Wikipedia

Negative responsiveness

(Redirected from Monotonicity criterion)

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]

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.
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 optimal solution (achieved by Condorcet or score voting).

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

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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

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Runoff voting

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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

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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 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

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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.[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

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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

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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.[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

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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

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Alaska 2022

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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

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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

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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 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

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Notes

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  1. ^ 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

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  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.
  2. ^ 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.
  3. ^ 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.
  4. ^ 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.
  5. ^ a b D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6, 1996
  6. ^ 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.
  7. ^ 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.
  8. ^ Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.
  9. ^ a b c dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN 0044-2070. Retrieved 2024-05-02.
  10. ^ 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.
  11. ^ 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.
  12. ^ 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%
  13. ^ 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.
  14. ^ 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.
  15. ^ 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.
  16. ^ 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.
  17. ^ 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.
  18. ^ 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.
  19. ^ 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%
  20. ^ Quas, Anthony (2004-03-01). "Anomalous Outcomes in Preferential Voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. ISSN 0219-4937.
  21. ^ 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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