View source for Participation criterion

{{Short description|Principle that voting for a candidate should help them}}
{{Electoral systems sidebar|expanded=Paradox}}
The '''participation criterion''' is a [[Comparison of electoral systems|voting system criterion]] that says candidates should never lose an election as a result of receiving too many votes in support.<ref name=":42">{{Cite journal |last1=Doron |first1=Gideon |last2=Kronick |first2=Richard |date=1977 |title=Single Transferrable Vote: An Example of a Perverse Social Choice Function |url=https://www.jstor.org/stable/2110496 |journal=American Journal of Political Science |volume=21 |issue=2 |pages=303–311 |doi=10.2307/2110496 |issn=0092-5853 |jstor=2110496|url-access=subscription }}</ref><ref name=":322">{{Cite journal |last=Ray |first=Depankar |date=1986-04-01 |title=On the practical possibility of a 'no show paradox' under the single transferable vote |url=https://dx.doi.org/10.1016/0165-4896%2886%2990024-7 |journal=Mathematical Social Sciences |volume=11 |issue=2 |pages=183–189 |doi=10.1016/0165-4896(86)90024-7 |issn=0165-4896|url-access=subscription }}</ref> More formally, it says that adding more voters who prefer ''Alice'' to ''Bob'' should not cause ''Alice'' to lose the election to ''Bob''.<ref>{{cite web |last=Woodall |first=Douglas |author-link=Douglas Woodall |date=December 1994 |title=Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994 |url=http://www.votingmatters.org.uk/ISSUE3/P5.HTM}}</ref>

Voting systems that fail the participation criterion exhibit the '''no-show paradox''',<ref name=":0222">{{Cite journal |last=Moulin |first=Hervé |date=1988-06-01 |title=Condorcet's principle implies the no show paradox |journal=Journal of Economic Theory |volume=45 |issue=1 |pages=53–64 |doi=10.1016/0022-0531(88)90253-0}}</ref> where a voter is effectively disenfranchised by the electoral system because turning out to vote could make the result worse for them; such voters are sometimes referred to as having [[negative vote weight]]s, particularly in the context of [[Basic Law for the Federal Republic of Germany|German constitutional law]], where courts have ruled such a possibility violates the principle of [[one man, one vote]].<ref name=":42222">{{Cite book |last=Pukelsheim |first=Friedrich |url=http://archive.org/details/proportionalrepr0000puke |title=Proportional representation : apportionment methods and their applications |date=2014 |publisher=Cham; New York : Springer |others=Internet Archive |isbn=978-3-319-03855-1}}</ref><ref name=":0322">{{Cite news |last=dpa |date=2013-02-22 |title=Bundestag beschließt neues Wahlrecht |url=https://www.zeit.de/politik/deutschland/2013-02/bundestag-wahlrecht-beschluss |access-date=2024-05-02 |work=Die Zeit |language=de-DE |issn=0044-2070}}</ref><ref name="bundesverfassungsgericht_de">[[Federal Constitutional Court|BVerfG]], Urteil des {{Interlanguage link|Federal Constitutional Court#Organization|lt=Zweiten Senats|de|Zweiter Senat}} 3 July 2008 - 2 BvC 1/07, 2 BvC 7/07 -, Rn. 1-145. {{Cite court|vol=121|reporter=BVerfGE|opinion=- 2 BvC 1/07, 2 BvC 7/07 -|court=[[BVerfG]]|date=03 July 2008|url=https://www.bundesverfassungsgericht.de/SharedDocs/Entscheidungen/DE/2008/07/cs20080703_2bvc000107.html}} {{ECLI|ECLI:DE:BVerfG:2008:cs20080703.2bvc000107}}</ref>

[[Positional voting|Positional methods]] and [[score voting]] satisfy the participation criterion. All deterministic voting rules that satisfy [[Condorcet method|pairwise majority-rule]]<ref name=":0222"/><ref name=":1222">{{Cite journal |last1=Brandt |first1=Felix |last2=Geist |first2=Christian |last3=Peters |first3=Dominik |date=2016-01-01 |title=Optimal Bounds for the No-Show Paradox via SAT Solving |url=http://dl.acm.org/citation.cfm?id=2936924.2936972 |journal=Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems |series=AAMAS '16 |location=Richland, SC |publisher=International Foundation for Autonomous Agents and Multiagent Systems |pages=314–322 |arxiv=1602.08063 |isbn=9781450342391}}</ref> can fail in situations involving four-way [[cyclic tie]]s, though such scenarios are empirically rare, and the [[Maximal lottery|randomized Condorcet rule]] is not affected by the pathology. The [[majority judgment]] rule fails as well.<ref>{{cite web |author=Markus Schulze |date=1998-06-12 |title=Regretted Turnout. Insincere = ranking. |url=http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-June/001727.html |access-date=2011-05-14}}</ref> [[Instant-runoff voting]] and the [[two-round system]] both fail the participation criterion with high frequency in competitive elections, typically as a result of a [[center squeeze]].<ref name=":42" /><ref name=":322" /><ref name=":2222">{{cite arXiv |eprint=2403.18857 |class=physics.soc-ph |first1=David |last1=McCune |first2=Jennifer |last2=Wilson |title=The Negative Participation Paradox in Three-Candidate Instant Runoff Elections |date=2024-04-07}}</ref>

The no-show paradox is similar to, but not the same as, the [[Mono-raise criterion|perverse response paradox]]. Perverse response happens when an ''existing'' voter can make a candidate win by ''de''creasing their rating of that candidate (or vice-versa). For example, under [[instant-runoff voting]], moving a candidate from first-place to last-place on a ballot can cause them to win.<ref name="Fishburn and Brams 1983">{{Cite journal |last1=Fishburn |first1=Peter C. |last2=Brams |first2=Steven J. |date=1983-01-01 |title=Paradoxes of Preferential Voting |journal=Mathematics Magazine |volume=56 |issue=4 |pages=207–214 |doi=10.2307/2689808 |jstor=2689808}}</ref>
{{TOC limit|limit=3}}

== Noncompliant methods ==
=== Instant-runoff (ranked-choice) voting ===
{{See also|Center squeeze}}
The most common cause of no-show paradoxes is the use of [[Instant-runoff voting|instant-runoff]] (often called [[ranked-choice voting in the United States]]). In instant-runoff voting, a no-show paradox can occur even in elections with only three candidates,<ref name="Fishburn and Brams 1983"/> and occur in 50%-60% of all 3-candidate elections where the results of IRV disagree with those of plurality.<ref name=":2222" /><ref name=":322" />

A notable example is given in the [[2009 Burlington mayoral election]], the United States' second [[Instant-runoff voting|instant-runoff election]] in the modern era, where [[Bob Kiss]] won the election as a result of 750 ballots ranking him in last place.<ref name=":5222">{{Cite journal |last1=Graham-Squire |first1=Adam T. |last2=McCune |first2=David |date=2023-06-12 |title=An Examination of Ranked-Choice Voting in the United States, 2004–2022 |journal=Representation |volume=61 |pages=1–19 |arxiv=2301.12075 |doi=10.1080/00344893.2023.2221689}}</ref>

An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Center party initially wins. However, say that a group of pro-Top voters joins the election, all of whom cast their ballots with the ranking Top > Center > Bottom:
{| class="wikitable"
|+
! colspan="3" |Baseline
!
! colspan="3" |More votes ranking Center > Bottom
|-
!
!Round 1
!Round 2
!
!
!Round 1
!Round 2
|-
!'''''Top'''''
|<s>25</s> {{Xmark}}
|
!+6
!'''''Top'''''
|31
|46
|-
!'''''Center'''''
|30
|'''55''' {{Tick}}
!
!'''''Center'''''
|<s>30</s> {{Xmark}}
|
|-
!'''''Bottom'''''
|39
|39
!
!'''''Bottom'''''
|39
|'''54 {{Tick}}'''
|}
The participation criterion says that the addition of voters who prefer Center over Bottom should not cause Bottom to win. Yet the increase in support for the Top party allows it to defeat the Center party in the first round. As a result, 15 votes that previously counted toward Center now count toward their second preference, Bottom, which is enough to give Bottom the victory. This makes the election an example of a [[Center squeeze|center-squeeze]], a class of elections where instant-runoff and [[Plurality voting|plurality]] have difficulty electing the majority-preferred candidate.<ref name="Handbook on Approval Voting2">{{Cite book |url=https://link.springer.com/10.1007/978-3-642-02839-7 |title=Handbook on Approval Voting |date=2010 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-02838-0 |editor-last=Laslier |editor-first=Jean-François |series=Studies in Choice and Welfare |location=Berlin, Heidelberg |pages=2 |doi=10.1007/978-3-642-02839-7 |quote=By eliminating the squeezing effect, Approval Voting would encourage the election of consensual candidates. The squeezing effect is typically observed in multiparty elections with a runoff. The runoff tends to prevent extremist candidates from winning, but a centrist candidate who would win any pairwise runoff (the "Condorcet winner") is also often "squeezed" between the left-wing and the right-wing candidates and so eliminated in the first round. |editor-last2=Sanver |editor-first2=M. Remzi}}</ref>

