COPELAND (Net Wins Method)

COPELAND is an aggregation method that reduces several ranking sources ("the prioritisation of three experts," "the ranking given by five criteria") to a single rank. Its input is a table in which every source ranks the alternatives; its output is a net score for each alternative, found by subtracting its number of losses from its number of wins, and a complete rank based on that score. Unlike CONDORCET's stance of "say nothing if there is no winner," COPELAND always produces a rank, even where a cycle exists. Arthur Copeland proposed it in a 1951 seminar note, and it is known in social choice theory as the simplest method extending the Condorcet idea into a complete ranking.

COPELAND's idea resembles a league table: every team plays every rival once, wins are added up, losses subtracted, and the ranking follows the remaining net score. Like CONDORCET, COPELAND looks only at pairwise comparisons; it rests on who wins a majority against whom. But where CONDORCET answers only the question "is there someone who beats every rival," COPELAND gives every alternative a number, and this number is enough to build a rank even where a cycle exists.

This simplicity carries a cost. COPELAND uses only the information of "who beat whom," not by how many votes. In a cycle-free comparison among three alternatives, the net scores always come out in the pattern plus two, zero, minus two; which alternative gets which score varies, but the pattern itself does not change. This shows the method is insensitive to margin, sensitive only to direction.

The method proceeds through four steps.

First, gathering the ranks. Every source ranks the alternatives; these ranks are combined into one table.

Second, counting pairwise majorities. For every pair of alternatives, how many sources place which one ahead is counted.

Third, calculating the net score. For every alternative, the number of rivals it loses to by majority is subtracted from the number of rivals it beats by majority. If there is a tie with a rival, that pairwise contribution counts as zero.

Fourth, ranking. Alternatives are arranged by net score from highest to lowest. The highest net score marks first place.

The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.

The net score is the difference between how many rivals an alternative beats and how many it loses to; it says nothing more. Scoring plus two does not mean "perfect," it means "beat both rivals"; scoring minus two means "lost to both rivals." Scoring zero can mean "beat one rival, lost to one rival," or "tied with every rival"; both situations appear as the same number.

The size of the score does not show by how many votes the win was secured. Whether an alternative beats a rival by nine votes to one, or by six votes to five, its contribution is the same, plus one, either way. If a cycle exists, COPELAND still gives a rank, but the ties within that rank (several alternatives receiving the same net score) must be resolved by an external preference; the method itself does not do this.

For that reason, instead of writing:

"According to COPELAND, this alternative is best because it has the highest net score"

it is correct to write:

"This alternative wins a majority in most of the pairwise comparisons; the size of the win is not reflected in this score, and an additional criterion is needed among alternatives with equal scores"

COPELAND works with crisp data: each source's rank information is a whole number. DecisionMind has no fuzzy, grey or similar data-type extension of this method; it works only with crisp rank data.

You need the following: a table in which every source ranks all the alternatives completely. A minimum of two alternatives and two ranking sources is required; three to twelve alternatives work comfortably. COPELAND neither asks for nor produces weights; every source's vote is counted equally.

COPELAND is suitable if you have several ranking sources and want a complete rank no matter what, even where a cycle turns up. Its typical territory includes vote aggregation, sports-league-style rankings, and combining the ranks produced by several criteria.

The case where it should not be used is one where the size of the win also matters. COPELAND looks only at direction; whether an alternative beats a rival by an overwhelming majority or by a very narrow margin is not reflected in the score. If cycles are frequent and ties leave the decision unclear, a method that also accounts for margin is needed instead.

A simple, universally understood net score is enough → COPELAND

Winning margin also matters, I want the least disagreement → KEMENY-YOUNG

I want to find a winner with the least change, even under a cycle → DODGSON

I want to see clearly when there is no winner, a complete rank is not essential → CONDORCET

COPELAND's greatest strength is its simplicity and the fact that it always produces a rank. Its calculation can be followed by hand, it reads like a league table, and it never leaves the decision-maker empty-handed even where a cycle turns up. The method does not depend on any choice of weighting or normalisation; it rests only on pairwise majority. This makes it a natural choice for combining sources of different scales and units, because it asks for nothing beyond rank information.

COPELAND's fundamental limitation is its insensitivity to margin; it looks only at who beat whom, not by how many votes (Saari and Merlin, 1996). Second, in the case of a cycle, several alternatives can receive the same net score, and the method cannot resolve this tie on its own; an external preference is required. Third, in a cycle-free comparison among three alternatives, the net scores always come out in the same pattern (plus two, zero, minus two); this limits how discriminating the score itself remains as the number of alternatives grows.

The most common mistake is reading the net score as though it were a percentage or a vote share; the score is only the difference between the number of pairwise contests won and lost. A second mistake is favouring one of two alternatives with equal net scores without justification; if a tie exists, how it was resolved must be stated clearly in the report. A third is using COPELAND in a decision where winning margin matters and presenting the result as "decisive superiority"; the method never sees margin at all. A fourth is allowing some sources to rank the alternatives incompletely; every source must rank all the alternatives.

The governing principle is this:

The COPELAND net score is only a count of who beat whom; the size of the win, or how ties were resolved, must be explained separately in the report.