf |title= The "Wisnumurti Guidelines" for Selecting a Candidate for Secretary-General |access-date=November 30, 2007 |archive-url= https://web.archive.org/web/20080227114317/http://www.unsgselection.org/files/WisnumurtiGuidelinesSelectingCandidateSecretary-General.pdf |archive-date= February 27, 2008 |url-status= dead }}</ref><ref>{{Cite web|url=http://www.openthemagazine.com/article/cover-story/the-inside-story-of-how-i-lost-the-race-for-the-un-secretary-generals-job-in-2006-shashi-tharoor|title=The inside Story of How I Lost the Race for the UN Secretary-General's Job in 2006|last=Tharoor|first=Shashi|date=October 21, 2016|website=OPEN Magazine|language=en|access-date=March 6, 2019 |archive-url=https://web.archive.org/web/20190721215655/http://www.openthemagazine.com/article/cover-story/the-inside-story-of-how-i-lost-the-race-for-the-un-secretary-generals-job-in-2006-shashi-tharoor |archive-date=July 21, 2019}}</ref>
*Approval was used in [[Greece|Greek]] legislative elections from 1864 to 1923, after which it was replaced with [[party-list proportional representation]].<ref name="mavrogordatos">{{Cite book|url=http://archive.org/details/bub_gb_MbSifRqxM1EC|title=Stillborn Republic: Social Coalitions and Party Strategies in Greece 1922{{endash}}1936|last=Mavrogordatos|first=George Th.|date=1983|publisher=University of California Press|pages=351–352|language=en}}</ref>
*[[Sequential proportional approval voting]] was used in [[Swedish elections]] in the early 20th century, prior to being replaced by [[party-list proportional representation]].

The idea of approval was adopted by X. Hu and [[Lloyd Shapley]] in 2003 in studying [[authority distribution]] in organizations.<ref>{{cite journal|last1=Hu|first1=Xingwei|last2=Shapley|first2=Lloyd S.|year=2003|title=On Authority Distributions in Organizations|journal=Games and Economic Behavior|volume=45|issue=1|pages=132–170|doi=10.1016/S0899-8256(03)00130-1}}</ref>

== Strategic voting ==
{{See also|Strategic voting#Cardinal single-winner voting}}

=== Overview ===
{{more citations needed section|date=June 2019}}
Approval voting allows voters to select all the candidates whom they consider to be reasonable choices.

''Strategic approval'' differs from [[ranked voting]] (aka preferential voting) methods where voters are generally forced to ''reverse'' the preference order of two options, which if done on a larger scale can cause an unpopular candidate to win. Strategic approval, with more than two options, involves the voter changing their approval threshold. The voter decides which options to give the ''same'' rating, even if they were to have a preference order between them. This leaves a tactical concern any voter has for approving their second-favorite candidate, in the case that there are three or more candidates. Approving their second-favorite means the voter harms their favorite candidate's chance to win. Not approving their second-favorite means the voter helps the candidate they least desire to beat their second-favorite and perhaps win.

Approval technically allows for but is strategically immune to [[Tactical voting#Push-over|push-over]] and [[Tactical voting#Burying|burying]].

Bullet voting occurs when a voter approves ''only'' candidate "a" instead of ''both'' "a" and "b" for the reason that voting for "b" can cause "a" to lose. The voter would be satisfied with either "a" or "b" but has a moderate preference for "a". Were "b" to win, this hypothetical voter would still be satisfied. If supporters of both "a" and "b" do this, it could cause candidate "c" to win. This creates the "[[chicken dilemma]]", as supporters of "a" and "b" are [[playing chicken]] as to which will stop strategic voting first, before both of these candidates lose.

Compromising occurs when a voter approves an ''additional'' candidate who is otherwise considered unacceptable to the voter to prevent an even worse alternative from winning.

=== Sincere voting ===
{{more citations needed section|date=June 2019}}
Approval experts describe sincere votes as those "... that directly reflect the true preferences of a voter, i.e., that do not report preferences 'falsely.{{'"}}{{sfn|Brams|Fishburn|1983|p=29}} They also give a specific definition of a sincere approval vote in terms of the voter's [[preference|ordinal preferences]] as being any vote that, if it votes for one candidate, it also votes for any more preferred candidate. This definition allows a sincere vote to treat strictly preferred candidates the same, ensuring that every voter has at least one sincere vote. The definition also allows a sincere vote to treat equally preferred candidates differently. When there are two or more candidates, every voter has at least three sincere approval votes to choose from. Two of those sincere approval votes do not distinguish between any of the candidates: vote for none of the candidates and vote for all of the candidates. When there are three or more candidates, every voter has more than one sincere approval vote that distinguishes between the candidates.

