Schulze method

From Wikipedia, the free encyclopedia

The Schulze method (/ˈʃʊltsə/) 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.

Description of the method[edit]


A sample ballot asking voters to order candidates by preference

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).[1]

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:

  • give the same preference to more than one candidate. This indicates that this voter is indifferent between these candidates.
  • use non-consecutive numbers to express preferences. This has no impact on the result of the elections, since only the order in which the candidates are ranked by the voter matters, and not the absolute numbers of the preferences.
  • keep candidates unranked. When a voter does not rank all candidates, then this is interpreted as if this voter (i) strictly prefers all ranked to all unranked candidates, and (ii) is indifferent among all unranked candidates.


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:

  1. and .
  2. For all .

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 all .

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 .[1]: §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:

Directed graph labeled with pairwise preferences d[*, *]
Matrix of pairwise preferences
20 26 30 22
25 16 33 18
19 29 17 24
15 12 28 14
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
Strengths of the strongest paths
28 28 30 24
25 28 33 24
25 29 29 24
25 28 28 24
25 28 28 31

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]
                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.

Ties and alternative implementations[edit]

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][citation needed] 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:[citation needed]

  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 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. 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 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.

Initial results table
E 1 −3 9 17
A −1 7 −5 15
C 3 −7 13 −11
B −9 5 −13 21
D −17 −15 11 −21

The first drop (A's loss to E by 1 vote) does not help shrink the Schwartz set.

First drop
E 1 −3 9 17
A −1 7 −5 15
C 3 −7 13 −11
B −9 5 −13 21
D −17 −15 11 −21

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.

Second drop, final
E 1 −3 9 17
A −1 7 −5 15
C 3 −7 13 −11
B −9 5 −13 21
D −17 −15 11 −21

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[edit]

Satisfied criteria[edit]

The Schulze method satisfies the following criteria:

Failed criteria[edit]

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[edit]

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

Comparison of preferential electoral systems
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
Schulze Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes No No No Yes Yes
Ranked pairs Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No Yes Yes
Tideman's Alternative No Yes Yes Yes Yes Yes Yes Yes No Yes No No No No Yes Yes
Kemeny–Young Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes No No No No Yes
Copeland Yes Yes Yes Yes Yes Yes Yes Yes No No Yes No No No Yes No
Nanson No Yes Yes Yes Yes Yes Yes No No No Yes No No No Yes Yes
Black Yes Yes Yes Yes Yes No No No No No Yes No No No Yes Yes
Instant-runoff voting No No Yes Yes Yes Yes No No No Yes No No Yes Yes Yes Yes
Smith/IRV No Yes Yes Yes Yes Yes Yes Yes No Yes No No No No Yes Yes
Borda Yes No No Yes Yes No No No No No Yes Yes No Yes Yes Yes
Baldwin No Yes Yes Yes Yes Yes Yes No No No No No No No Yes Yes
Bucklin Yes No Yes No Yes Yes No No No No No No No Yes Yes Yes
Plurality Yes No Yes No No No No No No No No Yes Yes Yes Yes Yes
Contingent voting No No Yes Yes Yes No No No No No No No Yes Yes Yes Yes
Coombs[4] No No Yes Yes Yes Yes No No No No No No No No Yes Yes
Mini­Max[specify] Yes Yes Yes No No No No No No No No No No No Yes Yes
Anti-plurality[4] Yes No No No Yes No No No No No No Yes No No Yes Yes
Sri Lankan contingent voting No No Yes No No No No No No No No No Yes Yes Yes Yes
Supplementary voting No No Yes No No No No No No No No No Yes Yes Yes Yes
Dodgson[4] No Yes Yes No No No No No No No No No No No No 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.[1]: §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.[citation needed][5] 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[6] and in 2000.[7]

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.[8][9] 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.[citation needed]

Political parties[edit]

Schulze was adopted by the Pirate Party of Sweden (2009),[10] and the Pirate Party of Germany (2010).[11] 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.[12]

Student government and associations[edit]


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,[40] and it has been used several times.[41][circular reference]

Organizations which currently use the Schulze method include:


  1. ^ a b c d e f g h i j k l m n o p q Markus Schulze, "A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method", Social Choice and Welfare, volume 36, number 2, page 267–303, 2011. Preliminary version in Voting Matters, 17:9-19, 2003.
  2. ^ Under reasonable probabilistic assumptions when the number of voters is much larger than the number of candidates
  3. ^ a b c Douglas R. Woodall, Properties of Preferential Election Rules, Voting Matters, issue 3, pages 8–15, December 1994
  4. ^ a b c Anti-plurality, Coombs and Dodgson are assumed to receive truncated preferences by apportioning possible rankings of unlisted alternatives equally; for example, ballot A > B = C is counted as 1/2 A > B > C and 1/2 A > C > B. If these methods are assumed not to receive truncated preferences, then later-no-harm and later-no-help are not applicable.
  5. ^ Tideman, T. Nicolaus, "Independence of clones as a criterion for voting rules", Social Choice and Welfare vol 4 #3 (1987), pp. 185–206.
  6. ^ See:
  7. ^ See:
  8. ^ Hortanoticias, Redacción (2016-02-23). "Al voltant de 2.000 participants en dos dies en la primera enquesta popular de Silla que decidirà sobre espectacles taurins". (in Spanish). Retrieved 2022-09-24.
  9. ^ Silla, ~ El Cresol de (2016-05-26). "Un any d'aprofundiment democràtic a Silla". El Cresol de Silla (in Catalan). Retrieved 2022-09-24.
  10. ^ a b See:
  11. ^ a b 11 of the 16 regional sections and the federal section of the Pirate Party of Germany are using LiquidFeedback for unbinding internal opinion polls. In 2010/2011, the Pirate Parties of Neukölln (link), Mitte (link), Steglitz-Zehlendorf (link), Lichtenberg (link), and Tempelhof-Schöneberg (link) adopted the Schulze method for its primaries. Furthermore, the Pirate Party of Berlin (in 2011) (link) and the Pirate Party of Regensburg (in 2012) (link) adopted this method for their primaries.
  12. ^ Chumich, Andrew. "DSA Special Election". Retrieved 2018-02-25.
  13. ^ Campobasso. Comunali, scattano le primarie a 5 Stelle, February 2014
  14. ^ Macaro, Mirko (2015-03-03). "Fondi, il punto sui candidati a sindaco. Certezze, novità e colpi di scena". h24 notizie - portale indipendente di news dalla provincia (in Italian). Retrieved 2022-09-24.
  15. ^ article 25(5) of the bylaws, October 2013
  16. ^ "MoVimento 5 Stelle - Montemurlo: 2° Step Comunarie di Montemurlo". November 2013. Archived from the original on 2015-04-02. Retrieved 2022-09-24.
  17. ^ article 12 of the bylaws, January 2015
  18. ^ Ridefinizione della lista di San Cesareo con Metodo Schulze, February 2014
  19. ^ "National Congress 2011 Results – Pirate Party Australia". Retrieved 2022-09-24.
  20. ^ §6(10) of the bylaws
  21. ^ Article III.3.4 of the Statutory Rules (french, dutch)
  22. ^ Píratar (2013-10-23). "Schulze aðferðin". Píratar (in Icelandic). Retrieved 2022-09-24.
  23. ^ Rules adopted on 18 December 2011
  24. ^ Pontier, Matthijs (2015-01-11). "Verslag ledenraadpleging 4 januari". Piratenpartij Noord Holland (in Dutch). Retrieved 2022-09-24.
  25. ^ Pankerl, Florian (2010-09-18). "Piratenversammlung der Piratenpartei Schweiz 2010 – Samstag" (in German). Retrieved 2022-09-24.
  26. ^ article IV section 3 of the bylaws, July 2012
  27. ^ §10 III of its bylaws, June 2013
  28. ^ "Some considerations on which group Volt Europe will join in the European Parliament".
  29. ^ Hajdu, Tekla (2017-09-24). "The Schulze Method – Agora 101". The AEGEEan - AEGEE's online magazine - AEGEE-Europe. Retrieved 2022-09-24.
  30. ^ Voting Details, January 2021
  31. ^ Référendum sur la réforme du thurnage, June 2021
  32. ^ article 57 of the statutory rules
  33. ^ "User Voting Instructions". Archived from the original on 2013-09-09. Retrieved 2010-05-08.
  34. ^ "Hillegass-Parker House Bylaws § 5. Elections". Hillegass-Parker House website. Retrieved 4 October 2015.
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  36. ^ article 9.4.5.h of the charter, November 2017
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  38. ^ §6 and §7 of its bylaws, May 2014
  39. ^ §6(6) of the bylaws
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  41. ^ "Pages liées à Méthode Schulze". (in French). Retrieved 2022-09-24.
  42. ^ Election of the Annodex Association committee for 2007, February 2007
  43. ^ §9a of the bylaws, October 2013
  44. ^ See:
    • 2013 Golden Geek Awards - Nominations Open, January 2014
    • 2014 Golden Geek Awards - Nominations Open, January 2015
    • 2015 Golden Geek Awards - Nominations Open, March 2016
    • 2016 Golden Geek Awards - Nominations Open, January 2017
    • 2017 Golden Geek Awards - Nominations Open, February 2018
    • 2018 Golden Geek Awards - Nominations Open, March 2019
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  46. ^ Adam Helman, Family Affair Voting Scheme - Schulze Method
  47. ^ Steering and Technical committee, November 2021
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  50. ^ Democratic election of the server admins Archived 2015-10-02 at the Wayback Machine, July 2010
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  52. ^ Project:Elections
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  54. ^ Haskell Logo Competition, March 2009
  55. ^ Article 6 Section 2 of the Constitution, February 2021
  56. ^ section of the Operating Procedures of the Address Council of the Address Supporting Organization
  57. ^ "A club by any other name..." Kanawha Valley Scrabble Club. 2009-04-02. Retrieved 2022-09-24.
  58. ^ section 3.4.1 of the Rules of Procedures for Online Voting
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  62. ^ article 8.3 of the bylaws
  63. ^ The Principles of LiquidFeedback. Berlin: Interaktive Demokratie e. V. 2014. ISBN 978-3-00-044795-2.
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  68. ^ "2009 Director Elections".
  69. ^ "Online Voting Policy".
  70. ^ ONNX Steering Committee election guideline
  71. ^ "OpenStack Election — OpenStack Governance". Retrieved 2022-09-24.
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  73. ^ "Committee Elections 2012". rllmuk. Retrieved 2022-09-24.
  74. ^ Squeak Oversight Board Election 2010, March 2010
  75. ^ See:
  76. ^ "[IAEP] Election status update". Retrieved 2022-09-24.
  77. ^ Minutes of the 2018 Annual Sverok Meeting, November 2018
  78. ^ "2007 TopCoder Collegiate Challenge". Retrieved 2022-09-24.
  79. ^ Bell, Alan (May 17, 2012). "Ubuntu IRC Council Position". Retrieved 2022-09-24.
  80. ^ "/v/GAs - Pairwise voting results".
  81. ^ See:
  82. ^ See e.g. here [1] (May 2009), here [2] (August 2009), and here [3] (December 2009).
  83. ^ See here and here.
  84. ^ Девятнадцатые выборы арбитров, второй тур [Result of Arbitration Committee Elections]. (in Russian). Archived from the original on 2015-02-22.
  85. ^ See here

External links[edit]