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''}}.<!-- |
-->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''}}.<!-- |
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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]. |
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]. |
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<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> |
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--> 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. |
--> 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. |
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Of the single-winner ranked voting systems, [[Borda count|Borda]], [[Schulze method|Schulze]], [[ranked pairs]], maximize affirmed majorities, |
Of the single-winner ranked voting systems, [[Borda count|Borda]], [[Schulze method|Schulze]], [[ranked pairs]], maximize affirmed majorities, [[Descending Solid Coalitions]], and [[Descending Acquiescing Coalitions]]<ref name="Woodall-Monotonicity" /> are monotone, while [[Coombs' method]], [[runoff voting]], and [[instant-runoff voting]] (IRV) are not. |
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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). |
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). |
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All [[plurality voting system]]s are |
All [[plurality voting system]]s are monotone if the ballots are treated as rankings where using ''more than two ranks is forbidden''. In this setting [[first past the post]] and [[approval voting]] as well as the multiple-winner systems [[single non-transferable vote]], [[plurality-at-large voting]] (multiple non-transferable vote, bloc voting) and [[cumulative voting]] are monotonic. [[Party-list proportional representation]] using [[D'Hondt method|D'Hondt]], [[Sainte-Laguë method|Sainte-Laguë]] or the [[largest remainder method]] is monotone in the same sense. |
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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. |
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=== |
===Estimated likelihood of IRV lacking monotonicity=== |
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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 |
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 name="Mathematical Social Sciences 1996 pp. 133–146">{{cite journal | last=Lepelley | first=Dominique | last2=Chantreuil | first2=Frédéric | last3=Berg | first3=Sven | title=The likelihood of monotonicity paradoxes in run-off elections | journal=Mathematical Social Sciences | volume=31 | issue=3 | date=1996-06-01 | issn=0165-4896 | doi=10.1016/0165-4896(95)00804-7 | pages=133–146 | url=https://www.sciencedirect.com/science/article/pii/0165489695008047 | access-date=2020-02-03}}</ref> found a probability of {{nowrap|397/6912 {{=}} 5.74%}} for 3-candidate elections. |
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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> |
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">{{cite web | title=Monotonicity failure under STV and related voting systems | url=https://userpages.umbc.edu/~nmiller/RESEARCH/MONOTONICITY.pdf | access-date=2020-02-03 |date=2002-03-22}}</ref> |
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==Real-life monotonicity violations== |
==Real-life monotonicity violations== |
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