Monotonicity: Difference between revisions
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(Add: monotonicity isn't participation, and no DPC methods have been proven monotone. Do some cleanup. Remove part with citation needed from 2011.) |
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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> |
<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. Nor does compliance tell anything about the effect of other candidates: it's possible for a method to change the winner from X to Y when Z is ranked higher on some ballots without failing the monotonicity criterion. |
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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. |
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]] |
Most variants of the [[single transferable vote]] (STV) [[proportional representation]] methods are not monotonic, especially all that are currently in use for public elections (which simplify to IRV when there is only one winner). No [[ranked voting]] method has been proven to pass both [[Droop proportionality criterion|Droop proportionality]] and monotonicity, though it is suspected that [[Schulze STV]] passes both. |
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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. |
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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Using an example that applies to [[instant-runoff voting]] (IRV) and to the [[two-round system]], it is shown that these voting systems violate the mono-raise criterion. |
Using an example that applies to [[instant-runoff voting]] (IRV) and to the [[two-round system]], it is shown that these voting systems violate the mono-raise criterion. |
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Suppose a |
Suppose a president were being elected among three candidates, a left, a right, and a center candidate, and 100 votes cast. The number of votes for an absolute majority is therefore 51. |
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Suppose the votes are cast as follows: |
Suppose the votes are cast as follows: |
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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 [[ |
Crispin Allard argued, based on a mathematical model that the probability of monotonicity failure actually changing the result of an election for any given [[single-member district|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). |
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). 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== |