=== {{Anchor|Moulin|Condorcet|Moulin impossibility}}Condorcet methods ===
When there are at most 3 major candidates, [[Minimax Condorcet]] and its variants (such as [[Ranked Pairs|ranked pairs]] and [[Schulze's method]]) satisfy the participation criterion.<ref name=":0222"/> However, with more than 3 candidates, [[Hervé Moulin]] proved that every deterministic Condorcet method can sometimes fail participation.<ref name=":0222"/><ref name=":1222"/> Similar incompatibilities have also been shown for [[Resolvability criterion|set-valued]] voting rules.<ref name=":1222"/><ref>{{Cite journal |last=Pérez |first=Joaquín |date=2001-07-01 |title=The Strong No Show Paradoxes are a common flaw in Condorcet voting correspondences |journal=Social Choice and Welfare |volume=18 |issue=3 |pages=601–616 |citeseerx=10.1.1.200.6444 |doi=10.1007/s003550000079 |issn=0176-1714 |s2cid=153489135}}</ref><ref>{{Cite journal |last1=Jimeno |first1=José L. |last2=Pérez |first2=Joaquín |last3=García |first3=Estefanía |date=2009-01-09 |title=An extension of the Moulin No Show Paradox for voting correspondences |journal=Social Choice and Welfare |volume=33 |issue=3 |pages=343–359 |doi=10.1007/s00355-008-0360-6 |issn=0176-1714 |s2cid=30549097}}</ref> The [[Maximal lottery|randomized Condorcet rule]] satisfies the criterion, but fails the closely related [[monotonicity criterion]] in situations with Condorcet cycles.<ref>{{Cite journal |last1=Brandl |first1=Florian |last2=Brandt |first2=Felix |last3=Stricker |first3=Christian |date=2022-01-01 |title=An analytical and experimental comparison of maximal lottery schemes |journal=Social Choice and Welfare |volume=58 |issue=1 |pages=5–38 |doi=10.1007/s00355-021-01326-x |issn=1432-217X|doi-access=free |hdl=10419/286729 |hdl-access=free}}</ref>

Studies suggest such failures may be empirically rare, however. One study surveying 306 publicly available election datasets found no participation failures for methods in the [[ranked pairs]]-[[Minimax Condorcet|minimax]] family.<ref>Mohsin, F., Han, Q., Ruan, S., Chen, P. Y., Rossi, F., & Xia, L. (2023, May). Computational Complexity of Verifying the Group No-show Paradox. In Proceedings of the 2023 International Conference on Autonomous Agents and Multiagent Systems (pp. 2877-2879).</ref>

In addition to participation, an even more severe paradox property can occur with Condorect's principle: it may be better for a voter to submit a completely reversed ballot than to submit a ballot that ranks all candidates honestly.<ref>{{Cite journal |last1=Sanver |first1=M. Remzi |last2=Zwicker |first2=William S. |date=2009-08-20 |title=One-way monotonicity as a form of strategy-proofness |journal=International Journal of Game Theory |volume=38 |issue=4 |pages=553–574 |doi=10.1007/s00182-009-0170-9 |issn=0020-7276 |s2cid=29563457}}</ref><ref>{{Cite journal |last=Duddy |first=Conal |date=2013-11-29 |title=Condorcet's principle and the strong no-show paradoxes |journal=Theory and Decision |volume=77 |issue=2 |pages=275–285 |doi=10.1007/s11238-013-9401-4 |issn=0040-5833 |doi-access=free |hdl-access=free |hdl=10379/11267}}</ref>