==== Examples ====

Based on the definition above, if there are four candidates, A, B, C, and D, and a voter has a strict preference order, preferring A to B to C to D, then the following are the voter's possible sincere approval votes:
*vote for A, B, C, and D
*vote for A, B, and C
*vote for A and B
*vote for A
*vote for no candidates

If the voter instead equally prefers B and C, while A is still the most preferred candidate and D is the least preferred candidate, then all of the above votes are sincere and the following combination is also a sincere vote:
*vote for A and C

The decision between the above ballots is equivalent to deciding an arbitrary "approval cutoff." All candidates preferred to the cutoff are approved, all candidates less preferred are not approved, and any candidates equal to the cutoff may be approved or not arbitrarily.

=== Sincere strategy with ordinal preferences ===

A sincere voter with multiple options for voting sincerely still has to choose which sincere vote to use. [[tactical voting|Voting strategy]] is a way to make that choice, in which case strategic approval includes sincere voting, rather than being an alternative to it.<ref name=probstrat>{{Cite journal | doi = 10.2307/1955800 | last = Niemi |first=R. G. | year = 1984 | title = The Problem of Strategic Behavior under Approval Voting | journal = American Political Science Review | volume = 78 | issue = 4| pages = 952–958 | jstor = 1955800 | s2cid = 146976380 }}</ref> This differs from other voting systems that typically have a unique sincere vote for a voter.

When there are three or more candidates, the winner of an approval election can change, depending on which sincere votes are used. In some cases, approval can sincerely elect any one of the candidates, including a [[Condorcet winner]] and a [[Condorcet loser]], without the voter preferences changing. To the extent that electing a Condorcet winner and not electing a Condorcet loser is considered desirable outcomes for a voting system, approval can be considered vulnerable to sincere, strategic voting.<ref>{{Cite journal | last = Yilmaz | first=M. R. | year = 1999 | title = Can we improve upon approval voting? | doi = 10.1016/S0176-2680(98)00043-3 | journal = European Journal of Political Economy | volume = 15 | issue = 1| pages = 89–100 }}</ref> In one sense, conditions where this can happen are robust and are not isolated cases.<ref>{{Cite journal | doi = 10.1007/BF00054447 |last1=Saari |first1=Donald G.  |last2=Van Newenhizen |first2=Jill | year = 2004 | title = The problem of indeterminancy in approval, multiple, and truncated voting systems | journal = Public Choice | volume = 59 | issue = 2| pages = 101–120 |jstor=30024954 |s2cid=154705078 }}</ref> On the other hand, the variety of possible outcomes has also been portrayed as a virtue of approval, representing the flexibility and responsiveness of approval, not just to voter ordinal preferences, but cardinal utilities as well.<ref name=unmitigated>{{Cite journal | doi = 10.1007/BF00054449 |last1=Saari |first1=Donald G.  |last2=Van Newenhizen |first2=Jill | year = 2004 | title = Is approval voting an 'unmitigated evil?' A response to Brams, Fishburn, and Merrill | journal = Public Choice | volume = 59 | issue = 2| pages = 133–147 |jstor=30024956  |s2cid=154007278 }}</ref>

==== Dichotomous preferences ====

Approval avoids the issue of multiple sincere votes in special cases when voters have [[dichotomous preferences]]. For a voter with dichotomous preferences, approval is [[Strategyproofness|strategyproof]].{{sfn|Brams|Fishburn|1983|p=31}} When all voters have dichotomous preferences and vote the sincere, strategy-proof vote, approval is guaranteed to elect a Condorcet winner.{{sfn|Brams|Fishburn|1983|p=38}} However, having dichotomous preferences when there are three or more candidates is not typical. It is an unlikely situation for all voters to have dichotomous preferences when there are more than a few voters.<ref name=probstrat/>

Having dichotomous preferences means that a voter has bi-level preferences for the candidates. All of the candidates are divided into two groups such that the voter is indifferent between any two candidates in the same group and any candidate in the top-level group is preferred to any candidate in the bottom-level group.{{sfn|Brams|Fishburn|1983|p=16–17}} A voter that has strict preferences between three candidates—prefers A to B and B to C—does not have dichotomous preferences.

