# Copeland's method

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Copeland's method

Copeland's method or Copeland's pairwise aggregation method is a Condorcet method in which candidates are ordered by the number of pairwise victories, minus the number of pairwise defeats.[1]

Proponents argue that this method is easily understood by the general populace, which is generally familiar with the sporting equivalent. In many round-robin tournaments, the winner is the competitor with the most victories. It is also easy to calculate.

When there is no Condorcet winner (i.e. when there are multiple members of the Smith set), this method often leads to ties. For example, if there is a three-candidate majority rule cycle, each candidate will have exactly one loss, and there will be an unresolved tie between the three.

Critics argue that it also puts too much emphasis on the quantity of pairwise victories and defeats rather than their magnitudes.[citation needed]

## Example of the Copeland Method

In an election with five candidates competing for one seat, the following votes were cast using a preferential voting method (100 votes with four distinct sets):

 31: A>E>C>D>B 30: B>A>E 29: C>D>B 10: D>A>E

The results of the 10 possible pairwise comparisons between the candidates are as follows:

Comparison Result Winner Comparison Result Winner
A v B 41 v 59 B B v D 30 v 70 D
A v C 71 v 29 A B v E 59 v 41 B
A v D 61 v 39 A C v D 60 v 10 C
A v E 71 v 0 A C v E 29 v 71 E
B v C 30 v 60 C D v E 39 v 61 E

No Condorcet winner (candidate who beats all other candidates in pairwise comparisons) exists.

Candidate Wins Losses Wins - Losses
A 3 1 2
B 2 2 0
C 2 2 0
D 1 3 -2
E 2 2 0

The table above shows the number of wins and losses for each candidate in the pairwise comparisons. Candidate A has the greatest number of wins minus losses, and is therefore the Copeland winner.

As a Condorcet completion method, Copeland requires a Smith set containing at least five candidates to give a clear winner unless two or more candidates tie in pairwise comparisons.

## References

1. ^ Pomerol, Jean-Charles; Sergio Barba-Romero (2000). Multicriterion decision in management: principles and practice. Springer. p. 122. ISBN 0792377567.

### Notes

1. E Stensholt, "Nonmonotonicity in AV"; 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.

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