Ranked pairs
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Ranked pairs (RP) or the Tideman method is an electoral system developed in 1987 by Nicolaus Tideman 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.
Contents
Procedure
The RP (Ranked Pair) 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 strength 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).
Tally
To tally the votes, consider each voter's 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.
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 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.^{[vs 1]}
Lock
The next step is to examine each pair in turn to determine the pairs to "lock in". This can be visualized by drawing an arrow from the pair's winner to the pair's loser in a directed graph. Using the sorted list above, lock in each pair in turn unless the pair will create a circularity in the graph (for example, where A is more than B, B is more than C, but C is more than A).
Winner
In the resulting graph, the source corresponds to the winner. A source is bound to exist because the graph is a directed acyclic graph by construction, and such graphs always have sources. In the absence of pairwise ties, the source is also unique (because whenever two nodes appear as sources, there would be no valid reason not to connect them, leaving only one of them as a source).
An example
The situation
Imagine that Tennessee is having an election 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 these four cities and that everyone wants to live as near to the capital 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, near the center of the state
 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 (wonlosttied):  030  300  210  120  
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
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 since it is the ratio of votes between two candidates that matters.
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% (a tie, unlikely in real life for this many voters). Since Chattanooga > Knoxville, and they are 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% 
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 RP, followed by Chattanooga, Knoxville, and Memphis in second, third, and fourth places respectively.
Ambiguity resolution example
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
 C > A is ignored as it creates an ambiguity or cycle.
Therefore, A is the winner.
Summary
In the example election, the winner is Nashville. This would be true for any Condorcet method.
Using the Firstpastthepost voting 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 Instantrunoff 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.
Independence of irrelevant alternatives
Ranked pairs fails independence of irrelevant alternatives. However, the method adheres to a less strict property, sometimes called independence of Smithdominated 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 preferential singlewinner election methods:
Comparison of Preferential Electoral Systems



Monotonic  Condorcet  Majority  Condorcet loser  Majority loser  Mutual majority  Smith  ISDA  LIIA  Clone independence  Reversal symmetry  Participation, Consistency  Laterno‑harm  Laterno‑help  Polynomial time  Resolvability  
Schulze  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  No  Yes  Yes  No  No  No  Yes  Yes 
Ranked pairs  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  No  No  No  Yes  Yes 
Alternative smith  No  Yes  Yes  Yes  Yes  Yes  Yes  Yes  No  Yes  No  No  No  No  Yes  Yes 
Alternative schwartz  No  Yes  Yes  Yes  Yes  Yes  Yes  Yes  No  Yes  No  No  No  No  Yes  Yes 
KemenyYoung  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  No  Yes  No  No  No  No  Yes 
Copeland  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes  No  No  Yes  No  No  No  Yes  No 
Nanson  No  Yes  Yes  Yes  Yes  Yes  Yes  No  No  No  Yes  No  No  No  Yes  Yes 
Instantrunoff voting  No  No  Yes  Yes  Yes  Yes  No  No  No  Yes  No  No  Yes  Yes  Yes  Yes 
Borda  Yes  No  No  Yes  Yes  No  No  No  No  No  Yes  Yes  No  Yes  Yes  Yes 
Baldwin  No  Yes  Yes  Yes  Yes  Yes  Yes  No  No  No  No  No  No  No  Yes  Yes 
Bucklin  Yes  No  Yes  No  Yes  Yes  No  No  No  No  No  No  No  Yes  Yes  Yes 
Plurality  Yes  No  Yes  No  No  No  No  No  No  No  No  Yes  Yes  Yes  Yes  Yes 
Contingent voting  No  No  Yes  Yes  Yes  No  No  No  No  No  No  No  Yes  Yes  Yes  Yes 
Coombs^{[1]}  No  No  Yes  Yes  Yes  Yes  No  No  No  No  No  No  No  No  Yes  Yes 
MiniMax  Yes  Yes  Yes  No  No  No  No  No  No  No  No  No  No  No  Yes  Yes 
Antiplurality^{[1]}  Yes  No  No  No  Yes  No  No  No  No  No  No  Yes  No  No  Yes  Yes 
Sri Lankan contingent voting  No  No  Yes  No  No  No  No  No  No  No  No  No  Yes  Yes  Yes  Yes 
Supplementary voting  No  No  Yes  No  No  No  No  No  No  No  No  No  Yes  Yes  Yes  Yes 
Dodgson^{[1]}  No  Yes  Yes  No  No  No  No  No  No  No  No  No  No  No  No  Yes 
References
 ^ In fact, there are different ways how the strength of a victory is measured. The approach used in this article is called winning votes. Another common approach also used by Tideman defining the ranked pairs method in 1987 is the variant using margins of a victory.
 Tideman, T.N. (1987) Independence of clones as a criterion for voting rules. Social Choice and Welfare 4: 185206.
External links
 Descriptions of rankedballot voting methods by Rob LeGrand
 Example JS implementation by Asaf Haddad
 Pair Ranking Ruby Gem by Bala Paranj
 A marginbased PHP Implementation of Tideman's Ranked Pairs
 ^ ^{a} ^{b} ^{c} Antiplurality, Coombs and Dodgson are assumed to receive truncated preferences by apportioning possible rankings of unlisted alternatives equally; for example, ballot A > B = C is counted as A > B > C and A > C > B. If these methods are assumed not to receive truncated preferences, then Laternoharm and Laternohelp are not applicable.