Kemeny–Young method

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. Unranked choices are usually interpreted as least-preferred.

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.

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.)

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:

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.

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.

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.

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.

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 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.

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:

This matrix summarizes the corresponding pairwise comparison counts:

The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:

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 score .

The largest ranking score is 393, and this score is associated with the following possible ranking, so this ranking is also the overall ranking.

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 order from most popular (top and left) to least popular (bottom and right).

In this arrangement the largest ranking score (393) equals the sum of the counts in bold, which are in the upper-right, triangular half of the matrix (with a green background).

Characteristics

In all cases that do not result in an exact tie, the Kemeny–Young method identifies a most-popular choice, second-most 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.

Satisfied criteria for all Condorcet methods

All Condorcet methods, including the Kemeny–Young method, satisfy these criteria:

Additional satisfied criteria

The Kemeny–Young method also satisfies these criteria:

Failed criteria for all Condorcet methods

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):

Additional failed criteria

The Kemeny–Young method also fails these criteria (which means the described criteria do not apply to the Kemeny–Young method):

Calculation methods and computational complexity

An algorithm for computing a Kemeny-Young ranking in time polynomial in the number of candidates is not known, and unlikely to exist since the problem is NP-hard even if there are just 4 voters.

It has been reported that calculation methods based on integer programming sometimes allowed the computation of full rankings for votes on as many as 40 candidates in seconds. However, certain 40-candidate 5-voter Kemeny elections generated at random were not solvable on a 3 GHz Pentium computer in a useful time bound in 2006.

Note that the complexity of computation scales linearly to the number of voters so the time needed to process a given set of votes is dominated by the number of candidates rather than the number of votes, limiting the importance of this constraint to elections where voters are able to effectively consider significantly more than the common seven items of working memory.

There exists a polynomial-time approximation scheme for computing a Kemeny-Young ranking, and there also exists a parameterized subexponential-time algorithm with running time O^*(2^O(√OPT)) for computing such a ranking.

History

The Kemeny–Young method was developed by John Kemeny in 1959.

In 1978 Peyton Young and Arthur Levenglick showed that this method was the unique neutral method satisfying reinforcement and a version of the Condorcet criterion. In other papers, Young adopted an epistemic approach to preference-aggregation: he supposed that there was an objectively 'correct', but unknown preference order over the alternatives, and voters receive noisy signals of this true preference order (cf. Condorcet's jury theorem.) Using a simple probabilistic model for these noisy signals, Young showed that the Kemeny–Young method was the maximum likelihood estimator of the true preference order. Young further argues that Condorcet himself was aware of the Kemeny-Young rule and its maximum-likelihood interpretation, but was unable to clearly express his ideas.

In the papers by John Kemeny and Peyton Young, the Kemeny scores use counts of how many voters oppose, rather than support, each pairwise preference, but the smallest such score identifies the same overall ranking.

Since 1991 the method has been promoted under the name "VoteFair popularity ranking" by Richard Fobes.