Copeland's method

Copeland's method is a Smith-efficient Condorcet method in which the winner is determined by finding the candidate with the most (pairwise victories minus pairwise defeats), known as their Copeland score. It was invented by Ramon Llull in his 1299 treatise Ars Electionis, but his form only counted pairwise victories and not defeats (which could lead to a different result in the case of a pairwise tie).

(Some alternative versions of Copeland don't count pairwise defeats, and others give each candidate in a pairwise tie half a point each.)

Proponents argue that this method is more understandable to the general populace, which is generally familiar with the sporting equivalent. In many team sports, the teams with the greatest number of victories in regular season matchups make it to the playoffs.

This method often leads to ties in cases when there are multiple members of the Smith set; specifically, there must be at least five or more candidates in the Smith set in order for Copeland to not produce a tie for winner unless there are pairwise ties. Critics argue that it also puts too much emphasis on the quantity of pairwise victories rather than the magnitude of those victories (or conversely, of the defeats).

Criteria

The reasoning for why Copeland's method is Smith-efficient is as follows: every candidate in the Smith set has a pairwise victory over every candidate not in the Smith set by definition, and at most has a pairwise defeat against all but one candidate other than themselves not in the Smith set (since, they can't be pairwise defeated by themselves, and if they had a pairwise defeat against all candidates other than themselves in the Smith set, then they themselves would not be in the Smith set by definition), so all candidates in the Smith set have a Copeland score of at least ((number of candidates not in the Smith set) - ((number of candidates in the Smith set) - 2). Every candidate not in the Smith set has a pairwise defeat against every candidate in the Smith set by definition, and can at most have pairwise victories against every candidate other than themselves not in the Smith set, thus their Copeland score can at most be ((number of candidates not in the Smith set) - 1) - (number of candidates in Smith set). Thus, the members of the Smith set will always have a Copeland score at least 3 points higher than the candidates not in the Smith set.

Copeland's method also passes ISDA; since the Copeland winner is always in the Smith set, 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 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; 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, 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 (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), 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 minimak/maximal Copeland scores of (K-J-N+2) and (K-J-N-1) respectively Therefore, the Copeland score of a candidate in the n-th Smith set is guaranteed to be at least 3 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 3 points higher than any candidate in the n-th Smith set.

See also

1. E Stensholt, "Nonmonotonicity in AV"; Electoral Reform Society Voting matters - Issue 15, June 2002 (online).
2. A.H. Copeland, A 'reasonable' social welfare function, Seminar on Mathematics in Social Sciences, University of Michigan, 1951.
3. 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.
4. 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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