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In social choice, the negative responsiveness,^[1]^[2] perversity,^[3] or additional support paradox^[4] is a pathological behavior of some voting rules, where a candidate loses as a result of having "too much support" from some voters, or wins because they had "too much opposition". In other words, increasing (decreasing) a candidate's ranking or rating causes that candidate to lose (win),^[4] contrary to common sense. Electoral systems that do not exhibit perversity are said to satisfy the positive response or monotonicity criterion.^[5]

Perversity is often described by social choice theorists as an exceptionally severe kind of electoral pathology.^[6]^[7] Systems that allow for perverse response can create situations where a voter's ballot has a reversed effect on the election, thus treating the well-being of some voters as "less than worthless".^[7] Similar arguments have led to constitutional prohibitions on such systems as violating the right to equal and direct suffrage.^[8]^[9] Negative response is often cited as an example of a perverse incentive, as voting rules with perverse response incentivize politicians to take unpopular or extreme positions in an attempt to shed excess votes.

Most ranked methods (including Borda and all common round-robin rules) satisfy positive response,^[5] as do all common rated voting methods (including approval, highest medians, and score).^{[note 1]}

Perversity occurs in instant-runoff voting,^[10] the single transferable vote,^[11] and quota-based apportionment methods.^[2] According to statistical culture models of elections, the paradox is especially common in ranked-choice voting (RCV-IRV) and the two-round system.^{[citation needed]} The randomized Condorcet method can violate monotonicity in the case of cyclic ties.

The participation criterion is a closely-related, but different, concept. While positive responsiveness deals with a voter changing their opinion (or vote), participation deals with situations where a voter choosing to cast a ballot has a reversed effect on the election.^[12]

By method

Runoff voting

Runoff-based voting systems such as ranked choice voting (RCV) are typically vulnerable to perverse response. A notable example is the 2009 Burlington mayoral election, the United States' second instant-runoff election in the modern era, where Bob Kiss won the election as a result of 750 ballots ranking him in last place.^[13] Another example is given by the 2022 Alaska at-large special election.

An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Bottom party initially loses. However, they are elected after running an unsuccessful campaign and adopting an unpopular platform, which pushes their supporters away from the party and into the Top party.


Popular Bottom				Unpopular Bottom
	Round 1	Round 2			Round 1	Round 2
Top	~~25%~~ N		+6%	Top	31%	46%
Center	30%	55% Y	↗	Center	~~30% N~~
Bottom	45%	45%	-6%	Bottom	39%	54% Y

This election is an example of a center-squeeze, a class of elections where instant-runoff and plurality have difficulties electing the majority-preferred candidate. Here, the loss of support for Bottom policies makes the Top party more popular, allowing it to defeat the Center party in the first round.

Proportional rules

Some proportional representation systems can exhibit negative responsiveness. These include the single transferable vote and some implementations of mixed-member proportional representation, generally as a result of poorly-designed overhang rules. An example can be found in the 2005 German federal election, where CDU supporters in Dresden were instructed to vote for the FDP, a strategy that allowed the CDU to win an additional seat.^[2] This led the Federal Constitutional Court to rule that negative responsiveness violates the German constitution's guarantee of equal and direct suffrage.^[14]

Frequency of violations

For electoral methods failing positive value, the frequency of less-is-more paradoxes will depend on the electoral method, the candidates, and the distribution of outcomes. Gallagher, in 2013, writes that for some social choice theorists, vulnerability to monotonicity violations is sufficient to disapprove of runoff based electoral methods, while political scientists and some other social choice theorists tend to be less concerned.^[15]

Theoretical models

Results using the impartial culture model estimate about 15% of elections with 3 candidates;^[16]^[17] however, the true probability may be much higher, especially when restricting observation to close elections.^[18] For moderate numbers of candidates, the probability of a less-is-more paradoxes quickly approaches 100%.^{[citation needed]}

A 2013 study using a two-dimensional spatial model of voting estimated at least 15% of IRV elections would be nonmonotonic in the best-case scenario (with only three equally-competitive candidates). The researchers concluded that "three-way competitive races will exhibit unacceptably frequent monotonicity failures" and "In light of these results, those seeking to implement a fairer multi-candidate election system should be wary of adopting IRV."^[10]

