:Schulze method

File:Preferential_ballot.svg

Schulze's method uses ranked ballots with equal ratings allowed. There are two common (equivalent) descriptions of Schulze's method.

The idea behind Schulze's method is that if Alice defeats Bob, and Bob beats Charlie, then Alice "indirectly" defeats Charlie. These chained sequences of "beats" are called 'beatpaths'.

Every beatpath is assigned a particular strength. The strength of a single-step beatpath from Alice to Bob is just the number of voters who rank Alice over Bob. For a longer beatpath, consisting of multiple beats, a beatpath is as strong as its weakest link (i.e. the beat with the smallest number of winning votes).

We say Alice has a "beatpath-win" over Bob if her strongest beatpath to Bob is stronger than all of Bob's strongest beatpaths to Alice. The winner is the candidate who has a beatpath-win over every other candidate.

Markus Schulze proved that this definition of a beatpath-win is transitive: in other words, if Alice has a beatpath-win over Bob, and Bob has a beatpath-win over Charlie, Alice has a beatpath-win over Charlie.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.{{rp|§4.1}} As a result, the Schulze method is a Condorcet method, providing a full extension of the majority rule to any set of ballots.

The Schulze winner can also be constructed iteratively, using a defeat-dropping method:

1. Draw a directed graph with all the candidates as nodes; label the edges with the number of votes supporting the winner.
2. If there is more than one candidate left:
3. * Check if any candidates are tied (and if so, break the ties by random ballot).
4. * Eliminate all candidates outside the majority-preferred set.
5. * Delete the edge closest to being tied.

The winner is the only candidate left at the end of the procedure.

In the following example 45 voters rank 5 candidates.

The pairwise preferences have to be computed first. For example, when comparing {{mvar|A}} and {{mvar|B}} pairwise, there are {{math|1=5+5+3+7=20}} voters who prefer {{mvar|A}} to {{mvar|B}}, and {{math|1=8+2+7+8=25}} voters who prefer {{mvar|B}} to {{mvar|A}}. So $d[A,\; B]\; =\; 20$ and $d[B,\; A]\; =\; 25$. The full set of pairwise preferences is:

File:Schulze_method_example1.svg labeled with pairwise preferences d[*, *] ]]

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.{{clear}}

Now the output of the Schulze method can be determined. For example, when comparing {{mvar|A}} and {{mvar|B}}, since $(28\; =)\; p[A,B]\; >\; p[B,A]\; (=\; 25)$, for the Schulze method candidate {{mvar|A}} is better than candidate {{mvar|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 {{mvar|E}} wins. In other words, {{mvar|E}} wins since $p[E,X]\; \backslash ge\; p[X,E]$ 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.

1. Input: d[i,j], the number of voters who prefer candidate i to candidate j.
2. 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.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.

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.

The Schulze method satisfies the following criteria:

{{div col|colwidth=30em}}{{rp}}

• Monotonicity criterionMarkus Schulze, "[https://doi.org/10.1007/s00355-010-0475-4 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.{{rp|§4.5}}
• Majority criterion
• Majority loser criterion
• Condorcet criterion
• Condorcet loser criterion
• Smith criterion{{rp|§4.7}}
• Independence of Smith-dominated alternatives{{rp|§4.7}}
• Mutual majority criterion
• Independence of clones{{rp|§4.6}}
• Reversal symmetry{{rp|§4.4}}
• Mono-appendDouglas R. Woodall, [http://www.votingmatters.org.uk/ISSUE3/P5.HTM Properties of Preferential Election Rules], Voting Matters, issue 3, pages 8–15, December 1994
• Mono-add-plump
• Resolvability criterion{{rp|§4.2}}
• Polynomial runtime{{rp|§2.3"}}
• prudence{{rp|§4.9"}}
• MinMax sets{{rp|§4.8"}}
• Woodall's plurality criterion if winning votes are used for d[X,Y]
• Symmetric-completion if margins are used for d[X,Y]

{{div col end}}

Since the Schulze method satisfies the Condorcet criterion, it automatically fails the following criteria:

• Participation{{rp|§3.4}}
• 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, meaning it can be vulnerable to the spoiler effect in some rare circumstances. The Schulze method also fails Peyton Young's criterion of Local Independence of Irrelevant Alternatives.

