Editing Negative responsiveness (section)

===Empirical analysis===
In the US, a 2021 analysis of [[Instant-runoff voting|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.<ref>{{cite journal |last1=Graham-Squire |first1=Adam |last2=Zayatz |first2=N. |title=Lack of Monotonicity Anomalies in Empirical Data of Instant-runoff Elections |journal=Representation |date=2 October 2021 |volume=57 |issue=4 |pages=565–573 |doi=10.1080/00344893.2020.1785536}}</ref><ref name="McCune"/> 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).<ref name=":522" /><ref name="McCune">{{cite journal |last1=McCune |first1=David |last2=Graham-Squire |first2=Adam |title=Monotonicity anomalies in Scottish local government elections |journal=Social Choice and Welfare |date=August 2024 |volume=63 |issue=1 |pages=69–101 |doi=10.1007/s00355-024-01522-5 |doi-access=free|arxiv=2305.17741 }}</ref> 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.<ref name="Gallagher">{{Cite conference |last=Gallagher |first=Michael |date=September 2013 |title=Monotonicity and non-monotonicity at PR-STV elections |url=https://www.lancaster.ac.uk/fass/events/epop2013/docs/MGallagherMonotonicityEPOP13.pdf |conference=Annual conference of the elections, public opinion and parties (EPOP) specialist group, University of Lancaster |volume=13}}</ref>

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<!--This should be the 111 elections where PL support >25% but might need to check that, also probably get consensus that it's correct per CALC-->, 40.2% appeared vulnerable (33% upward, 7.1% downward), and further increasing with closer competition, a result closer to the simulations.<ref name=":3">{{Cite journal |last=Miller |first=Nicholas R. |date=October 2017 |title=Closeness matters: monotonicity failure in IRV elections with three candidates |url=https://userpages.umbc.edu/~nmiller/MONFAILURE.R2.NRM.pdf |journal=Public Choice |volume=173 |issue=1–2 |pages=91–108 |doi=10.1007/s11127-017-0465-5 |quote=Impartial Culture Profiles: All, TMF: 15.1% |hdl-access=free |hdl=11603/20938}}</ref>

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 French presidential election|2002]] and [[2007 French presidential election|2007]]) where an upward monotonicity violation was probable and likely respectively.<ref name=":4">{{cite journal |last1=Keskin |first1=Umut |last2=Sanver |first2=M. Remzi |last3=Tosunlu |first3=H. Berkay |url=https://hal.science/hal-03413280/document |title=Monotonicity violations under plurality with a runoff: the case of French presidential elections |journal=Social Choice and Welfare |date=August 2022 |volume=59 |issue=2 |pages=305–333 |doi=10.1007/s00355-022-01397-4}}</ref>