Negative responsiveness: Difference between revisions
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[[Category:Electoral system criteria]] |
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{{about|a voting system criterion|the mathematical notion of an order preserving mapping|monotonic function}} |
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{{Short description|Property of electoral systems}}{{Electoral systems sidebar|expanded=Paradox}} |
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[[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 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).]] |
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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 name="jstor.org">{{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" /> 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> |
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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 name="Felsenthal-severe">{{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 |quote=it is generally agreed among social choice theorists that a voting method that is susceptible to any type of monotonicity failure suffers from a particularly serious defect.}}</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="Arrow">{{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 |p=25 |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> 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. |
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The '''monotonicity criterion''' is a [[voting system criterion]] used to evaluate both single and multiple winner [[ranked voting system]]s. A ranked voting system is '''monotonic''' if it is neither possible to prevent the election of a candidate by ranking them higher on some of the ballots, nor possible to elect an otherwise unelected candidate by ranking them lower on some of the ballots (while nothing else is altered on any ballot). |
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<ref name="Woodall-Monotonicity">D R Woodall, [http://www.votingmatters.org.uk/ISSUE6/P4.HTM "Monotonicity and Single-Seat Election Rules"], ''[[Voting matters]]'', Issue 6, 1996</ref> That is to say, in single winner elections no winner is harmed by up-ranking and no loser is helped by down-ranking. Douglas R. Woodall called the criterion '''mono-raise'''. |
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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.}} |
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Raising a candidate {{math|''x''}} on some ballots ''while changing'' the orders of other candidates does ''not'' constitute a failure of monotonicity. E.g., harming candidate {{math|''x''}} by changing some ballots from {{math|''z'' > ''x'' > ''y''}} to {{math|''x'' > ''y'' > ''z''}} isn't a violation of the monotonicity criterion. |
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Perversity occurs in [[instant-runoff voting]] (IRV),<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 name="jstor.org"/> and [[Quota method|quota-based apportionment methods]].<ref name=":4222" /> According to statistical culture models of elections, the paradox is especially common in 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]]. |
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The monotonicity criterion renders the intuition that there should be neither need to worry about harming a candidate by (nothing else than) up-ranking nor it should be possible to support a candidate by (nothing else than) counter-intuitively down-ranking. <!-- |
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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> |
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(Something similar (down-ranking CDU '''and''' additionally up-ranking another party, e.g. FDP) happened in the German federal election of 2005, in which conservative voters in Dresden deliberately voted against their party of choice (the CDU) in order to maximize that party's number of seats in the federal parliament. This was possible due to Germany's voting system (mixed member proportional with overhang seats computed independently for each federal state) and the fact that the vote in Dresden took place a week after the rest of the country due to the death of a candidate, enabling voters in Dresden to vote tactically in full knowledge of the results already achieved elsewhere. As a result of this, the German Constitutional Court ruled on July 3 2008 that the German voting system must be reformed to eliminate its non-monotonicity.<ref> See e.g. [http://fruitsandvotes.com/blog/?p=117]</ref> |
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==Definition== |
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-->There are several variations of that criterion; e.g., what Douglas R. Woodall called ''mono-add-plump'': A candidate {{math|''x''}} should not be harmed if further ballots are added that have {{math|''x''}} top with no second choice. Agreement with such rather special properties is the best any ranked voting system may fulfill: The [[Gibbard–Satterthwaite theorem]] shows, that any meaningful ranked voting system is susceptible to some kind of [[tactical voting]], and [[Arrow's impossibility theorem]] shows that individual rankings can't be meaningfully translated into a community-wide ranking where the order of candidates {{math|''x''}} and {{math|''y''}} is always [[Independence of irrelevant alternatives|independent of irrelevant alternatives]] {{math|''z''}}.<!-- |
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Miller defined two main classes of monotonicity failure in 2012, which have been repeated in later papers:<ref name="Miller2012"><!--Kinda think we should replace this one with the PDF version, but I can't tell if it's too different-->{{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><ref name="Felsenthal-severe"/> |
