# Ranked pairs

Ranked Pairs (RP) is a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.[1][2]

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.[3]

## 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).[note 1]

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.

## Example

### The situation

Suppose that Tennessee is holding an election on the location of its capital. 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:

42% of voters
Far-West
26% of voters
Center
15% of voters
Center-East
17% of voters
Far-East
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

The results are tabulated as follows:

 AB Memphis Nashville Chattanooga Knoxville Memphis [A] 58% [B] 42% [A] 58% [B] 42% [A] 58% [B] 42% Nashville [A] 42% [B] 58% [A] 32% [B] 68% [A] 32% [B] 68% Chattanooga [A] 42% [B] 58% [A] 68% [B] 32% [A] 17% [B] 83% Knoxville [A] 42% [B] 58% [A] 68% [B] 32% [A] 83% [B] 17%
• [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:

Pair 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:

Pair Winner
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%

### 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[broken anchor] 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:

Comparison of single-winner voting systems
Criterion

Method
Majority Majority loser Mutual majority Condorcet winner[Tn 1] Condorcet loser Smith[Tn 1] Smith-IIA[Tn 1] IIA/LIIA[Tn 1] Clone­proof Mono­tone Participation Later-no-harm[Tn 1] Later-no-help[Tn 1] No favorite betrayal[Tn 1] Ballot
type
Anti-plurality No Yes No No No No No No No Yes Yes No No Yes Single mark
Approval Yes No No No No No No Yes[Tn 2] Yes Yes Yes No Yes Yes Appr­ovals
Baldwin Yes Yes Yes Yes Yes Yes No No No No No No No No Ran­king
Black Yes Yes No Yes Yes No No No No Yes No No No No Ran­king
Borda No Yes No No Yes No No No No Yes Yes No Yes No Ran­king
Bucklin Yes Yes Yes No No No No No No Yes No No Yes No Ran­king
Coombs Yes Yes Yes No Yes No No No No No No No No Yes Ran­king
Copeland Yes Yes Yes Yes Yes Yes Yes No No Yes No No No No Ran­king
Dodgson Yes No No Yes No No No No No No No No No No Ran­king
Highest median Yes Yes[Tn 3] No[Tn 4] No No No No Yes[Tn 2] Yes Yes No[Tn 5] No Yes Yes Scores
Instant-runoff Yes Yes Yes No Yes No No No Yes No No Yes Yes No Ran­king
Kemeny–Young Yes Yes Yes Yes Yes Yes Yes LIIA Only No Yes No No No No Ran­king
Minimax Yes No No Yes[Tn 6] No No No No No Yes No No[Tn 6] No No Ran­king
Nanson Yes Yes Yes Yes Yes Yes No No No No No No No No Ran­king
Plurality Yes No No No No No No No No Yes Yes Yes Yes No Single mark
Random ballot[Tn 7] No No No No No No No Yes Yes Yes Yes Yes Yes Yes Single mark
Ranked pairs Yes Yes Yes Yes Yes Yes Yes LIIA Only Yes Yes No[Tn 5] No No No Ran­king
Runoff Yes Yes No No Yes No No No No No No Yes Yes No Single mark
Schulze Yes Yes Yes Yes Yes Yes Yes No Yes Yes No[Tn 5] No No No Ran­king
Score No No No No No No No Yes[Tn 2] Yes Yes Yes No Yes Yes Scores
Sortition[Tn 8] No No No No No No No Yes No Yes Yes Yes Yes Yes None
STAR No Yes No No Yes No No No No Yes No No No No Scores
Tideman alternative Yes Yes Yes Yes Yes Yes Yes No Yes No No No No No Ran­king
Table Notes
1. Condorcet's criterion is incompatible with the consistency, participation, later-no-harm, later-no-help, and sincere favorite criteria.
2. ^ a b c Approval voting, score voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates independently using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates.
3. ^ Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters.
4. ^ Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade.
5. ^ a b c In Highest median, Ranked Pairs, and Schulze voting, there is always a regret-free, semi-honest ballot for any voter, holding all other ballots constant and assuming they know enough about how others will vote. Under such circumstances, there is always at least one way for a voter to participate without grading any less-preferred candidate above any more-preferred one.
6. ^ a b A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.
7. ^ A randomly chosen ballot determines winner. This and closely related methods are of mathematical interest and included here to demonstrate that even unreasonable methods can pass voting method criteria.
8. ^ Where a winner is randomly chosen from the candidates, sortition is included to demonstrate that even non-voting methods can pass some criteria.

## Notes

1. ^ Rather than crossing out near-ties, step 3 is sometimes described as going down the list and confirming ("locking in") the largest victories that do not create a cycle, then ignoring any victories that are not locked-in.

## References

1. ^ Tideman, T. N. (1987-09-01). "Independence of clones as a criterion for voting rules". Social Choice and Welfare. 4 (3): 185–206. doi:10.1007/BF00433944. ISSN 1432-217X. S2CID 122758840.
2. ^ Schulze, Markus (October 2003). "A New Monotonic and Clone-Independent Single-Winner Election Method". Voting matters (www.votingmatters.org.uk). 17. McDougall Trust. Archived from the original on 2020-07-11. Retrieved 2021-02-02.
3. ^ Munger, Charles T. (2022). "The best Condorcet-compatible election method: Ranked Pairs". Constitutional Political Economy. 34 (3): 434–444. doi:10.1007/s10602-022-09382-w.