Ranked pairs

 ( Electoral Systems ) 

Ranked Pairs ( RP), also known as the Tideman method, is a tournament-style system of ranked voting first proposed by Nicolaus Tideman in 1987.

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.

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 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) 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.

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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 value) margin of victory, going from largest to smallest.
3. Going down the list, check whether adding each matchup would create a Condorcet 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 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.

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Example

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The situation The results are tabulated as follows:

+Pairwise election results !	Memphis	Nashville	Chattanooga	Knoxville
Memphis
Nashville
Chattanooga
Knoxville

• 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

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Tally First, list every pair, and determine the 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:

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%

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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.

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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.

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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."

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Independence of irrelevant alternatives Ranked pairs fails independence of irrelevant alternatives, like all other Ranked voting. 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.

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Comparison table The following table compares ranked pairs with other single-winner election methods:

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Notes

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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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