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

{{Comparison of Schulze to preferential voting systems}}

Ranked pairs is another Condorcet method which is very similar to Schulze's rule, and typically produces the same outcome. There are slight differences, however. The main difference between the beatpath method and ranked pairs is that Schulze retains behavior closer to minimax. Say that the minimax 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 the winner is always a candidate of the set with minimum minimax score.{{rp|§4.8}} This is the sense in which 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.Tideman, T. Nicolaus, "Independence of clones as a criterion for voting rules", Social Choice and Welfare vol 4 #3 (1987), pp. 185–206. 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 1997–1998See:

• Markus Schulze, [http://lists.electorama.com/pipermail/election-methods-electorama.com/1997-October/001570.html Condorect sub-cycle rule], October 1997
• Mike Ossipoff, [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-July/099943.html Party List P.S.], July 1998
• Markus Schulze, [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-August/100045.html Tiebreakers, Subcycle Rules], August 1998
• Markus Schulze, [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-August/100131.html Maybe Schulze is decisive], August 1998
• Norman Petry, [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-September/067383.html Schulze Method - Simpler Definition], September 1998
• Markus Schulze, [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-November/068099.html Schulze Method], November 1998 and in 2000.See:

• Anthony Towns, [https://lists.debian.org/debian-vote/2000/11/msg00121.html Disambiguation of 4.1.5], November 2000
• Norman Petry, [https://lists.debian.org/debian-vote/2000/12/msg00045.html Constitutional voting, definition of cumulative preference], December 2000 In 2011, Schulze published the method in the academic journal Social Choice and Welfare.

File:Voting2.png elections]]