Real-world situations

Survey of nonmonotonic elections

A survey of 185 American instant-runoff elections where no candidate was ranked first by a majority of voters found five additional perverse elections (a total of seven).^[13]

A 2013 survey of Irish elections using IRV and PR-STV found non-monotonicity in 20 out of 1326 elections between 1922 and 2011.^[15]

Alaska 2022

Alaska's first-ever instant-runoff election resulted in a victory for Mary Peltola, but had many Republican voters ranked Peltola first, Peltola would have lost.^[19]

Burlington, Vermont

In Burlington's second IRV election, incumbent Bob Kiss was re-elected, despite losing in a head-to-head matchup with Democrat Andy Montroll (the Condorcet winner). However, if Kiss had gained more support from Wright voters, Kiss would have lost.^[13]

2005 German Election in Dresden

Some proportional representation systems can exhibit negative responsiveness. These include the single transferable vote and some implementations of mixed-member proportional representation, generally as a result of poorly-designed overhang rules. An example can be found in the 2005 German federal election, where CDU supporters in Dresden were instructed to vote for the FDP, a strategy that allowed the CDU to win an additional seat.^[2] This led the Federal Constitutional Court to rule that negative responsiveness violates the German constitution's guarantee of equal and direct suffrage.^[14]

Notes

^ Apart from majority judgment, these systems satisfy an even stronger form of positive responsiveness: if there is a tie, any increase in a candidate's rating will break the tie in that candidate's favor.

References

^ May, Kenneth O. (1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. ISSN 0012-9682. JSTOR 1907651.
^ ^a ^b ^c ^d Pukelsheim, Friedrich (2014). Proportional representation: apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.
^ Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN 0092-5853. JSTOR 2110496.
^ ^a ^b Felsenthal, Dan S. (April 2010). "Review of paradoxes afflicting various voting procedures where one out of m candidates (m ≥ 2) must be elected". GBR. pp. 1–52.
^ ^a ^b D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6, 1996
^ Felsenthal, Dan S.; Tideman, Nicolaus (2014-01-01). "Interacting double monotonicity failure with direction of impact under five voting methods". Mathematical Social Sciences. 67: 57–66. doi:10.1016/j.mathsocsci.2013.08.001. ISSN 0165-4896.
^ ^a ^b Arrow, Kenneth J. (2017-12-13). Social Choice and Individual Values. doi:10.12987/9780300186987. ISBN 978-0-300-18698-7. Since we are trying to describe social welfare and not some sort of illfare, we must assume that the social welfare function is such that the social ordering responds positively to alterations in individual values, or at least not negatively. Hence, if one alternative social state rises or remains still in the ordering of every individual without any other change in those orderings, we expect that it rises, or at least does not fall, in the social ordering.
^ Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.
^ dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN 0044-2070. Retrieved 2024-05-02.
^ ^a ^b Ornstein, Joseph T.; Norman, Robert Z. (2014-10-01). "Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections". Public Choice. 161 (1–2): 1–9. doi:10.1007/s11127-013-0118-2. ISSN 0048-5829. S2CID 30833409.
^ Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN 0092-5853. JSTOR 2110496.
^ Dančišin, Vladimír (2017-01-01). "No-show paradox in Slovak party-list proportional system". Human Affairs. 27 (1): 15–21. doi:10.1515/humaff-2017-0002. ISSN 1337-401X.
^ ^a ^b ^c Graham-Squire, Adam T.; McCune, David (2023-06-12). "An Examination of Ranked-Choice Voting in the United States, 2004–2022". Representation: 1–19. arXiv:2301.12075. doi:10.1080/00344893.2023.2221689.
^ ^a ^b dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN 0044-2070. Retrieved 2024-05-02.
^ ^a ^b Gallagher, Michael (September 2013). Monotonicity and non-monotonicity at PR-STV elections (PDF). Annual conference of the elections, public opinion and parties (EPOP) specialist group, University of Lancaster. Vol. 13.
^ Miller, Nicholas R. (2016). "Monotonicity Failure in IRV Elections with Three Candidates: Closeness Matters" (PDF). University of Maryland Baltimore County (2nd ed.). Table 2. Retrieved 2020-07-26. Impartial Culture Profiles: All, TMF: 15.1%
^ Miller, Nicholas R. (2012). Monotonicity Failure in IRV Elections With Three Candidates (PowerPoint). p. 23. Impartial Culture Profiles: All, Total MF: 15.0%
^ Quas, Anthony (2004-03-01). "Anomalous Outcomes in Preferential Voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. ISSN 0219-4937.
^ Graham-Squire, Adam; McCune, David (2024-01-02). "Ranked Choice Wackiness in Alaska". Math Horizons. 31 (1): 24–27. doi:10.1080/10724117.2023.2224675. ISSN 1072-4117.

