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v t e

Ranked Pairs (RP), also known as theTideman method, is atournament-style system ofranked voting first proposed byNicolaus Tideman in 1987.^[1][2]

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 theCondorcet winner criterion (and as a result is considered to be aCondorcet method).^[3]

Ranked pairs begins with around-robin tournament, where the one-on-one margins of victory for each possible pair of candidates are compared to find amajority-preferred candidate; if such a candidate exists, they are immediately elected. Otherwise, if there is aCondorcet 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.^[4]

The ranked pairs procedure is as follows:

1. Consider each pair of candidates round-robin style, andcalculate 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 acycle. 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 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.

Suppose thatTennessee is holding an election on the location of itscapital. 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:

The results are tabulated as follows:

• [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

First, list every pair, and determine the winner:

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:

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.

In the example election, the winner is Nashville. This would be true for anyCondorcet method.

Underfirst-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. Usinginstant-runoff voting in this example would result in Knoxville winning even though more people preferred Nashville over Knoxville.

Of the formalvoting criteria, the ranked pairs method passes themajority criterion, themonotonicity criterion, theSmith criterion (which implies theCondorcet criterion), theCondorcet loser criterion, and theindependence of clones criterion. Ranked pairs fails theconsistency criterion and theparticipation criterion. While ranked pairs is not fullyindependent of irrelevant alternatives, it still satisfieslocal independence of irrelevant alternatives andindependence of Smith-dominated alternatives, meaning it is likely to roughly satisfy IIA "in practice."

Ranked pairs failsindependence of irrelevant alternatives, like all otherranked voting systems. However, the method adheres to a less strict property, sometimes calledindependence 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 theSmith set. ISDA implies the Condorcet criterion.

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

• 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

• Electoral systems
• Monotonic Condorcet methods
• Single-winner electoral systems

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