The Schulze method is used by the city of Silla, Spain for all referendums.{{Cite web |last=Hortanoticias |first=Redacción |date=2016-02-23 |title=Al voltant de 2.000 participants en dos dies en la primera enquesta popular de Silla que decidirà sobre espectacles taurins |url=https://www.hortanoticias.com/al-voltant-de-2-000-participants-en-dos-dies-en-la-primera-enquesta-popular-de-silla-que-decidira-sobre-espectacles-taurins/ |access-date=2022-09-24 |website=Hortanoticias.com |language=es |quote=S'utilitzarà el sistema de recompte Schulze perquè és un sistema de vot preferencial que compleix amb el criteri de Condorcet, la qual cosa permet identificar l’opció que, comparada amb qualsevol de les altres, és preferida per la majoria absoluta dels votants |trans-quote=The Schulze counting system will be used because it is a preferential voting system that complies with the Condorcet criterion, which makes it possible to identify the option that, compared to any other option, is preferred by the absolute majority of voters}}{{Cite web |date=2016-02-23 |title=22-26 de febrer: La primera Enquesta Popular de la història de Silla - Notícies - Ajuntament de Silla |url=http://www.silla.es/noticies/i/82830/1342/22-26-de-febrer-la-primera-enquesta-popular-de-la-historia-de-silla |url-status=dead |archive-url=https://web.archive.org/web/20160223122017/http://www.silla.es/noticies/i/82830/1342/22-26-de-febrer-la-primera-enquesta-popular-de-la-historia-de-silla |archive-date=2016-02-23 |access-date=2024-12-13 |website=Silla Town Council |language=Catalan |quote=S'utilitzarà el sistema de recompte Schulze perquè és un sistema de vot preferencial que compleix amb el criteri de Condorcet, la qual cosa permet identificar l’opció que, comparada amb qualsevol de les altres, és preferida per la majoria absoluta dels votants. |trans-quote=The Schulze counting system will be used because it is a preferential voting system that complies with the Condorcet criterion, which makes it possible to identify the option that, compared to any of the others, is preferred by the absolute majority of voters.}}{{Cite web |date=2017-03-22 |title=UN AÑO DE PROFUNDIZACIÓN DEMOCRÁTICA EN SILLA {{!}} Silla en Democracia |url=http://www.sillaendemocracia.es/un-ano-profundizacion-democratica-silla/ |url-status=dead |archive-url=https://web.archive.org/web/20170322161112/http://www.sillaendemocracia.es/un-ano-profundizacion-democratica-silla/ |archive-date=2017-03-22 |access-date=2024-12-13 |website=Silla in Democracy |language=Spanish |quote=Ha sido un rotundo éxito organizativo que repetiremos y que se gestó desde meses antes, desarrollando un know-how propio y siendo pioneros en la aplicación del sistema de recuento Condorcet-Schulze para las preguntas con respuesta de ordenación preferencial. Logramos un récord impresionante de participación que pulverizó todas las marcas existentes en procesos similares en municipios españoles (la media usual ronda el 10% de participación mientras que en Silla alcanzamos el 27%). |trans-quote=It has been a resounding organizational success that we will repeat, developed over months with our own know-how, and pioneering the application of the Condorcet-Schulze counting system for questions with preferential ranking responses. We achieved an impressive participation record that shattered all existing benchmarks in similar processes in Spanish municipalities (the usual average is around 10% participation, while in Silla, we reached 27%).}}{{Cite web |last=Silla |first=~ El Cresol de |date=2016-05-26 |title=Un any d'aprofundiment democràtic a Silla |url=https://elcresoldesilla.wordpress.com/2016/05/26/un-any-daprofundiment-democratic-a-silla/ |archive-url=https://web.archive.org/web/20220708140022/https://elcresoldesilla.wordpress.com/2016/05/26/un-any-daprofundiment-democratic-a-silla/ |url-status=dead |archive-date=2022-07-08 |access-date=2022-09-24 |website=El Cresol de Silla |language=ca}} It is also used by the cities of Turin and San Donà di Piave in Italy and by the London Borough of Southwark through their use of the WeGovNow platform, which in turn uses the LiquidFeedback decision tool.{{Cite web |title=LiquidFeedback in London, Turin and San Donà di Piave {{!}} Interaktive Demokratie |url=https://interaktive-demokratie.org/news/2016/20160701-LiquidFeedback-in-London-Turin-and-San-Dona-di-Piave.en.html |access-date=2024-12-13 |website=interaktive-demokratie.org |quote=WeGovNow! … is an OpenStreetMap based E-Government solution to be validated and evaluated in London Southwark (UK), Turin (Italy) and San Donà di Piave (Metropolitan City of Venice, Italy)}}{{Cite web |title=Algorithms for Good (The Liquid Democracy Journal, Issue 7) |url=https://liquid-democracy-journal.org/issue/7/The_Liquid_Democracy_Journal-Issue007-02-Algorithms_for_Good.html |access-date=2024-12-13 |website=liquid-democracy-journal.org |language=en |quote=To determine the winner from a set of alternatives, LiquidFeedback implements Clone-Proof Schwartz Sequential Dropping, known as the Schulze Method.}}

Schulze was adopted by the Pirate Party of Sweden (2009),See:

