Negative responsiveness

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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). IRV Yee.svg
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).

In social choice, the negative response, [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 (or wins because of increased opposition). In other words, increasing (decreasing) a candidate's ranking or rating causes that candidate to lose (win), respectively. [4] Electoral systems that do not exhibit perversity are sometimes called monotonic. [5]

Contents

Perversity is often described by social choice theorists as an exceptionally severe kind of electoral pathology, [6] as such rules can have "backwards" responses to voters' opinions, where popularity causes defeat while unpopularity leads to a win. [7] Similar rules treat the well-being of some voters as "less than worthless". [8] These issues have led to constitutional prohibitions on such systems as violating the right to equal and direct suffrage. [9] [10] Negative response is often cited as an example of perverse incentives, as rules with perverse response can incentivize politicians to take extreme or unpopular positions in an attempt to shed excess votes. [11]

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]

Negative responsiveness occurs in instant-runoff voting (IRV), [12] the single transferable vote, [3] and the two-round system. [11] Some quota-based apportionment methods also violate the rule, [13] as can the randomized Condorcet method in cases of cyclic ties.

The participation criterion is closely-related, but different. While positive responsiveness deals with a voter changing their opinion (or vote), participation deals with situations where a voter choosing to cast a ballot at all has a backwards effect on the election. [13]

Definition

Miller defined two main classes of monotonicity failure in 2012, which have been repeated in later papers: [14] [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.

Views

Social choice theorists generally agree that negative responsiveness is an especially severe issue for a voting rule. [6] Some have argued the mere possibility should be enough to disqualify runoff-based electoral methods, while others argue this is only true if it occurs in "easy" or "common" cases. [15]

Gallagher notes some political scientists are less concerned, arguing we should not worry about negative response because voters will not notice it, making it appear random from their perspective and preventing exploitation by strategic voters. [15] By contrast, other researchers have argued voters will easily predict negative response and respond by strategically down-ranking their preferred candidates, keeping negative response from affecting the results. [16]

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. [17] 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 BottomUnpopular Bottom
Round 1Round 2Round 1Round 2
Top25%X mark.svg+6%Top31%46%
Center30%55%Yes check.svg↗Center30% X mark.svg
Bottom45%45%-6%Bottom39%54% Yes check.svg

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. [10]

Frequency of violations

The frequency of negative response will depend on the electoral method, the candidates, and the distribution of outcomes.

Empirical analysis

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. [18] [19] 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). [17] [19] Two of those elections are also noted as specific examples below.

Semi-empirical

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. [15]

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. [20]

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. [11]

Theoretical models

Results using the impartial culture model estimate about 15% of elections with 3 candidates; [21] [14] however, the true probability may be much higher, especially when restricting observation to close elections. [22]

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." [12]

Specific examples

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. [23]

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. [17]

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. [10]

See also

Notes

  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.

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<span class="mw-page-title-main">Two-round system</span> Voting system

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<span class="mw-page-title-main">Spoiler effect</span> Losing candidate affecting election result

In social choice theory and politics, a spoiler effect happens when a losing candidate affects the results of an election simply by participating. Voting rules that are not affected by spoilers are said to be spoilerproof

<span class="mw-page-title-main">Coombs' method</span> Single-winner ranked voting rule

Coombs' method is a ranked voting system. Like instant-runoff (IRV-RCV), Coombs' method is a sequential-loser method, where the last-place finisher according to one method is eliminated in each round. However, unlike in instant-runoff, each round has electors voting against their least-favorite candidate; the candidate ranked last by the most voters is eliminated.

<span class="mw-page-title-main">Condorcet winner criterion</span> Property of electoral systems

A Condorcet winner is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condorcet winner criterion. The Condorcet winner criterion extends the principle of majority rule to elections with multiple candidates.

<span class="mw-page-title-main">No-show paradox</span> When voting for a candidate makes them lose

In social choice, a no-show paradox is a surprising behavior in some voting rules, where a candidate loses an election as a result of having too many supporters. More formally, a no-show paradox occurs when adding voters who prefer Alice to Bob causes Alice to lose the election to Bob. Voting systems without the no-show paradox are said to satisfy the participation criterion.

<span class="mw-page-title-main">Nanson's method</span> Single-winner electoral system

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<span class="mw-page-title-main">FairVote</span> U.S. electoral reform organization

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Instant-runoff voting (IRV) is a ranked voting method used in single-winner elections. IRV is also known outside the US as the alternative vote (AV). Today it is in use at a national level to elect the Australian House of Representatives, the National Parliament of Papua New Guinea, the President of Ireland and President of India. In Australia it is also used for elections to the legislative assemblies of all states and territories except Tasmania and the Australian Capital Territory, and for the Tasmanian Legislative Council.

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<span class="mw-page-title-main">Ranked-choice voting in the United States</span> Electoral system used in some cities and states

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<span class="mw-page-title-main">Instant-runoff voting</span> Single-winner ranked-choice electoral system

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<span class="mw-page-title-main">2009 Burlington mayoral election</span> American municipal election in Vermont

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<span class="mw-page-title-main">Electoral system</span> Method by which voters make a choice between options

An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, non-profit organisations and informal organisations. These rules govern all aspects of the voting process: when elections occur, who is allowed to vote, who can stand as a candidate, how ballots are marked and cast, how the ballots are counted, how votes translate into the election outcome, limits on campaign spending, and other factors that can affect the result. Political electoral systems are defined by constitutions and electoral laws, are typically conducted by election commissions, and can use multiple types of elections for different offices.

<span class="mw-page-title-main">Ranked voting</span> Voting systems that use ranked ballots

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There are a number of different criteria which can be used for voting systems in an election, including the following

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<span class="mw-page-title-main">STAR voting</span> Single-winner electoral system

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<span class="mw-page-title-main">Mixed ballot transferable vote</span>

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<span class="mw-page-title-main">Top-four primary</span> Nonpartisan blanket primary

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<span class="mw-page-title-main">Center squeeze</span> Type of independence of irrelevant alternatives violation

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References

  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. 1 2 3 Pukelsheim, Friedrich (2014). Proportional representation: apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN   978-3-319-03855-1.
  3. 1 2 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. 1 2 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. 1 2 D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters , Issue 6, 1996
  6. 1 2 3 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. 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. ↑ 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.
  9. ↑ Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN   978-3-319-03855-1.
  10. 1 2 3 dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN   0044-2070 . Retrieved 2024-05-02.
  11. 1 2 3 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.
  12. 1 2 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.
  13. 1 2 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.
  14. 1 2 Miller, Nicholas R. (2012). Monotonicity Failure in IRV Elections With Three Candidates (PowerPoint). p. 23. Impartial Culture Profiles: All, Total MF: 15.0%
  15. 1 2 3 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.
  16. ↑ 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.
  17. 1 2 3 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.
  18. ↑ 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.
  19. 1 2 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 .
  20. ↑ 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%
  21. ↑ 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%
  22. ↑ Quas, Anthony (2004-03-01). "Anomalous Outcomes in Preferential Voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. ISSN   0219-4937.
  23. ↑ 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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