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{{bq| |
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''Upward monotonicity failure'': Given the use of voting method V and a ballot profile B in which candidate X is the winner, X may nevertheless lose in ballot profile B' that differs from B only in that some voters rank X higher in B' than in B{{pb}} |
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''Downward monotonicity failure'': Given the use of voting method V and a ballot profile B in which candidate X is a loser, X may nevertheless win in ballot profile B' that differs from B only in that some voters rank X lower in B' than in B. |
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}} |
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In simpler terms, an upward failure occurs when a winner loses from more support, and a downward failure occurs when a loser wins with less support. |
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== By method == |
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The result of David Austen-Smith and Jeffrey Banks that monotonicity in individual preferences is impossible is a nonissue: For given voter preferences v=v_1...v_n and a winner x under voting scheme alpha, they investigate changes in v, where e.g. altering v_i from a,b,c,d,x to d,c,x,b,a is allowed, which can't be seriously named a monotonicity property. That allows random permutations even ''ahead'' of x, and is therefore even more rigid than Woodall's mono-raise-random, which is already incompatible with [majority AND later-no-help AND later-no-harm].<ref>''[http://www.jstor.org/stable/1963173 Monotonicity in Electoral Systems]''. The American Political Science Review Vol. 85, No. 2, June 1991; p. 533.</ref> |
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=== Runoff voting === |
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[[Instant-runoff voting|Runoff-based voting]] systems such as [[Instant-runoff voting|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 voting|instant-runoff election]] in the modern era, where [[Bob Kiss]] won the election as a result of 750 ballots ranking him in last place.<ref name=":522">{{Cite journal |last1=Graham-Squire |first1=Adam T. |last2=McCune |first2=David |date=2023-06-12 |title=An Examination of Ranked-Choice Voting in the United States, 2004–2022 |journal=Representation |language=en |pages=1–19 |arxiv=2301.12075 |doi=10.1080/00344893.2023.2221689}}</ref> Another example is given by the [[2022 Alaska's at-large congressional district special election|2022 Alaska at-large special election]]. |
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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 ''un''successful campaign and adopting an ''un''popular platform, which pushes their supporters away from the party and into the Top party. |
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--> Noncompliance with the monotonicity criterion doesn't tell anything about the likelihood of monotonicity violations, failing in one of a million possible elections would be as well a violation as missing the criterion in any possible election. |
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{| class="wikitable" |
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|+ |
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Of the single-winner ranked voting systems, [[Borda count|Borda]], [[Schulze method|Schulze]], [[ranked pairs]], maximize affirmed majorities, descending solid coalitions,<ref>[https://electowiki.org/wiki/Descending_Solid_Coalitions Electowiki:Descending Solid Coalitions].</ref> and descending acquiescing coalitions<ref name="Woodall-Monotonicity" /><ref>[https://electowiki.org/wiki/Descending_Acquiescing_Coalitions Electowiki:Descending Acquiescing Coalitions].</ref> are monotonic, while [[Coombs' method]], [[Two-round system|runoff voting]], and [[instant-runoff voting]] (IRV) are not. |
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! colspan="3" |Popular Bottom |
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! |
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Most variants of the [[single transferable vote]] (STV) [[proportional representation]]s are not monotonic, especially all that are currently in use for public elections (which simplify to IRV when there is only one winner). |
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! colspan="3" |Unpopular Bottom |
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All [[plurality voting system]]s are monotonic if the ballots are treated as rankings where using ''more than two ranks is forbidden''. In this setting [[first past the post]] and [[approval voting]] as well as the multiple-winner systems [[single non-transferable vote]], [[plurality-at-large voting]] (multiple non-transferable vote, bloc voting) and [[cumulative voting]] are monotonic. [[Party-list proportional representation]] using [[D'Hondt method|D'Hondt]], [[Sainte-Laguë method|Sainte-Laguë]] or the [[largest remainder method]] is monotonic in the same sense. |
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In elections via the single-winner methods [[range voting]] and [[majority judgment]] nobody can help a candidate by reducing or removing support for them. The definition of the monotonicity criterion with regard to these methods is disputed. Some voting theorists argue that this means these methods pass the monotonicity criterion; others say that, as these are not ''ranked'' voting systems, they are out of the monotonicity criterion's scope. |
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==Instant-runoff voting and the two-round system are not monotonic== |
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Using an example that applies to [[instant-runoff voting]] (IRV) and to the [[two-round system]], it is shown that these voting systems violate the mono-raise criterion. |
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Suppose a [[President (government title)|president]] were being elected among three candidates, a left, a right, and a center candidate, and 100 votes cast. The number of votes for an absolute majority is therefore 51. |
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Suppose the votes are cast as follows: |
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{| class="wikitable" border="1" |
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|- |
|- |
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! |
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! colspan=2 | Preference |
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!Round 1 |
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! rowspan=2 | Voters |
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!Round 2 |
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! |
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! |
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!Round 1 |
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!Round 2 |
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|- |
|- |
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!'''''Top''''' |
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! 1st |
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|<s>25%</s> {{Xmark}} |
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! 2nd |
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| |
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!+6% |
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!'''''Top''''' |
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|31% |
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|46% |
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|- |
|- |
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!'''''Center''''' |
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| Right |
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|30% |
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| Center |
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|'''55%''' {{Tick}} |
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| 28 |
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!↗ |
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!'''''Center''''' |
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|<s>30% {{Xmark}}</s> |
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| |
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|- |
|- |
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!'''''Bottom''''' |
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| Right |
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|45% |
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| Left |
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| |
|45% |
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!-6% |
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!'''''Bottom''''' |
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| Left |
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|39% |
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| Center |
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|'''54% {{Tick}}''' |
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| 30 |
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|- |
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| Left |
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| Right |
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| 5 |
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|- |
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| Center |
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| Left |
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| 16 |
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|- |
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| Center |
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| Right |
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| 16 |
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|} |
|} |
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This election is an example of a [[center-squeeze]], a class of elections where instant-runoff and [[Plurality voting|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. |
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=== Proportional rules === |
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According to the 1st preferences, Left finishes first with 35 votes, Right gets 33 votes, and Center 32 votes, thus all candidates lack an absolute majority of first preferences. |
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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 seat|overhang rules]]. An example can be found in the [[2005 German federal election]], where [[Christian Democratic Union of Germany|CDU]] supporters in [[Dresden]] were instructed to vote for the [[Free Democratic Party (Germany)|FDP]], a strategy that allowed the CDU to win an additional seat.<ref name=":4222" /> This led the [[Federal Constitutional Court]] to rule that negative responsiveness violates the [[Basic Law for the Federal Republic of Germany|German constitution]]'s guarantee of [[One man, one vote|equal and direct suffrage]].<ref name=":0322"/> |
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In an actual runoff between the top two candidates, Left would win against Right with 30+5+16=51 votes. The same happens (in this example) under IRV, Center gets eliminated, and Left wins against Right with 51 to 49 votes. |
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== Frequency of violations == |
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But if at least two of the five voters who ranked Right first, and Left second, would raise Left, and vote 1st Left, 2nd Right; then Right would be defeated by these votes in favor of Center. |