• {{cite web |date=8 October 2009 |title=Inför primärvalen |trans-title=Before the primary elections |url=https://forum.piratpartiet.se/showthread.php?p=172218 |archive-url=https://archive.today/20121224194036/https://forum.piratpartiet.se/showthread.php?p=172218 |archive-date=24 December 2012 |language=sv}}
• {{cite web |date=17 October 2009 |title=Dags att kandidera till riksdagen |trans-title=Time to run for the Riksdag |url=https://forum.piratpartiet.se/showthread.php?p=173792 |archive-url=https://archive.today/20140723081035/https://forum.piratpartiet.se/showthread.php?p=173792 |archive-date=23 July 2014 |language=sv}}
• {{cite web |date=19 January 2010 |title=Råresultat primärvalet |trans-title=The raw result of the primary election |url=https://forum.piratpartiet.se/showthread.php?p=190840 |archive-url=https://archive.today/20121224195649/https://forum.piratpartiet.se/showthread.php?p=190840 |archive-date=24 December 2012 |language=sv}} and the Pirate Party of Germany (2010).11 of the 16 regional sections and the federal section of the Pirate Party of Germany are using [https://liquidfeedback.org/ LiquidFeedback] for unbinding internal opinion polls. In 2010/2011, the Pirate Parties of Neukölln ([http://wiki.piratenpartei.de/BE:Neuk%C3%B6lln/Gebietsversammlungen/2010.3/Protokoll link]), Mitte ([https://berlin.piratenpartei.de/innerparteiliches/kandidaten-der-piraten-in-mitte-aufgestellt/ link]), Steglitz-Zehlendorf ([http://wiki.piratenpartei.de/wiki/images/d/da/BE_Gebietsversammlung_Steglitz_Zehlendorf_2011_01_20_Protokoll.pdf link]), Lichtenberg ([http://piraten-lichtenberg.de/2011/02/02/groser-schritt-in-richtung-wahl/ link]), and Tempelhof-Schöneberg ([http://wiki.piratenpartei.de/BE:Gebietsversammlungen/Tempelhof-Schoeneberg/Protokoll_2011.1 link]) adopted the Schulze method for its primaries. Furthermore, the Pirate Party of Berlin (in 2011) ([http://wiki.piratenpartei.de/BE:Parteitag/2011.1/Protokoll link]) and the Pirate Party of Regensburg (in 2012) ([http://wiki.piratenpartei.de/BY:Regensburg/Gr%C3%BCndung/Gesch%C3%A4ftsordnung#Anlage_A link]) adopted this method for their primaries. The Boise, Idaho chapter of the Democratic Socialists of America in February chose this method for their first special election held in March 2018.{{Cite web |last=Chumich |first=Andrew |title=DSA Special Election |url=https://www.boisedsa.org/statements/yjmtibx8okazm7i7042c10tuld0gm5 |access-date=2018-02-25}}

