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The Kemeny–Young method is a voting system that uses preferential ballots, pairwise comparison counts, and sequence scores to identify the most popular choice, and also identify the secondmost popular choice, the thirdmost popular choice, and so on down to the leastpopular choice.
This is a Condorcet method because if there is a Condorcet winner, it will always be ranked as the most popular choice.
The Kemeny–Young method is also known as VoteFair popularity ranking, the linear ordering problem, the maximum likelihood method, and the median relation.
Contents 
The Kemeny–Young method uses preferential ballots on which a voter ranks the choices according to their order of preference. A voter is allowed to rank more than one choice at the same preference level. Unranked choices are usually interpreted as leastpreferred.
Kemeny–Young calculations are usually done in two steps. The first step is to create a matrix or table that counts pairwise voter preferences. The second step is to test all possible orderofpreference sequences, calculate a sequence score for each sequence, and compare the scores. Each sequence score equals the sum of the pairwise counts that apply to the sequence. The sequence with the highest score is identified as the overall ranking, from most popular to least popular.
A tally table, which arranges all the pairwise counts in three columns, is useful for counting (tallying) ballot preferences and calculating Kemeny scores. The center column tracks voters who indicate more than one choice at the same preference level.
All possible pairs of choice names 
Number of votes with indicated preference  
Prefer X over Y  Equal preference  Prefer Y over X  
X = Selden Y = Meredith 
0  +1 vote  0 
X = Selden Y = Elliot 
0  0  +1 vote 
X = Selden Y = Roland 
0  0  +1 vote 
X = Meredith Y = Elliot 
0  0  +1 vote 
X = Meredith Y = Roland 
0  0  +1 vote 
X = Elliot Y = Roland 
+1 vote  0  0 
The above tally table summarizes the following preferences in which
a voter has ranked two choices at the same preference level.
Preference order 
Choice 
First  Elliot 
Second  Roland 
Third  Meredith or Selden (equal preference) 
After all ballots have been counted, the same tally table is used
to summarize all the preferences of all the voters. Here is an
example for a case that has 100 voters. (It includes 10 ballots
marked as above.)
All possible pairs of choice names 
Number of votes with indicated preference  
Prefer X over Y  Equal preference  Prefer Y over X  
X = Selden Y = Meredith 
50  10  40 
X = Selden Y = Elliot 
40  0  60 
X = Selden Y = Roland 
40  0  60 
X = Meredith Y = Elliot 
40  0  60 
X = Meredith Y = Roland 
30  0  70 
X = Elliot Y = Roland 
30  0  70 
The sum of the counts in each row must equal the total number of
votes.
Calculating a Kemeny score for a specified sequence is done by adding one count from each row of this tally table. The counts in the center column do not contribute to the score, so they are ignored in the score calculations. The choice between using the count in the left (Prefer X over Y) or right (Prefer Y over X) column depends on which choice name appears first in the specified sequence.
After the sequence with the highest Kemeny score has been identified, the pairwise comparison counts can be arranged in a summary matrix, as shown below, in which the choices appear in the winning sequence from most popular (top and left) to least popular (bottom and right). This matrix layout does not include the equalpreference pairwise counts that appear in the tally table.
... over Roland  ... over Elliot  ... over Selden  ... over Meredith  
Prefer Roland ...    70  60  70 
Prefer Elliot ...  30    60  60 
Prefer Selden ...  40  40    50 
Prefer Meredith ...  30  40  40   
In this summary matrix, the winning Kemeny score equals the sum of
the counts in the upperright, triangular half of the matrix (shown
here in bold, with a green background). No other possible sequence
can have a summary matrix that yields a higher sum of numbers in
the upperright, triangular half. (If it did, that would be the
winning sequence.)
In this summary matrix, the sum of the numbers in the lowerleft, triangular half of the matrix (shown here with a red background) are a minimum. The academic papers by John Kemeny and Peyton Young^{[1]}^{[2]} refer to finding this minimum sum, which is based on how many voters oppose each pairwise order.
The numbers in the above example refer to a case[1] in which different voting methods identify different firstplace winners.
Method  Firstplace winner 
Kemeny–Young  Roland 
Condorcet  Roland 
Instant runoff voting  Elliot or Selden (depending on how the secondround tie is handled) 
Plurality  Selden 
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 the capital as possible.
The candidates for the capital are:
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) 





This matrix summarizes the corresponding pairwise comparison counts:
... over Memphis 
... over Nashville 
... over Chattanooga 
... over Knoxville 

Prefer Memphis ... 
  42%  42%  42% 
Prefer Nashville ... 
58%    68%  68% 
Prefer Chattanooga ... 
58%  32%    83% 
Prefer Knoxville ... 