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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. Social choice theorists generally agree that non-monotonicity is a particularly serious defect.<ref name="Felsenthal-severe"/> 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"/> |
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Let's assume that two voters change their preferences in that way, which changes two rows of the table: |
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===Empirical analysis=== |
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{| class="wikitable" border="1" |
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In the US, a 2021 analysis of [[Instant-runoff voting|instant-runoff]] elections in California between 2008 and 2016, as well as the [[2009 Burlington, Vermont mayoral election]], found an upward monotonicity anomaly rate of 0.74% (1/135) in all elections, 2.71% (1/37) when limited to elections going to a second round of counting and 7.7% (1/13) of elections with three competitive candidates.<ref>{{cite journal |last1=Graham-Squire |first1=Adam |last2=Zayatz |first2=N. |title=Lack of Monotonicity Anomalies in Empirical Data of Instant-runoff Elections |journal=Representation |date=2 October 2021 |volume=57 |issue=4 |pages=565–573 |doi=10.1080/00344893.2020.1785536}}</ref><ref name="McCune"/> A more comprehensive 2023 survey of 182 American IRV elections where no candidate was ranked first by a majority of voters found seven total examples of non-monotonicity (3.8%), broken down into 2.2% (4/182) examples of upward monotonicity, 1.6% (3/182) of downward montonicity and 0.5% (1/182) of no-show or truncation (one example was both an upward and downward monotonicity failure).<ref name=":522" /><ref name="McCune">{{cite journal |last1=McCune |first1=David |last2=Graham-Squire |first2=Adam |title=Monotonicity anomalies in Scottish local government elections |journal=Social Choice and Welfare |date=August 2024 |volume=63 |issue=1 |pages=69–101 |doi=10.1007/s00355-024-01522-5 |doi-access=free}}</ref> Two of those elections are also noted as specific examples below. |
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|- |
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! colspan=2 | Preference |
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! rowspan=2 | Voters |
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|- |
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! 1st |
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! 2nd |
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|- |
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| Right |
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| Left |
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| 3 |
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|- |
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| Left |
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| Right |
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| 7 |
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|} |
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====Semi-empirical==== |
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Now Left gets 37 first preferences, Right only 31 first preferences, and Center still 32 first preferences, and there is again no candidate with an absolute majority of first preferences. |
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Some empirical research do not have access to full ballot preference data, and thus make probabilistic estimates of transfer patterns. A 2013 survey of Irish elections using IRV and [[PR-STV]] found plausible 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> |
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But now Right gets eliminated, and Center remains in round 2 of IRV (or the actual runoff in the Two-round system). And Center beats its opponent Left with a remarkable majority of 60 to 40 votes. |
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Data from the five UK general elections between 1992 and 2010 showed 2642 three candidate elections in English constituencies. With second preferences imputed from survey data, 1.7% of all elections appeared vulnerable to monotonicity anomalies (1.4% upward, 0.3% downward), significantly lower than simulated datasets from the same paper. However, when limited to the 4.2% of elections considered three-way competitive<!--This should be the 111 elections where PL support >25% but might need to check that, also probably get consensus that it's correct per CALC-->, 40.2% appeared vulnerable (33% upward, 7.1% downward), and further increasing with closer competition, a result closer to the simulations.<ref>{{Cite journal |last=Miller |first=Nicholas R. |date=October 2017 |title=Closeness matters: monotonicity failure in IRV elections with three candidates |journal=Public Choice |volume=173 |issue=1-2 |pages=91–108 |doi=10.1007/s11127-017-0465-5 |hdl=11603/20938 |hdl-access=free}}</ref> |
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===Estimated likelihood of IRV lacking monotonicity=== |
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A 2022 analysis out of the 10 French presidential elections (conducted under the [[two-round system]]) 2 had results where monotonicity violations were not mathematically possible, another 6 where violations were unlikely given the evidence, leaving 2 elections ([[2002 French presidential election|2002]] and [[2007 French presidential election|2007]]) where an upward monotonicity violation was probable and likely respectively.<ref>{{cite journal |last1=Keskin |first1=Umut |last2=Sanver |first2=M. Remzi |last3=Tosunlu |first3=H. Berkay |url=https://hal.science/hal-03413280/document |title=Monotonicity violations under plurality with a runoff: the case of French presidential elections |journal=Social Choice and Welfare |date=August 2022 |volume=59 |issue=2 |pages=305–333 |doi=10.1007/s00355-022-01397-4}}</ref> |
