DODGSON (Least-Change Method)

DODGSON is an aggregation method that reduces multiple ranking sources to a single result. It rests, like CONDORCET, on the idea of a majority winner, but unlike CONDORCET it does not leave you empty-handed when a cycle occurs: for every alternative, it counts the smallest change that would need to be made in the rankings to make that alternative the majority winner. The alternative requiring the fewest changes is the winner. The method was proposed by the mathematician and writer Charles Lutwidge Dodgson (also known by the pen name Lewis Carroll) in a pamphlet in 1876.

The question DODGSON asks is this: "How far would this community's opinion have to be moved from its true view for this alternative to become the majority winner?" Every ranking source's opinion is treated as something that can be altered in small steps, by swapping two adjacent alternatives. The fewer such changes in total are needed to make an alternative the majority winner, the closer that alternative is considered to the true majority view.

This turns CONDORCET's question of "is there a winner or not" into a question of distance: if there is no winner, who is closest to being one? DODGSON's philosophical consequence is a completion logic: the way to resolve a cycle is to look for the alternative that disturbs the majority view the least. The price of this is computational burden; finding the smallest change is a search problem that quickly becomes harder as the number of alternatives grows.

The method proceeds through three steps.

First, aggregate the rankings and count pairwise majorities. Every source's ranking is combined into one table; for every pair of alternatives, how many sources place one ahead of the other is counted.

Second, find the smallest change for every alternative. For an alternative to be the majority winner means it beats the majority against every one of its rivals. If this is not yet the case, that alternative's position in the rankings must be moved upward by swapping it with its adjacent neighbours. Every such move is counted as one "change"; the total number of changes needed for an alternative is its Dodgson score.

Third, choose the lowest score. The alternative with the lowest Dodgson score is the winner; if it is already the majority winner, its score is zero and no change is needed. The remaining alternatives are ranked by their increasing scores.

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

The Dodgson score is the smallest number of changes needed in the rankings for an alternative to become the majority winner. A score of zero means that alternative is already the majority winner; no further interpretation is needed. A positive score shows how many adjacent swaps the community's opinion would need to be disturbed by for that alternative to become the winner; a small score means the alternative is considered close to the true majority view, a large one means it is considered far from it.

This score is neither a percentage nor a probability; it is purely a count. Moreover, DODGSON's ranking of the non-winning alternatives rests on each one's own distance from becoming a winner; this is not the same question as how the alternatives should be ranked relative to one another. Two alternatives can share the same Dodgson score; this means both are equally distant from becoming the winner, and a further criterion may be needed to choose between them.

Thus instead of writing:

"DODGSON found the best alternative, its score is such-and-such"

it is correct to write:

"This alternative needed the fewest changes to become the majority winner; the further the score sits from zero, the further it is from the community's true view"

DODGSON works with crisp data: every source's ranking information is a whole number. DecisionMind has no fuzzy, grey or other data-type extension of this method; it works only with crisp ranking data.

You need: a table in which every source has ranked all alternatives completely. A minimum of two alternatives and two ranking sources is required. As the number of alternatives grows (beyond twelve), finding the smallest change becomes computationally difficult; this stems from the method being NP-hard (Bartholdi, Tovey and Trick, 1989). DODGSON requires no weights and produces none.

DODGSON is suitable if you have more than one ranking source, want to rest on the idea of a majority, and need to choose a winner even when a cycle occurs. It works with a small number of alternatives (twelve or fewer), and its result can be interpreted as the alternative reaching the majority view "with the least intervention".

It should not be used where the number of alternatives is large or where a fast result is needed for every decision. Finding the smallest change is an NP-hard problem; practical computation becomes difficult beyond twelve alternatives. In addition, according to an observation by Orakçı (2024), DODGSON fails to produce a complete ranking in eighty to ninety-nine per cent of random samples; this is not a problem if only the winner needs to be determined, but caution is needed where a complete ranking is required.

A small number of alternatives, wanting the winner closest to the majority view → DODGSON

A large number of alternatives, wanting a fast, always-complete ranking → COPELAND

Wanting a complete ranking that globally minimises total disagreement → KEMENY-YOUNG

Wanting to see clearly when there is no winner, not wanting to calculate a distance → CONDORCET

DODGSON's greatest strength is that it finds the alternative closest to the majority view even when a cycle occurs; it fills the gap that CONDORCET leaves open. Because a score of zero directly indicates the majority winner, the method always finds the CONDORCET winner when one exists; this makes DODGSON a Condorcet-completion method. The score itself is intuitively understandable: the question "how many changes are needed" is easily explained to a decision-maker.

DODGSON's best-known limitation is computational difficulty; determining the winner with certainty is an NP-hard problem and becomes impractical as the number of alternatives grows (Bartholdi, Tovey and Trick, 1989). Second, the method frequently fails to produce a complete ranking; according to an observation by Orakçı (2024), in the large majority of random samples DODGSON can only determine the winner, leaving ties among the remaining alternatives. Third, the Dodgson score rests on each alternative's own distance from becoming the winner; this does not guarantee a consistent, complete ranking of alternatives relative to one another (Black, 1958).

The most common mistake is trying to run DODGSON with full precision when the number of alternatives is large; in that case approximate methods or restriction to small candidate sets are needed. A second mistake is reading a non-zero Dodgson score as a percentage or a probability; the score is purely a count of the changes required. A third mistake is making an unjustified choice between two alternatives with equal scores; a tie means both are equally distant from becoming the winner. A fourth mistake is assuming DODGSON will always give a complete ranking; in most cases the method can only reliably determine the winner.

The governing principle is this:

The number DODGSON gives is how hard the majority view would have to be strained to make this alternative the winner; the closer it sits to zero, the less contestable the alternative.