[10] Apart from majority judgment, these systems satisfy an even stronger form of positive responsiveness: if there is a tie, any increase in a candidate's rating will break the tie in that candidate's favor.

[1] May, Kenneth O. (1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. ISSN 0012-9682. JSTOR 1907651.

[:4222-2] Pukelsheim, Friedrich (2014). Proportional representation: apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.

[3] Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN 0092-5853. JSTOR 2110496.

[:0-4] Felsenthal, Dan S. (April 2010). "Review of paradoxes afflicting various voting procedures where one out of m candidates (m ≥ 2) must be elected". GBR. pp. 1–52.

[Woodall-Monotonicity222-5] D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6, 1996

[6] Felsenthal, Dan S.; Tideman, Nicolaus (2014-01-01). "Interacting double monotonicity failure with direction of impact under five voting methods". Mathematical Social Sciences. 67: 57–66. doi:10.1016/j.mathsocsci.2013.08.001. ISSN 0165-4896.

[:12-7] Arrow, Kenneth J. (2017-12-13). Social Choice and Individual Values. doi:10.12987/9780300186987. ISBN 978-0-300-18698-7. Since we are trying to describe social welfare and not some sort of illfare, we must assume that the social welfare function is such that the social ordering responds positively to alterations in individual values, or at least not negatively. Hence, if one alternative social state rises or remains still in the ordering of every individual without any other change in those orderings, we expect that it rises, or at least does not fall, in the social ordering.

[:42322-8] Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.

[:0322-9] (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN 0044-2070. Retrieved 2024-05-02.

[Ornstein-11] Ornstein, Joseph T.; Norman, Robert Z. (2014-10-01). "Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections". Public Choice. 161 (1–2): 1–9. doi:10.1007/s11127-013-0118-2. ISSN 0048-5829. S2CID 30833409.

[12] Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN 0092-5853. JSTOR 2110496.

[13] Dančišin, Vladimír (2017-01-01). "No-show paradox in Slovak party-list proportional system". Human Affairs. 27 (1): 15–21. doi:10.1515/humaff-2017-0002. ISSN 1337-401X.

[:522-14] Graham-Squire, Adam T.; McCune, David (2023-06-12). "An Examination of Ranked-Choice Voting in the United States, 2004–2022". Representation: 1–19. arXiv:2301.12075. doi:10.1080/00344893.2023.2221689.

[:022-15] (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN 0044-2070. Retrieved 2024-05-02.

[Gallagher-16] Gallagher, Michael (September 2013). Monotonicity and non-monotonicity at PR-STV elections (PDF). Annual conference of the elections, public opinion and parties (EPOP) specialist group, University of Lancaster. Vol. 13.

[17] Miller, Nicholas R. (2016). "Monotonicity Failure in IRV Elections with Three Candidates: Closeness Matters" (PDF). University of Maryland Baltimore County (2nd ed.). Table 2. Retrieved 2020-07-26. Impartial Culture Profiles: All, TMF: 15.1%

[18] Miller, Nicholas R. (2012). Monotonicity Failure in IRV Elections With Three Candidates (PowerPoint). p. 23. Impartial Culture Profiles: All, Total MF: 15.0%

[19] Quas, Anthony (2004-03-01). "Anomalous Outcomes in Preferential Voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. ISSN 0219-4937.

[20] Graham-Squire, Adam; McCune, David (2024-01-02). "Ranked Choice Wackiness in Alaska". Math Horizons. 31 (1): 24–27. doi:10.1080/10724117.2023.2224675. ISSN 1072-4117.

