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Insocial choice, thenegative response,[1][2]perversity,[3] oradditional support paradox[4] is apathological behavior of somevoting 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'sranking orrating causes that candidate to lose (win), respectively.[4] Electoral systems that do not exhibit perversity are sometimes said to satisfy themonotonicitycriterion.[5]
Perversity is often described bysocial choice theorists as an exceptionally severe kind ofelectoral 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 toequal and direct suffrage.[9][10] Negative response is often cited as an example of aperverse incentive, as rules with negative response can incentivize politicians to takeextreme or unpopular positions in an attempt to shed excess votes.[11]
Mostranked methods (includingBorda and all commonround-robin rules) satisfy positive response,[5] as do all commonrated voting methods (includingapproval,highest medians, andscore).[note 1]
Negative responsiveness occurs ininstant-runoff voting (IRV),[12] thesingle transferable vote,[3] and thetwo-round system.[11] Somequota-based apportionment methods also violate the rule,[13] as can therandomized Condorcet method in cases ofcyclic ties.
Theparticipation 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]
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.
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 somepolitical 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 bystrategic 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]
Runoff-based voting systems such asranked choice voting (RCV) are typically vulnerable to perverse response. A notable example is the2009 Burlington mayoral election, the United States' secondinstant-runoff election in the modern era, whereBob Kiss won the election as a result of 750 ballots ranking him in last place.[17] Another example is given by the2022 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 areelected after running anunsuccessful campaign and adopting anunpopular 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 |  N | +6% | Top | 31% | 46% | |
| Center | 30% | 55% Y | ↗ | Center | N | |
| Bottom | 45% | 45% | -6% | Bottom | 39% | 54%Y |
This election is an example of acenter-squeeze, a class of elections where instant-runoff andplurality 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.
Someproportional representation systems can exhibit negative responsiveness. These include thesingle transferable vote and some implementations ofmixed-member proportional representation, generally as a result of poorly-designedoverhang rules. An example can be found in the2005 German federal election, whereCDU supporters inDresden were instructed to vote for theFDP, a strategy that allowed the CDU to win an additional seat.[2] This led theFederal Constitutional Court to rule that negative responsiveness violates theGerman constitution's guarantee ofequal and direct suffrage.[10]
The frequency of negative response will depend on the electoral method, the candidates, and the distribution of outcomes.
In the US, a 2021 analysis ofinstant-runoff elections in California between 2008 and 2016, as well as the2009 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.
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 andPR-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 thetwo-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 and2007) where an upward monotonicity violation was probable and likely respectively.[11]
Results using theimpartial 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-dimensionalspatial 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]
Alaska's first-ever instant-runoff election resulted in a victory forDemocratMary Peltola, but had many voters forRepublicanSarah Palin instead ranked Peltola first, Peltola would have lost.[23]
InBurlington's second IRV election, incumbentBob Kiss was re-elected, despite losing in a head-to-head matchup with Democrat Andy Montroll (theCondorcet winner). However, if Kiss had gained more support from Wright voters, Kiss would have lost.[17]
Someproportional representation systems can exhibit negative responsiveness. These include thesingle transferable vote and some implementations ofmixed-member proportional representation, generally as a result of poorly-designedoverhang rules. An example can be found in the2005 German federal election, whereCDU supporters inDresden were instructed to vote for theFDP, a strategy that allowed the CDU to win an additional seat.[2] This led theFederal Constitutional Court to rule that negative responsiveness violates theGerman constitution's guarantee ofequal and direct suffrage.[10]
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.
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.
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.
Impartial Culture Profiles: All, Total MF: 15.0%
Impartial Culture Profiles: All, TMF: 15.1%
Impartial Culture Profiles: All, TMF: 15.1%
