SCHULZE (Schulze Method)

The Schulze method takes several rankings at once, whether produced by several experts, several voters or several methods, and turns them into a single common ranking. Its distinguishing feature is that it does not do this by simply summing scores; it first compares every pair of alternatives separately, starting from the answer to "in how many sources did A finish ahead of B." These pairwise superiorities are then treated as a network of paths, and the strongest indirect path is found for every pair. Its output is a score for every alternative and the ranking that score produces. The method was proposed by Markus Schulze in 2011 and has been favoured by some open-source software communities and internet-based voting systems for its strong mathematical properties, such as clone independence, meaning that adding or removing candidates very similar to one another does not disturb the result.

The idea comes from taking the majority principle seriously at the pairwise level: if one alternative is preferred to another in more sources when compared directly, that counts as a "win." Yet real data sets can contain cycles, A beats B, B beats C, and C beats A, in which case a simple ranking is not possible. Schulze resolves such cycles as follows: even where there is no direct superiority between two alternatives, if an indirect "chain of superiority" runs between them, the strength of that chain's weakest link is what counts, and the strongest of all possible chains between two alternatives is taken as decisive.

This carries a philosophical consequence: Schulze is a consistency method, not a compensatory or conciliatory one. The method seeks an answer to "who beats whom, directly or indirectly"; a high average score does not save an alternative, what matters is genuine superiority in the pairwise comparisons. This is what sets Schulze apart from position-summing methods such as Borda.

The method proceeds through four steps.

First, gathering the source rankings. Every ranking produced for the same set of alternatives, whether by experts, votes or methods, is brought together.

Second, counting pairwise superiority. For every pair of alternatives, it is counted in how many sources one alternative finished ahead of the other. This gives, for each pair, a one-directional count of "how many sources preferred A to B."

Third, finding the strongest indirect path. Whether or not a direct superiority exists between two alternatives, every possible chain running through intermediate alternatives is scanned. The strength of a chain is the strength of its weakest link, that is, its lowest pairwise superiority count. The strongest of all chains between two alternatives is recorded as the "strongest path" for that pair. This is repeated for every pair.

Fourth, determining the winner. If the strongest path from A to B is stronger than the strongest path from B to A, A beats B. This pairwise beats-relation is derived for every pair, and an overall ranking is obtained from it.

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

The score Schulze gives summarises an alternative's net pairwise superiority over the others; it is derived from the difference between how many alternatives it beats and how many it is beaten by. This score is not a percentage, and it does not show "how good" the alternative is, only its net position in the pairwise comparisons. A positive score shows that the alternative beats more than it is beaten by; a negative score shows the reverse.

Schulze's strongest feature is that, if the data set contains an alternative that wins every pairwise comparison (a Condorcet winner), it will always find it. But where a genuine cycle exists (three alternatives beating one another in turn), the strongest path between two alternatives can come out equal; this is a genuine tie, not a computational error, and the report must state this clearly.

Thus instead of writing:

"The Schulze method found the best alternative"

the report should read:

"Based on the pairwise comparisons within this set of sources, this is the alternative with the strongest indirect superiority; this result prioritises pairwise consistency, a different logic from methods that prioritise average score"

Schulze works with a crisp ranking matrix: alternatives in rows, different ranking sources (experts, votes or the output of other methods) in columns, and in every cell an integer rank position from 1 to m, where 1 denotes the best position. Every source must have ranked all the alternatives. The method does not produce weights; every source is given equal weight by default, though DecisionMind also allows different weights to be assigned to sources. There is currently no additional member alongside this base method; a minimum of two alternatives and two separate ranking sources is required.

Schulze is suitable if you have several rankings and want, when combining them, a ranking that is consistent in pairwise comparisons and resistant to cycles, and if you also care that adding or removing an alternative afterwards (adding a clone) does not disturb the result. If the number of sources is very small, such as only two, the strength of the pairwise comparisons is reduced and the result readily falls into ties. If there is no pairwise disagreement at all among the sources, that is, everyone gives the same ranking, Schulze's elaborate path-finding logic becomes an unnecessary burden and a simple aggregation suffices.

Combine several rankings, prioritise pairwise consistency and resistance to cycles → Schulze

A simple average score is sufficient → Borda / average ranking

Find the middling position most resistant to outlier views → median ranking

Keep Borda-style scoring but search for a Condorcet winner through elimination rounds → Nanson

Schulze's core advantage is that, if a Condorcet winner exists, it will always find one, a property many simple aggregation methods cannot guarantee. It also carries clone independence, meaning that adding or removing two very similar alternatives at the same time does not disturb the ranking between the others. Its computation, finding the strongest indirect path, can be carried out in polynomial time with a classical graph algorithm (the widest-path problem, a variant of Floyd-Warshall); this is an advantage compared with the computational difficulty of some other pairwise comparison methods (Bartholdi, Tovey and Trick, 1989).

Its limitations stem mainly from intelligibility. First, the concept of the strongest indirect path is not intuitive; the answer to a decision-maker's question of "why did this alternative win" cannot be conveyed as easily as a simple total such as a Borda score. Second, where a genuine cycle exists, the strongest path between two alternatives can come out equal, and in that case the method cannot make the distinction on its own. Third, different Condorcet methods (Schulze, ranked pairs, Kemeny-Young) can give different results in cyclic cases, and which is "most correct" remains debated, a debate that Schulze himself has continued (Schulze, 2024).

The most common mistake is confusing the pairwise superiority count with a shortcut such as "is A better on average," rather than "in how many sources was A preferred to B"; Schulze's strength comes precisely from this pairwise count, and this step cannot be skipped. A second mistake is limiting the strongest-path calculation to direct comparisons only, leaving indirect chains out of account; this reduces Schulze to a simple majority method. A third mistake is confusing a genuine tie, where a cycle leaves both directions equally strong, with a computational error. A fourth mistake is presenting Schulze's result as the single "correct" answer without stating in the report that another Condorcet method, such as ranked pairs, could give a different result.

The governing principle is this:

Schulze looks for the strongest indirect superiority in pairwise comparisons; if a Condorcet winner exists, it finds it, and where a cycle exists the result depends on path strength, a fact the report must show.