• Five Star Movement of Campobasso,[http://www.quotidianomolise.com/campobasso-comunali-scattano-le-primarie-a-5-stelle/ Campobasso. Comunali, scattano le primarie a 5 Stelle], February 2014 Fondi,{{Cite web |last=Macaro |first=Mirko |date=2015-03-03 |title=Fondi, il punto sui candidati a sindaco. Certezze, novità e colpi di scena |url=https://www.h24notizie.com/2015/03/03/fondi-punto-sui-candidati-sindaco-certezze-novita-colpi-scena/ |access-date=2022-09-24 |website=h24 notizie - portale indipendente di news dalla provincia |language=it-IT}} Monte Compatri,article 25(5) of the [http://www.5stellemontecompatri.it/wp-content/uploads/2013/11/REGOLAMENTO-M5S-Montecompatri1.doc bylaws], October 2013 Montemurlo,{{Cite web |date=November 2013 |title=MoVimento 5 Stelle - Montemurlo: 2° Step Comunarie di Montemurlo |url=http://www.montemurlo5stelle.net/2013/11/2-step-comunarie-di-montemurlo.html |archive-url=https://web.archive.org/web/20150402230827/http://www.montemurlo5stelle.net/2013/11/2-step-comunarie-di-montemurlo.html |archive-date=2015-04-02 |access-date=2022-09-24}} Pescara,article 12 of the [http://movimento5stellepescara.it/wp-content/uploads/2015/02/Regolamento-m5S-pescara_rev.03_del_30_01_-2015.pdf bylaws], January 2015 and San Cesareo[https://www.meetup.com/MoVimento-5-Stelle-San-Cesareo/events/166108572/ Ridefinizione della lista di San Cesareo con Metodo Schulze], February 2014
• Pirate Parties of Australia,{{Cite web |date=18 November 2011 |title=National Congress 2011 Results – Pirate Party Australia |url=https://pirateparty.org.au/2011/11/18/national-congress-2011-results/ |access-date=2022-09-24 |website=pirateparty.org.au |language=en-US}} Austria,§6(10) of the [http://wiki.piratenpartei.at/wiki/Satzung bylaws] Belgium,Article III.3.4 of the Statutory Rules ([https://wiki.pirateparty.be/images/9/9f/Ppbe-statutes-fr_be.pdf french], [https://wiki.pirateparty.be/images/e/e7/Ppbe-statutes-nl_be.pdf dutch]) Brazil, Germany, Iceland,{{Cite web |last=Píratar |date=2013-10-23 |title=Schulze aðferðin |url=https://piratar.is/frettir/piratafraedarinn/schulze-adferdin/ |access-date=2022-09-24 |website=Píratar |language=is}} Italy,[https://www.partito-pirata.it/statute/ Rules adopted on 18 December 2011] the Netherlands,{{Cite web |last=Pontier |first=Matthijs |date=2015-01-11 |title=Verslag ledenraadpleging 4 januari |url=https://noord-holland.piratenpartij.nl/711/ |access-date=2022-09-24 |website=Piratenpartij Noord Holland |language=nl}} Sweden, Switzerland,{{Cite web |last=Pankerl |first=Florian |date=2010-09-18 |title=Piratenversammlung der Piratenpartei Schweiz 2010 – Samstag |url=https://blog.florian-pankerl.de/2010/piratenversammlung-der-piratenpartei-schweiz-2010-samstag/ |access-date=2022-09-24 |language=de}} and the United Statesarticle IV section 3 of the [https://wiki.uspirates.org/w/index.php?title=Pirate_National_Committee_(PNC)/Bylaws bylaws], July 2012
• SustainableUnion§10 III of its [http://sustainableunion.yolasite.com/resources/130614_Satzung_sud_fBayern.pdf bylaws], June 2013
• Volt Europe{{cite web |author=The Board of Directors of Volt Europe in Spain |title=Algunas consideraciones sobre en qué grupo estará Volt Europa en el Parlamento Europeo |trans-title=Some considerations on which group Volt Europe will join in the European Parliament |url=https://medium.com/volt-espa%C3%B1a/algunas-consideraciones-sobre-en-qu%C3%A9-grupo-estar%C3%A1-volt-europa-en-el-parlamento-europeo-b10b7431d947 |archive-url=https://archive.today/20240820145337/https://web.archive.org/web/20211113112113/https://medium.com/volt-espa%C3%B1a/algunas-consideraciones-sobre-en-qu%C3%A9-grupo-estar%C3%A1-volt-europa-en-el-parlamento-europeo-b10b7431d947 |archive-date=20 August 2024 |website=Medium |language=es}}

• AEGEE – European Students' Forum{{Cite web |last=Hajdu |first=Tekla |date=2017-09-24 |title=The Schulze Method – Agora 101 |url=https://www.zeus.aegee.org/magazine/2017/09/24/the-schulze-method-agora-101/ |access-date=2022-09-24 |website=The AEGEEan - AEGEE's online magazine - AEGEE-Europe |language=en-GB}}
• Club der Ehemaligen der Deutschen SchülerAkademien e. V.[https://db.cde-ev.de/doc/Realm_Assembly_Voting-Details.html Voting Details], January 2021
• Associated Student Government at École normale supérieure de Paris[https://www.dg.ens.fr/documents/148/note_explicative.pdf Référendum sur la réforme du thurnage], June 2021
• Flemish Society of Engineering Students Leuvenarticle 57 of the [https://www.vtk.be/page/file/ef87370c1d5798758d1730a14f7410d96783301e/ statutory rules]
• Graduate Student Organization at the State University of New York: Computer Science (GSOCS){{cite web |title=User Voting Instructions |url=http://gso.cs.binghamton.edu/index.php/Voting |url-status=dead |archive-url=https://web.archive.org/web/20130909055056/http://gso.cs.binghamton.edu/index.php/Voting |archive-date=2013-09-09 |access-date=2010-05-08 |publisher=Gso.cs.binghamton.edu}}
• Hillegass Parker House{{cite web |title=Hillegass-Parker House Bylaws § 5. Elections |url=https://sites.google.com/a/bsc.coop/hip-house/bylaws-and-policies/bylaws-bylaw-appendix |access-date=4 October 2015 |work=Hillegass-Parker House website}}
• Kingman HallSee:
• Konglig Datasektionen KTH

