Kemeny-Young Maximum Likelihood Method
Each possible complete ranking of the candidates is given a "distance" score. For each pair of candidates, find the number of ballots that order them the the opposite way as the given ranking. The distance is the sum across all such pairs. The ranking with the least distance wins.
The winning candidate is the top candidate in the winning ranking.
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:
- Memphis on Wikipedia, the state's largest city, with 42% of the voters, but located far from the other cities
- Nashville on Wikipedia, with 26% of the voters, near the center of Tennessee
- Knoxville on Wikipedia, with 17% of the voters
- Chattanooga on Wikipedia, 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)
Consider the ranking Nashville>Chattanooga>Knoxville>Memphis. This ranking contains 6 orderings of pairs of candidates:
- Nashville>Chattanooga, for which 32% of the voters disagree.
- Nashville>Knoxville, for which 32% of the voters disagree.
- Nashville>Memphis, for which 42% of the voters disagree.
- Chattanooga>Knoxville, for which 17% of the voters disagree.
- Chattanooga>Memphis, for which 42% of the voters disagree.
- Knoxville>Memphis, for which 42% of the voters disagree.
The distance score for this ranking is 32+32+42+17+42+42=207.
It can be shown that this ranking is the one with the lowest distance score. Therefore, the winning ranking is Nashville>Chattanooga>Knoxville>Memphis, and so the winning candidate is Nashville.
Some text of this article is derived with permission from Electoral Methods: Single Winner.