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@@ Line 3: / Line 3: @@
 {{Short description|Property of electoral systems}}{{Electoral systems sidebar|expanded=Paradox}}
 [[File:IRV_Yee.svg|alt=A diagram showing who would win an IRV election for different electorates. The win region for each candidate is erratic, with random pixels dotting the image and jagged, star-shaped (convex) regions occupying much of the image. Moving the electorate to the left can cause a right-wing candidate to win, and vice versa.|thumb|300x300px|A 4-candidate [https://electowiki.org/wiki/Yee_diagram Yee diagram] under IRV. The diagram shows who would win an IRV election if the electorate is centered at a particular point. Moving the electorate to the left can cause a right-wing candidate to win, and vice versa. Black lines show the [[Voronoi diagram|optimal solution]] (achieved by [[Condorcet method|Condorcet]] or [[Score voting|score]] voting).]]
-In [[Social choice theory|social choice]], the '''negative responsiveness''',<ref>{{Cite journal |last=May |first=Kenneth O. |date=1952 |title=A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision |url=https://www.jstor.org/stable/1907651 |journal=Econometrica |volume=20 |issue=4 |pages=680–684 |doi=10.2307/1907651 |issn=0012-9682 |jstor=1907651}}</ref>'''<ref name=":4222">{{Cite book |last=Pukelsheim |first=Friedrich |url=http://archive.org/details/proportionalrepr0000puke |title=Proportional representation: apportionment methods and their applications |date=2014 |publisher=Cham; New York : Springer |others=Internet Archive |isbn=978-3-319-03855-1}}</ref>''' '''perversity''',<ref>{{Cite journal |last1=Doron |first1=Gideon |last2=Kronick |first2=Richard |date=1977 |title=Single Transferrable Vote: An Example of a Perverse Social Choice Function |url=https://www.jstor.org/stable/2110496 |journal=American Journal of Political Science |volume=21 |issue=2 |pages=303–311 |doi=10.2307/2110496 |issn=0092-5853 |jstor=2110496}}</ref> or '''additional support paradox<ref name=":0">{{Cite journal |last=Felsenthal |first=Dan S. |date=April 2010 |title=Review of paradoxes afflicting various voting procedures where one out of m candidates (m ≥ 2) must be elected |url=https://eprints.lse.ac.uk/27685/ |language=en |location=GBR |pages=1–52}}</ref>''' is a [[Pathological (mathematics)#In voting and social choice|pathological behavior]] of some [[Electoral system|voting rules]], where a candidate loses as a result of having "too much support" from some voters, or wins because they had "too much opposition". In other words, increasing (decreasing) a candidate's [[Ranked voting|ranking]] or [[Rated voting|rating]] causes that candidate to lose (win),<ref name=":0" /> contrary to common sense. Electoral systems that do not exhibit perversity are said to satisfy the '''positive response''' or [[Monotonic function|'''monotonicity''']] '''criterion'''.<ref name="Woodall-Monotonicity222">D R Woodall, [http://www.votingmatters.org.uk/ISSUE6/P4.HTM "Monotonicity and Single-Seat Election Rules"], ''[[Voting matters]]'', Issue 6, 1996</ref>
+In [[Social choice theory|social choice]], the '''negative responsiveness''',<ref>{{Cite journal |last=May |first=Kenneth O. |date=1952 |title=A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision |url=https://www.jstor.org/stable/1907651 |journal=Econometrica |volume=20 |issue=4 |pages=680–684 |doi=10.2307/1907651 |issn=0012-9682 |jstor=1907651}}</ref>'''<ref name=":4222">{{Cite book |last=Pukelsheim |first=Friedrich |url=http://archive.org/details/proportionalrepr0000puke |title=Proportional representation: apportionment methods and their applications |date=2014 |publisher=Cham; New York : Springer |others=Internet Archive |isbn=978-3-319-03855-1}}</ref>''' '''perversity''',<ref>{{Cite journal |last1=Doron |first1=Gideon |last2=Kronick |first2=Richard |date=1977 |title=Single Transferrable Vote: An Example of a Perverse Social Choice Function |url=https://www.jstor.org/stable/2110496 |journal=American Journal of Political Science |volume=21 |issue=2 |pages=303–311 |doi=10.2307/2110496 |issn=0092-5853 |jstor=2110496}}</ref> or '''additional support paradox<ref name=":0">{{Cite web <!