Ranked pairs

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 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.

The situation
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

Tally
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:

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: