|Part of the Politics series|
The Schulze method (//) is an electoral system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners. The Schulze method is also known as Schwartz Sequential dropping (SSD), cloneproof Schwartz sequential dropping (CSSD), the beatpath method, beatpath winner, path voting, and path winner. The Schulze method is a Condorcet method, which means that if there is a candidate who is preferred by a majority over every other candidate in pairwise comparisons, then this candidate will be the winner when the Schulze method is applied.
The output of the Schulze method gives an ordering of candidates. Therefore, if several positions are available, the method can be used for this purpose without modification, by letting the k top-ranked candidates win the k available seats. Furthermore, for proportional representation elections, a single transferable vote (STV) variant known as Schulze STV has been proposed. The Schulze method is used by several organizations including Wikimedia, Debian, Ubuntu, Gentoo, Pirate Party political parties and many others.
The input for the Schulze method is the same as for other ranked single-winner electoral systems: each voter must furnish an ordered preference list on candidates where ties are allowed (a strict weak order).
One typical way for voters to specify their preferences on a ballot is as follows. Each ballot lists all the candidates, and each voter ranks this list in order of preference using numbers: the voter places a '1' beside the most preferred candidate, a '2' beside the second-most preferred, and so forth. Each voter may optionally:
Let be the number of voters who prefer candidate to candidate .
A path from candidate to candidate is a sequence of candidates with the following properties:
In other words, in a pairwise comparison, each candidate in the path will beat the following candidate.
The strength of a path from candidate to candidate is the smallest number of voters in the sequence of comparisons:
For a pair of candidates and that are connected by at least one path, the strength of the strongest path is the maximum strength of the paths connecting them. If there is no path from candidate to candidate at all, then .
Candidate is better than candidate if and only if .
Candidate is a potential winner if and only if for every other candidate .
It can be proven that and together imply .: §4.1 Therefore, it is guaranteed (1) that the above definition of "better" really defines a transitive relation and (2) that there is always at least one candidate with for every other candidate .
In the following example 45 voters rank 5 candidates.
|Number of voters||Order of preference|
The pairwise preferences have to be computed first. For example, when comparing A and B pairwise, there are 5+5+3+7=20 voters who prefer A to B, and 8+2+7+8=25 voters who prefer B to A. So and . The full set of pairwise preferences is:
The cells for d[X, Y] have a light green background if d[X, Y] > d[Y, X], otherwise the background is light red. There is no undisputed winner by only looking at the pairwise differences here.
Now the strongest paths have to be identified. To help visualize the strongest paths, the set of pairwise preferences is depicted in the diagram on the right in the form of a directed graph. An arrow from the node representing a candidate X to the one representing a candidate Y is labelled with d[X, Y]. To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d[X, Y] > d[Y, X] (i.e. the table cells with light green background), omitting the one in the opposite direction (the table cells with light red background).
One example of computing the strongest path strength is p[B, D] = 33: the strongest path from B to D is the direct path (B, D) which has strength 33. But when computing p[A, C], the strongest path from A to C is not the direct path (A, C) of strength 26, rather the strongest path is the indirect path (A, D, C) which has strength min(30, 28) = 28. The strength of a path is the strength of its weakest link.
For each pair of candidates X and Y, the following table shows the strongest path from candidate X to candidate Y in red, with the weakest link underlined.
Now the output of the Schulze method can be determined. For example, when comparing A and B, since , for the Schulze method candidate A is better than candidate B. Another example is that , so candidate E is better than candidate D. Continuing in this way, the result is that the Schulze ranking is , and E wins. In other words, E wins since for every other candidate X.
The only difficult step in implementing the Schulze method is computing the strongest path strengths. However, this is a well-known problem in graph theory sometimes called the widest path problem. One simple way to compute the strengths, therefore, is a variant of the Floyd–Warshall algorithm. The following pseudocode illustrates the algorithm.
