Kemeny–Young 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 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.

Statistical interpretation
The Kemeny-Young method produces the maximum likelihood estimate for a voting model where the voters know a best order of the candidates, and for each pair of candidates, ranks that pair correct with some probability $$p > \frac 1 2 $$, or reversed with probability $$1-p$$. This is called a Mallows model.

Strategic vulnerability
Kemeny-Young is vulnerable to compromising, burying, and crowding. It fails clone independence because adding a clone can cause a non-clone to be elected, and this effect increases as the number of clones increases.

Example
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 (this is because this is the Condorcet ranking, and therefore switching any pair of candidates would require overturning the majority of voters in that pairing rather than the minority). Therefore, the winning ranking is Nashville>Chattanooga>Knoxville>Memphis, and so the winning candidate is Nashville.

Example with a Condorcet cycle
A>B: 60>40, B>C: 65>35, C>A:75>25. There are 6 main rankings to consider here:
 * A>B>C: A>B opposed by 40, A>C by 75, and B>C by 35. Score is 150. So the minimum score so far is 150.
 * A>C>B: A>C by 75, A>B by 40, C>B by 65. Score is 180. Since this is greater than the minimum (150) this is disqualified.
 * B>A>C: B>A by 60, B>C by 35, A>C by 75. Score is 170. Disqualified by 150.
 * B>C>A: B>C by 35, B>A by 60, C>A by 25. Score is 120. This is the new minimum, so A>B>C is now disqualified.
 * C>A>B: C>A by 25, C>B by 65, A>B by 40. Score is 130. Disqualified by 120.
 * C>B>A: C>B by 65, C>A by 25, B>A by 60. Score is 150. Disqualified by 120.

So the final ranking is B>C>A, with B winning.

Approximate methods
Some other voting methods have the property that the social order they return has a Kemeny distance at most k times the optimum (the one that Kemeny finds). They do not necessarily pass the same criteria as Kemeny but can be interesting methods in their own right. Some results are:


 * The Borda count is a 5-approximation.
 * The footrule method is a 2-approximation.
 * The "best fit" method, which returns the ranked ballot with the least Kemeny distance, is a 2-approximation.
 * The random Kwiksort method and its deterministic version CC-Pivot are both 3-approximations.
 * Choosing the best order of best fit and Kwiksort is a 6/5-approximation.