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Crispin Allard argued, based on a mathematical model that the probability of monotonicity failure actually changing the result of an election for any given [[European Parliament constituency|constituency]] would be 1 in 4000;<ref>{{Cite web|url=http://www.votingmatters.org.uk/ISSUE5/P1.HTM|title=Estimating the Probability of Monotonicity Failure in a UK General Election|last=Allard|first=Crispin|date=January 1996|website=Voting matters - Issue 5|archive-url=|archive-date=|access-date=2017-03-14}}</ref> however, Lepelley ''et al.''<ref>{{Cite journal|last=Lepelley|first=Dominique|last2=Chantreuil|first2=Frédéric|last3=Berg|first3=Sven|year=1996|title=The likelihood of monotonicity paradoxes in run-off elections|journal=Mathematical Social Sciences|language=en|volume=31|issue=3|pages=133–146|doi=10.1016/0165-4896(95)00804-7}}</ref> found a probability of {{nowrap|397/6912 {{=}} 5.74%}} for 3-candidate elections. |
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=== Theoretical models=== |
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Another probability model, the "[[impartial culture]]", yields about 15% probability.{{Citation needed|date=November 2017}} In elections with more than 3 candidates, these probabilities tend to increase eventually toward 100% (in some models this limit has been proven, in others it is only conjectured). Estimates of 5–15% order are easily confirmed in any probability model with [[Monte_Carlo_method|Monte Carlo]] experiments and the aid of the "was it monotonic?" tests stated in the Lepelley paper.{{Citation needed|date=June 2011}} Nicholas Miller also disputed Allard's conclusion and provided a different mathematical model.<ref>{{Cite web|url=https://www.researchgate.net/publication/228435339|title=Monotonicity Failure Under STV and Related Voting Systems|last=Miller|first=Nicholas R.|date=|website=ResearchGate|language=en|archive-url=|archive-date=|access-date=2017-03-14}}</ref> |
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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 name="Miller2012"/> 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> |
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A 2013 study using a [[ |
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" /> |
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=== Specific examples === |
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==Real-life monotonicity violations== |
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==== Alaska 2022 ==== |
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[[2022 Alaska's at-large congressional district special election|Alaska's first-ever instant-runoff election]] resulted in a victory for [[Mary Peltola]], but had many [[Republican Party (United States)|Republican]] voters ranked Peltola first, Peltola would have lost.<ref>{{Cite journal |last1=Graham-Squire |first1=Adam |last2=McCune |first2=David |date=2024-01-02 |title=Ranked Choice Wackiness in Alaska |url=https://www.tandfonline.com/doi/full/10.1080/10724117.2023.2224675 |journal=Math Horizons |language=en |volume=31 |issue=1 |pages=24–27 |doi=10.1080/10724117.2023.2224675 |issn=1072-4117}}</ref> |
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==== Burlington, Vermont ==== |
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If the ballots of a real election are released, it is fairly easy to prove if |
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In [[2009 Burlington, Vermont mayoral election|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.<ref name=":522" /> |
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* election of a candidate could have been circumvented by raising them on some of the ballots, or |
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* election of an otherwise unelected candidate by lowering them on some of the ballots |
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==== 2005 German Election in Dresden ==== |
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would have been possible (nothing else is altered on any ballot). Both events can be considered as real-life monotonicity violations. |
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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 seat|overhang rules]]. An example can be found in the [[2005 German federal election]], where [[Christian Democratic Union of Germany|CDU]] supporters in [[Dresden]] were instructed to vote for the [[Free Democratic Party (Germany)|FDP]], a strategy that allowed the CDU to win an additional seat.<ref name=":4222" /> This led the [[Federal Constitutional Court]] to rule that negative responsiveness violates the [[Basic Law for the Federal Republic of Germany|German constitution]]'s guarantee of [[One man, one vote|equal and direct suffrage]].<ref name=":0322"/> |
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== See also == |
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However, the ballots (or information allowing them to be reconstructed) are rarely released for ranked voting elections, which means there are few recorded monotonicity violations for real elections. |
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* [[Participation criterion]], a closely-related concept |
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===2009 Burlington, Vermont mayoral election=== |
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* [[Voting system]] |
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* [[Voting system criterion]] |
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* [[Monotone preferences]] in consumer theory |
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* [[Monotonicity (mechanism design)]] |
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* [[Maskin monotonicity]] |
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== Notes == |