=== Implementations of proportional representation ===

Some [[proportional representation]] systems allow for no-show paradoxes. Rules based on the [[Droop quota]] (including all current implementations of the [[single transferable vote]]) allow for no-show paradoxes.<ref name=":42222" /><ref>{{Cite journal |last=Ray |first=Dipankar |date=1983-07-01 |title=Hare's voting scheme and negative responsiveness |url=https://linkinghub.elsevier.com/retrieve/pii/016548968390032X |journal=Mathematical Social Sciences |volume=4 |issue=3 |pages=301–303 |doi=10.1016/0165-4896(83)90032-X |issn=0165-4896|url-access=subscription }}</ref>

[[Hamilton's method]] exhibits a milder form of the paradox called the [[population paradox]], but does not have a true no-show paradox. However, adding an [[electoral threshold]] produces a no-show paradox.<ref>{{Cite journal |last=Jones |first=Michael A. |last2=McCune |first2=David |last3=Wilson |first3=Jennifer |date=2019-01-01 |title=The elimination paradox: apportionment in the Democratic Party |url=https://doi.org/10.1007/s11127-018-0608-3 |journal=Public Choice |language=en |volume=178 |issue=1 |pages=53–65 |doi=10.1007/s11127-018-0608-3 |issn=1573-7101|url-access=subscription }}</ref>

==== In Germany ====
In Germany, situations where a voter's ballot has the opposite of its intended effect (e.g. a vote ''for'' a party or candidate causes them to ''lose'') are called ''negatives Stimmgewicht'' ({{Literal translation|negative voting weights}}). An infamous example occurred in the [[2005 German federal election]], when an article in ''[[Der Spiegel]]'' laid out how [[Christian Democratic Union of Germany|CDU]] voters in [[Dresden I]] would have to vote ''against'' their own party if they wished to avoid losing a seat in the [[Bundestag]].<ref name=":42222" /> This led to a lawsuit by [[electoral reform]] organization {{Interlanguage link|Mehr Demokratie|de}} and [[Alliance 90/The Greens]], joined by the neo-Nazi [[National Democratic Party of Germany|NPD of Germany]], who argued the election law was unconstitutionally undemocratic.<ref name=":0">[http://www.bundestag.de/dokumente/textarchiv/2011/35830868_kw39_angenommen_abgelehnt/index.html ''Die Beschlüsse des Bundestages 29. und 30. September'']{{Dead link|date=August 2025 |bot=InternetArchiveBot |fix-attempted=yes }}. In: ''bundestag.de'', retrieved 25 July 2012. Das Gesetz trat 3 December 2011 in Kraft: [http://www.buzer.de/gesetz/9944/index.htm ''Neunzehntes Gesetz zur Änderung des Bundeswahlgesetzes (19. BWahlGÄndG)'']. In: ''buzer.de'', retrieved 25 July 2012.</ref>

The [[Federal Constitutional Court]] agreed with the plaintiffs, ruling that negative vote weights violate the [[Basic Law for the Federal Republic of Germany|German constitution]]'s guarantee of [[One man, one vote|equal and direct suffrage]]. The majority wrote that:<ref name=":0322" /><ref name="bundesverfassungsgericht_de" />{{Blockquote|A seat allocation procedure that allows an increase in votes to lead to a loss of seats, or results in more seats being won if [proportionally] fewer votes are cast for it, contradicts the meaning and purpose of a democratic election[...]

Such nonsensical relationships between voting and electoral success not only impair the equality of the right to vote and the equal opportunities of the parties, but also the principle of a popular election, as it is no longer apparent to the voter how their vote results in the success or failure of a candidate.[...]

Negative vote weights cannot be accepted as constitutional on the premise that they cannot be predicted or planned, and thus can hardly be influenced by the individual voter. To what extent this is true can be set aside, as such arbitrary results make a mockery of the democratic competition for support from the electorate.|source=}}The ruling forced the [[Bundestag]] to abandon its old practice of ignoring [[overhang seat]]s, and instead adopt a new system of compensation involving [[leveling seat]]s.<ref name=":0" />

=== Quorum requirements ===
A common cause of no-show paradoxes is the use of a [[quorum]]. For example, if a public [[referendum]] requires 50% turnout to be binding, additional "no" votes may push turnout above 50%, causing the measure to pass. A referendum that instead required a minimum number of yes votes (e.g. >25% of the population voting "yes") would pass the participation criterion.<ref>{{Cite journal |last1=Aguiar-Conraria |first1=Luís |last2=Magalhães |first2=Pedro C. |date=2009-09-19 |title=Referendum design, quorum rules and turnout |url=http://dx.doi.org/10.1007/s11127-009-9504-1 |journal=Public Choice |volume=144 |issue=1–2 |pages=63–81 |doi=10.1007/s11127-009-9504-1 |issn=0048-5829|hdl=1822/7721 |hdl-access=free}}</ref>

