Monotonicity: Difference between revisions
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-->There are several variations of that criterion; e.g., what Douglas R. Woodall called ''mono-add-plump'': A candidate {{math|''x''}} should not be harmed if further ballots are added that have {{math|''x''}} top with no second choice. Agreement with such rather special properties is the best any ranked voting system may fulfill: The [[Gibbard–Satterthwaite theorem]] shows, that any meaningful ranked voting system is susceptible to some kind of [[tactical voting]], and [[Arrow's impossibility theorem]] shows that individual rankings can't be meaningfully translated into a community-wide ranking where the order of candidates {{math|''x''}} and {{math|''y''}} is always [[Independence of irrelevant alternatives|independent of irrelevant alternatives]] {{math|''z''}}.<!--
The result of David Austen-Smith and Jeffrey Banks that monotonicity in individual preferences is impossible is a nonissue: For given voter preferences v=v_1...v_n and a winner x under voting scheme alpha, they investigate changes in v, where e.g. altering v_i from a,b,c,d,x to d,c,x,b,a is allowed, which can't be seriously named a monotonicity property. That allows random permutations even ''ahead'' of x, and is therefore even more rigid than Woodall's mono-raise-random, which is already incompatible with [majority AND later-no-help AND later-no-harm].
<ref name="Austen-Smith Banks 2014 pp. 531–537">{{cite journal | last=Austen-Smith | first=David | last2=Banks | first2=Jeffrey | title=Monotonicity in Electoral Systems - American Political Science Review | journal=American Political Science Review | volume=85 | issue=2 | date=2014-08-01 | issn=1537-5943 | doi=10.2307/1963173 | pages=531–537 | url=http://www.jstor.org/stable/1963173 | access-date=2020-02-03}}</ref>
--> Noncompliance with the monotonicity criterion doesn't tell anything about the likelihood of monotonicity violations, failing in one of a million possible elections would be as well a violation as missing the criterion in any possible election.
Of the single-winner ranked voting systems, [[Borda count|Borda]], [[Schulze method|Schulze]], [[ranked pairs]], maximize affirmed majorities,
Most variants of the [[single transferable vote]] (STV) [[proportional representation]]s are not monotonic, especially all that are currently in use for public elections (which simplify to IRV when there is only one winner).
All [[plurality voting system]]s are
In elections via the single-winner methods [[range voting]] and [[majority judgment]] nobody can help a candidate by reducing or removing support for them, but as they are not ''ranked'' voting systems, they are out of the monotonicity criterion's scope.
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===Estimated likelihood of IRV lacking monotonicity===
Crispin Allard argued, based on a mathematical model that the probability of monotonicity failure actually changing the result of an election for any given [[European Parliament constituency|constituency]] would be 1 in 4000;<ref>[http://www.mcdougall.org.uk/VM/ISSUE5/P1.HTM Estimating the Probability of Monotonicity Failure in a UK General Election]</ref> however, Lepelley ''et al.''<ref
Another probability model, the "impartial culture", yields about 15% probability. In elections with more than 3 candidates, these probabilities tend to increase eventually toward 100% (in some models this limit has been proven, in others it is only conjectured). Estimates of 5–15% order are easily confirmed in any probability model with "Monte Carlo experiments" and the aid of the "was it monotonic?" tests stated in the Lepelley paper.{{Citation needed|date=June 2011}} Nicholas Miller also disputed Allard's conclusion and provided a different mathematical model.<ref name="Annual Meeting of the Public Choice Society 2002">
==Real-life monotonicity violations==
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