Copeland's method
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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]} 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).^{[2]}
Proponents argue that this method is easily understood by the general populace, which is generally familiar with the sporting equivalent. In many roundrobin 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 threecandidate 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.
Examples of the Copeland Method
Example with Condorcet winner
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities and that everyone wants to live as near to the capital as possible.
The candidates for the capital are:
 Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
 Nashville, with 26% of the voters, near the center of the state
 Knoxville, with 17% of the voters
 Chattanooga, with 15% of the voters
The preferences of the voters would be divided like this:
42% of voters (close to Memphis) 
26% of voters (close to Nashville) 
15% of voters (close to Chattanooga) 
17% of voters (close to Knoxville) 





To find the Condorcet winner, every candidate must be matched against every other candidate in a series of imaginary oneonone 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:
Comparison  Result  Winner 

Memphis vs Nashville  42 v 58  Nashville 
Memphis vs Knoxville  42 v 58  Knoxville 
Memphis vs Chattanooga  42 v 58  Chattanooga 
Nashville vs Knoxville  68 v 32  Nashville 
Nashville vs Chattanooga  68 v 32  Nashville 
Knoxville vs Chattanooga  17 v 83  Chattanooga 
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 (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 
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 (candidate who beats all other candidates in pairwise comparisons) 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.
Secondorder Copeland method
The secondorder 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 firstorder Copeland method described above.
The secondorder Copeland method has a particularly beneficial feature: manipulation of the voting is more difficult because it requires NPcomplete complexity calculations to compute the manipulation.
External links
 Condorcet Class PHP library supporting multiple Condorcet methods, including Copeland method.
See also
References
 ↑ Pomerol, JeanCharles; Sergio BarbaRomero (2000). Multicriterion decision in management: principles and practice. Springer. p. 122. ISBN 0792377567.
 ↑ Colomer, Josep (2013). "Ramon Llull: From Ars Electionis to Social Choice Theory". Social Choice and Welfare. 40 (2): 317328. doi:10.1007/s0035501105982.
Notes
 E Stensholt, "Nonmonotonicity in AV"; 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; 148172.
 D.G. Saari. and V.R. Merlin, 'The Copeland Method. I. Relationships and the Dictionary'; Economic Theory; Vol. 8, No. l; June, 1996; 5176.