Being strategy-proof for a voter means that there is a unique way for the voter to vote that is a strategically best way to vote, regardless of how others vote. In approval, the strategy-proof vote, if it exists, is a sincere vote.{{sfn|Brams|Fishburn|1983|p=29}}

==== Approval threshold ====

Another way to deal with multiple sincere votes is to augment the ordinal preference model with an approval or acceptance threshold. An approval threshold divides all of the candidates into two sets, those the voter approves of and those the voter does not approve of. A voter can approve of more than one candidate and still prefer one approved candidate to another approved candidate. Acceptance thresholds are similar. With such a threshold, a voter simply votes for every candidate that meets or exceeds the threshold.<ref name=probstrat/>

With threshold voting, it is still possible to not elect the Condorcet winner and instead elect the Condorcet loser when they both exist. However, according to Steven Brams, this represents a strength rather than a weakness of approval. Without providing specifics, he argues that the pragmatic judgments of voters about which candidates are acceptable should take precedence over the [[Condorcet criterion]] and other social choice criteria.<ref name=critstrats>{{Cite journal |last1=Brams |first1=S. J.  |author2=Remzi Sanver, M. | year = 2005 | title = Critical strategies under approval voting: Who gets ruled in and ruled out | doi = 10.1016/j.electstud.2005.05.007 | journal = Electoral Studies | volume = 25 | issue = 2| pages = 287–305 }}</ref>

=== Strategy with cardinal utilities ===

Voting strategy under approval is guided by two competing features of approval. On the one hand, approval fails the [[later-no-harm criterion]], so voting for a candidate can cause that candidate to win instead of a candidate more preferred by that voter. On the other hand, approval satisfies the [[monotonicity criterion]], so not voting for a candidate can never help that candidate win, but can cause that candidate to lose to a less preferred candidate. Either way, the voter can risk getting a less preferred election winner. A voter can balance the risk-benefit trade-offs by considering the voter's cardinal utilities, particularly via the [[von Neumann–Morgenstern utility theorem]], and the probabilities of how others vote.

A [[Tactical voting#Myerson-Weber strategy|rational voter model]] described by [[Roger Myerson|Myerson]] and Weber specifies an approval strategy that votes for those candidates that have a positive prospective rating.<ref name=":2">{{Cite journal | doi = 10.2307/2938959 |last1=Myerson |first1=R. |last2=Weber |first2=R. J. | year = 1993 | title = A theory of Voting Equilibria | jstor = 2938959| journal = American Political Science Review | volume = 87 | issue = 1| pages = 102–114 |url=http://www.kellogg.northwestern.edu/research/math/papers/782.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.kellogg.northwestern.edu/research/math/papers/782.pdf |archive-date=October 9, 2022 |url-status=live | hdl = 10419/221141 |s2cid=143828854 | hdl-access = free }}</ref> This strategy is optimal in the sense that it maximizes the voter's [[Utility#Expected utility|expected utility]], subject to the constraints of the model and provided the number of other voters is sufficiently large.

An optimal approval vote always votes for the most preferred candidate and not for the least preferred candidate, which is a [[dominant strategy]]. An optimal vote can require supporting one candidate and not voting for a more preferred candidate if there 4 candidates or more, e.g. the third and fourth choices are correlated to gain or lose decisive votes together; however, such situations are inherently unstable, suggesting such strategy should be rare.<ref>{{Cite journal | last1=Dutta |first1=B |last2=De Sinopoli |first2=F. | last3=Laslier |first3= J.-F.| year = 2006 | title = Approval voting: three examples | journal = International Journal of Game Theory | volume = 35 | pages = 27–38 |doi=10.1007/s00182-006-0053-2 |s2cid=801286 | citeseerx=10.1.1.365.8090 }}</ref>

Other strategies are also available and coincide with the optimal strategy in special situations. For example:
* Vote for the candidates that have above average utility. This strategy coincides with the optimal strategy if the voter thinks that all pairwise ties are equally likely.{{sfn|Brams|Fishburn|1983|p=85}}
* Vote for any candidate that is more preferred than the expected winner and also vote for the expected winner if the expected winner is more preferred than the expected runner-up. This strategy coincides with the optimal strategy if there are three or fewer candidates or if the pivot probability for a tie between the expected winner and expected runner-up is sufficiently large compared to the other pivot probabilities. This strategy, if used by all voters, implies at equilibrium the election of the Condorcet winner whenever it exists.<ref name=":3">{{Cite journal  | last1=Laslier |first1= J.-F.| year = 2009 | title = The Leader rule: a model of strategic approval voting in a large electorate | journal = Journal of Theoretical Politics | volume = 21 | issue=1 | pages = 113–136 |doi= 10.1177/0951629808097286|s2cid= 153790214}}</ref>
*Vote for the most preferred candidate only. This strategy coincides with the optimal strategy when the best candidate is either much better than all others (i.e. is the only one with a positive expected value).{{sfn|Brams|Fishburn|1983|p=74, 81}}
*If all voters are rational and cast a strategically optimal vote based on a common knowledge of how all other voters vote except for small-probability, statistically independent errors, then the winner will be the Condorcet winner, if one exists.<ref name=":4">Laslier, J.-F. (2006) [http://halshs.archives-ouvertes.fr/docs/00/12/17/51/PDF/stratapproval4.pdf "Strategic approval voting in a large electorate,"] ''IDEP Working Papers'' No. 405 (Marseille, France: Institut D'Economie Publique)</ref>