Many representative bodies have quorum requirements where the same dynamic can be at play. For example, the requirement for a two-thirds [[quorum]] in the [[Oregon Legislative Assembly]] effectively creates an unofficial two-thirds [[supermajority]] requirement for passing bills, and can result in a law passing if too many senators turn out to oppose it.<ref>{{Cite web |last=Shumway |first=Julia |date=2023-06-14 |title=Oregon Democrats float new proposal to change quorum laws as Senate Republican walkout drags on • Oregon Capital Chronicle |url=https://oregoncapitalchronicle.com/2023/06/14/oregon-democrats-float-new-proposal-to-change-quorum-laws-as-senate-republican-walkout-drags-on/ |access-date=2024-08-15 |website=Oregon Capital Chronicle}}</ref> Deliberate ballot-spoiling strategies have been successful in ensuring referendums remain non-binding, as in the [[2023 Polish referendum]].

== Manipulation ==
The participation criterion can also be justified as a weaker form of [[strategyproofness]]: while it is impossible for honesty to always be the ''best'' strategy (by [[Gibbard's theorem]]), the participation criterion guarantees honesty will always be an effective, rather than actively counterproductive, strategy (i.e. a voter can always safely cast a sincere vote).<ref name=":0222" /> This can be particularly effective for encouraging honest voting if voters exhibit [[loss aversion]]. Rules with no-show paradoxes do not always allow voters to cast a sincere vote; for example, a sincere Palin > Begich > Peltola voter in the [[2022 Alaska's at-large congressional district special election|2022 Alaska special election]] would have been better off if they had not shown up at all, rather than casting an honest vote.

While no-show paradoxes can be deliberately exploited as a kind of [[strategic voting]], systems that fail the participation criterion are typically considered to be undesirable because they expose the underlying system as logically incoherent or "[[Spite (sentiment)|spiteful]]" (actively seeking to violate the preferences of some voters).<ref>{{Cite journal |last1=Holliday |first1=Wesley H. |last2=Pacuit |first2=Eric |date=2023-08-29 |title=Split Cycle: a new Condorcet-consistent voting method independent of clones and immune to spoilers |journal=Public Choice |volume=197 |issue=1–2 |pages=1–62 |doi=10.1007/s11127-023-01042-3 |issn=0048-5829 |quote=Of course, a method not satisfying participation will incentivize some strategic non-voting, as the voters in question will have an incentive not to vote (sincerely). But again, all voting methods incentivize strategic behavior[...] By contrast, we ''are'' troubled by failures of positive or negative involvement, as this shows that the method responds in the wrong way to unequivocal support for (resp. rejection of) a candidate.|doi-access=free |arxiv=2004.02350}}</ref>

=== Quorum-busting ===
{{Excerpt|Quorum|section=Quorum-busting|bold=no}}

== Examples ==
=== Majority judgment ===
{{Main|Majority judgment}}
This example shows that majority judgment violates the participation criterion. Assume two candidates A and B with 5 potential voters and the following ratings:
{| class="wikitable"
! colspan="2" |Candidates
! rowspan="2" |# of 
voters
|-
!A
!B
|-
| bgcolor="green" |Excellent
| bgcolor="YellowGreen" |Good
|2
|-
| bgcolor="yellow" |Fair
| bgcolor="orangered" |Poor
|2
|-
| bgcolor="orangered" |Poor
| bgcolor="YellowGreen" |Good
|1
|}
The two voters rating A "Excellent" are unsure whether to participate in the election.

==== Voters not participating ====
Assume the 2 voters would not show up at the polling place.