--Is this a conference?--> |last=Felsenthal |first=Dan S. |date=April 2010 |title=Review of paradoxes afflicting various voting procedures where one out of m candidates (m ≥ 2) must be elected |url=https://eprints.lse.ac.uk/27685/ |language=en |location=GBR |pages=1–52}}</ref>''' is a [[Pathological (mathematics)#In voting and social choice|pathological behavior]] of some [[Electoral system|voting rules]], where a candidate loses as a result of having "too much support" from some voters, or wins because they had "too much opposition". In other words, increasing (decreasing) a candidate's [[Ranked voting|ranking]] or [[Rated voting|rating]] causes that candidate to lose (win),<ref name=":0" /> contrary to common sense. Electoral systems that do not exhibit perversity are said to satisfy the '''positive response''' or [[Monotonic function|'''monotonicity''']] '''criterion'''.<ref name="Woodall-Monotonicity222">D R Woodall, [http://www.votingmatters.org.uk/ISSUE6/P4.HTM "Monotonicity and Single-Seat Election Rules"], ''[[Voting matters]]'', Issue 6, 1996</ref>
 Perversity is often described by [[Social choice theory|social choice theorists]] as an exceptionally severe kind of [[Pathological (mathematics)#In voting and social choice|electoral pathology]].<ref>{{Cite journal |last1=Felsenthal |first1=Dan S. |last2=Tideman |first2=Nicolaus |date=2014-01-01 |title=Interacting double monotonicity failure with direction of impact under five voting methods |url=https://www.sciencedirect.com/science/article/pii/S0165489613000723 |journal=Mathematical Social Sciences |volume=67 |pages=57–66 |doi=10.1016/j.mathsocsci.2013.08.001 |issn=0165-4896}}</ref><ref name=":12">{{Cite book |last=Arrow |first=Kenneth J. |url=http://dx.doi.org/10.12987/9780300186987 |title=Social Choice and Individual Values |date=2017-12-13 |isbn=978-0-300-18698-7 |doi=10.12987/9780300186987 |quote=Since we are trying to describe social welfare and not some sort of illfare, we must assume that the social welfare function is such that the social ordering responds positively to alterations in individual values, or at least not negatively. Hence, if one alternative social state rises or remains still in the ordering of every individual without any other change in those orderings, we expect that it rises, or at least does not fall, in the social ordering.}}</ref> Systems that allow for perverse response can create situations where a voter's ballot has a reversed effect on the election, thus treating the [[Social welfare function|well-being of some voters]] as "less than worthless".<ref name=":12" /> Similar arguments have led to constitutional prohibitions on such systems as violating the right to [[One man, one vote|equal and direct suffrage]].<ref name=":42322">{{Cite book |last=Pukelsheim |first=Friedrich |url=http://archive.org/details/proportionalrepr0000puke |title=Proportional representation : apportionment methods and their applications |date=2014 |publisher=Cham; New York : Springer |others=Internet Archive |isbn=978-3-319-03855-1}}</ref><ref name=":0322">{{Cite news |last=dpa |date=2013-02-22 |title=Bundestag beschließt neues Wahlrecht |url=https://www.zeit.de/politik/deutschland/2013-02/bundestag-wahlrecht-beschluss |access-date=2024-05-02 |work=Die Zeit |language=de-DE |issn=0044-2070}}</ref> Negative response is often cited as an example of a [[perverse incentive]], as voting rules with perverse response incentivize politicians to take unpopular or [[Center squeeze|extreme]] positions in an attempt to shed excess votes.
@@ Line 9: / Line 9: @@
 Most [[Ranked voting|ranked methods]] (including [[Borda count|Borda]] and all common [[Round-robin voting|round-robin rules]]) satisfy positive response,<ref name="Woodall-Monotonicity222" /> as do all common [[rated voting]] methods (including [[Approval voting|approval]], [[Highest median voting rules|highest medians]], and [[Score voting|score]]).{{NoteTag|Apart from majority judgment, these systems satisfy an even stronger form of positive responsiveness: if there is a tie, any increase in a candidate's rating will break the tie in that candidate's favor.}}