=== Strategy examples ===
{{more citations needed section|date=June 2019}}In the example election described [[Condorcet method#Example: Voting on the location of Tennessee's capital|here]], assume that the voters in each faction share the following [[Von Neumann–Morgenstern utility theorem|von Neumann–Morgenstern utilities]], fitted to the interval between 0 and 100. The utilities are consistent with the rankings given earlier and reflect a strong preference each faction has for choosing its city, compared to weaker preferences for other factors such as the distance to the other cities.<!-- This seems like a very weird choice of utilities. Wouldn't it be better to make the disutility equal to the distance between the two cities? -->

{| class="wikitable" style="text-align: center"
|+Voter utilities for each candidate city
|-
! scope="col" rowspan="2" | Fraction of voters <small>(living close to)</small>
! scope="colgroup" colspan="4" | Candidates
! scope="col" rowspan=2 | Average
|-
! scope="col" style="width:6em;"| Memphis
! scope="col" style="width:6em;"| Nashville
! scope="col" style="width:6em;"| Chattanooga
! scope="col" style="width:6em;"| Knoxville
|-
| style="text-align:left;"| Memphis (42%)
| 100 || 15 || 10 || 0 || 31.25
|-
| style="text-align:left;"| Nashville (26%)
| 0 || 100 || 20 || 15 || 33.75
|-
| style="text-align:left;"| Chattanooga (15%)
| 0 || 15 || 100 || 35 || 37.5
|-
| style="text-align:left;"| Knoxville (17%)
| 0 || 15 || 40 || 100 || 38.75
|}

Using these utilities, voters choose their optimal strategic votes based on what they think the various pivot probabilities are for pairwise ties. In each of the scenarios summarized below, all voters share a common set of pivot probabilities.

{| class="wikitable" style="text-align: center"
|+Approval Voting results for scenarios using optimal strategic voting
|-
! scope="col" rowspan="2" | Strategy scenario
! scope="col" rowspan="2" | Winner
! scope="col" rowspan="2" | Runner-up
! scope="colgroup" colspan="4" | Candidate vote totals
|-
! scope="col" style="width:6em" | Memphis
! scope="col" style="width:6em" | Nashville
! scope="col" style="width:6em" | Chattanooga
! scope="col" style="width:6em" | Knoxville
|-
| style="text-align: left" | Zero-info
| Memphis || Chattanooga || 42 || 26 || 32 || 17
|-
| style="text-align: left" | Memphis leading Chattanooga
| colspan="2" | Three-way tie || 42 || 58 || 58 || 58
|-
| style="text-align: left" | Chattanooga leading Knoxville
| Chattanooga || Nashville || 42 || 68 || 83 || 17
|-
| style="text-align: left" | Chattanooga leading Nashville
| Nashville || Memphis || 42 || 68 || 32 || 17
|-
| style="text-align: left" | Nashville leading Memphis
| Nashville || Memphis || 42 || 58 || 32 || 32
|}

In the first scenario, voters all choose their votes based on the assumption that all pairwise ties are equally likely. As a result, they vote for any candidate with an above-average utility. Most voters vote for only their first choice. Only the Knoxville faction also votes for its second choice, Chattanooga. As a result, the winner is Memphis, the Condorcet loser, with Chattanooga coming in second place. In this scenario, the winner has minority approval (more voters disapproved than approved) and all the others had even less support, reflecting the position that no choice gave an above-average utility to a majority of voters.

