Ranked pairs

Last updated June 25, 2024

Ranked Pairs (RP) is a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.^[1]^[2]

Ranked pairs begins with a round-robin tournament, where the one-on-one margins of victory for each candidate are compared to find a majority-preferred candidate; if such a candidate exists, they are immediately elected. Otherwise, if there is a Condorcet cycle—a rock-paper-scissors-like sequence A > B > C > A—the cycle is broken by dropping the "weakest" elections in the cycle, i.e. the ones that are closest to being tied.^[3]

Procedure

The ranked pairs procedure is as follows:

Consider each pair of candidates round-robin style, and calculate the pairwise margin of victory for each in a one-on-one matchup.
Sort the pairs by the (absolute) margin of victory, going from largest to smallest.
Going down the list, check whether adding each matchup would create a 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).^{[note 1]}

At the end of this procedure, all cycles will be eliminated, leaving a unique winner who wins every one-on-one matchup (that has not been crossed out). The lack of cycles means that candidates can be ranked directly based on the matchups that have been left behind.

Example

The situation

Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:

Memphis, the largest city, but far from the others (42% of voters)
Nashville, near the center of the state (26% of voters)
Chattanooga, somewhat east (15% of voters)
Knoxville, far to the northeast (17% of voters)

The preferences of each region's voters are:

42% of voters Far-West	26% of voters Center	15% of voters Center-East	17% of voters Far-East
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville Memphis

The results are tabulated as follows:

Pairwise election results
A B	Memphis	Nashville	Chattanooga	Knoxville
Memphis		[A] 58% [B] 42%	[A] 58% [B] 42%	[A] 58% [B] 42%
Nashville	[A] 42% [B] 58%		[A] 32% [B] 68%	[A] 32% [B] 68%
Chattanooga	[A] 42% [B] 58%	[A] 68% [B] 32%		[A] 17% [B] 83%
Knoxville	[A] 42% [B] 58%	[A] 68% [B] 32%	[A] 83% [B] 17%

[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:

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:

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).

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 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 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 independent of irrelevant alternatives, it still satisfies local independence of irrelevant alternatives ^{[ broken anchor ]} and independence of Smith-dominated alternatives, meaning it is likely to roughly satisfy IIA "in practice."

Independence of irrelevant alternatives

Ranked pairs fails independence of irrelevant alternatives, like all other 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 single-winner voting systems
Criterion Method	Majority	Majority loser	Mutual majority	Condorcet winner ^{[Tn 1]}	Condorcet loser	Smith ^{[Tn 1]}	Smith-IIA ^{[Tn 1]}	IIA/LIIA ^{[Tn 1]}	Cloneproof	Monotone	Participation	Later-no-harm ^{[Tn 1]}	Later-no-help ^{[Tn 1]}	No favorite betrayal ^{[Tn 1]}	Ballot type
Anti-plurality	No	Yes	No	No	No	No	No	No	No	Yes	Yes	No	No	Yes	Single mark
Approval	Yes	No	No	No	No	No	No	Yes^{[Tn 2]}	Yes	Yes	Yes	No	Yes	Yes	Approvals
Baldwin	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	No	Ranking
Black	Yes	Yes	No	Yes	Yes	No	No	No	No	Yes	No	No	No	No	Ranking
Borda	No	Yes	No	No	Yes	No	No	No	No	Yes	Yes	No	Yes	No	Ranking
Bucklin	Yes	Yes	Yes	No	No	No	No	No	No	Yes	No	No	Yes	No	Ranking
Coombs	Yes	Yes	Yes	No	Yes	No	No	No	No	No	No	No	No	Yes	Ranking
Copeland	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	No	Yes	No	No	No	No	Ranking
Dodgson	Yes	No	No	Yes	No	No	No	No	No	No	No	No	No	No	Ranking
Highest median	Yes	Yes^{[Tn 3]}	No^{[Tn 4]}	No	No	No	No	Yes^{[Tn 2]}	Yes	Yes	No^{[Tn 5]}	No	Yes	Yes	Scores
Instant-runoff	Yes	Yes	Yes	No	Yes	No	No	No	Yes	No	No	Yes	Yes	No	Ranking
Kemeny–Young	Yes	Yes	Yes	Yes	Yes	Yes	Yes	LIAA Only	No	Yes	No	No	No	No	Ranking
Minimax	Yes	No	No	Yes^{[Tn 6]}	No	No	No	No	No	Yes	No	No^{[Tn 6]}	No	No	Ranking
Nanson	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	No	Ranking
Plurality	Yes	No	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	No	Single mark
Random ballot ^{[Tn 7]}	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Single mark
Ranked pairs	Yes	Yes	Yes	Yes	Yes	Yes	Yes	LIAA Only	Yes	Yes	No^{[Tn 5]}	No	No	No	Ranking
Runoff	Yes	Yes	No	No	Yes	No	No	No	No	No	No	Yes	Yes	No	Single mark
Schulze	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	No^{[Tn 5]}	No	No	No	Ranking
Score	No	No	No	No	No	No	No	Yes^{[Tn 2]}	Yes	Yes	Yes	No	Yes	Yes	Scores
Sortition ^{[Tn 8]}	No	No	No	No	No	No	No	Yes	No	Yes	Yes	Yes	Yes	Yes	None
STAR	No	Yes	No	No	Yes	No	No	No	No	Yes	No	No	No	No	Scores
Tideman alternative	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	No	No	No	No	No	Ranking
Table Notes	1 2 3 4 5 6 7 \|Condorcet, Smith and Independence of Smith-dominated alternatives criteria are incompatible with Independence of irrelevant alternatives, Consistency, Participation, Later-no-harm, Later-no-help, and Favorite betrayal ^{[ clarification needed ]} criteria. 1 2 3 Approval voting, score voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates independently using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates. ↑ Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters. ↑ Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade. 1 2 3 In Highest median, Ranked Pairs, and Schulze voting, there is always a regret-free, semi-honest ballot for any voter, holding all other ballots constant and assuming they know enough about how others will vote. Under such circumstances, there is always at least one way for a voter to participate without grading any less-preferred candidate above any more-preferred one. 1 2 A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm. ↑ A randomly chosen ballot determines winner. This and closely related methods are of mathematical interest and included here to demonstrate that even unreasonable methods can pass voting method criteria. ↑ Where a winner is randomly chosen from the candidates, sortition is included to demonstrate that even non-voting methods can pass some criteria.