-Perversity occurs in [[instant-runoff voting]],<ref name=":3222">{{Cite journal |last1=Ornstein |first1=Joseph T. |last2=Norman |first2=Robert Z. |date=2014-10-01 |title=Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections |journal=Public Choice |language=en |volume=161 |issue=1–2 |pages=1–9 |doi=10.1007/s11127-013-0118-2 |issn=0048-5829 |s2cid=30833409}}</ref> the [[single transferable vote]],<ref>{{Cite journal |last1=Doron |first1=Gideon |last2=Kronick |first2=Richard |date=1977 |title=Single Transferrable Vote: An Example of a Perverse Social Choice Function |url=https://www.jstor.org/stable/2110496 |journal=American Journal of Political Science |volume=21 |issue=2 |pages=303–311 |doi=10.2307/2110496 |issn=0092-5853 |jstor=2110496}}</ref> and [[Quota method|quota-based apportionment methods]].<ref name=":4222" /> According to statistical culture models of elections, the paradox is especially common in [[Ranked-choice voting|ranked-choice voting (RCV-IRV)]] and the [[two-round system]].<ref name=":332">{{Cite journal |last1=Ornstein |first1=Joseph T. |last2=Norman |first2=Robert Z. |date=2014-10-01 |title=Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections |journal=Public Choice |language=en |volume=161 |issue=1–2 |pages=1–9 |doi=10.1007/s11127-013-0118-2 |issn=0048-5829 |s2cid=30833409}}</ref> The [[Maximal lotteries|randomized Condorcet method]] can violate monotonicity in the case of [[Condorcet cycle|cyclic ties]].
+Perversity occurs in [[instant-runoff voting]],<ref name="Ornstein">{{Cite journal |last1=Ornstein |first1=Joseph T. |last2=Norman |first2=Robert Z. |date=2014-10-01 |title=Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections |journal=Public Choice |language=en |volume=161 |issue=1–2 |pages=1–9 |doi=10.1007/s11127-013-0118-2 |issn=0048-5829 |s2cid=30833409}}</ref> the [[single transferable vote]],<ref>{{Cite journal |last1=Doron |first1=Gideon |last2=Kronick |first2=Richard |date=1977 |title=Single Transferrable Vote: An Example of a Perverse Social Choice Function |url=https://www.jstor.org/stable/2110496 |journal=American Journal of Political Science |volume=21 |issue=2 |pages=303–311 |doi=10.2307/2110496 |issn=0092-5853 |jstor=2110496}}</ref> and [[Quota method|quota-based apportionment methods]].<ref name=":4222" /> According to statistical culture models of elections, the paradox is especially common in [[Ranked-choice voting|ranked-choice voting (RCV-IRV)]] and the [[two-round system]].{{Citation needed|date=October 2024}} The [[Maximal lotteries|randomized Condorcet method]] can violate monotonicity in the case of [[Condorcet cycle|cyclic ties]].
 The [[participation criterion]] is a closely-related, but different, concept. While positive responsiveness deals with a voter changing their opinion (or vote), participation deals with situations where a voter choosing to ''cast'' a ballot has a reversed effect on the election.<ref>{{Cite journal |last=Dančišin |first=Vladimír |date=2017-01-01 |title=No-show paradox in Slovak party-list proportional system |url=https://www.degruyter.com/document/doi/10.1515/humaff-2017-0002/html?lang=en |journal=Human Affairs |language=en |volume=27 |issue=1 |pages=15–21 |doi=10.1515/humaff-2017-0002 |issn=1337-401X}}</ref>
@@ Line 62: / Line 62: @@
 == Frequency of violations ==
-For electoral methods failing positive value, the frequency of less-is-more paradoxes will depend on the electoral method, the candidates, and the distribution of outcomes. Negative voting weights tend to be most common with [[Instant-runoff voting|instant-runoff]], with what {{ww|text=some researchers|date=October 2024}} have described as an "unacceptably high" frequency.<ref name=":332" />