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A monotonicity violation could have occurred in the [[2009 Burlington, Vermont mayoral election|2009 Burlington, Vermont mayor election]] under instant-runoff voting (IRV), where the necessary information is available. In this election, the winner [[Bob Kiss]] could have been defeated by raising him on some of the ballots. For example, if all voters who ranked Republican [[Kurt Wright]] over Progressive Bob Kiss over Democrat Andy Montroll, would have ranked Kiss over Wright over Montroll, and additionally some people who ranked Wright but not Kiss or Montroll, would have ranked Kiss over Wright, then these votes in favor of Kiss would have defeated him.<ref>[http://www.rangevoting.org/Burlington.html Burlington Vermont 2009 IRV mayor election]</ref> The winner in this scenario would have been Andy Montroll, who was also the [[Concordet method|Condorcet winner]] according to the original ballots, i.e. for any other running candidate, a majority ranked Montroll above the competitor. This hypothetical monotonicity violating scenario, however, would require that right-leaning voters switch to the most left-wing candidate. |
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<references group="note" responsive="1"></references> |
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===Australian elections and by-elections=== |
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{{More citations needed section|date=April 2011}} |
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Since every or almost every IRV election in Australia has been conducted in the black (i.e. not releasing enough information to reconstruct the ballots), non-monotonicity is difficult to detect in Australia. |
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However in 2009, the theoretical disadvantage of non-monotonicity worked out in practice in the [[2009 Frome state by-election]], in an election between the [[National Party of Australia]], the [[Australian Labor Party]], the [[Australian Liberal Party]] and Independent candidate [[Geoff Brock]]. The eventual winner was Brock who was a town mayor who scored only third on the primaries, with about 24% of the vote. But since the National scored fourth place, their preferences were distributed beforehand. This allowed the independent to overtake the Labor candidate by 31 votes. Thus Labor was pushed into third place, and most of its votes where transferred to the independent, who overtook the leading Liberal candidate to win the election. However, if a number of voters who preferred Liberal had voted Labor, the independent would have been removed from the race, and it would have allowed the Liberal to win the IRV election. For this to happen, between 31 and 321 voter who voted Liberal would have had to have voter Labor. Less than that, the independent would have won anyway. More than that, the Labor candidate would have won. This is classic monotonicity violation: part of those who voted for the Liberals took part in hurting their own candidate.<ref>{{Cite web|url=http://blogs.abc.net.au/antonygreen/2011/05/an-example-of-non-monotonicity-and-opportunites-for-tactical-voting-at-an-australian-election.html|title=An Example of Non-Monotonicity and Opportunites [sic] for Tactical Voting at an Australian Election|last=|first=|date=2011-05-04|website=Antony Green's Election Blog|archive-url=https://web.archive.org/web/20110508002558/http://blogs.abc.net.au/antonygreen/2011/05/an-example-of-non-monotonicity-and-opportunites-for-tactical-voting-at-an-australian-election.html|archive-date=2011-05-08|access-date=2017-03-14}}</ref> |
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== See also == |
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*[[Voting system]] |
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*[[Voting system criterion]] |
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*[[Monotone preferences]] in consumer theory. |
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*[[Monotonicity (mechanism design)]] |
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*[[Maskin monotonicity]] |
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== References == |
== References == |
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<references responsive="1"></references> |
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{{reflist}} |
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{{voting systems}} |
{{voting systems}} |
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{{DEFAULTSORT:Monotonicity Criterion}} |
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[[Category:Electoral system criteria]] |
[[Category:Electoral system criteria]] |
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Latest revision as of 22:18, 18 December 2024
| A joint Politics and Economics series |
| Social choice and electoral systems |
|---|
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 Mathematics portal |
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] 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] 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 (IRV),[10] the single transferable vote,[3] and quota-based apportionment methods.[2] According to statistical culture models of elections, the paradox is especially common in 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.[11]
Definition
[edit]Miller defined two main classes of monotonicity failure in 2012, which have been repeated in later papers:[12][6]
Upward monotonicity failure: Given the use of voting method V and a ballot profile B in which candidate X is the winner, X may nevertheless lose in ballot profile B' that differs from B only in that some voters rank X higher in B' than in B
Downward monotonicity failure: Given the use of voting method V and a ballot profile B in which candidate X is a loser, X may nevertheless win in ballot profile B' that differs from B only in that some voters rank X lower in B' than in B.
In simpler terms, an upward failure occurs when a winner loses from more support, and a downward failure occurs when a loser wins with less support.
