Ranked Pairs
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Ranked Pairs (RP) or Tideman (named after its developer Nicolaus Tideman) is a voting method that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners.
If there is a candidate who is preferred over the other candidates, when compared in turn with each of the others, RP guarantees that candidate will win. Because of this property, RP is (by definition) a Condorcet method. It is closely related to another Condorcet method, the Schulze method.
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[edit] Procedure
The RP procedure is as follows:
- Tally the vote count comparing each pair of candidates, and determine the winner of each pair (provided there is not a tie)
- Sort (rank) each pair, by the largest margin of victory first to smallest last.
- "Lock in" each pair, starting with the one with the largest number of winning votes, and add one in turn to a graph as long as they do not create a cycle (which would create an ambiguity). The completed graph shows the winner.
RP can also be used to create a sorted list of preferred candidates. To create a sorted list, repeatedly use RP to select a winner, remove that winner from the list of candidates, and repeat (to find the next runner up, and so forth).
[edit] Tally
To tally the votes, consider each voters' preferences. For example, if a voter states "A > B > C" (A is better than B, and B is better than C), the tally should add one for A in A vs. B, one for A in A vs. C, and one for B in B vs. C. Voters may also express indifference (e.g., A = B), and unstated candidates are assumed to be equally worse than the stated candidates.
Once tallied the majorities can be determined. If "Vxy" is the number of Votes that rank x over y, then "x" wins if Vxy > Vyx, and "y" wins if Vyx > Vxy.
[edit] Sort
The pairs of winners, called the "majorities", are then sorted from the largest majority to the smallest majority. A majority for x over y precedes a majority for z over w if and only if at least one of the following conditions holds:
- Vxy > Vzw. In other words, the majority having more support for its alternative is ranked first.
- Vxy = Vzw and Vwz > Vyx. Where the majorities are equal, the majority with the smaller minority opposition is ranked first.
[edit] Lock
The next step is to examine each pair in turn to determine which pairs to "lock in". Using the sorted list above, lock in each pair in turn unless the pair will create a circularity in a graph (e.g., where A is more than B, B is more than C, but C is more than A).
[edit] An example
[edit] The situation
Imagine that the population of Tennessee, a state in the United States, is voting on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in one of these four cities, and that they would like the capital to be established as close to their city as possible.
The candidates for the capital are:
- Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
- Nashville, with 26% of the voters
- Knoxville, with 17% of the voters
- Chattanooga, with 15% of the voters
The preferences of the voters would be divided like this:
| 42% of voters (close to Memphis) |
26% of voters (close to Nashville) |
15% of voters (close to Chattanooga) |
17% of voters (close to Knoxville) |
|---|---|---|---|
|
|
|
|
The results would be tabulated as follows:
| A | |||||
|---|---|---|---|---|---|
| Memphis | Nashville | Chattanooga | Knoxville | ||
| B | 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% |
||
| Pairwise election results (won-lost-tied): | 0-3-0 | 3-0-0 | 2-1-0 | 1-2-0 | |
| Votes against in worst pairwise defeat: | 58% | N/A | 68% | 83% | |
- [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
- [NP] indicates voters who expressed no preference between either candidate
[edit] 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% |
Note that absolute counts of votes can be used, or percentages of the total number of votes; it makes no difference.
[edit] Sort
The votes are then sorted. The largest majority is "Chattanooga over Knoxville"; 83% of the voters prefer Chattanooga. Nashville (68%) beats both Chattanooga and Knoxville by a score of 68% over 32% (an exact tie, which is unlikely in real life for this many voters). Since Chattanooga > Knoxville, and they're the losers, Nashville vs. Knoxville will be added first, followed by Nashville vs. 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% |
[edit] 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 from the winner).
In this example, Nashville is the winner using RP.
[edit] Ambiguity resolution example
Let's say there was an ambiguity. For a simple situation involving candidates A, B, and C.
- A > B 68%
- B > C 72%
- C > A 52%
In this situation we "lock in" the majorities starting with the greatest one first.
- Lock B > C
- Lock A > B
- We don't lock in the final C > A as it creates an ambiguity or cycle.
Therefore, A is the winner.
[edit] Summary
In the example election, the winner is Nashville. This would be true for any Condorcet method. Using the first-past-the-post system 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.
[edit] Criteria
Of the formal voting system criteria, the Ranked Pairs method passes the majority criterion, the monotonicity criterion, 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 does satisfy local independence of irrelevant alternatives.
[edit] Independence of irrelevant alternatives
Ranked Pairs fails independence from irrelevant alternatives. However, the method adheres to a less strict property, sometimes called local independence from irrelevant alternatives ("local IIA"). 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. Local IIA implies the Condorcet criterion.
[edit] Use of Ranked Pairs
[edit] See also
[edit] References
Tideman, T.N. (1987) Independence of clones as a criterion for voting rules. Social Choice and Welfare 4: 185-206.
[edit] External resources
- Ranked Pairs by Blake Cretney
- Voting methods survey by James Green-Armytage
- Descriptions of ranked-ballot voting methods by Rob LeGrand
