Monotonicity: Difference between revisions
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{{wikipedia|Monotonicity criterion}}
The '''monotonicity criterion''' (sometimes referred to as the "'''mono-raise criterion'''") is a [[voting system criterion]] used to evaluate both single and multiple winner [[
== Mono-raise criterion==
▲The '''monotonicity criterion''' is a [[voting system criterion]] used to evaluate both single and multiple winner [[ranked voting system]]s. A ranked voting system is '''monotonic''' if it is neither possible to prevent the election of a candidate by ranking them higher on some of the ballots, nor possible to elect an otherwise unelected candidate by ranking them lower on some of the ballots (while nothing else is altered on any ballot).
<ref name="Woodall-Monotonicity">D R Woodall, [http://www.votingmatters.org.uk/ISSUE6/P4.HTM "Monotonicity and Single-Seat Election Rules"], ''[[Voting matters]]'', Issue 6, 1996</ref> In deterministic single winner elections that is to say no winner is harmed by up-ranking and no loser can win by down-ranking. If the method relies on chance, then up-ranking a candidate can not decrease that candidate's chance of winning, nor can down-ranking the candidate increase it. Douglas R. Woodall called the criterion '''mono-raise'''.▼
The '''mono-raise criterion''' is one of several sub-cases of the monotonicity family of criteria. A [[voting system]] satisfies the ''Mono-raise criterion'':
{{Definition|If an alternative X loses, and the ballots are changed only by placing X in lower positions, without changing the relative position of other candidates, then X must still lose.}}
A looser way of phrasing this is that in a non-monotonic system, voting for a candidate can cause that candidate to lose. Systems which fail the monotonicity criterion suffer a form of [[tactical voting]] where voters might try to elect their candidate by voting against that candidate.
[[Plurality voting]], [[Majority Choice Approval]], [[Borda count]], [[Schulze method|Schulze]], [[Maximize Affirmed Majorities]], and [[Descending Solid Coalitions]] are monotonic, while [[Coombs' method]] and [[Instant-runoff voting]] are not. [[Approval voting]] is monotonic, using a slightly different definition, because it is not a preferential system: You can never help a candidate by not voting for them.<ref>Some parts of this article are derived from text at https://web.archive.org/web/20090610060543/http://condorcet.org/emr/criteria.shtml which was released to the [[project:public domain|public domain]].</ref><ref>Mono-raise test copied from https://electowiki.org/w/index.php?title=Mono-raise_criterion&oldid=15949</ref>
=== Details ===
▲
Raising a candidate {{math|''x''}} on some ballots ''while changing'' the orders of other candidates does ''not'' constitute a failure of monotonicity. E.g., harming candidate {{math|''x''}} by changing some ballots from {{math|''z'' > ''x'' > ''y''}} to {{math|''x'' > ''y'' > ''z''}} isn't a violation of the monotonicity criterion.
The monotonicity criterion renders the intuition that there should be neither need to worry about harming a candidate by (nothing else than) up-ranking nor it should be possible to support a candidate by (nothing else than) counter-intuitively down-ranking.
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>
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.
Most variants of the [[single transferable vote]] (STV) [[proportional representation]]
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. The definition of the monotonicity criterion with regard to these methods is disputed. Some voting theorists argue that this means these methods pass the monotonicity criterion; others say that, as these are not ''ranked'' voting systems, they are out of the monotonicity criterion's scope.
=== Implications ===
==Definition of monotonicity criteria==▼
It's impossible for a method to pass all of monotonicity, [[later-no-harm]], [[later-no-help]], and [[mutual majority]],<ref name="Woodall-Monotonicity"/> but there do exist methods that pass three of the four. [[First past the post]] passes the first three, [[instant-runoff voting]] passes the last three, and [[Descending Acquiescing Coalitions]] and [[Descending Solid Coalitions]] pass one of the later-no-help/harm criteria as well as monotonicity and mutual majority.
▲== Definition of monotonicity criteria==
The general pattern of monotonicity criteria is:
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{{definition|If X is a winner under a voting rule, and one or more voters change their preferences by ranking or rating X higher without otherwise changing their ballots, then X should still be a winner.}}
==Instant-runoff voting and the two-round system are not monotonic ==
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.
Suppose a
Suppose the votes are cast as follows:
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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 [[
==Real-life monotonicity violations==
If the ballots of a real election are released, it is fairly easy to prove if
*
*
would have been possible (nothing else is altered on any ballot). Both events can be considered as real-life monotonicity violations.
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A party-list strategy exploiting something similar (down-ranking CDU '''and''' additionally up-ranking another party, e.g. FDP) happened in the German federal election of 2005, in which conservative voters in Dresden deliberately voted against the CDU, their party of choice, in order to maximize that party's number of seats in the federal parliament. This was possible due to Germany's voting system (mixed member proportional with overhang seats computed independently for each federal state) and the fact that the vote in Dresden took place a week after the rest of the country due to the death of a candidate, enabling voters in Dresden to vote tactically in full knowledge of the results already achieved elsewhere. As a result of this, the German Constitutional Court ruled on July 3 2008 that the German voting system must be reformed to eliminate its non-monotonicity.<ref> See e.g. [http://fruitsandvotes.com/blog/?p=117]</ref>
=== 2009 Burlington, Vermont mayoral election ===
{{seealso|[[2009 Burlington mayoral election]]}}
A real-life monotonicity violation was detected in the [[2009 Burlington
===Australian elections and by-elections ===▼
▲A real-life monotonicity violation was detected in the [[Burlington, Vermont mayoral election, 2009|2009 Burlington, Vermont mayor election]] under instant-runoff voting (IRV), where the necessary information is available. In this election, the winner Bob Kiss could have been defeated by raising him on some of the ballots. For example, if all voters who ranked Kurt Wright over Bob Kiss over Andy Montroll, would have ranked Kiss over Wright over Montroll, and additionally some people who ranked Wright but not Kiss or Montroll, would have ranked Kiss over Wright, then these votes in favor of Kiss would have defeated him.<ref>[http://www.rangevoting.org/Burlington.html Burlington Vermont 2009 IRV mayor election]</ref> The winner in this scenario would have been Andy Montroll, who was also the [[Condorcet winner]] according to the original ballots, i.e. for any other running candidate, a majority ranked Montroll above the competitor.
Since every or almost every IRV election in Australia has been conducted in the black (i.e. not releasing enough information to reconstruct the ballots), nonmonotonicity is difficult to detect in Australia, even though thanks to the Lepelley ''et al'' probability estimates it seems safe to say that it must have occurred in over 100 of their elections.<ref
▲===Australian elections and by-elections===
▲has been conducted in the black (i.e. not releasing enough information to reconstruct the ballots), nonmonotonicity is difficult to detect in Australia, even though thanks to the Lepelley ''et al'' probability estimates it seems safe to say that it must have occurred in over 100 of their elections. (The policy of Australia's election authorities not to release this data
However, for the [[Australian federal election, 2010]], one article was aware of the non-monotonicity possibility: [http://andrewnorton.info/2010/08/16/why-labor-voters-in-melbourne-need-to-vote-liberal/ Why Labor Voters In Melbourne Need To Vote Liberal]. In 2009, the theoretical disadvantage of non-monotonicity worked out in practice in a state [[by-election]] in the [[South Australia]]n seat of [[Electoral district of Frome|Frome]]. The eventual winner, an Independent who was a town mayor, scored only third on the primaries with about 21% of the vote. But since the [[National Party of Australia]] scored 4th place, their preferences were distributed beforehand, allowing the Independent to overtake the [[Australian Labor Party]] Candidate by 31 votes. Thus Labor was pushed into third place, and their preference distribution favoured the Independent, who overtook the leading [[Australian Liberal Party]] candidate to win the election. However, had anywhere between 31 and 321 of the voters who preferred Liberal over Labor and Independent switched their support from Liberal to Labor, it would have allowed the Liberal to win the IRV election. This is classic monotonicity violation: the 321 who voted for the Liberals took part in hurting their own candidate.<ref>http://blogs.abc.net.au/antonygreen/2011/05/an-example-of-non-monotonicity-and-opportunites-for-tactical-voting-at-an-australian-election.html</ref>
==Other forms of
▲There are several variations of
===
{{seealso|Douglas Woodall
}}
* ([[Mono-raise criterion|'''mono-raise''']]) x is raised on some ballots without changing the orders of the other candidates;
▲There are several variations or types of monotonicity: <blockquote>MONOTONICITY. A candidate x should not be harmed if:
* ('''mono-raise-delete''') x is raised on some ballots and all candidates now below x on those ballots are deleted from them;
*
*
*
*
*
*
*
=== Multi-winner monotonicity ===
Monotonicity would be more aptly named ''endorsement monotonicity'' since it is the preservation of monotonicity relative to endorsement. Since it is the most important form of monotonicity is bears the simple naming. There are however two other important forms of monotonicity for multi-winner voting systems, [[Population monotonicity]] and [[House monotonicity criterion |House monotonicity]].
Multi-winner monotonicity could also be considered in a weaker and stronger sense: the weak form is satisfied whenever, if A is one of the winners, ranking A higher does not kick A out of the winning set; whereas the stronger form is satisfied whenever, if A is one of the winners, ranking A higher does not kick ''anyone'' out of the winning set. In a single winner election, these criteria are the same, but the stronger form is harder to satisfy for multi-winner. Woodall's definition of mono-raise corresponds to the weak form.
==
{{reflist}}▼
===Notes===
{{reflist| group="nb"}}
===Videos===
*[https://www.youtube.com/watch?v=OI232JSDwDg "Voting Theory: Monotonicity Criterion Using Instant Runoff Voting" - Mathispower4u] - posted to YouTube on August 22, 2013. "''This video explains the Monotonicity Criterion and how it can affect the outcome of an election when using instant runoff voting.''"
===References===
▲{{reflist}}
[[Category:Voting system criteria]]
[[Category:Monotonic electoral systems|*]]
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