XIV

Strengths of the strongest paths

	${\displaystyle p※}$	${\displaystyle p※}$	${\displaystyle p※}$	${\displaystyle p※}$	${\displaystyle p※}$
${\displaystyle p※}$		28	28	30	24
${\displaystyle p※}$	25		28	33	24
${\displaystyle p※}$	25	29		29	24
${\displaystyle p※}$	25	28	28		24
${\displaystyle p※}$	25	28	28	31

Now the output of the Schulze method can be determined. For example, when comparing A and B, since ${\displaystyle (28=)p※>p※(=25)}$, for the Schulze method candidate A is better than candidate B. Another example is that ${\displaystyle (31=)p※>p※(=24)}$, so candidate E is better than candidate D. Continuing in this way, the result is that the Schulze ranking is ${\displaystyle E>A>C>B>D}$, and E wins. In other words, E wins since ${\displaystyle p※\geq p※}$ 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.

# Input: d※, the number of voters who prefer candidate i to candidate j.
# Output: p※, 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※ > d※ then
                p※ := d※
            else
                p※ := 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※ := max (p※, min (p※, p※))

This algorithm is efficient and has running time O(C) 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※ represents either the number of voters who strictly prefer A to B (A>B),/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 recommended breaking ties by random ballot.

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.

1. Make the results table, called the "matrix of pairwise preferences", such as used above in the example. Then, every positive number is a pairwise win for the candidate on that row (and marked green), ties are zeroes, and losses are negative (marked red). Order the candidates by how long they last in elimination.
2. If there is a candidate with no red on their line, they win.
3. Otherwise, draw a square box around the Schwartz set in the upper left corner. It can be described as the minimal "winner's circle" of candidates who do not lose to anyone outside the circle. Note that to the right of the box there is no red, which means it is a winner's circle, and note that within the box there is no reordering possible that would produce a smaller winner's circle.
4. Cut away every part of the table outside the box.
5. If there is still no candidate with no red on their line, something needs to be compromised on; every candidate lost some race, and the loss we tolerate the best is the one where the loser obtained the most votes. So, take the red cell with the highest number (if going by margins, the least negative), make it green—or any color other than red—and go back step 2.

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.

Satisfied and failed criteria※

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:

• Participation
• Consistency
• Invulnerability to burying
• Later-no-harm

Likewise, since the Schulze method is not a dictatorship and is a ranked voting system (not rated), Arrow's Theorem implies it fails:

• Independence of irrelevant alternatives

The Schulze method also fails

• Peyton Young's criterion of Local Independence of Irrelevant Alternatives.

Comparison table※

The following table compares the Schulze method with other 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.

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

Usage※

Government※

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.

Political parties※

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.

• Five Star Movement of Campobasso, Fondi, Monte Compatri, Montemurlo, Pescara, and San Cesareo
• Pirate Parties of Australia, Austria, Belgium, Brazil, Germany, Iceland, Italy, the Netherlands, Sweden, Switzerland, and the United States
• SustainableUnion
• Volt Europe

Student government and associations※

• AEGEE – European Students' Forum
• Club der Ehemaligen der Deutschen SchülerAkademien e. V.
• Associated Student Government at École normale supérieure de Paris
• Flemish Society of Engineering Students Leuven
• Graduate Student Organization at the State University of New York: Computer Science (GSOCS)
• Hillegass Parker House
• Kingman Hall
• Associated Students of Minerva Schools at KGI
• Associated Student Government at Northwestern University
• Associated Student Government at University of Freiburg
• Associated Student Government at the Computer Sciences Department of the University of Kaiserslautern

Organizations※

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.

Organizations which currently use the Schulze method include:

Notes※

External links※

• The Schulze Method of Voting by Markus Schulze
• The Schulze Method by Hubert Bray
• Spieltheorie (in German) by Bernhard Nebel
• Accurate Democracy by Rob Loring
• Christoph Börgers (2009), Mathematics of Social Choice: Voting, Compensation, and Division, SIAM, ISBN 0-89871-695-0
• Nicolaus Tideman (2006), Collective Decisions and Voting: The Potential for Public Choice, Burlington: Ashgate, ISBN 0-7546-4717-X
• preftools by the Public Software Group
• Arizonans for Condorcet Ranked Voting
• Condorcet PHP Command line application and PHP library, supporting multiple Condorcet methods, including Schulze.
• Implementation in Java
• Implementation in Ruby
• Implementation in Python 2
• Implementation in Python 3