Copeland's method is a ranked voting method based on a scoring system of pairwise "wins", "losses", and "ties". The method has a long history:
Ramon Llull described the system in 1299, so it is sometimes referred to as "Llull's method" The Marquis de Condorcet described a similar system in the 1780s, so the method could be referred to as "Condorcet's method", but instead other systems were subsequently devised that choose the Condorcet winner. Arthur Herbert Copeland described the system in the 1950s, so it has been frequently been called "Copeland's method".
Each voter is asked to rank candidates in order of preference. A candidate A is said to have majority preference over another candidate B if more voters prefer A to B than prefer B to A; if the numbers are equal then there is a preference tie. The Copeland score for a candidate is the number of other candidates over whom they have a majority preference plus half the number of candidates with whom they have a preference tie. The winner of the election under Copeland's method is the candidate with the highest Copeland score; under Condorcet's method this candidate wins only if they have the maximum possible score of where is the number of candidates. Hence victory under this system amounts to satisfying the Condorcet criterion.
Any voting method satisfying the Condorcet winner criterion may sometimes be referred to as "a Condorcet method". Other methods that satisfy the Condorcet winner criterion include the Kemeny–Young method, the Schulze method, and Minimax.
History
Copeland's method was devised by Ramon Llull in his 1299 treatise Ars Electionis and discussed by Nicholas of Cusa in the fifteenth century and by the Marquis de Condorcet in the eighteenth (who drew attention to the related criterion). However, it is frequently named after Arthur Herbert Copeland who advocated it independently in a 1951 lecture.
Voting mechanism
Ballot
The input is the same as for other ranked voting systems: each voter must furnish an ordered preference list on candidates where ties are allowed (a strict weak order).
This can be done by providing each voter with a list of candidates on which to write a "1" against the most preferred candidate, a "2" against the second preference, and so forth. A voter who leaves some candidates' rankings blank is assumed to be indifferent between them but to prefer all ranked candidates to them.
Computation
A results matrix r is constructed as follows: rij is
1 if more voters strictly prefer candidate i to candidate j than prefer j to i
if the numbers are equal
0 if more voters prefer j to i than prefer i to j.
This may be called the "1//0" method (one number for wins, ties, and losses, respectively).
By convention, rii is 0.
The Copeland score for candidate i is the sum over j of the rij. If there is a candidate with a score of (where n is the number of candidates) then this candidate is the (necessarily unique) Condorcet and Copeland winner. Otherwise the Condorcet method produces no decision and the candidate with greatest score is the Copeland winner (but may not be unique).
An alternative (and equivalent) way to construct the results matrix is by letting rij be 1 if more voters strictly prefer candidate i to candidate j than prefer j to i, 0 if the numbers are equal, and −1 if more voters prefer j to i than prefer i to j. In this case the matrix r is antisymmetric.
Tied preferences
The method as initially described above is sometimes called the "1//0" method. Llull himself put forward a 1/1/0 method, so that two candidates with equal support would both get the same credit as if they had beaten the other.
Preference ties become increasingly unlikely as the number of voters increases.
Use in sporting tournaments
A method related to Copeland's is commonly used in round-robin tournaments. Generally it is assumed that each pair of competitors plays the same number of games against each other. rij is the number of times competitor i won against competitor j plus half the number of draws between them.
It was adopted in precisely this form in international chess in the middle of the nineteenth century. It was adopted in the first season of the English Football League (1888–1889), the organisers having initially considered using a 1/0/0 system. For convenience the numbers were doubled, i.e. the system was written as 2/1/0 rather than as 1//0.
Sporting use differs from politics in that the scoring system is seen as one of the rules of the game with less emphasis on objective truth. For this reason modified Copeland systems using 3/1/0 scoring are commonly adopted.