The ratings of the remaining 3 voters would be:
{| class="wikitable"
! colspan="2" |Candidates
! rowspan="2" |# of 
voters
|-
!A
!B
|-
| bgcolor="yellow" |Fair
| bgcolor="orangered" |Poor
|2
|-
| bgcolor="orangered" |Poor
| bgcolor="YellowGreen" |Good
|1
|}
The sorted ratings would be as follows:
{|
| align="right" |Candidate&nbsp;&nbsp;&nbsp;
|
{| cellspacing="0" border="0" width="500" cellpadding="0"
| width="49%" |&nbsp;
| width="2%" |↓
| width="49%" |Median point
|}
|-
| align="right" |A
|
{| cellspacing="0" border="0" width="500" cellpadding="0"
| width="67%" bgcolor="yellow" |&nbsp;
| width="33%" bgcolor="orangered" |
|}
|-
| align="right" |B
|
{| cellspacing="0" border="0" width="500" cellpadding="0"
| width="33%" bgcolor="yellowgreen" |&nbsp;
| width="67%" bgcolor="orangered" |
|}
|-
|&nbsp;
|&nbsp;
|-
|&nbsp;
|
{| cellspacing="1" border="0" cellpadding="1"
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
| bgcolor="green" |&nbsp;
|&nbsp;Excellent&nbsp;&nbsp;
| bgcolor="YellowGreen" |&nbsp;
|&nbsp;Good&nbsp;&nbsp;
| bgcolor="Yellow" |&nbsp;
|&nbsp;Fair&nbsp;&nbsp;
| bgcolor="Orangered" |&nbsp;
|&nbsp;Poor&nbsp;&nbsp;
|}
|}
'''Result''': A has the median rating of "Fair" and B has the median rating of "Poor". Thus, '''A''' is elected majority judgment winner.

==== Voters participating ====
Now, consider the 2 voters decide to participate:
{| class="wikitable"
! colspan="2" |Candidates
! rowspan="2" |# of 
voters
|-
!A
!B
|-
| bgcolor="green" |Excellent
| bgcolor="YellowGreen" |Good
|2
|-
| bgcolor="yellow" |Fair
| bgcolor="orangered" |Poor
|2
|-
| bgcolor="orangered" |Poor
| bgcolor="YellowGreen" |Good
|1
|}
The sorted ratings would be as follows:
{|
| align="right" |Candidate&nbsp;&nbsp;&nbsp;
|
{| cellspacing="0" border="0" width="500" cellpadding="0"
| width="49%" |&nbsp;
| width="2%" |↓
| width="49%" |Median point
|}
|-
| align="right" |A
|
{| cellspacing="0" border="0" width="500" cellpadding="0"
| width="40%" bgcolor="green" |&nbsp;
| width="40%" bgcolor="yellow" |
| width="20%" bgcolor="orangered" |
|}
|-
| align="right" |B
|
{| cellspacing="0" border="0" width="500" cellpadding="0"
| width="60%" bgcolor="yellowgreen" |&nbsp;
| width="40%" bgcolor="orangered" |
|}
|-
|&nbsp;
|&nbsp;
|-
|&nbsp;
|
{| cellspacing="1" border="0" cellpadding="1"
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
| bgcolor="green" |&nbsp;
|&nbsp;Excellent&nbsp;&nbsp;
| bgcolor="YellowGreen" |&nbsp;
|&nbsp;Good&nbsp;&nbsp;
| bgcolor="Yellow" |&nbsp;
|&nbsp;Fair&nbsp;&nbsp;
| bgcolor="Orangered" |&nbsp;
|&nbsp;Poor&nbsp;&nbsp;
|}
|}
'''Result''': A has the median rating of "Fair" and B has the median rating of "Good". Thus, '''B''' is the majority judgment winner.