• Ka-Ping Yee, [https://zestyping.livejournal.com/102718.html Condorcet elections], March 2005
• Ka-Ping Yee, [https://zestyping.livejournal.com/111588.html Kingman adopts Condorcet voting], April 2005
• Associated Students of Minerva Schools at KGIarticle 9.4.5.h of the [https://associatedstudents.minerva.community/wp-content/uploads/2017/11/ASM-Charter-November-2017.pdf charter], November 2017
• Associated Student Government at Northwestern University[http://www.northbynorthwestern.com/story/ajith-van-atta-win-asg-election/ Ajith, Van Atta win ASG election], April 2013
• Associated Student Government at University of Freiburg§6 and §7 of its [https://www.stura.uni-freiburg.de/gremien/studierendenrat/go/at_download/file/Stura-GO_Stand_06_06_2019.pdf bylaws], June 2019
• Associated Student Government at the Computer Sciences Department of the University of Kaiserslautern-Landau§6(6) of the [https://www.fachschaft.informatik.uni-kl.de/assets/documents/go.pdf bylaws]

It is used by the Institute of Electrical and Electronics Engineers, by the Association for Computing Machinery, and by USENIX{{Cn|date=October 2024}} through their use of the HotCRP decision tool.{{Technical inline|date=October 2024}}

Organizations which currently use the Schulze method include:

{{div col|colwidth=30em}}

• Annodex Association[https://civs.cs.cornell.edu/cgi-bin/results.pl?id=E_50cfc592ae8f13d9 Election of the Annodex Association committee for 2007], February 2007
• {{ill|Berufsverband der Kinder- und Jugendärzte|de}} (BVKJ)§9a of the [https://www.bvkj.de/mitgliedschaft/satzung bylaws], October 2013
• BoardGameGeekSee:
• 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
• Cloud Foundry Foundationarticle 7(e)(iii)(2) of the [https://www.cloudfoundry.org/wp-content/uploads/CFF-Charter-September-2023.pdf charter], September 2023
• County HighpointersAdam Helman, [http://www.cohp.org/records/votes/family_affair_voting_scheme.html Family Affair Voting Scheme - Schulze Method]
• Dapr[https://github.com/dapr/community/blob/master/steering-and-technical-committee-charter.md Steering and Technical committee], November 2021
• DebianSee:

• [https://www.debian.org/vote/2003/vote_0002 Constitutional Amendment: Condorcet/Clone Proof SSD Voting Method], June 2003
• [https://www.debian.org/devel/constitution Constitution for the Debian Project], appendix A6
• [https://www.debian.org/vote/ Debian Voting Information]
• EuroBillTracker
• European Democratic Education Community (EUDEC){{cite web|url=https://eudec.org/about-us/guidance-document/|title=Guidance Document |publisher=Eudec.org |date=2009-11-15 |access-date=2010-05-08}}
• FFmpeg[http://article.gmane.org/gmane.comp.video.ffmpeg.devel/113026/match=%22schulze%20method%22+%22Cloneproof%20schwartz%20sequential%20droping%22+Condorcet Democratic election of the server admins] {{Webarchive|url=https://web.archive.org/web/20151002053259/http://article.gmane.org/gmane.comp.video.ffmpeg.devel/113026/match=%22schulze%20method%22+%22Cloneproof%20schwartz%20sequential%20droping%22+Condorcet |date=2015-10-02 }}, July 2010
• Free Geek[http://wiki.freegeek.org/images/7/7a/Voters_guide.pdf Voters Guide], September 2011
• Free Hardware Foundation of Italy
• Gentoo FoundationProject:Elections
• GNU Privacy Guard (GnuPG){{Cite web |date=2013-10-03 |title=CIVS Election Results: GnuPG Logo Vote |url=http://logo-contest.gnupg.org/results.html |archive-url=https://web.archive.org/web/20131003164018/http://logo-contest.gnupg.org/results.html |archive-date=2013-10-03 |access-date=2022-09-24 }}
• Haskell[https://civs.cs.cornell.edu/cgi-bin/results.pl?num_winners=1&id=E_d21b0256a4fd5ed7&algorithm=beatpath Haskell Logo Competition], March 2009
• Homebrew[https://github.com/Homebrew/brew/blob/30c80dec155df6cc8603e2b70966943723911d27/docs/Homebrew-Governance.md#6-project-leader Article 6 Section 2 of the Constitution], February 2021
• Internet Corporation for Assigned Names and Numbers (ICANN) (until 2023)section 9.4.7.3 of the [https://web.archive.org/web/20230606191239/https://aso.icann.org/documents/operational-documents/operating-procedures-of-the-address-council-of-the-address-supporting-organization/ Operating Procedures of the Address Council of the Address Supporting Organization] (archived from [https://aso.icann.org/documents/operational-documents/operating-procedures-of-the-address-council-of-the-address-supporting-organization/#A_7.2.3._Ranked_Choice_Instant_Runoff source] 2023-06-06)
• Kanawha Valley Scrabble Club{{Cite web |date=2009-04-02 |title=A club by any other name... |url=http://wvscrabble.blogspot.com/2009/04/club-by-any-other-name.html |access-date=2022-09-24 |website=Kanawha Valley Scrabble Club}}
• KDE e.V.section 3.4.1 of the [https://ev.kde.org/rules/online_voting.php Rules of Procedures for Online Voting]
• Knight Foundation[https://civic.mit.edu/2009/6/23/knight-foundation-awards-5000-to-best-created-on-the-spot-projects/ Knight Foundation awards $5000 to best created-on-the-spot projects], June 2009
• Kubernetes{{Citation |title=Kubernetes Community |date=2022-09-24 |url=https://github.com/kubernetes/community/blob/dd18326394391b67662e9e062d8f3321a776f4dd/committee-code-of-conduct/election.md |publisher=Kubernetes |access-date=2022-09-24}}
• Kumoricon{{Cite web |title=Kumoricon – Mascot Contest |url=https://www.kumoricon.org/ |access-date=2022-09-24 |website=Kumoricon}}
• League of Professional System Administrators (LOPSA)article 8.3 of the [http://governance.lopsa.org/LOPSA_Bylaws bylaws]
• LiquidFeedback{{Cite book

| title = The Principles of LiquidFeedback

| publisher = Interaktive Demokratie e. V.