Notes

↑ 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.

Related Research Articles

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Instant-runoff voting (IRV), also known as ranked-choice voting or the alternative vote (AV), combines ranked voting together with a system for choosing winners from these rankings by repeatedly eliminating the candidate with the fewest first-place votes and reassigning their votes until only one candidate is left. It can be seen as a modified form of a runoff election or exhaustive ballot in which, after eliminating some candidates, the choice among the rest is made from already-given voter rankings rather than from a separate election. Many sources conflate this system of choosing winners with ranked-choice voting more generally, for which several other systems of choosing winners have also been used.

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<span class="mw-page-title-main">Ranked voting</span> Family of electoral systems

The term ranked voting, also known as preferential voting or ranked-choice voting, pertains to any voting system where voters indicate a rank to order candidates or options—in a sequence from first, second, third, and onwards—on their ballots. Ranked voting systems vary based on the ballot marking process, how preferences are tabulated and counted, the number of seats available for election, and whether voters are allowed to rank candidates equally.

STAR voting is an electoral system for single-seat elections. The name stands for "Score then Automatic Runoff", referring to the fact that this system is a combination of score voting, to pick two finalists with the highest total scores, followed by an "automatic runoff" in which the finalist who is preferred on more ballots wins. It is a type of cardinal voting electoral system.

Tideman's Alternative Method, also called Alternative Smith or Alternative Schwartz, is an electoral system developed by Nicolaus Tideman which selects a single winner using votes that express preferences.

Round-robin voting refers to a set of ranked voting systems that elect winners by comparing all candidates in a round-robin tournament. Every candidate is matched up against every other candidate, where their point total is equal to the number of votes they receive; the method then selects a winner based on the results of these paired matchups.

References

↑ Tideman, T. N. (1987-09-01). "Independence of clones as a criterion for voting rules". Social Choice and Welfare. 4 (3): 185–206. doi:10.1007/BF00433944. ISSN 1432-217X. S2CID 122758840.
↑ Schulze, Markus (October 2003). "A New Monotonic and Clone-Independent Single-Winner Election Method". Voting matters (www.votingmatters.org.uk). 17. McDougall Trust. Archived from the original on 2020-07-11. Retrieved 2021-02-02.
↑ Munger, Charles T. (2022). "The best Condorcet-compatible election method: Ranked Pairs". Constitutional Political Economy . 34 (3): 434–444. doi: 10.1007/s10602-022-09382-w .

External links

Descriptions of ranked-ballot voting methods by Rob LeGrand
Example JS implementation by Asaf Haddad
Pair Ranking Ruby Gem by Bala Paranj
A margin-based PHP Implementation of Tideman's Ranked Pairs
Rust implementation of Ranked Pairs by Cory Dickson

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[condorcet-iia-incompatibility-5] 1 2 3 4 5 6 7 |Condorcet, Smith and Independence of Smith-dominated alternatives criteria are incompatible with Independence of irrelevant alternatives, Consistency, Participation, Later-no-harm, Later-no-help, and Favorite betrayal ^{[ clarification needed ]} criteria.

[IIA_rating_methods-6] 1 2 3 Approval voting, score voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates independently using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates.

[7] Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters.

[mjmmc-8] Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade.

[semipc-9] 1 2 3 In Highest median, Ranked Pairs, and Schulze voting, there is always a regret-free, semi-honest ballot for any voter, holding all other ballots constant and assuming they know enough about how others will vote. Under such circumstances, there is always at least one way for a voter to participate without grading any less-preferred candidate above any more-preferred one.

[MinimaxVarient-10] 1 2 A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.

[11] A randomly chosen ballot determines winner. This and closely related methods are of mathematical interest and included here to demonstrate that even unreasonable methods can pass voting method criteria.

[12] Where a winner is randomly chosen from the candidates, sortition is included to demonstrate that even non-voting methods can pass some criteria.

[4] 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.

[Tideman2-1] Tideman, T. N. (1987-09-01). "Independence of clones as a criterion for voting rules". Social Choice and Welfare. 4 (3): 185–206. doi:10.1007/BF00433944. ISSN 1432-217X. S2CID 122758840.

[2] Schulze, Markus (October 2003). "A New Monotonic and Clone-Independent Single-Winner Election Method". Voting matters (www.votingmatters.org.uk). 17. McDougall Trust. Archived from the original on 2020-07-11. Retrieved 2021-02-02.

[3] Munger, Charles T. (2022). "The best Condorcet-compatible election method: Ranked Pairs". Constitutional Political Economy . 34 (3): 434–444. doi: 10.1007/s10602-022-09382-w .

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