Copeland's method or Copeland's pairwiseaggregation method is a Smith-efficientCondorcet method in which candidates are ordered by the number of pairwise victories, minus the number of pairwise defeats. It was invented by Ramon Llull in his 1299 treatise Ars Electionis, but his form only counted pairwise victories and not defeats. It is named after Arthur Herbert Copeland, who independently proposed it in a 1951 lecture. 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, this method often leads to ties. For example, if there is a three-candidate majority rule cycle, each candidate will have exactly oneloss, 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.
To find the Condorcet winner, every candidate must be matched against every other candidate in a series of imaginary one-on-one contests. In each pairing, each voter will choose the city physically closest to their location. In each pairing the winner is the candidate preferred by a majority of voters. When results for every possible pairing have been found they are as follows:
The wins and losses of each candidate sum as follows:
Candidate
Wins
Losses
Net
Memphis
0
3
−3
Nashville
3
0
3
Knoxville
1
2
−1
Chattanooga
2
1
1
Nashville, with no defeats, is a Condorcet winner and, with the greatest number of net wins, is a Copeland winner.
Example without Condorcet winner
In an election with five candidates competing for one seat, the following votes were cast using a ranked voting method : The results of the 10 possible pairwise comparisons between the candidates are as follows:
The wins and losses of each candidate sum as follows:
Candidate
Wins
Losses
Net
A
3
1
2
B
2
2
0
C
2
2
0
D
1
3
−2
E
2
2
0
No Condorcet winner exists. Candidate A is the Copeland winner, with the greatest number of wins minus losses. 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.
Second-order Copeland method
The second-order Copeland method uses the sum of the Copeland scores of the defeated opponents as the means of determining a winner. This is useful in breaking ties when using the first-order Copeland method described above. The second-order Copeland method has a particularly beneficial feature: manipulation of the voting is more difficult because it requires NP-complete complexity calculations to compute the manipulation.