+For electoral methods failing positive value, the frequency of less-is-more paradoxes will depend on the electoral method, the candidates, and the distribution of outcomes. Gallagher, in 2013, writes that for some social choice theorists, vulnerability to monotonicity violations is sufficient to disapprove of runoff based electoral methods, while political scientists and some other social choice theorists tend to be less concerned.<ref name="Gallagher"/>
 === Theoretical models ===
 Results using the [[impartial culture]] model estimate about 15% of elections with 3 candidates;<ref>{{Cite web |last=Miller |first=Nicholas R. |date=2016 |title=Monotonicity Failure in IRV Elections with Three Candidates: Closeness Matters |url=https://userpages.umbc.edu/~nmiller/MONFAILURE.R2.NRM.pdf |access-date=2020-07-26 |website=University of Maryland Baltimore County |at=Table 2 |language=en |quote=Impartial Culture Profiles: All, TMF: 15.1% |edition=2nd}}</ref><ref>{{Cite book |last=Miller |first=Nicholas R. |url=https://userpages.umbc.edu/~nmiller/RESEARCH/MF&IRV.pptx |title=Monotonicity Failure in IRV Elections With Three Candidates |year=2012 |pages=23 |type=PowerPoint |quote=Impartial Culture Profiles: All, Total MF: 15.0%}}</ref> however, the true probability may be much higher, especially when restricting observation to close elections.<ref>{{Cite journal |last=Quas |first=Anthony |date=2004-03-01 |title=Anomalous Outcomes in Preferential Voting |url=https://www.worldscientific.com/doi/abs/10.1142/S0219493704000912 |journal=Stochastics and Dynamics |language=en |volume=04 |issue=1 |pages=95–105 |doi=10.1142/S0219493704000912 |issn=0219-4937}}</ref> For moderate numbers of candidates, the probability of a less-is-more paradoxes quickly approaches 100%.{{citation needed|date=March 2024}}
-A 2013 study using a two-dimensional [[spatial model of voting]] estimated at least 15% of IRV elections would be nonmonotonic in the best-case scenario (with only three equally-competitive candidates). The researchers concluded that "three-way competitive races will exhibit unacceptably frequent monotonicity failures" and "In light of these results, those seeking to implement a fairer multi-candidate election system should be wary of adopting IRV."<ref name=":332" />
+A 2013 study using a two-dimensional [[spatial model of voting]] estimated at least 15% of IRV elections would be nonmonotonic in the best-case scenario (with only three equally-competitive candidates). The researchers concluded that "three-way competitive races will exhibit unacceptably frequent monotonicity failures" and "In light of these results, those seeking to implement a fairer multi-candidate election system should be wary of adopting IRV."<ref name="Ornstein" />
 === Real-world situations ===
@@ Line 73: / Line 73: @@
 A survey of 185 American [[Instant-runoff voting|instant-runoff]] elections where no candidate was ranked first by a majority of voters found five additional perverse elections (a total of seven).<ref name=":522" />
-A 2013 survey of Irish elections using IRV and [[PR-STV]] found non-monotonicity in 20 out of 1326 elections between 1922 and 2011.<ref>{{Cite conference |last=Gallagher |first=Michael |date=September 2013 |title=Monotonicity and non-monotonicity at PR-STV elections |url=https://www.lancaster.ac.uk/fass/events/epop2013/docs/MGallagherMonotonicityEPOP13.pdf |conference=Annual conference of the elections, public opinion and parties (EPOP) specialist group, University of Lancaster |volume=13}}</ref>
+A 2013 survey of Irish elections using IRV and [[PR-STV]] found non-monotonicity in 20 out of 1326 elections between 1922 and 2011.<ref name="Gallagher">{{Cite conference |last=Gallagher |first=Michael |date=September 2013 |title=Monotonicity and non-monotonicity at PR-STV elections |url=https://www.lancaster.ac.uk/fass/events/epop2013/docs/MGallagherMonotonicityEPOP13.pdf |conference=Annual conference of the elections, public opinion and parties (EPOP) specialist group, University of Lancaster |volume=13}}</ref>
 ==== Alaska 2022 ====