In the second scenario, all of the voters expect that Memphis is the likely winner, that Chattanooga is the likely runner-up, and that the pivot probability for a Memphis-Chattanooga tie is much larger than the pivot probabilities of any other pair-wise ties. As a result, each voter votes for any candidate they prefer more than the leading candidate, and also vote for the leading candidate if they prefer that candidate more than the expected runner-up. Each remaining scenario follows a similar pattern of expectations and voting strategies.

In the second scenario, there is a three-way tie for first place. This happens because the expected winner, Memphis, was the Condorcet loser and was also ranked last by any voter that did not rank it first.

Only in the last scenario does the actual winner and runner-up match the expected winner and runner-up. As a result, this can be considered a stable strategic voting scenario. In the language of [[game theory]], this is an "equilibrium." In this scenario, the winner is also the Condorcet winner.

===Dichotomous cutoff===
{{more citations needed section|date=June 2019}}
Modeling voters with a 'dichotomous cutoff' assumes a voter has an immovable approval cutoff, while having meaningful cardinal preferences. This means that rather than voting for their top 3 candidates, or all candidates above the average approval, they instead vote for all candidates above a certain approval 'cutoff' that they have decided. This cutoff does not change, regardless of which and how many candidates are running, so when all available alternatives are either above or below the cutoff, the voter votes for all or none of the candidates, despite preferring some over others. This could be imagined to reflect a case where many voters become disenfranchised and apathetic if they see no candidates they approve of. In a case such as this, many voters may have an internal cutoff, and would not simply vote for their top 3, or the above average candidates.

For example, in this scenario, voters are voting for candidates with approval above 50% (bold signifies that the voters voted for the candidate):

{| class="wikitable" style="text-align:center; width:600px;"
|-
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| '''90%''' || '''60%''' || 40% || 10% || ''50%''
|-
! 35%
| 10% || '''90%''' || '''60%''' || 40% || ''50%''
|-
! 30%
| 40% || 10% || '''90%''' || '''60%''' || ''50%''
|-
! 10%
| '''60%''' || 40% || 10% || '''90%''' || ''50%''
|}

C wins with 65% of the voters' approval, beating B with 60%, D with 40% and A with 35%

If voters' threshold for receiving a vote is that the candidate has an above average approval, or they vote for their two most approved of candidates, this is not a dichotomous cutoff, as this can change if candidates drop out. On the other hand, if voters' threshold for receiving a vote is fixed (say 50%), this is a dichotomous cutoff, and satisfies IIA as shown below:

{| class="wikitable" style="text-align:center; width:600px;"
|+ A drops out, candidates voting for above average approval
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| – || '''60%''' || '''40%''' || 10% || ''37%''
|-
! 35%
| – || '''90%''' || 60% || 40% || ''63%''
|-
! 30%
| – || 10% || '''90%''' || '''60%''' || ''53%''
|-
! 10%
| – || 40% || 10% || '''90%''' || ''47%''
|}
B now wins with 60%, beating C with 55% and D with 40%
{| class="wikitable" style="text-align:center; width:600px;"
|+ A drops out, candidates voting for approval > 50%
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| – || '''60%''' || 40% || 10% || ''37%''
|-
! 35%
| – || '''90%''' || '''60%''' || 40% || ''63%''
|-
! 30%
| – || 10% || '''90%''' || '''60%''' || ''53%''
|-
! 10%
| – || 40% || 10% || '''90%''' || ''47%''
|}
With dichotomous cutoff, C still wins.

{| class="wikitable" style="text-align:center; width:600px;"
|+ D drops out, candidates voting for top 2 candidates
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| '''90%''' || '''60%''' || 40% || – || ''63%''
|-
! 35%
| 10% || '''90%''' || '''60%''' || – || ''53%''
|-
! 30%
| '''40%''' || 10% || '''90%''' || – || ''47%''
|-
! 10%
| '''60%''' || '''40%''' || 10% || – || ''37%''
|}
B now wins with 70%, beating C and A with 65%
{| class="wikitable" style="text-align:center; width:600px;"
|+ D drops out, candidates voting for approval > 50%
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| '''90%''' || '''60%''' || 40% || – || ''63%''
|-
! 35%
| 10% || '''90%''' || '''60%''' || – || ''53%''
|-
! 30%
| 40% || 10% || '''90%''' || – || ''47%''
|-
! 10%
| '''60%''' || 40% || 10% || – || ''37%''
|}
With dichotomous cutoff, C still wins.

== Compliance with voting system criteria ==
{{Incomplete table|date=October 2024}}
Most of the mathematical criteria by which voting systems are compared were formulated for voters with ordinal preferences. In this case, approval voting requires voters to make an additional decision of where to put their approval cutoff (see examples above). Depending on how this decision is made, approval satisfies different sets of criteria.

