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The Kemeny–Young method is an electoral system that uses ranked ballots and pairwise comparison counts to identify the most popular choices in an election. It is a Condorcet method because if there is a Condorcet winner, it will always be ranked as the most popular choice.

This method assigns a score for each possible sequence, where each sequence considers which choice might be most popular, which choice might be second-most popular, which choice might be third-most popular, and so on down to which choice might be least-popular. The sequence that has the highest score is the winning sequence, and the first choice in the winning sequence is the most popular choice. (As explained below, ties can occur at any ranking level.)

The Kemeny–Young method is also known as the Kemeny rule, VoteFair popularity ranking, the maximum likelihood method, and the median relation.

Description

The Kemeny–Young method uses preferential ballots on which voters rank choices according to their order of preference. A voter is allowed to rank more than one choice at the same preference level.^{[citation needed]} Unranked choices are usually interpreted as least-preferred.

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 rankings, calculate a score for each such ranking, and compare the scores. Each ranking score equals the sum of the pairwise counts that apply to that ranking.

The ranking that has the largest score is identified as the overall ranking. (If more than one ranking has the same largest score, all these possible rankings are tied, and typically the overall ranking involves one or more ties.)

Another way to view the ordering is that it is the one which minimizes the sum of the Kendall tau distances (bubble sort distance) to the voters' lists.

In order to demonstrate how an individual preference order is converted into a tally table, it is worth considering the following example. Suppose that a single voter has a choice among four candidates (i.e. Elliot, Meredith, Roland, and Selden) and has the following preference order:

Preference order	Choice
First	Elliot
Second	Roland
Third	Meredith or Selden (equal preference)

These preferences can be expressed in a tally table. A tally table, which arranges all the pairwise counts in three columns, is useful for counting (tallying) ballot preferences and calculating ranking scores. The center column tracks when a voter indicates more than one choice at the same preference level. The above preference order can be expressed as the following tally table:^{[citation needed]}

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

Now suppose that multiple voters had voted on those four candidates. After all ballots have been counted, the same type of tally table can be used to summarize all the preferences of all the voters. Here is an example for a case that has 100 voters:

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.

After the tally table has been completed, each possible ranking of choices is examined in turn, and its ranking score is calculated by adding the appropriate number from each row of the tally table. For example, the possible ranking:

1. Elliot
2. Roland
3. Meredith
4. Selden

satisfies the preferences Elliot > Roland, Elliot > Meredith, Elliot > Selden, Roland > Meredith, Roland > Selden, and Meredith > Selden. The respective scores, taken from the table, are

• Elliot > Roland: 30
• Elliot > Meredith: 60
• Elliot > Selden: 60
• Roland > Meredith: 70
• Roland > Selden: 60
• Meredith > Selden: 40

giving a total ranking score of 30 + 60 + 60 + 70 + 60 + 40 = 320.

Calculating the overall ranking

After the scores for every possible ranking have been calculated, the ranking that has the largest score can be identified, and becomes the overall ranking. In this case, the overall ranking is:

1. Roland
2. Elliot
3. Selden
4. Meredith

with a ranking score of 370.

If there are cycles or ties, more than one possible ranking can have the same largest score. Cycles are resolved by producing a single overall ranking where some of the choices are tied.^{[clarification needed]}

Summary matrix

After the overall ranking has been calculated, the pairwise comparison counts can be arranged in a summary matrix, as shown below, in which the choices appear in the winning order from most popular (top and left) to least popular (bottom and right). This matrix layout does not include the equal-preference pairwise counts that appear in the tally table:^[1]

	... 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 largest ranking score equals the sum of the counts in the upper-right, triangular half of the matrix (shown here in bold, with a green background). No other possible ranking can have a summary matrix that yields a higher sum of numbers in the upper-right, triangular half. (If it did, that would be the overall ranking.)

In this summary matrix, the sum of the numbers in the lower-left, triangular half of the matrix (shown here with a red background) are a minimum. The academic papers by John Kemeny and Peyton Young^[2][3] refer to finding this minimum sum, which is called the Kemeny score, and which is based on how many voters oppose (rather than support) each pairwise order:

Method	First-place winner
Kemeny–Young	Roland
Condorcet	Roland
Instant runoff voting	Elliot or Selden (depending on how the second-round tie is handled)
Plurality	Selden

Example

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
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville Memphis

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 ranking score for the possible ranking of Memphis first, Nashville second, Chattanooga third, and Knoxville fourth equals (the unit-less number) 345, which is the sum of the following annotated numbers.

42% (of the voters) prefer Memphis over Nashville

42% prefer Memphis over Chattanooga

42% prefer Memphis over Knoxville

68% prefer Nashville over Chattanooga

68% prefer Nashville over Knoxville

83% prefer Chattanooga over Knoxville

This table lists all the ranking scores:

Zdroj:https://en.wikipedia.org?pojem=Kemeny-Young_method
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First choice	Second choice	Third choice	Fourth choice	Ranking 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