NANSON (Nanson's Method)

Nanson's method, like median ranking, reduces multiple existing rankings to a single order once you have them in hand. But it works differently: it first computes a classical score (a Borda score, the total number of alternatives that an alternative outranks across every ranking), then eliminates entirely whichever alternatives fall below that score's mean, and rescores the remainder from scratch. Unlike methods that look at a single figure, this is an elimination that proceeds in rounds. Its output is an order determined by which round an alternative was eliminated in; the alternative left standing last comes first. The mathematician Edward J. Nanson proposed the method in 1882, and it is known in the social-choice (vote-aggregation) literature as a version of Borda scoring strengthened with an elimination logic.

Classical Borda scoring scores every alternative in a single round and picks the highest-scoring one; but this method has a known weakness, the alternative preferred by the majority in a pairwise comparison (the Condorcet winner) may not come first under Borda scoring. Nanson tries to solve this by eliminating in rounds: an alternative is eliminated for as long as it stays below the mean, and the calculation is repeated with the alternatives above the mean. This keeps Borda's simple scoring logic while progressively removing the "weak" alternatives that fall below the mean.

The philosophical consequence is that Nanson carries a pursuit of consistency: if a Condorcet winner exists, that is, an alternative that wins every pairwise comparison, the method aims to find it. This sets Nanson apart from methods that merely sum positions (such as Borda); Nanson also sums positions, but confirms the result through a staged elimination rather than in a single round.

The method proceeds through a repeated round cycle.

In every round, a Borda score is first computed. Among the surviving alternatives, each is given a score according to its position in every ranking; it earns as many points as the number of alternatives it outranks in that ranking. The scores across all rankings are summed.

Then the round's mean score is found. The mean is taken across all the alternatives surviving in that round.

Every alternative falling below the mean is eliminated. Not just one, but every alternative below the mean is eliminated at once; this is a detail that sets Nanson apart from methods that eliminate only the single worst alternative.

Once the eliminated alternatives are removed, the process repeats from the start with those remaining. Scoring is redone among the surviving alternatives, because an alternative's relative position can change once the number of rivals shrinks. This cycle continues until a single alternative remains, or until no alternative falls below the mean. In the latter case (nobody is eliminated), the remaining alternatives are ranked by their current scores.

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

The value Nanson reports shows how many elimination rounds an alternative survived; a higher value means the alternative survived to a later round, and so was stronger. This value is not a percentage and is not a direct measure of "how good" the alternative is; it only summarises the elimination order. The gap in rounds between two alternatives does not show the size of the underlying score gap; an alternative may have been eliminated in the final round by a very narrow margin.

Nanson's result gives the Condorcet winner, if one exists in the dataset, that is, an alternative that wins every pairwise comparison. But if no such alternative exists (there is a cyclical preference among the sources), which alternative Nanson picks depends on the elimination order, and in that case the result should not be claimed as the single "correct" alternative on its own.

Therefore, instead of writing:

"Nanson's method found the best alternative"

the report should read:

"Among this set of rankings, the alternative that survives the elimination rounds longest is this one; this is consistent with the majority's preference in pairwise comparisons"

Nanson works with a crisp rank matrix: alternatives in rows, different ranking sources (experts, votes, or the outputs of other methods) in columns, with each cell holding an integer rank position from 1 to m; 1 denotes the best position. Every source must have ranked all the alternatives. The method neither produces nor requires weights. DecisionMind currently holds no additional member alongside this base method; a minimum of two alternatives and two separate ranking sources is required.

If you hold more than one ranking and want to combine them with a method that, where possible, preserves the majority winner of the pairwise comparisons (the Condorcet winner), Nanson is suitable. If the number of sources is very small (say, two), the elimination rounds lose their meaning, because very few, or no, alternatives are eliminated in each round. If there is a strong cyclical contradiction among the sources (no Condorcet winner exists), Nanson's result becomes sensitive to the elimination order, and this sensitivity should be shown in the report.

Combine several rankings, preserve the pairwise majority winner → Nanson

Simple average scoring is enough, no elimination needed → Borda / average ranking

Find the position most resistant to an outlying view → median ranking

Examine every pairwise comparison individually, find the strongest indirect path → Schulze

Nanson's core advantage is that it keeps the simplicity of classical Borda scoring while adding a guarantee of finding the Condorcet winner, if one exists. The elimination logic is intuitive: weak alternatives drop out step by step, and the survivors are reassessed in every round, giving a more reliable result than single-round methods. The calculation can be followed by hand and requires no complex optimisation.

Its limitations arise from the elimination mechanism. First, if no Condorcet winner exists (there is a cyclical preference among the sources), the method's result becomes sensitive to the order of the rounds. Second, eliminating every alternative below the mean at once can cost an alternative very close to the mean an early exit, one that might have come to the fore in a subsequent round. Third, how positional methods and elimination-based voting rules behave under computation and manipulation has been discussed in a separate literature; for some elimination-based rules, the computational burden of determining the winner is higher than for classical scoring (Bartholdi, Tovey and Trick, 1989).

The most frequent mistake is eliminating only the single worst alternative in each round; in fact Nanson eliminates every alternative below the mean at once, and skipping this detail produces the wrong result. A second mistake is scoring only in the first round and failing to recompute scores after removing the eliminated alternatives; the remaining alternatives' relative scores change once the number of rivals falls. A third mistake is treating Nanson's result as "definitely correct"; if no Condorcet winner exists, the result depends on the order of the rounds, and this dependence must be stated in the report. A fourth mistake is including a source with an incomplete ranking without correcting it first.

The governing principle is this:

Nanson eliminates Borda scoring in rounds to find the pairwise majority winner, if one exists; if no such winner exists, the result depends on the elimination order, and this dependence must be reported.