By method
[edit]Runoff voting
[edit]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 | 
|
+6% | Top | 31% | 46% | |
| Center | 30% | 55% 
|
↗ | Center | |
|
| Bottom | 45% | 45% | -6% | Bottom | 39% | 54%
|
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
[edit]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.[9]
Frequency of violations
[edit]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. Social choice theorists generally agree that non-monotonicity is a particularly serious defect.[6] 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.[14]
Empirical analysis
[edit]In the US, a 2021 analysis of instant-runoff elections in California between 2008 and 2016, as well as the 2009 Burlington, Vermont mayoral election, found an upward monotonicity anomaly rate of 0.74% (1/135) in all elections, 2.71% (1/37) when limited to elections going to a second round of counting and 7.7% (1/13) of elections with three competitive candidates.[15][16] A more comprehensive 2023 survey of 182 American IRV elections where no candidate was ranked first by a majority of voters found seven total examples of non-monotonicity (3.8%), broken down into 2.2% (4/182) examples of upward monotonicity, 1.6% (3/182) of downward montonicity and 0.5% (1/182) of no-show or truncation (one example was both an upward and downward monotonicity failure).[13][16] Two of those elections are also noted as specific examples below.
Semi-empirical
[edit]Some empirical research do not have access to full ballot preference data, and thus make probabilistic estimates of transfer patterns. A 2013 survey of Irish elections using IRV and PR-STV found plausible non-monotonicity in 20 out of 1326 elections between 1922 and 2011.[14]
Data from the five UK general elections between 1992 and 2010 showed 2642 three candidate elections in English constituencies. With second preferences imputed from survey data, 1.7% of all elections appeared vulnerable to monotonicity anomalies (1.4% upward, 0.3% downward), significantly lower than simulated datasets from the same paper. However, when limited to the 4.2% of elections considered three-way competitive, 40.2% appeared vulnerable (33% upward, 7.1% downward), and further increasing with closer competition, a result closer to the simulations.[17]
A 2022 analysis out of the 10 French presidential elections (conducted under the two-round system) 2 had results where monotonicity violations were not mathematically possible, another 6 where violations were unlikely given the evidence, leaving 2 elections (2002 and 2007) where an upward monotonicity violation was probable and likely respectively.[18]
Theoretical models
[edit]Results using the impartial culture model estimate about 15% of elections with 3 candidates;[19][12] however, the true probability may be much higher, especially when restricting observation to close elections.[20]
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]
Specific examples
[edit]Alaska 2022
[edit]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.[21]
Burlington, Vermont
[edit]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
[edit]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.[9]
See also
[edit]- Participation criterion, a closely-related concept
- Voting system
- Voting system criterion
- Monotone preferences in consumer theory
- Monotonicity (mechanism design)
- Maskin monotonicity
Notes
[edit]- ^ 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
[edit]- ^ 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.
- ^ a b 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
- ^ a b c 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.
it is generally agreed among social choice theorists that a voting method that is susceptible to any type of monotonicity failure suffers from a particularly serious defect.
- ^ Arrow, Kenneth J. (2017-12-13). Social Choice and Individual Values. p. 25. 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.
- ^ a b c 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.
- ^ 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 Miller, Nicholas R. (2012). Monotonicity Failure in IRV Elections With Three Candidates (PowerPoint). p. 23.
Impartial Culture Profiles: All, Total MF: 15.0%
- ^ 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 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.
- ^ Graham-Squire, Adam; Zayatz, N. (2 October 2021). "Lack of Monotonicity Anomalies in Empirical Data of Instant-runoff Elections". Representation. 57 (4): 565–573. doi:10.1080/00344893.2020.1785536.
- ^ a b McCune, David; Graham-Squire, Adam (August 2024). "Monotonicity anomalies in Scottish local government elections". Social Choice and Welfare. 63 (1): 69–101. doi:10.1007/s00355-024-01522-5.
- ^ Miller, Nicholas R. (October 2017). "Closeness matters: monotonicity failure in IRV elections with three candidates". Public Choice. 173 (1–2): 91–108. doi:10.1007/s11127-017-0465-5. hdl:11603/20938.
- ^ Keskin, Umut; Sanver, M. Remzi; Tosunlu, H. Berkay (August 2022). "Monotonicity violations under plurality with a runoff: the case of French presidential elections". Social Choice and Welfare. 59 (2): 305–333. doi:10.1007/s00355-022-01397-4.
- ^ 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%
- ^ 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.