# Copeland's method

Copeland's method or Copeland's pairwise aggregation method is a Smith-efficient 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] It is named after Arthur Herbert Copeland, who independently proposed it in a 1951 lecture.[3]

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]

## 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)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

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:

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.

## 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.

## Copeland Star Voting method

This method utilizes the Star Voting Method (Score Then Automatic Runoff), except the final round extends to the top three scoring candidates, and a pairwise count is used to determine the winner. By extending the final round to three, it improves the chances of choosing the overall Condorcet winner.

If the final round leads to a three way tie, then the final round becomes a simple approval vote between the top two highest scoring candidates. This alleviates the possibility of a three way tie that the Copeland Method may create.