Monotonicity criterion

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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).

The positive response, [1] [2] monotonicity, or nonperversitycriterion [3] is a principle of social choice theory that says that increasing a candidate's ranking or rating should not cause them to lose. [4] Positive response rules out cases where a candidate loses an election as a result of receiving too much support from voters (i.e. being "too popular to win"); rules that violate positive response are called perverse. [5]

Contents

Systems that violate positive response (such as instant-runoff and the two-round system) can create situations where a voter's ballot has a reversed effect on the election, making it "less than worthless". This runs counter to the basic principle that increasing an option's popularity in a democratic election should only improve the chances of that option winning; as a result, German courts have previously struck down nonmonotonic systems for violating the right to equal and direct suffrage. [2] [6]

Most voting systems (including Borda and all common tournament solutions) satisfy positive response, [4] as do all commonly-used rated voting methods (including approval, score, and their proportional counterparts). [note 1]

However, the criterion is violated by instant-runoff voting, [7] the single transferable vote, [8] and Hamilton's apportionment method. [2]

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 can have a reversed effect on the election.

By method

Runoff voting

Runoff-based voting systems, such as ranked choice voting (instant-runoff) fail the monotonicity criterion. 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. [9]

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, because the first-round vote is split between an extremist and a moderate. 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.

A famous example of a monotonicity failure is the 2022 Alaska at-large special election.

Quota rules

Proportional representation systems using largest remainders for apportionment do not pass the monotonicity criterion. This happened in the 2005 German federal election, when CDU voters in Dresden were instructed to vote for the FDP, a strategy that allowed the party an additional seat. [2] As a result, the Federal Constitutional Court ruled that negative voting weights violate the German constitution's guarantee of equal and direct suffrage. [6]

Frequency of violations

For electoral methods failing positive value, the frequency of monotonicity violations will depend on the electoral method, the candidates, and the distribution of outcomes. Negative voting weights tend to be most common with instant-runoff, with what some researchers have described as an "unacceptably high" frequency. [10]

Theoretical models

Results using the impartial culture model estimate about 15% of elections with 3 candidates; [11] [12] however, the true probability may be much higher, especially when restricting observation to close elections. [13] For moderate numbers of candidates, the probability of a monotonicity failure quickly approaches 100%.[ citation needed ]

A 2013 study using a 2D spatial model with various voter distributions estimated at least 15% of IRV elections are nonmonotonic in the best-case scenario (when only three candidates run). The researchers concluded that "three-way competitive races will exhibit unacceptably frequent monotonicity failures" and "In light of these results, those seeking to implement a fairer multi-candidate election system should be wary of adopting IRV." [10]

Real-world situations

Alaska 2022

Alaska's first-ever instant-runoff election resulted in negative vote weights for many Republican supporters of Sarah Palin, who could have defeated Mary Peltola by placing her first on their ballots. [14]

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. [9]

Survey of nonmonotonic elections

A survey of 185 American instant-runoff elections where no candidate was ranked first by a majority of voters found five additional elections containing monotonicity failures. [9]

2005 German Election in Dresden

A negative voting weight event famously resulted in the abolition of Hamilton's method for apportionment in Germany after the 2005 federal election. CDU voters in Dresden were instructed to strategically vote for the FDP, a strategy that allowed the party to earn an additional seat, causing substantial controversy. As a result, the Federal Constitutional Court ruled that negative voting weights violate the German constitution's guarantee of equal and direct suffrage. [2]

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.

Related Research Articles

Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approval voting, but also lets voters give partial (in-between) approval ratings to candidates.

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

The two-round system is a voting method used to elect a single winner. In the United States, it is often called a jungle or nonpartisan primary. The system is also called runoff voting, though this term often means the closely-related exhaustive ballot and ranked-choice runoff systems.

In social choice theory and politics, the spoiler effect or Arrow's paradox refers to a situation where a losing candidate affects the results of an election. A voting system that is not affected by spoilers satisfies independence of irrelevant alternatives or independence of spoilers.

<span class="mw-page-title-main">Condorcet method</span> Pairwise-comparison electoral system

A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.

