Copeland's method

Copeland’s method is a ranked voting methods based on the 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"
  • 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". [1]
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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 he or she has a majority preference plus half the number of candidates with whom he or she has 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 he or she has the maximum possible score of n –1 where n is the number of candidates. Hence victory under this system amounts to satisfying the Condorcet criterion.[2]

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 the "Minimax Condorcet method".

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[3] 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.[1]

Ties

Some people claim that Copeland's method is liable to leave elections undetermined, but realistic examples are difficult to produce. One way is because of a tie, which is a problem for every voting method (even the "first-past-the-post" electoral system that many are used to). Another way for pairwise methods to have a tie is because of Condorcet's paradox. While theoreticians come up with examples, there has not been a published example of a large, publicly-administered, ranked ballot election that had a circular preference as described by Marquis de Condorcet. Regardless, people who work on game theory have created systems to deal with possible ties (e.g. Tideman's Ranked Pairs, the Schulze method, and the Minimax Condorcet method). In case of no candidate meeting the necessary condition for complying with the Condorcet criterion (i.e. which is equivalent to an n -way tie), some form of tie-breaker is needed to deal with these cases. The use of the Borda count to resolve a tie in Copeland's method gives rise respectively to "Black's method" and the "Dasgupta-Maskin method".

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, ½ if the numbers are equal, and 0 if more voters prefer j to i than prefer i to j. 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 n –1 (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 ‘1/½/0’ 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.[4]

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.[5] 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 ‘1/½/0’.

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.

(The Borda count is also analogous to sporting tournaments. Copeland‘s method is analogous to a tournament in which each pair of competitors play a single game whose result is determined by the entire electorate whereas the Borda count is analogous to a tournament in which every completed ballot determines the result of a game between every pair of competitors.)

Properties

Copeland’s method has many of the standard desirable properties (see the table below). In particular it satisfies the Condorcet criterion, i.e. if there is a candidate who would win against each of his or her rivals in a binary vote, then this candidate is the winner. It follows that the Copeland method satisfies the median voter theorem which states that if views lie along a spectrum, then the winning candidate will be the one preferred by the median voter.

In this it differs from instant runoff voting. Suppose that there are 3 candidates: A on the left, B in the centre, and C on the right, and that A has 36% support (voting A-B-C), B has 30% support (split equally between B-C-A and B-A-C) and C has 34% support (voting C-B-A). Then B wins under any Condorcet method but is eliminated on the first round under instant runoff voting.

Critics argue that Copeland’s method puts too much emphasis on the number of pairwise victories and defeats rather than their magnitudes.

Tied results

Copeland’s method is particularly prone to giving tied results owing to the fact that each candidate’s score is a small integer multiple of ½, and that the multiples do not become larger as the number of voters increases.

A simple tie can be constructed with 5 voters and 3 candidates. One voter votes A-B-C, two vote B-C-A, and two vote C-A-B. This is a 3-way Copeland tie, with A being preferred to B by 3:2, B to C by 3:2, and C to A by 4:1. It is a Condorcet cycle but it is not symmetric and under some voting methods (including the Borda count) C would be the winning candidate.

Comparison with other systems
Comparison of preferential electoral systems
Sys­tem Mono­tonic Cond­orcet Majo­rity Condorcet loser Majority loser Mutual majority Smith ISDA LIIA Independence of clones Reversal symmetry Participation, consistency Later-no‑harm Later-no‑help Polynomial time Resol­vability
Schulze YesYesYesYesYesYesYesYesNoYesYesNoNoNoYesYes
Ranked pairs YesYesYesYesYesYesYesYesYesYesYesNoNoNoYesYes
Tideman's Alternative NoYesYesYesYesYesYesYesNoYesNoNoNoNoYesYes
Kemeny–Young YesYesYesYesYesYesYesYesYesNoYesNoNoNoNoYes
Copeland YesYesYesYesYesYesYesYesNoNoYesNoNoNoYesNo
Nanson NoYesYesYesYesYesYesNoNoNoYesNoNoNoYesYes
Black YesYesYesYesYesNoNoNoNoNoYesNoNoNoYesYes
Instant-runoff voting NoNoYesYesYesYesNoNoNoYesNoNoYesYesYesYes
Smith/IRV NoYesYesYesYesYesYesYesNoYesNoNoNoNoYesYes
Borda YesNoNoYesYesNoNoNoNoNoYesYesNoYesYesYes
Baldwin NoYesYesYesYesYesYesNoNoNoNoNoNoNoYesYes
Bucklin YesNoYesNoYesYesNoNoNoNoNoNoNoYesYesYes
Plurality YesNoYesNoNoNoNoNoNoNoNoYesYesYesYesYes
Contingent voting NoNoYesYesYesNoNoNoNoNoNoNoYesYesYesYes
Coombs[6] NoNoYesYesYesYesNoNoNoNoNoNoNoNoYesYes
MiniMax YesYesYesNoNoNoNoNoNoNoNoNoNoNoYesYes
Anti-plurality[6] YesNoNoNoYesNoNoNoNoNoNoYesNoNoYesYes
Sri Lankan contingent voting NoNoYesNoNoNoNoNoNoNoNoNoYesYesYesYes
Supplementary voting NoNoYesNoNoNoNoNoNoNoNoNoYesYesYesYes
Dodgson[6] NoYesYesNoNoNoNoNoNoNoNoNoNoNoNoYes

Rationale

In many cases decided by Copeland’s method the winner is the unique candidate satisfying the Condorcet criterion; in these cases, the arguments for that criterion (which are powerful but not universally accepted[7]) apply equally to Copeland’s method.

When there is no Condorcet winner Copeland’s method seeks to make a decision by a natural extension of the Condorcet method, combining preferences by simple addition. The justification for this lies more in its intuitive appeal than in any logical arguments.

The Borda count is another method which combines preferences additively. The salient difference is that a voter’s preference for one candidate over another has a weight in the Borda system which increases with the number of candidates ranked between them. The argument from the viewpoint of the Borda count is that the number of intervening candidates gives an indication of the strength of the preference; the counter-argument is that it depends to a worrying degree on which candidates stood in the election.

Partha Dasgupta and Eric Maskin sought to justify Copeland’s method in a popular journal, where they compare it with the Borda count and plurality voting.[8] Their argument turns on the merits of the Condorcet criterion, paying particular attention to opinions lying on a spectrum. The use of Copeland’s method in the first instance, and then of a tie-break, to decide elections with no Condorcet winner is presented as ‘perhaps the simplest modification’ to the Condorcet method.

Tie breaks

As mentioned earlier, the Copeland method has a realistic chance of producing a tie, regardless of the number of voters – i.e it is not ‘resolvable’. This is a significant weakness, not least because Copeland’s method itself seeks to resolve the Condorcet criterion in cases in which it does not identify a winner. In order to be considered for practical use, the Copeland method needs to be supplemented by a tie-break. This may be a coin-toss or a secondary voting method, in which case it can be applied either to the entire list of candidates or to subsets involved in ties.

Dasgupta and Maskin proposed the Borda count as a backup to a Copeland primary, seemingly applying it to the entire list of candidates.[9] This is known as the Dasgupta-Maskin method. It had previously been used in figure-skating under the name of the ‘OBO’ (=one-by-one) rule.[4] Duncan Black used a Borda secondary in conjunction with the Condorcet criterion as primary; this is Black's method.

Certain dangers need to be taken into consideration:

  • The adoption of a secondary voting method with less risk of ties may reduce acceptance of the legitimacy of the primary method, particularly if the secondary method is applied to the entire list and if the public does not understand the reasons for preferring the primary one.
  • Whenever the tie-break is invoked, the combined system suffers the drawbacks of the secondary method, admitted to be less fair than the primary.
  • Voters who cannot obtain a win for their own party by tactical voting may be able to force a tie in which it participates. If they can do this by means which allow them to profit from the tie-break, then they can falsify the result in their own favour.

Illustration

Suppose that there are 4 candidates, A, B, C and D, and 5 voters, of whom two vote A-B-C-D, two vote B-C-D-A, and one votes D-A-B-C. The results between pairs of candidates are shown in the main part of the following table, with the Copeland score for the first candidate in the additional column.

2nd
1st
 ABCD score
A —3:23:22:32
B 2:3—5:04:12
C 2:30:5—4:11
D 3:21:41:4—1

(If you think of the pairwise results as football matches, then a team’s Copeland score is the number of wins it obtains along a row, and its Borda score is the number of its goals.)

No candidate satisfies the Condorcet criterion, and there is a Copeland tie between A and B. The Borda scores are (8,11,6,5), so would accord victory to B in a tie-break. In an instant runoff (IRV) tie-break, on the other hand, C and D would be pre-eliminated, reducing the ballots to 3 A-Bs and 2 B-As, so A would be elected.

Using Black’s method there is no Condorcet decision so the election reduces to a four-way Borda tie-break won by B.

If we were to use the Condorcet method with an IRV tie-break, then again we would have to resolve a four-way tie. C is eliminated in the first round through having no first-place preferences, reducing the ballots to 2 A-B-Ds, 2 B-D-As, and a D-A-B. Now D is eliminated, reducing the ballots as before to 3 A-Bs and 2 B-As with victory going to A.

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:

ComparisonResultWinner
Memphis vs Nashville42 v 58Nashville
Memphis vs Knoxville42 v 58Knoxville
Memphis vs Chattanooga42 v 58Chattanooga
Nashville vs Knoxville68 v 32Nashville
Nashville vs Chattanooga68 v 32Nashville
Knoxville vs Chattanooga17 v 83Chattanooga

The wins and losses of each candidate sum as follows:

CandidateWinsLossesNetr
Memphis03−3 0 0 0 0
Nashville303 1 0 1 1
Knoxville12−1 1 0 0 0
Chattanooga211 1 0 1 0

Nashville, with no defeats, is the Condorcet winner. The Copeland score under the ‘1/0/–1’ method is the number of net wins, maximized by Nashville. Since the voters expressed a preference one way or the other between every pair of candidates, the score under the ‘1/½/0’ method is just the number of wins, likewise maximized by Nashville. The r matrix for this scoring system is shown in the final column.

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

In this example there are some tied votes: for instance 10% of the voters assigned no position to B or C in their rankings; they are therefore considered to have tied these candidates with each other while ranking them below D, A and 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 Netr
A 3 1 2 0 0 1 1 1
B 2 2 0 1 0 0 0 1
C 2 2 0 0 1 0 1 0
D 1 3 −2 0 1 0 0 0
E 2 2 0 0 0 1 1 0

No Condorcet winner (candidate who beats all other candidates in pairwise comparisons) exists. Candidate A is the Copeland winner. Again there is no pair of candidates between whom the voters express no preference.

Use for producing a tabulation in other methods

Since Copeland's method produces a total ordering of candidates by score and is simple to compute, it is often useful for producing a sorted list of candidates in conjunction with another voting method which does not produce a total order. For example, the Schulze and Ranked pairs methods produce a transitive partial ordering of candidates, which generally produces a single winner, but not a unique way of tabulating runner-ups. Applying Copeland's method according to the respective method's partial ordering will yield a total order (topological ordering) guaranteed to be compatible with the method’s partial order, and is simpler than a depth-first search when the partial order is given by an adjacency matrix.

More generally, the Copeland score has the useful property that if there is a subset S of candidates such that every candidate in S will beat every candidate not in S, then there exists a threshold θ such that every candidate with a Copeland score above θ is in S while every candidate with a Copeland score below θ is not in S. This makes the Copeland score practical for finding various subsets of candidates that may be of interest, such as the Smith set or the dominant mutual third set.

See also

References
  1. Copeland, Arthur Herbert (1951), A 'reasonable' social welfare function, Seminar on Mathematics in Social Sciences, University of Michigan (unpublished).
  2. Pomerol, Jean-Charles; Sergio Barba-Romero (2000). Multicriterion decision in management: principles and practice. Springer. p. 122. ISBN 0-7923-7756-7.
  3. George G. Szpiro, ‘Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present’ (2010).
  4. Balinski, Michel, and Rida Laraki, “Judge: Don’t vote !” (2014), esp. footnote 4.
  5. Scoring Systems in Chess Tournaments.
  6. Anti-plurality, Coombs and Dodgson are assumed to receive truncated preferences by apportioning possible rankings of unlisted alternatives equally; for example, ballot A > B = C is counted as  A > B > C and  A > C > B. If these methods are assumed not to receive truncated preferences, then later-no-harm and later-no-help are not applicable.
  7. Eric Pacuit, “Voting Methods”, The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.)
  8. P. Dasgupta and E. Maskin, ‘The fairest vote of all’ (2004).
  9. P. Dasgupta and E. Maskin, ‘The fairest vote of all’ (2004). The specification of their method is on p. 97, where they write “If no one [candidate] obtains a majority against all opponents, then among those candidates who defeat the most opponents in head-to-head comparisons, select as winner the one with the highest rank-order score”.

Notes
  1. E Stensholt, "Nonmonotonicity in AV"; Voting matters; Issue 15, June 2002 (online).
  2. 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.
  3. 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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