Schulze method

In the following example 45 voters rank 5 candidates.

${\displaystyle {\begin{array}{|c|c|}{\text{number of voters}}&{\text{order of preference}}\\\hline 5&ACBED\\5&ADECB\\8&BEDAC\\3&CABED\\7&CAEBD\\2&CBADE\\7&DCEBA\\8&EBADC\end{array}}}$

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 ${\displaystyle d[A,B]=20}$ and ${\displaystyle d[B,A]=25}$. The full set of pairwise preferences is:

Directed graph labeled with pairwise preferences d[*, *]

Matrix of pairwise preferences

	${\displaystyle d[*,A]}$	${\displaystyle d[*,B]}$	${\displaystyle d[*,C]}$	${\displaystyle d[*,D]}$	${\displaystyle d[*,E]}$
${\displaystyle d[A,*]}$		20	26	30	22
${\displaystyle d[B,*]}$	25		16	33	18
${\displaystyle d[C,*]}$	19	29		17	24
${\displaystyle d[D,*]}$	15	12	28		14
${\displaystyle d[E,*]}$	23	27	21	31

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.

Strongest paths

To From	A	B	C	D	E
A	N/A	A-(30)-D-(28)-C-(29)-B	A-(30)-D-(28)-C	A-(30)-D	A-(30)-D-(28)-C-(24)-E	A
B	B-(25)-A	N/A	B-(33)-D-(28)-C	B-(33)-D	B-(33)-D-(28)-C-(24)-E	B
C	C-(29)-B-(25)-A	C-(29)-B	N/A	C-(29)-B-(33)-D	C-(24)-E	C
D	D-(28)-C-(29)-B-(25)-A	D-(28)-C-(29)-B	D-(28)-C	N/A	D-(28)-C-(24)-E	D
E	E-(31)-D-(28)-C-(29)-B-(25)-A	E-(31)-D-(28)-C-(29)-B	E-(31)-D-(28)-C	E-(31)-D	N/A	E
	A	B	C	D	E	From To

Strengths of the strongest paths

	${\displaystyle p[*,A]}$	${\displaystyle p[*,B]}$	${\displaystyle p[*,C]}$	${\displaystyle p[*,D]}$	${\displaystyle p[*,E]}$
${\displaystyle p[A,*]}$		28	28	30	24
${\displaystyle p[B,*]}$	25		28	33	24
${\displaystyle p[C,*]}$	25	29		29	24
${\displaystyle p[D,*]}$	25	28	28		24
${\displaystyle p[E,*]}$	25	28	28	31

Now the output of the Schulze method can be determined. For example, when comparing A and B, since ${\displaystyle (28=)p[A,B]>p[B,A](=25)}$, for the Schulze method candidate A is better than candidate B. Another example is that ${\displaystyle (31=)p[E,D]>p[D,E](=24)}$, so candidate E is better than candidate D. Continuing in this way, the result is that the Schulze ranking is ${\displaystyle E>A>C>B>D}$, and E wins. In other words, E wins since ${\displaystyle p[E,X]\geq p[X,E]}$ 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.[1]

Although ties in the Schulze ranking are unlikely,[2] they are possible. Schulze's original paper[1] 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:

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 x which cannot reach all others who reach x) and [b] delete the graph edge with the smallest value (if by margins, smallest margin; if by votes, fewest votes).
3. the winner is the last non-deleted candidate.

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 you go through it, not for computation.

1. Make the results table, called the "matrix of pairwise preferences," such as used above in the example. If using margins rather than raw vote totals, subtract it from its transpose. 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's 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. You can describe it 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's 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 that isn't in 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) doesn't 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 you make the table in such a way that you can conveniently and reliably rearrange the order of the candidates on both row and column (use the same order on both at all times).

The Schulze method was developed by Markus Schulze in 1997. It was first discussed in public mailing lists in 1997–1998[6] and in 2000.[7] Subsequently, Schulze method users included Debian (2003),[8] Gentoo (2005),[9] Topcoder (2005),[10] Wikimedia (2008),[11] KDE (2008),[12] the Pirate Party of Sweden (2009),[13] and the Pirate Party of Germany (2010).[14] In the French Wikipedia, the Schulze method was one of two multi-candidate methods approved by a majority in 2005,[15] and it has been used several times.[16] 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.[17]

In 2011, Schulze published the method in the academic journal Social Choice and Welfare.[1]

sample ballot for Wikimedia's Board of Trustees elections

The Schulze method is used by the city of Silla for all referendums. 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. The Schulze method is 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. Organizations which currently use the Schulze method include:

Wikimedia Commons has media related to Schulze method.

• The Schulze Method of Voting by Markus Schulze
• Condorcet Computations by Johannes Grabmeier
• 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