Mono-raise criterion
| A joint Politics and Economics series |
| Social choice and electoral systems |
|---|
 |
 Mathematics portal |
The mono-raise or monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. 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). That is to say, in single winner elections no winner is harmed by up-ranking and no loser is helped by down-ranking. Douglas Woodall called the criterion "mono-raise" in order to distinguish it from other monotonicity-related criteria that he had defined.[1]
The mono-raise criterion renders the intuition that there should be no need to worry about harming a candidate by (nothing else than) up-ranking nor should it be possible to support a candidate by (nothing else than) counter-intuitively down-ranking.
Voting systems that violate monotonicity can be said to exhibit the negative or perverse response paradox. Perversity is often described by social choice theorists as an exceptionally severe kind of electoral pathology,[2] as such rules can have "backwards" responses to voters' opinions, where popularity causes defeat while unpopularity leads to a win.[3] Similar rules treat the well-being of some voters as "less than worthless".[4] These issues have led to constitutional prohibitions on such systems as violating the right to equal and direct suffrage.[5][6]
Of the single-winner ranked voting systems, Borda, Schulze, ranked pairs, maximize affirmed majorities, descending solid coalitions,[7] and descending acquiescing coalitions[1][8] are monotonic, while Coombs' method, runoff voting, and instant-runoff voting (IRV) are not. The multi-winner single transferable vote (STV) system is also non-monotonic.
While Woodall articulated monotonicity in the context of ordinal voting systems, the property can be generalized to plurality and cardinal voting systems by evaluating whether reducing or removing support for a candidate can help that candidate win an election. In this context, first past the post is monotonic, as are all common rated voting methods (including approval, highest medians, and score).[note 1]. Party-list proportional representation using D'Hondt, Sainte-Laguë or the largest remainder method is monotonic in the same sense.
Definition
[edit]Miller defined two main classes of monotonicity failure in 2012, which have been repeated in later papers:[9][2]
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.
Views
[edit]Social choice theorists generally agree that non-monotonicity is an especially severe issue for a voting rule.[2] Some have argued that the mere possibility should be enough to disqualify runoff-based electoral methods, while others argue that this is only true if it occurs in "easy" or "common" cases.[10]
As Gallagher notes, some political scientists are less concerned, arguing that we should not worry about non-monotonicity because voters will not notice it, making it appear random from their perspective and preventing exploitation by strategic voters.[10] By contrast, other researchers have argued that voters will easily predict non-monotonicity and respond by strategically down-ranking their preferred candidates.[11]
Instant-runoff voting and the two-round system are not monotonic
[edit]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. Suppose a president were being elected among three candidates, a left, a right, and a center candidate, and 100 votes were cast. The number of votes for an absolute majority is therefore 51.
Suppose the votes are cast as follows in two scenarios:
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||
According to the 1st preferences in Scenario 1, Left finishes first with 35 votes, Right gets 33 votes, and Center 32 votes, thus all candidates lack an absolute majority of first preferences. 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.
Scenario 2 differs from Scenario 1 only by two voters: instead of ranking Right first, Left second, they rank Left first, Right second. Now Left gets 37 first preferences, Right receives 31 first preferences, and Center still receives 32 first preferences, and there is again no candidate with an absolute majority of first preferences. But now Right is 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.
Frequency of violations
[edit]The frequency of non-monotonicity will depend on the electoral method, the candidates, and the distribution of outcomes.
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.[12][13] 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] Two of those elections are also noted as specific examples below.
Semi-empirical
[edit]Some empirical researchers 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.[10]
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.[14]
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.[15]
Theoretical models
[edit]Results using the impartial culture model estimate about 15% of elections with 3 candidates;[16][9] however, the true probability may be much higher, especially when restricting observation to close elections.[17]
A 2013 study using a two-dimensional spatial model of voting estimated at least 15% of IRV elections would be non-monotonic 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."[18]
Specific examples
[edit]Alaska 2022
[edit]Alaska's first-ever instant-runoff election resulted in a victory for Democrat Mary Peltola, but had many voters for Republican Sarah Palin instead ranked Peltola first, Peltola would have lost.[19]
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.[20]
2005 German Election in Dresden
[edit]Some proportional representation systems can exhibit non-monotonity. 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.[21] This led the Federal Constitutional Court to rule that non-monotonicity violates the German constitution's guarantee of equal and direct suffrage.[5][6]
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 rated systems satisfy an even stronger form of monotonicity: if there is a tie, any increase in a candidate's rating will break the tie in that candidate's favor.
References
[edit]- ^ 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. 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.
- ^ 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.
- ^ a b Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.
- ^ a b dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN 0044-2070. Retrieved 2024-05-02.
- ^ Electowiki:Descending Solid Coalitions.
- ^ Electowiki:Descending Acquiescing Coalitions.
- ^ 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 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.
- ^ Austen-Smith, David; Banks, Jeffrey (1991). "Monotonicity in Electoral Systems". The American Political Science Review. 85 (2): 531–537. doi:10.2307/1963173. JSTOR 1963173.
- ^ 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. arXiv:2305.17741. doi:10.1007/s00355-024-01522-5.
- ^ Miller, Nicholas R. (October 2017). "Closeness matters: monotonicity failure in IRV elections with three candidates" (PDF). Public Choice. 173 (1–2): 91–108. doi:10.1007/s11127-017-0465-5. hdl:11603/20938.
Impartial Culture Profiles: All, TMF: 15.1%
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Pukelsheim, Friedrich (2014). Proportional representation: apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.