Copeland's method: Difference between revisions

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(Simplify criterion compliance, and add multi-armed bandit reference.)
(Simplify ISDA proof)
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(Example showing Smith members having only 2 points more than non-Smith members: Suppose there are two candidates, one of whom is the Condorcet winner, and thus the only candidate in the Smith set. The CW has one victory and no defeats for a Copeland score of 1, while the other candidate has no victories and one defeat for a score of -1.)
(Example showing Smith members having only 2 points more than non-Smith members: Suppose there are two candidates, one of whom is the Condorcet winner, and thus the only candidate in the Smith set. The CW has one victory and no defeats for a Copeland score of 1, while the other candidate has no victories and one defeat for a score of -1.)


Copeland's method also passes [[ISDA]]; the first paragraph proves that all candidates in the Smith set must have higher Copeland scores than all candidates not in the Smith set, and since by definition candidates in the Smith set have a pairwise victory (and thus no pairwise defeat) against every candidate not in the Smith set, adding or removing any number of candidates not in the Smith set will only result in every candidate in the Smith set having that number of pairwise victories added or subtracted from their total (with no change to their number of pairwise defeats); since the original Copeland winner must have had a higher Copeland score than all other Smith set candidates in order to win, they will still have a higher Copeland score and thus still win.
Copeland's method also passes [[ISDA]]: since every candidate in the Smith set beats everybody outside it, eliminating a candidate outside of the Smith set will subtract one win from the score of every candidate in that Smith set. Thus eliminating a Smith-dominated candidate can never change the relative Copeland scores of candidates in the Smith set, and thus not change the winner either.


Further, Copeland always elects from the [[uncovered set]], and the Copeland ranking is an uncovered set ranking. This is because when one candidate covers another, the former candidate pairwise beats all candidates pairwise beaten by the latter candidate, and also either pairwise beats the latter candidate or beats someone who beats the latter candidate. Because of this, the covering candidate will have a minimum Copeland score of ((number of candidates beaten by latter candidate) + 1) - (number of candidates beating former candidate)), and the covered candidate will have a maximal Copeland score of ((number of candidates beaten by latter candidate) - ((number of candidates beating former candidate) + 1), resulting in the covering candidate having at least 2 more points than the covered candidate. This type of logic can be used to simplify the above Smith set-related proofs too.
Further, Copeland always elects from the [[uncovered set]], and the Copeland ranking is an uncovered set ranking. This is because when one candidate covers another, the former candidate pairwise beats all candidates pairwise beaten by the latter candidate, and also either pairwise beats the latter candidate or beats someone who beats the latter candidate. Because of this, the covering candidate will have a minimum Copeland score of ((number of candidates beaten by latter candidate) + 1) - (number of candidates beating former candidate)), and the covered candidate will have a maximal Copeland score of ((number of candidates beaten by latter candidate) - ((number of candidates beating former candidate) + 1), resulting in the covering candidate having at least 2 more points than the covered candidate. This type of logic can be used to simplify the above Smith set-related proofs too.
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