Arrow's impossibility theorem is a key result in social choice showing that no rank-order method for collective decision-making can behave rationally or coherently. Specifically, any such rule violates independence of irrelevant alternatives, the principle that a choice between and should not depend on the quality of a third, unrelated option .

Coombs' method is a ranked voting system popularized by Clyde Coombs. It was described by Edward J. Nanson as the "Venetian method", but should not be confused with the Republic of Venice's use of score voting in elections for Doge. Coombs' method can be thought of as a cross between instant-runoff voting and anti-plurality voting.

<span class="mw-page-title-main">Pathological (mathematics)</span> Mathematical phenomena whose properties are counterintuitive

In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or nice. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved.

Ranked Pairs (RP) is a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.

In an election, a candidate is called a majority winner or majority-preferred candidate if more than half of all voters would support them 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 majority-rule principle, because they extend the principle of majority rule to elections with multiple candidates.

The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election as a result of receiving too many votes in support. More formally, it says that adding more voters who prefer Alice to Bob should not cause Alice to lose the election to Bob.

In political science and social choice theory, Black'smedian voter theorem states that if voters and candidates are distributed along a one-dimensional spectrum and voters have single peaked preferences, any voting method satisfying the Condorcet criterion will elect the candidate preferred by the median voter.

The Borda count electoral system can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method. Both methods are designed to satisfy the Condorcet criterion, and allow for incomplete ballots and equal rankings.

Later-no-harm is a property of some ranked-choice voting systems, first described by Douglas Woodall. In later-no-harm systems, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause this higher-ranked candidate to lose.

In social choice theory, the independence of clones criterion says that adding a clone, i.e. a new candidate very similar to an already-existing candidate, should not spoil the results. It can be considered a very weak form of the independence of irrelevant alternatives (IIA) criterion.

Schulze STV is a draft single transferable vote (STV) ranked voting system designed to achieve proportional representation. It was invented by Markus Schulze, who developed the Schulze method for resolving ties using a Condorcet method. Schulze STV is similar to CPO-STV in that it compares possible winning candidate pairs and selects the Condorcet winner. It is not used in parliamentary elections.

Instant-runoff voting (IRV), also known as ranked-choice voting or the alternative vote (AV), combines ranked voting together with a system for choosing winners from these rankings by repeatedly eliminating the candidate with the fewest first-place votes and reassigning their votes until only one candidate is left. It can be seen as a modified form of a runoff election or exhaustive ballot in which, after eliminating some candidates, the choice among the rest is made from already-given voter rankings rather than from a separate election. Many sources conflate this system of choosing winners with ranked-choice voting more generally, for which several other systems of choosing winners have also been used.

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

The 2009 Burlington mayoral election was held in March 2009 for the city of Burlington, Vermont. This was the second mayoral election since the city's 2005 change to instant-runoff voting (IRV), after the 2006 mayoral election. In the 2009 election, incumbent Burlington mayor won reelection as a member of the Vermont Progressive Party, defeating Kurt Wright in the final round with 48% of the vote.

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

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The sincere favorite or no favorite-betrayal criterion is a property of some voting systems, that says voters should have no incentive to vote for someone else over their favorite. It protects voters from having to engage in a kind of strategy called lesser evil voting or decapitation.

A top-four primary or top-four ranked-choice voting is an election method using a nonpartisan blanket primary as the first step.

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 4 5 Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN   978-3-319-03855-1.
  3. ↑ Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN   0092-5853.
  4. 1 2 D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters , Issue 6, 1996
  5. ↑ 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.
  6. 1 2 dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN   0044-2070 . Retrieved 2024-05-02.
  7. ↑ 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.
  8. ↑ 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.
  9. 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.
  10. 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.
  11. ↑ 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%
  12. ↑ Miller, Nicholas R. (2012). MONOTONICITY FAILURE IN IRV ELECTIONS WITH THREE ANDIDATES (PowerPoint). p. 23. Impartial Culture Profiles: All, Total MF: 15.0%
  13. ↑ Quas, Anthony (2004-03-01). "Anomalous Outcomes in Preferential Voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. ISSN   0219-4937.
  14. ↑ 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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