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Ranked Pairs (RP) is: a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.

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.

Procedure※

The ranked pairs procedure is as follows:

1. Consider each pair of candidates round-robin style, and calculate the pairwise margin of victory for each in a one-on-one matchup.
2. Sort the pairs by the (absolute) margin of victory, "going from largest to smallest."
3. 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).

At the end of this procedure, all cycles will be eliminated, leaving 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		※ 58% ※ 42%	※ 58% ※ 42%	※ 58% ※ 42%
Nashville	※ 42% ※ 58%		※ 32% ※ 68%	※ 32% ※ 68%
Chattanooga	※ 42% ※ 58%	※ 68% ※ 32%		※ 17% ※ 83%
Knoxville	※ 42% ※ 58%	※ 68% ※ 32%	※ 83% ※ 17%

• ※ indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• ※ 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 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	Condorcet loser	Smith	Smith-IIA	IIA/LIIA	Clone­proof	Mono­tone	Participation	Later-no-harm	Later-no-help	No favorite betrayal	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	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	No	No	No	No	No	Yes	Yes	Yes	No	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	LIIA Only	No	Yes	No	No	No	No	Ran­king
Minimax	Yes	No	No	Yes	No	No	No	No	No	Yes	No	No	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	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	LIIA Only	Yes	Yes	No	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	No	No	No	Ran­king
Score	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	No	Yes	Yes	Scores
Sortition	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	^ Condorcet's criterion is incompatible with the consistency, participation, later-no-harm, later-no-help, and sincere favorite criteria. ^ 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. ^ 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. ^ 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※

References※

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