Condorcet (Pairwise Majority Method)

Condorcet is an aggregation method that reduces several ranking sources ("three jurors' rankings," "five experts' priorities") into a single order. Its input is a rank list from each source placing the alternatives from best to worst; its output is how many rivals each alternative beats and, where one exists, an outright winner. Condorcet does not sum votes or average scores; it looks only at pairwise comparisons. The idea was put forward in 1785 by the Marquis de Condorcet and took its modern form in Duncan Black's 1958 study. It is one of the cornerstones of social choice theory, and the other four aggregation methods in DecisionMind (Copeland, Dodgson, Kemeny-Young, Cook-Seiford) answer the same question, "who represents the majority's preference," by different routes.

The idea behind Condorcet is simple: if an alternative is genuinely the best, it should also win when compared one by one against every rival. If alternative A is preferred over B, and also preferred over C, A is a strong candidate. An alternative that achieves this against every one of its rivals is called the Condorcet winner, and this is intuitively the most defensible result, because anyone wanting to overturn the winner must argue against a majority every single time.

This idea has an uncomfortable consequence. With three or more alternatives, a majority preferring A over B, B over C, and C over A can all exist at once. This is called a Condorcet cycle, and in this situation no alternative beats every rival. Condorcet does not manufacture a false winner in such a case; it says "no winner." This is the method's honesty, but also its limit: it does not always deliver a complete ranking.

The method proceeds through four steps.

First, gathering the rankings. Every source (expert, juror, criterion) ranks the alternatives from first to worst. These rankings are combined into a table: alternatives in rows, sources in columns.

Second, counting the pairwise majority. For every pair of alternatives, the method counts how many sources prefer one over the other. Between A and B, if most sources rank A higher, A wins this pairwise contest.

Third, building the dominance network. Once the winner of every pairwise contest is determined, the "who beats whom" relationship is built like a network. If an alternative beats another by pairwise majority, an arrow is drawn between them.

Fourth, searching for a winner. If there is an alternative that turns the arrows in its favour against every rival, it is the Condorcet winner. If no such alternative exists (a cycle is present), the method returns a partial result and a cycle warning instead of a complete ranking.

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

Condorcet's output has two parts: a number showing how many rivals each alternative beats, and a cycle flag. If an alternative beats every rival, this number equals the number of alternatives minus one, and that alternative is the winner; this means "it lost no pairwise contest by majority vote," not "it received the highest score." The win count does not show by how large a margin the wins came; winning with two sources out of three looks the same as winning with all three.

If a cycle is present, the numbers Condorcet returns are not a suggested ranking, only a record of who prevailed over whom. In that case it is wrong to say "according to Condorcet, the best alternative is this one"; the correct statement is "among these sources there is no single winner the majority would accept."

For this reason:

"Condorcet ranked the alternatives, and this came first"

should be written as:

"Among these sources, the single alternative that beats every rival by pairwise majority is this one; if no such alternative exists, the method states this plainly"

Condorcet works with crisp data: each source's ranking is expressed as whole numbers, first, second, third, and so on. DecisionMind holds no fuzzy, grey or intuitionistic data-type extension of this method; it works only with crisp ranking data.

You need a table in which every source ranks all the alternatives completely, with no missing ranks or skipped alternatives. A minimum of two alternatives and two ranking sources is required; three to twelve alternatives work comfortably. Condorcet neither asks for nor produces weights; every source's vote counts equally.

Condorcet is suitable when you hold complete rankings produced by several experts or methods, and the goal is an honest answer to the question "is there a winner the majority would accept." Its typical territory is vote aggregation, expert-panel consensus, and reducing the rankings produced by several methods into a single order.

The case where it should not be used is one where a complete ranking is strictly required. As the number of alternatives grows, so does the likelihood of a cycle; if Condorcet returns a cycle, it tells you only "no winner," not a ranking. In that case, either switch to a completion method that resolves the cycle, or make the decision on some other justification.

I want an honest view of whether a winner exists → Condorcet

I want a complete ranking even if a cycle occurs → Copeland, Dodgson, Kemeny-Young

A numerical score is needed, not a ranking → scoring methods such as TOPSIS, VIKOR

The size of the winning margin also matters → Kemeny-Young (measures total disagreement) or Cook-Seiford (measures positional difference)

Condorcet's greatest strength is its honesty. When it declares a winner, that winner is a result the majority would accept; it has not lost a majority contest against any rival. The method is simple, every step can be followed through pairwise comparisons, and the result does not depend on a choice of weighting or normalisation. Where a winner exists, any objection wanting to overturn it must argue against a majority; this makes the result robust to dispute.

Condorcet's fundamental limitation is that a result does not always exist. With three or more alternatives, majority preferences can be cyclical; this is called the Condorcet paradox and was examined in detail by Duncan Black (Black, 1958). Second, the method uses only "who beats whom" information and does not take into account the margin of a win; winning by two votes looks the same as winning unanimously. Third, as the number of alternatives grows, the number of pairwise comparisons rises rapidly and the likelihood of a cycle increases.

The most common mistake is forcing the method to declare a winner anyway when a cycle occurs. A cycle is a situation arising from the data itself; Condorcet saying "no winner" is not an error but a correct finding. A second mistake is reading the win count as a score and interpreting it as, say, "beat two rivals, so is two-thirds good"; this number only counts how many pairwise contests were won. A third mistake is filling in the sources' rankings incompletely or inconsistently and blaming the result on Condorcet's error; every source must rank all the alternatives. A fourth mistake is ignoring the cycle warning and reporting a single winner regardless.

The governing principle is this:

Condorcet answers only the question "is there a winner the majority would accept"; if the answer is no, the method does not hide this, and neither should the report.