=== Condorcet methods ===
This example shows how Condorcet methods can violate the participation criterion when there is a [[preference paradox]]. Assume four candidates A, B, C and D with 26 potential voters and the following preferences:
{| class="wikitable"
!Preferences
!# of voters
|-
|A > D > B > C
|8
|-
|B > C > A > D
|7
|-
|C > D > B > A
|7
|}
This gives the [[pairwise counting]] method:
{| class="wikitable" border="1"
|+Pairwise election results
! {{diagonal split header|2=X|1=Y}}
!A
!B
!C
!D
|-
! scope="row" |A
| {{CNone|}}
| {{diagonal split header|3=#e0e0ff|2=14|1=8}}
| {{diagonal split header|3=#e0e0ff|2=14|1=8}}
| {{diagonal split header|3=#ffe0e0|2=7|1=15}}
|-
! scope="row" |B
| {{diagonal split header|3=#ffe0e0|2=8|1=14}}
| {{CNone|}}
| {{diagonal split header|3=#ffe0e0|2=7|1=15}}
| {{diagonal split header|3=#e0e0ff|2=15|1=7}}
|-
! scope="row" |C
| {{diagonal split header|3=#ffe0e0|2=8|1=14}}
| {{diagonal split header|3=#e0e0ff|2=15|1=7}}
| {{CNone|}}
| {{diagonal split header|3=#ffe0e0|2=8|1=14}}
|-
! scope="row" |D
| {{diagonal split header|3=#e0e0ff|2=15|1=7}}
| {{diagonal split header|3=#ffe0e0|2=7|1=15}}
| {{diagonal split header|3=#e0e0ff|2=14|1=8}}
| {{CNone|}}
|-
! Pairwise results<br />for X,<br />won-tied-lost
|1-0-2
|2-0-1
|2-0-1
|1-0-2
|}
The sorted list of victories would be:
{| class="wikitable"
!Pair
!Winner
|-
|A (15) vs. D (7)
|A 15
|-
|B (15) vs. C (7)
|B 15
|-
|B (7) vs. D (15)
|D 15
|-
|A (8) vs. B (14)
|B 14
|-
|A (8) vs. C (14)
|C 14
|-
|C (14) vs. D (8)
|C 14
|}
'''Result''': A > D, B > C and D > B are locked in (and the other three can't be locked in after that), so the full ranking is A > D > B > C. Thus, '''A''' is elected ranked pairs winner.

==== Voters participating ====
Now, assume an extra 4 voters, in the top row, decide to participate:
{| class="wikitable"
!Preferences
!# of voters
|-
|A > B > C > D
|4
|-
|A > D > B > C
|8
|-
|B > C > A > D
|7
|-
|C > D > B > A
|7
|}
The results would be tabulated as follows:
{| class="wikitable" border="1"
|+Pairwise election results
! {{diagonal split header|2=X|1=Y}}
!A
!B
!C
!D
|-
! scope="row" |A
| {{CNone}}
| {{diagonal split header|3=#e0e0ff|2=14|1=12}}
| {{diagonal split header|3=#e0e0ff|2=14|1=12}}
| {{diagonal split header|3=#ffe0e0|2=7|1=19}}
|-
! scope="row" |B
| {{diagonal split header|3=#ffe0e0|2=12|1=14}}
| {{CNone}}
| {{diagonal split header|3=#ffe0e0|2=7|1=19}}
| {{diagonal split header|3=#e0e0ff|2=15|1=11}}
|-
! scope="row" |C
| {{diagonal split header|3=#ffe0e0|2=12|1=14}}
| {{diagonal split header|3=#e0e0ff|2=19|1=7}}
| {{CNone}}
| {{diagonal split header|3=#ffe0e0|2=8|1=18}}
|-
! scope="row" |D
| {{diagonal split header|3=#e0e0ff|2=19|1=7}}
| {{diagonal split header|3=#ffe0e0|2=11|1=15}}
| {{diagonal split header|3=#e0e0ff|2=18|1=8}}
| {{CNone}}
|-
! Pairwise results<br />for X,<br />won-tied-lost
|1-0-2
|2-0-1
|2-0-1
|1-0-2
|}
The sorted list of victories would be:
{| class="wikitable"
!Pair
!Winner
|-
|A (19) vs. D (7)
|A 19
|-
|B (19) vs. C (7)
|B 19
|-
|C (18) vs. D (8)
|C 18
|-
|B (11) vs. D (15)
|D 15
|-
|A (12) vs. B (14)
|B 14
|-
|A (12) vs. C (14)
|C 14
|}
'''Result''': A > D, B > C and C > D are locked in first. Now, D > B can't be locked in since it would create a cycle B > C > D > B. Finally, B > A and C > A are locked in. Hence, the full ranking is B > C > A > D. Thus, '''B''' is elected ranked pairs winner by adding a set of voters who prefer A to B.

== See also ==

* {{anl|Consistency criterion}}
* {{anl|Electoral system}}
* {{anl|Non-negative responsiveness}}

== References ==
<references ></references>

==Further reading==

* [[Douglas Woodall|Woodall, Douglas R]], "[http://www.votingmatters.org.uk/ISSUE6/P4.HTM Monotonicity and Single-Seat Election Rules]" ''Voting matters'', Issue 6, 1996
{{voting systems}}

[[Category:Electoral system criteria]]