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{{Short description|Single-winner electoral system}}
{{Electoral systems sidebar|expanded=Single-winner}}
'''Ranked Pairs''' ('''RP'''), also known as the '''Tideman method''', is a [[Round-robin voting|tournament-style]] system of [[ranked voting]] first proposed by [[Nicolaus Tideman]] in 1987.<ref name="Tideman2">{{Cite journal |last=Tideman |first=T. N. |date=1987-09-01 |title=Independence of clones as a criterion for voting rules |url=https://doi.org/10.1007/BF00433944 |journal=Social Choice and Welfare |language=en |volume=4 |issue=3 |pages=185–206 |doi=10.1007/BF00433944 |issn=1432-217X |s2cid=122758840|url-access=subscription }}</ref><ref>{{Cite journal |last=Schulze |first=Markus |date=October 2003 |title=A New Monotonic and Clone-Independent Single-Winner Election Method |url=http://www.votingmatters.org.uk/ISSUE17/INDEX.HTM |journal=Voting matters (www.votingmatters.org.uk) |publisher=McDougall Trust |volume=17 |pages= |archive-url=https://web.archive.org/web/20200711055629/www.votingmatters.org.uk/ISSUE17/INDEX.HTM |archive-date=2020-07-11 |access-date=2021-02-02 |via=}}</ref>

If there is a candidate who is preferred over the other candidates, when compared in turn with each of the others, the ranked-pairs procedure guarantees that candidate will win. Therefore, the ranked-pairs procedure complies with the [[Condorcet winner criterion]]—that is, it is a [[Condorcet method]].<ref>{{cite journal |last1=Munger |first1=Charles T. |title=The best Condorcet-compatible election method: Ranked Pairs |journal=[[Constitutional Political Economy]] |date=2022 |doi=10.1007/s10602-022-09382-w |doi-access=free}}</ref>

Ranked pairs begins with a [[round-robin tournament]], where the one-on-one margins of victory for each possible pair of candidates are compared to find a [[Condorcet winner criterion|majority-preferred candidate]]; if such a candidate exists, they are immediately elected. Otherwise, if there is a [[Condorcet paradox|Condorcet cycle]] (a rock-paper-scissors-like sequence A > B > C > A) of three or more candidates then the cycle is broken by dropping the "weakest" election in the cycle, i.e. the one that is closest to being tied.<ref>{{cite journal |last1=Munger |first1=Charles T. |date=2022 |title=The best Condorcet-compatible election method: Ranked Pairs |journal=[[Constitutional Political Economy]] |volume=34 |issue=3 |pages=434–444 |doi=10.1007/s10602-022-09382-w |doi-access=free}}</ref>

== Procedure ==
The ranked pairs procedure is as follows:

# Consider each pair of candidates round-robin style, and [[Pairwise counting|calculate the pairwise margin of victory]] for each in a one-on-one matchup.
# Sort the pairs by the ([[Absolute value|absolute]]) margin of victory, going from largest to smallest.
# Going down the list, check whether adding each matchup would create a [[Condorcet cycle|cycle]]. If it would, [[cross out]] the election; this will be the election(s) in the cycle with the smallest margin of victory (near-ties).{{NoteTag|Rather than crossing out near-ties, step 3 is sometimes described as going down the list and confirming ("locking in") the largest victories that do not create a cycle, then ignoring any victories that are not locked-in.}}

At the end of this procedure, all cycles will be eliminated, leaving a unique winner who wins all of the remaining one-on-one matchups. The lack of cycles means that candidates can be ranked directly based on the matchups that have been left behind.

== Example ==

=== The situation ===
{{Tenn voting example}}

The results are tabulated as follows:
{| class="wikitable"
|+Pairwise election results
!{{Diagonal split header|B|A|#ffe0e0|#e0e0ff}}
| bgcolor="#c0c0ff" |'''Memphis'''
| bgcolor="#c0c0ff" |'''Nashville'''
| bgcolor="#c0c0ff" |'''Chattanooga'''
| bgcolor="#c0c0ff" |'''Knoxville'''
|-
| bgcolor="#ffc0c0" |'''Memphis'''
|
| {{Diagonal split header|42%|58%|#e0e0ff}}
| {{Diagonal split header|42%|58%|#e0e0ff}}
| {{Diagonal split header|42%|58%|#e0e0ff}}
|-
| bgcolor="#ffc0c0" |'''Nashville'''
| {{Diagonal split header|58%|42%|#ffe0e0}}
|
| {{Diagonal split header|68%|32%|#ffe0e0}}
| {{Diagonal split header|68%|32%|#ffe0e0}}
|-
| bgcolor="#ffc0c0" |'''Chattanooga'''
| {{Diagonal split header|58%|42%|#ffe0e0}}
| {{Diagonal split header|32%|68%|#e0e0ff}}
|
| {{Diagonal split header|83%|17%|#ffe0e0}}
|-
| bgcolor="#ffc0c0" |'''Knoxville'''
| {{Diagonal split header|58%|42%|#ffe0e0}}
| {{Diagonal split header|32%|68%|#e0e0ff}}
| {{Diagonal split header|17%|83%|#e0e0ff}}
|
|}

* [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
* [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption

=== Tally ===
First, list every pair, and determine the winner:
{| class="wikitable sortable"
!Pair
!Winner
|-
|Memphis (42%) vs. Nashville (58%)
|Nashville 58%
|-
|Memphis (42%) vs. Chattanooga (58%)
|Chattanooga 58%
|-
|Memphis (42%) vs. Knoxville (58%)
|Knoxville 58%
|-
|Nashville (68%) vs. Chattanooga (32%)
|Nashville 68%
|-
|Nashville (68%) vs. Knoxville (32%)
|Nashville 68%
|-
|Chattanooga (83%) vs. Knoxville (17%)
|Chattanooga 83%
|}
The votes are then sorted. The largest majority is "Chattanooga over Knoxville"; 83% of the voters prefer Chattanooga. Thus, the pairs from above would be sorted this way:
{| class="wikitable"
!Pair
!Winner
|-
|Chattanooga (83%) vs. Knoxville (17%)
|Chattanooga 83%
|-
|Nashville (68%) vs. Knoxville (32%)
|Nashville 68%
|-
|Nashville (68%) vs. Chattanooga (32%)
|Nashville 68%
|-
|Memphis (42%) vs. Nashville (58%)
|Nashville 58%
|-
|Memphis (42%) vs. Chattanooga (58%)
|Chattanooga 58%
|-
|Memphis (42%) vs. Knoxville (58%)
|Knoxville 58%
|}

=== Lock ===
The pairs are then locked in order, skipping any pairs that would create a cycle:

* Lock Chattanooga over Knoxville.
* Lock Nashville over Knoxville.
* Lock Nashville over Chattanooga.
* Lock Nashville over Memphis.
* Lock Chattanooga over Memphis.
* Lock Knoxville over Memphis.

In this case, no cycles are created by any of the pairs, so every single one is locked in.

Every "lock in" would add another arrow to the graph showing the relationship between the candidates. Here is the final graph (where arrows point away from the winner).

[[File:Tennessee-vote.svg]]

In this example, Nashville is the winner using the ranked-pairs procedure.  Nashville is followed by Chattanooga, Knoxville, and Memphis in second, third, and fourth places respectively.

=== Summary ===
In the example election, the winner is Nashville. This would be true for any [[Condorcet method]].

Under [[First-past-the-post voting|first-past-the-post]] and some other systems, Memphis would have won the election by having the most people, even though Nashville won every simulated pairwise election outright. Using [[instant-runoff voting]] in this example would result in Knoxville winning even though more people preferred Nashville over Knoxville.

== Criteria ==
Of the formal [[voting criteria]], the ranked pairs method passes the [[Majority favorite criterion|majority criterion]], the [[monotonicity criterion]], the [[Smith criterion]] (which implies the [[Condorcet criterion]]), the [[Condorcet loser criterion]], and the [[independence of clones criterion]]. Ranked pairs fails the [[consistency criterion]] and the [[participation criterion]]. While ranked pairs is not fully [[Independence of irrelevant alternatives|independent of irrelevant alternatives]], it still satisfies [[Independence of irrelevant alternatives#Local independence|local independence of irrelevant alternatives]] and [[independence of Smith-dominated alternatives]], meaning it is likely to roughly satisfy IIA "in practice."

=== Independence of irrelevant alternatives ===<!--This section is linked from [[Condorcet method]]-->
Ranked pairs fails [[independence of irrelevant alternatives]], like all other [[Ranked voting|ranked voting systems]]. However, the method adheres to a less strict property, sometimes called [[independence of Smith-dominated alternatives]] (ISDA). It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the [[Smith set]]. ISDA implies the Condorcet criterion.
=== Comparison table ===
The following table compares ranked pairs with other single-winner election methods:{{Comparison of Schulze to preferential voting systems}}

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}

== External links ==

* [https://accuratedemocracy.com/download/software/rbvote/desc.html Descriptions of ranked-ballot voting methods] by Rob LeGrand
* [https://gist.github.com/asafh/a8e9af7a3e5282cbba27 Example JS implementation] by Asaf Haddad
* [https://bitbucket.org/bparanj/rasam Pair Ranking Ruby Gem] by Bala Paranj
* [https://github.com/pivot-libre/tideman/ A margin-based PHP Implementation of Tideman's Ranked Pairs]
* [https://github.com/corydickson/pairwise-voting Rust implementation of Ranked Pairs] by Cory Dickson
{{voting systems}}
[[Category:Electoral systems]]
[[Category:Monotonic Condorcet methods]]
[[Category:Single-winner electoral systems]]