58%  32%  17%   
The Kemeny–Young method arranges the pairwise comparison counts in
the following tally table:
All possible pairs of choice names 
Number of votes with indicated preference  
Prefer X over Y  Equal preference  Prefer Y over X  
X = Memphis Y = Nashville 
42%  0  58% 
X = Memphis Y = Chattanooga 
42%  0  58% 
X = Memphis Y = Knoxville 
42%  0  58% 
X = Nashville Y = Chattanooga 
68%  0  32% 
X = Nashville Y = Knoxville 
68%  0  32% 
X = Chattanooga Y = Knoxville 
83%  0  17% 
The Kemeny sequence score for the sequence Memphis first, Nashville
second, Chattanooga third, and Knoxville fourth equals (the
unitless number) 345, which is the sum of the following annotated
numbers.
This table lists all the sequence scores.
First choice 
Second choice 
Third choice 
Fourth choice 
Sequence score 
Memphis  Nashville  Chattanooga  Knoxville  345 
Memphis  Nashville  Knoxville  Chattanooga  279 
Memphis  Chattanooga  Nashville  Knoxville  309 
Memphis  Chattanooga  Knoxville  Nashville  273 
Memphis  Knoxville  Nashville  Chattanooga  243 
Memphis  Knoxville  Chattanooga  Nashville  207 
Nashville  Memphis  Chattanooga  Knoxville  361 
Nashville  Memphis  Knoxville  Chattanooga  295 
Nashville  Chattanooga  Memphis  Knoxville  377 
Nashville  Chattanooga  Knoxville  Memphis  393 
Nashville  Knoxville  Memphis  Chattanooga  311 
Nashville  Knoxville  Chattanooga  Memphis  327 
Chattanooga  Memphis  Nashville  Knoxville  325 
Chattanooga  Memphis  Knoxville  Nashville  289 
Chattanooga  Nashville  Memphis  Knoxville  341 
Chattanooga  Nashville  Knoxville  Memphis  357 
Chattanooga  Knoxville  Memphis  Nashville  305 
Chattanooga  Knoxville  Nashville  Memphis  321 
Knoxville  Memphis  Nashville  Chattanooga  259 
Knoxville  Memphis  Chattanooga  Nashville  223 
Knoxville  Nashville  Memphis  Chattanooga  275 
Knoxville  Nashville  Chattanooga  Memphis  291 
Knoxville  Chattanooga  Memphis  Nashville  239 
Knoxville  Chattanooga  Nashville  Memphis  255 
The highest sequence score is 393, and this score is associated
with the following sequence, so this is the winning preference
order.
Preference order 
Choice 
First  Nashville 
Second  Chattanooga 
Third  Knoxville 
Fourth  Memphis 
If a single winner is needed, the first choice, Nashville, is
chosen. (In this example Nashville is the Condorcet winner.)
The summary matrix below arranges the pairwise counts in sequence from most popular (top and left) to least popular (bottom and right).
... over Nashville ...  ... over Chattanooga ...  ... over Knoxville ...  ... over Memphis ...  
Prefer Nashville ...    68%  68%  58% 
Prefer Chattanooga ...  32%    83%  58% 
Prefer Knoxville ...  32%  17%    58% 
Prefer Memphis ...  42%  42%  42%   
In this arrangement the winning Kemeny score (393) equals the sum
of the counts in bold, which are in the upperright, triangular
half of the matrix (with a green background).
In all cases that do not result in an exact tie, the Kemeny–Young method identifies a mostpopular choice, secondmost popular choice, and so on.
A tie can occur at any preference level. Except in some cases where circular ambiguities are involved, the Kemeny–Young method only produces a tie at a preference level when the number of voters with one preference exactly matches the number of voters with the opposite preference.
All Condorcet methods, including the Kemeny–Young method, satisfy these criteria:
The Kemeny–Young method also satisfies these criteria:
In common with all Condorcet methods, the Kemeny–Young method fails these criteria (which means the described criteria do not apply to the Kemeny–Young method):
The Kemeny–Young method also fails these criteria (which means the described criteria do not apply to the Kemeny–Young method):
Calculating all sequence scores requires time proportional to N!, where N is the number of choices. Although one need not compute scores to find the winner, any algorithm finding the winner requires superpolynomial time (unless P = NP). Nevertheless, fast calculation methods based on integer programming allow the computation of full rankings for votes with as many as 40 choices in seconds.^{[4]}
The Kemeny–Young method was developed by John Kemeny in 1959.^{[1]}
In 1978 Peyton Young and Arthur Levenglick showed^{[2]} that this method was the unique neutral method satisfying reinforcement and the Condorcet criterion. In another paper^{[5]}, Young argued that the Kemeny–Young method was a possible interpretation of one of Condorcet's proposals.
In the papers by John Kemeny and Peyton Young, the Kemeny sequence scores use counts of how many voters oppose, rather than support, each pairwise preference,^{[1]}^{[2]} but the smallest such sequence score identifies the same overall ranking.
Since 1991 the method has been promoted under the name "VoteFair popularity ranking" by Richard Fobes.^{[6]}