# Input: d[i,j], the number of voters who prefer candidate i to candidate j. # Output: p[i,j], the strength of the strongest path from candidate i to candidate j. for i from 1 to C for j from 1 to C if i ≠ j then if d[i,j] > d[j,i] then p[i,j] := d[i,j] else p[i,j] := 0 for i from 1 to C for j from 1 to C if i ≠ j then for k from 1 to C if i ≠ k and j ≠ k then p[j,k] := max (p[j,k], min (p[j,i], p[i,k]))
This algorithm is efficient and has running time O(C3) where C is the number of candidates.
When allowing users to have ties in their preferences, the outcome of the Schulze method naturally depends on how these ties are interpreted in defining d[*,*]. Two natural choices are that d[A, B] represents either the number of voters who strictly prefer A to B (A>B), or the margin of (voters with A>B) minus (voters with B>A). But no matter how the ds are defined, the Schulze ranking has no cycles, and assuming the ds are unique it has no ties.
Although ties in the Schulze ranking are unlikely, they are possible. Schulze's original paper proposed breaking ties in accordance with a voter selected at random, and iterating as needed.
An alternative way to describe the winner of the Schulze method is the following procedure:
There is another alternative way to demonstrate the winner of the Schulze method. This method is equivalent to the others described here, but the presentation is optimized for the significance of steps being visually apparent as a human goes through it, not for computation.
Here is a margins table made from the above example. Note the change of order used for demonstration purposes.
The first drop (A's loss to E by 1 vote) does not help shrink the Schwartz set.
So we get straight to the second drop (E's loss to C by 3 votes), and that shows us the winner, E, with its clear row.
This method can also be used to calculate a result, if the table is remade in such a way that one can conveniently and reliably rearrange the order of the candidates on both the row and the column, with the same order used on both at all times.
The Schulze method satisfies the following criteria:
Since the Schulze method satisfies the Condorcet criterion, it automatically fails the following criteria:
Likewise, since the Schulze method is not a dictatorship and agrees with unanimous votes, Arrow's Theorem implies it fails the criterion
The Schulze method also fails
The following table compares the Schulze method with other preferential single-winner election methods:
|System||Monotonic||Condorcet winner||Majority||Condorcet loser||Majority loser||Mutual majority||Smith||ISDA||LIIA||Independence of clones||Reversal symmetry||Participation, consistency||Laternoharm||Laternohelp||Polynomial time||Resolvability|
|Sri Lankan contingent voting||No||No||Yes||No||No||No||No||No||No||No||No||No||Yes||Yes||Yes||Yes|
The main difference between the Schulze method and the ranked pairs method can be seen in this example:
Suppose the MinMax score of a set X of candidates is the strength of the strongest pairwise win of a candidate A ∉ X against a candidate B ∈ X. Then the Schulze method, but not Ranked Pairs, guarantees that the winner is always a candidate of the set with minimum MinMax score.: §4.8 So, in some sense, the Schulze method minimizes the largest majority that has to be reversed when determining the winner.
On the other hand, Ranked Pairs minimizes the largest majority that has to be reversed to determine the order of finish, in the MinLexMax sense. In other words, when Ranked Pairs and the Schulze method produce different orders of finish, for the majorities on which the two orders of finish disagree, the Schulze order reverses a larger majority than the Ranked Pairs order.
The Schulze method was developed by Markus Schulze in 1997. It was first discussed in public mailing lists in 1997–1998 and in 2000.
In 2011, Schulze published the method in the academic journal Social Choice and Welfare.
The Schulze method is used by the city of Silla for all referendums. It is also used by the cities of Turin and San Donà di Piave and by the London Borough of Southwark through their use of the WeGovNow platform, which in turn uses the LiquidFeedback decision tool.
Schulze was adopted by the Pirate Party of Sweden (2009), and the Pirate Party of Germany (2010). The newly formed Boise, Idaho, chapter of the Democratic Socialists of America in February chose this method for their first special election held in March 2018.
It is used by the Institute of Electrical and Electronics Engineers, by the Association for Computing Machinery, and by USENIX through their use of the HotCRP decision tool.
In the French Wikipedia, the Schulze method was one of two multi-candidate methods approved by a majority in 2005, and it has been used several times.[circular reference]
Organizations which currently use the Schulze method include: