Schulze method

Ballot

The input for the Schulze method is the same as for other ranked single-winner election methods: 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 (see right) 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(s), a '2' beside the second-most preferred, and so forth. Each voter may optionally:

Computation

Let d [ V , W ] be the number of voters who prefer candidate V to candidate W .

A path from candidate X to candidate Y of strength p is a sequence of candidates C ( 1 ) , ⋯ , C ( n ) with the following properties:

1. C ( 1 ) = X and C ( n ) = Y .
2. For all i = 1 , ⋯ , ( n − 1 ) : d [ C ( i ) , C ( i + 1 ) ] > d [ C ( i + 1 ) , C ( i ) ] .
3. For all i = 1 , ⋯ , ( n − 1 ) : d [ C ( i ) , C ( i + 1 ) ] ≥ p .

Let p [ A , B ] , the strength of the strongest path from candidate A to candidate B , be the maximum value such that there is a path from candidate A to candidate B of that strength (the strength of a path is the strength of its weakest link). If there is no path from candidate A to candidate B at all, then p [ A , B ] = 0 .

Candidate D is better than candidate E if and only if p [ D , E ] > p [ E , D ] .

Candidate D is a potential winner if and only if p [ D , E ] ≥ p [ E , D ] for every other candidate E .

It can be proven that p [ X , Y ] > p [ Y , X ] and p [ Y , Z ] > p [ Z , Y ] together imply p [ X , Z ] > p [ Z , X ] . 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 D with p [ D , E ] ≥ p [ E , D ] for every other candidate E .

Example

In the following example 45 voters rank 5 candidates.

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 d [ A , B ] = 20 and d [ B , A ] = 25 . 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 ( 28 = ) p [ A , B ] > p [ B , A ] ( = 25 ) , for the Schulze method candidate A is better than candidate B. Another example is that ( 31 = ) p [ E , D ] > p [ D , E ] ( = 24 ) , so candidate E is better than candidate D. Continuing in this way, the result is that the Schulze ranking is E > A > C > B > D , and E wins. In other words, E wins since p [ E , X ] ≥ p [ X , E ] for every other candidate X.

Implementation

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.

This algorithm is efficient, and has running time O(C³) where C is the number of candidates.

Ties and alternative implementations

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, slower, way to describe the winner of the Schulze method is the following procedure:

1. draw a complete directed graph with all candidates, and all possible edges between candidates
2. iteratively [a] delete all candidates not in the Schwartz set (i.e. any candidate which cannot reach all others) and [b] delete the weakest link
3. the winner is the last non-deleted candidate.

Satisfied criteria

The Schulze method satisfies the following criteria:

Failed 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

Comparison table

The following table compares the Schulze method with other preferential single-winner election methods:

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. 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.

History

The Schulze method was developed by Markus Schulze in 1997. It was first discussed in public mailing lists in 1997–1998 and in 2000. Subsequently, Schulze method users included Software in the Public Interest (2003), Debian (2003), Gentoo (2005), TopCoder (2005), Wikimedia (2008), KDE (2008), the Free Software Foundation Europe (2008), the Pirate Party of Sweden (2009), and the Pirate Party of Germany (2010). 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.

In 2011, Schulze published the method in the academic journal Social Choice and Welfare.

Users

The Schulze method is not currently used in parliamentary elections. However, it has been used for parliamentary primaries in the Swedish Pirate Party. It is also starting to receive support in other public organizations. Organizations which currently use the Schulze method are: