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== Criteria ==
The reasoning for why Copeland's method is Smith-efficient is as follows: every candidate inpasses the [[Smith setcriterion]] hasbecause a pairwise victory over everyany candidate not in the Smith set by definition, andbeats ateverybody mostoutside has a pairwise defeat against all but one candidate other than themselves inof the Smith set, which is reduced tobut no pairwisecandidate defeatsoutside whenof theit Smithdoes setso. has fewer than 3 candidates (sinceThus, theyevery can't be pairwise defeated by themselves, and if they had a pairwise defeat against all candidates other than themselvescandidate in the Smith set, thenmust theyhave themselvesa wouldgreater notCopeland be in the Smith set by definition, and when there are fewerscore than 3 Smith candidates, either there is only one Smithany candidate, whooutside thusof can't be beaten byit. anyoneFurthermore, or 2 Smith candidates, who must pairwise tie each other), so all candidates in the Smith set have a Copeland scoreranking of at leastcandidates (NS - S + 1), fully written out as ((number of candidates not in the Smith set) - ((numberordering of candidates inbased theon SmithCopeland setscore) -is 1). Every candidate not in thea [[Smith set hasranking]], a pairwise defeat againstsince every candidate in the nth Smith set by definition, and can at most have pairwise victories againstdefeats every candidate other than themselves not in the Smith set, thus their Copeland score can at most be (NS - S - n+1), which is ((number of candidates not in the-th Smith set), -and 1)thus -a (numbercandidate of candidates in Smith set). Thus,from the members of the Smithformer set will always havehas a greater Copeland score at least 2 points higher than thea candidatescandidate not infrom the Smith setlatter.
(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 previous 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]]; the previousfirst 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.
The Copeland ranking of candidates (the ordering of candidates based on Copeland score) is a [[Smith set ranking]]. This is because in general, a candidate in the n-th Smith set (if n is 1, this is the Smith set. If n is 2, this is the secondary Smith set, which is the set of candidates that would be in the Smith set if all the candidates in the Smith set were eliminated from the election. If n is 3, this is the tertiary Smith set, the set of candidates that would be in the Smith set ignoring all candidates in the Smith set and secondary Smith set, etc.) will have pairwise victories against at least all candidates in k-th Smith sets (for any value of k which is greater than n), and have pairwise defeats against at most all candidates in j-th Smith sets (for any non-negative value of j which is smaller than n) and all but two candidates in the n-th Smith set, which reduces to no defeats against any candidates in the n-th Smith set when the n-th Smith set is smaller than 3 candidates (since they can't be beaten by themselves, and must not be beaten by everyone else in their Smith set in order to be in it, and there are guaranteed to be no pairwise defeats for candidates in a given Smith set against other candidates in that same Smith set when that Smith set is either one or two candidates large), while a candidate in any k-th Smith set will have pairwise victories against at most all candidates other than themselves in k-th Smith sets, and will have pairwise defeats against at least all candidates in n-th or j-th Smith sets. Mathematically, this can be represented as minimal/maximal Copeland scores of (K-J-N+1) and (K-J-N-1) respectively, with K being the number of candidates in all Smith sets "after" the n-th Smith set, J for all Smith sets "before" the n-th Smith set, and N for the candidates in the n-th Smith set. Therefore, the Copeland score of a candidate in the n-th Smith set is guaranteed to be at least 2 points higher than a candidate in any of the k-th (after or below n) Smith sets. Similar reasoning shows that a candidate in any of the j-th (before or above n) Smith sets is at least 2 points higher than any candidate in the n-th Smith set.
Further, Copeland always elects from the [[ Uncovered set|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. ▼▲(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.)
== Generalizations==
▲Further, Copeland always elects from the [[Uncovered set|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.
Zoghi et al have developed a multi-armed bandit variant of Copeland's method. It can be used to determine a winner in a multi-armed bandit setting, even if a Condorcet winner does not necessarily exist.<ref>{{Cite journal|last=Zoghi|first=Masrour|last2=Karnin|first2=Zohar|last3=Whiteson|first3=Shimon|last4=de Rijke|first4=Maarten|date=2015-05-31|title=Copeland Dueling Bandits|url=http://arxiv.org/abs/1506.00312|journal=arXiv:1506.00312 [cs]}}</ref>
== See also ==
# E Stensholt, "[http://www.electoral-reform.org.uk/publications/votingmatters/P2.HTM Nonmonotonicity in AV]"; Electoral Reform Society ''Voting matters'' - Issue 15, June 2002 (online). ▼# A.H. Copeland, A 'reasonable' social welfare function, Seminar on Mathematics in Social Sciences, University of Michigan, 1951. ▼# V.R. Merlin, and D.G. Saari, "Copeland Method. II. Manipulation, Monotonicity, and Paradoxes"; Journal of Economic Theory; Vol. 72, No. 1; January, 1997; 148-172. ▼# D.G. Saari. and V.R. Merlin, 'The Copeland Method. I. Relationships and the Dictionary'; Economic Theory; Vol. 8, No. l; June, 1996; 51-76. ▼
==See References also==
▲# E Stensholt, "[http://www.electoral-reform.org.uk/publications/votingmatters/P2.HTM Nonmonotonicity in AV]"; Electoral Reform Society ''Voting matters'' - Issue 15, June 2002 (online).
▲# A.H. Copeland, A 'reasonable' social welfare function, Seminar on Mathematics in Social Sciences, University of Michigan, 1951.
▲# V.R. Merlin, and D.G. Saari, "Copeland Method. II. Manipulation, Monotonicity, and Paradoxes"; Journal of Economic Theory; Vol. 72, No. 1; January, 1997; 148-172.
▲# D.G. Saari. and V.R. Merlin, 'The Copeland Method. I. Relationships and the Dictionary'; Economic Theory; Vol. 8, No. l; June, 1996; 51-76.
==References==
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