Ranked pairs · Lablypedia

The ballots are used to determine the winner in any race with just two candidates, based upon which of the two candidates is ranked higher on each ballot. If there is a candidate who wins regardless of whom they are paired against then that candidate is elected the winner. If there is no candidate who wins every pairing then the pairings with a more decisive win dominate those that are less decisive. For example, if Paper beats Rock, Rock beats Scissors, and Scissors beats Paper; and it is the case that the first two wins are more decisive than the third, then the third is ignored and Paper is elected the winner by virtue of winning their remaining pairings.

This system of ranked voting was first proposed by Nicolaus Tideman in 1987. Unlike Instant Runoff Voting, Ranked Pairs is guaranteed to satisfy the Condorcet winner criterion, meaning that any candidate who beats every other candidate, in a one-on-one race between the two, will be elected the winner.

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 pairing.
Sort the pairs by the absolute margin of victory, going from largest to smallest.
Going down the list, check whether adding each pairing 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 all of their remaining one-on-one pairings. The lack of cycles means that candidates can be ranked directly based on the pairings that have been left behind.

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Example

The situation

Suppose Tennessee is holding an election on the location of its capital. The population is split between four cities, and all the voters want the capital to be as close to them as possible. The options are:

Memphis, large but far to the west
Nashville, medium, near the center
Chattanooga, small and in the east
Knoxville, small and isolated

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: