Ranked pairs
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Ranked pairs (sometimes abbreviated "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.^{[1]}^{[2]} The rankedpairs procedure 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, the rankedpairs procedure guarantees that candidate will win. Because of this property, the rankedpairs procedure complies with the Condorcet winner criterion (and is a Condorcet method).
Procedure[edit]
The rankedpairs procedure operates 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 strength of victory, from largest first to smallest last.^{[vs 1]}
 "Lock in" each pair, starting with the one with the largest strength of victory and, continuing through the sorted pairs, add each one in turn to a graph if it does not create a cycle in the graph with the existing locked in pairs. The completed graph shows the final ranking.
The procedure can be illustrated using a simple example. Suppose that there are 27 voters and 4 candidates w, x, y and z such that the votes are cast as shown in the table of ballots.
w>x>z>y  7 ballots 
w>y>x>z  2 ballots 
x>y>z>w  4 ballots 
x>z>w>y  5 ballots 
y>w>x>z  1 ballots 
y>z>w>x  8 ballots 
Tally[edit]
w  x  y  z  

w  0  9  1  –7 
x  –9  0  5  11 
y  –1  –5  0  3 
z  7  –11  –3  0 
The vote tally can be expressed as a table in which the (w, x) entry is the number of ballots in which w comes higher than x minus the number in which x comes higher than w. In the example w comes higher than x in the first two rows and the last two rows of the ballot table (total 18 ballots) while x comes higher than w in the middle two rows (total 9), so the entry in the (w, x) cell is 18–9 = 9.
Notice the skew symmetry of the table.
Sort[edit]
The positive majorities are then sorted in decreasing order of magnitude.
Lock[edit]
The next stage is to examine the majorities in turn to determine which pairs to "lock in". This can be done by building up a matrix in which the (x, y) entry is initially 0, and is set to 1 if we decide that x is preferred to y and to –1 if we decide that y is preferred to x. These preferences are decided by the list of sorted majorities, simply skipping over any which are inconsistent with previous decisions.
The first two majorities tell us that x is preferred to z and w to x, from which it follows by transitivity that w is preferred to z. Once these facts have been incorporated into the table it takes the form as shown after two steps. Notice again the skew symmetry. The third majority tells us that z is preferred to w, but since we have already decided that w is preferred to z we ignore it, which is why our bank of knowledge looks the same after both 2 and 3 steps.
w  x  y  z  

w  0  1  0  1 
x  –1  0  0  1 
y  0  0  0  0 
z  –1  –1  0  0 
w  x  y  z  

w  0  1  1  1 
x  –1  0  1  1 
y  –1  –1  0  0 
z  –1  –1  0  0 
w  x  y  z  

w  0  1  1  1 
x  –1  0  1  1 
y  –1  –1  0  1 
z  –1  –1  –1  0 
The fourth majority tells us that x is preferred to y, and since we know that w is preferred to x we infer that w is preferred to y, giving us the table after 4 steps.
The fifth majority tells us that y is preferred to z, and this completes the table.
Winner[edit]
In the resulting graph for the locked pairs, the source corresponds to the winner. In this case w is preferred to all other candidates and is therefore identified as the winner.
Tied majorities[edit]
In the example the majorities are all different, and this is what will usually happen when the number of voters is large. If ties are unlikely, then it does not matter much how they are resolved, so a random choice can be made. However this is not Tideman's procedure, which is considerably more complicated. See his paper for details.^{[1]}
An example[edit]
The situation[edit]
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[edit]
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[edit]
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[edit]
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 rankedpairs procedure. Nashville is followed by Chattanooga, Knoxville, and Memphis in second, third, and fourth places respectively.
Summary[edit]
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[edit]
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[edit]
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[edit]
The following table compares Ranked Pairs with other preferential singlewinner election methods:
System  Monotonic  Condorcet winner  Majority  Condorcet loser  Majority loser  Mutual majority  Smith  ISDA  LIIA  Independence of clones  Reversal symmetry  Participation, consistency  Laternoharm  Laternohelp  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 
Tideman's Alternative  No  Yes  Yes  Yes  Yes  Yes  Yes  Yes  No  Yes  No  No  No  No  Yes  Yes 
Kemeny–Young  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 
Black  Yes  Yes  Yes  Yes  Yes  No  No  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 
Smith/IRV  No  Yes  Yes  Yes  Yes  Yes  Yes  Yes  No  Yes  No  No  No  No  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^{[3]}  No  No  Yes  Yes  Yes  Yes  No  No  No  No  No  No  No  No  Yes  Yes 
MiniMax^{[specify]}  Yes  Yes  Yes  No  No  No  No  No  No  No  No  No  No  No  Yes  Yes 
Antiplurality^{[3]}  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^{[3]}  No  Yes  Yes  No  No  No  No  No  No  No  No  No  No  No  No  Yes 
Notes[edit]
 ^ In fact, there are different ways how the strength of a victory is measured. This article uses Tideman's original method based on margins of victory.
References[edit]
 ^ ^{a} ^{b} Tideman, T. N. (19870901). "Independence of clones as a criterion for voting rules". Social Choice and Welfare. 4 (3): 185–206. doi:10.1007/BF00433944. ISSN 1432217X. S2CID 122758840.
 ^ Schulze, Markus (October 2003). "A New Monotonic and CloneIndependent SingleWinner Election Method". Voting matters (www.votingmatters.org.uk). McDougall Trust. 17. Archived from the original on 20200711. Retrieved 20210202.
 ^ ^{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 1/2 A > B > C and 1/2 A > C > B. If these methods are assumed not to receive truncated preferences, then laternoharm and laternohelp are not applicable.
External links[edit]
 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
 Rust implementation of Ranked Pairs by Cory Dickson