There is no ultimate authority on which criteria should be considered, but the following are criteria that many voting theorists accept and consider desirable:
{| class="wikitable" style="text-align:center"
|- style="font-size:90%;"
! '''Voting model:'''!! [[Majority winner criterion|Majority]]!! [[Monotonicity criterion|Monotone]] and [[Participation criterion|Participation]] !! [[Condorcet criterion|Condorcet]] and [[Smith set|Smith]] !! [[Independence of irrelevant alternatives|IIA]] !! [[Clone independence]] !! [[Reversal symmetry]]
![[Sincere favorite criterion|Sincere favorite]]
![[Gibbard's theorem|Strategyproof]]
|-
! [[Impartial anonymous culture|Zero information]]<ref name=":2" />
|{{No}} 
|{{Yes}} 
|{{No}} 
|{{No}} 
|{{No}} 
|{{Yes}}
|{{Yes}}
|{{No}}
|-
!Leader rule<ref name=":3" />{{Explain|date=May 2025}}
|{{Yes}}
|{{Yes}}
|{{Yes}}
|{{No}}
| 
| |
|{{Yes}}
|{{No}}
|-
![[Trembling hand perfect equilibrium|Trembling ballots]]<ref name=":4" />
|{{Yes}}
|{{Yes}}
|{{Yes}}
|{{No}}
|
| |
|{{Yes}}
|{{No}}
|-
! [[Dichotomous preferences|Binary preferences]]<ref name=":0" />{{NoteTag|Assuming voters have only two categories of preference ("approve" and "disapprove"). Voters are indifferent between any two candidates they approve, and between any two they disapprove.{{sfn|Brams|Fishburn|1983|p=16–17}} As a result, a voter's approval of a candidate does not depend on what the other alternatives are.}}
|{{Yes}} 
|{{Yes}} 
|{{Yes}} 
|{{Yes}} 
|{{Yes}} 
|{{Yes}}
|{{Yes}} 
|{{Yes}}
|}

== See also ==
{{portal|Politics}}Some variants and generalizations of approval voting are:
* [[Multiwinner approval voting]] — multiple candidates may be elected, instead of just one.
* [[Fractional approval voting]] — the election outcome is a distribution - assigning a fraction to each candidate.
* [[Score voting]] (also called range voting) — is simply approval voting where voters can give a wider range of scores than 0 or 1 (e.g. 0-5 or 0–7).
* [[Combined approval voting]] — form of score voting with three levels that uses a scale of (-1, 0, +1) or (0, .5, 1).
* [[D21 – Janeček method]] — limited to two approval and one negative vote per voter.
* [[Unified primary]] — a nonpartisan primary that uses approval voting for the first round.

== Notes ==
<references group="note" />
== References ==
{{Reflist|30em}}

=== Sources ===
* {{cite book |last1=Brams |first1=Steven J. |last2=Fishburn |first2=Peter C. |date=1983 |title=Approval Voting |location=Boston |publisher=Birkhäuser |isbn=3764331240 |url-access=registration |url=https://archive.org/details/approvalvoting00bram }}

== External links ==
{{Wiktionary|approval}}
* [https://electionscience.org/library/approval-voting/ Approval Voting] Article by [http://www.electology.org/ The Center for Election Science]
* [https://www.youtube.com/watch?v=BZiS3U7EG0M Could approval Voting Prevent Electoral Disaster?] Video by [[Big Think]]
* [http://pareto.uab.cat/wp/2004/61904.pdf Approval Voting on Dichotomous Preferences] Article by Marc Vorsatz.
* [http://pareto.uab.cat/wp/2004/61704.pdf Scoring Rules on Dichotomous Preferences] Article by Marc Vorsatz.
* [http://www.universalworkshop.com/ARVOfull.htm The Arithmetic of Voting] article by Guy Ottewell
* [https://www.sciencedirect.com/science/article/abs/pii/S0261379405000570 Critical Strategies Under approval Voting: Who Gets Ruled In And Ruled Out] Article by Steven J. Brams and M. Remzi Sanver.
* [https://www.youtube.com/watch?v=orybDrUj4vA Quick and Easy Voting for Normal People] YouTube video

{{voting systems}}

[[Category:Electoral systems]]
[[Category:Single-winner electoral systems]]
[[Category:Cardinal electoral systems]]
[[Category:Monotonic electoral systems]]
[[Category:Approval voting]]
[[Category:Rating systems]]