Ranked pairs
Ranked Pairs, also known as the Tideman method, is a ranked voting method that determines a single winner from ballots that rank candidates in order of preference. The method is like a round-robin tournament in that it examines every possible pairing of one candidate against another.
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 in the cycle with the smallest margin of victory.
Example
The situation
The results are tabulated as follows:| Memphis | Nashville | Chattanooga | Knoxville | |
| Memphis | ||||
| Nashville | ||||
| Chattanooga | ||||
| Knoxville |
- 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
| Pair | Winner |
| Memphis vs. Nashville | Nashville 58% |
| Memphis vs. Chattanooga | Chattanooga 58% |
| Memphis vs. Knoxville | Knoxville 58% |
| Nashville vs. Chattanooga | Nashville 68% |
| Nashville vs. Knoxville | Nashville 68% |
| Chattanooga vs. Knoxville | 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 vs. Knoxville | Chattanooga 83% |
| Nashville vs. Knoxville | Nashville 68% |
| Nashville vs. Chattanooga | Nashville 68% |
| Memphis vs. Nashville | Nashville 58% |
| Memphis vs. Chattanooga | Chattanooga 58% |
| Memphis vs. Knoxville | 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.
Every "lock in" would add another arrow to the graph showing the relationship between the candidates. Here is the final graph.
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.