Kemeny–Young method: Difference between revisions

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== Notes ==

~~The~~ Kemeny-Young ranking is a [[Smith set ranking]]. This is because any candidate in the n-th Smith set will always be ranked higher than any candidate in a lower Smith set by more voters than vice versa by definition (because the n-th Smith set candidate pairwise beats all candidates in lower Smith sets), so if you take any non-Smith set ranking and minimally modify it to become a Smith set ranking, this will always reduce the distance score. In other words, if there is some ranking which puts a candidate in the n-th Smith set after some candidate in a lower Smith set, then modifying it to swap the two will reduce the distance created by that pair of candidates.

If, when the distance score of a ranking A>B>C is being calculated, a voter who ranked B but not A is treated as ordering A and B the opposite way as the ranking, then the Kemeny-Young ranking is a [[Smith set ranking]]. This is because any candidate in the n-th Smith set will always be ranked higher than any candidate in a lower Smith set by more voters than vice versa by definition (because the n-th Smith set candidate pairwise beats all candidates in lower Smith sets), so if you take any non-Smith set ranking and minimally modify it to become a Smith set ranking, this will always reduce the distance score. In other words, if there is some ranking which puts a candidate in the n-th Smith set after some candidate in a lower Smith set, then modifying it to swap the two will reduce the distance created by that pair of candidates.

See [[Pairwise sorted methods]], which do what is called "Local Kemenization" to produce a ranking, while being cloneproof.