| isbn = 978-3-00-044795-2

| url = http://principles.liquidfeedback.org/

| location = Berlin

| year = 2014

}}

• Madisonium{{cite web|url=https://docs.google.com/document/d/1UhAGouwOEbzM6Uv3fQd78e5P1VfuYldtjjEK7XO_B3Y/edit|title=Madisonium Bylaws - Adopted|work=Google Docs}}
• Metalab{{cite web|url=https://metalab.at/wiki/Generalversammlung_2007/Wahlmodus |title=Wahlmodus |language=de |publisher=Metalab.at |access-date=2010-05-08}}
• MTVDavid Chandler, [https://news.mit.edu/newsoffice/2008/techtalk52-19.pdf Voting for more than just either-or], MIT Tech Talk, volume 52, number 19, page 2, 12 March 2008
• NeoSee:
• [http://wiki.neo-layout.org/wiki/Neo-2-Freeze/Wahl?version=10#a7.Wahlverfahren Wahlen zum Neo-2-Freeze: Formalitäten] {{Webarchive|url=https://web.archive.org/web/20110727115047/http://wiki.neo-layout.org/wiki/Neo-2-Freeze/Wahl?version=10#a7.Wahlverfahren |date=2011-07-27 }}, February 2010
• Hinweise zur Stimmabgabe, March 2010
• Ergebnisse, March 2010
• Noisebridge{{cite web|url=https://www.noisebridge.net/index.php?title=2009_Director_Elections&oldid=8951|title=2009 Director Elections|work=noisebridge.net}}
• OpenEmbedded{{cite web|url=http://www.openembedded.org/wiki/Online_Voting_Policy|title=Online Voting Policy|work=openembedded.org}}
• Open Neural Network Exchange[https://github.com/onnx/onnx/blob/master/community/sc-election-guidelines.md ONNX Steering Committee election guideline]
• OpenStack{{Cite web |title=OpenStack Election — OpenStack Governance |url=https://governance.openstack.org/election/#election-system |access-date=2022-09-24 |website=governance.openstack.org}}
• OpenSwitch{{Cite web |last=Mark |first=Atwood |date=May 25, 2016 |title=[Partners] text of OpenSwitch Project Charter 2016-05-03 |url=https://lists.linuxfoundation.org/pipermail/ops-partners/2016-May/000030.html |access-date=2022-09-24}}
• RLLMUK{{Cite web |title=Committee Elections 2012 |url=https://www.rllmukforum.com/index.php?/topic/260622-committee-elections-2012/ |access-date=2022-09-24 |website=rllmuk |date=10 April 2012 |language=en-GB}}
• Squeak[https://civs.cs.cornell.edu/cgi-bin/results.pl?num_winners=7&id=E_716d8c257e6cf36b&algorithm=beatpath Squeak Oversight Board Election 2010], March 2010
• Students for Free CultureSee:
• [http://wiki.freeculture.org/Bylaws Bylaws of the Students for Free Culture] {{Webarchive|url=https://web.archive.org/web/20130318164643/http://wiki.freeculture.org/Bylaws |date=2013-03-18 }}, article V, section 1.1.1
• [https://blog.selectricity.org/2008/02/15/free-culture-student-board-elected-using-selectricity/ Free Culture Student Board Elected Using Selectricity], February 2008
• Sugar Labs{{Cite web |title=[IAEP] Election status update |url=http://lists.sugarlabs.org/archive/iaep/2009-September/008620.html |access-date=2022-09-24 |website=lists.sugarlabs.org}}
• Sverok[https://medlem.sverok.se/2019/01/18/vad-hande-pa-riksmotet-2018/ Minutes of the 2018 Annual Sverok Meeting], November 2018
• TopCoder

{{Cite web |title=2007 TopCoder Collegiate Challenge |url=https://community.topcoder.com/tc?module=Static&d1=tournaments&d2=tccc07&d3=blog&d4=description |access-date=2022-09-24 |website=community.topcoder.com}}

• Ubuntu{{Cite web |last=Bell |first=Alan |date=May 17, 2012 |title=Ubuntu IRC Council Position |url=https://lists.ubuntu.com/archives/ubuntu-irc/2012-May/001538.html |access-date=2022-09-24}}
• Vidya Gaem Awards{{cite web|url=https://2012.vidyagaemawards.com/voting/results/pairwise|title=/v/GAs - Pairwise voting results|work=vidyagaemawards.com}}
• Wikimedia (2008)See:

• 2008 Board Elections, June 2008
• 2009 Board Elections, August 2009
• 2011 Board Elections, June 2011
• Wikipedia in French,{{Citation |title=Wikipédia:Prise de décision/Choix dans les votes |date=2019-08-22 |url=https://fr.wikipedia.org/w/index.php?title=Wikip%C3%A9dia:Prise_de_d%C3%A9cision/Choix_dans_les_votes&oldid=162034454 |work=Wikipédia |language=fr |access-date=2022-09-24}} Hebrew,{{cite web|url=https://he.wikipedia.org/wiki/ויקיפדיה:פרלמנט/הכרעה|title=ויקיפדיה:פרלמנט/הכרעה|trans-title=Wikipedia:Parliament/Decisionmaking|website=he.wikipedia.org|language=he}}{{Circular reference|date=August 2024}}See e.g. here [https://he.wikipedia.org/w/index.php?title=ויקיפדיה:פרלמנט&oldid=7014412] (May 2009), here [https://he.wikipedia.org/w/index.php?title=ויקיפדיה:סדנה_לגרפיקה/סמליל&oldid=7388447] (August 2009), and here [https://he.wikipedia.org/w/index.php?title=ויקיפדיה:סדנה_לגרפיקה/100,000/הצבעה&oldid=8057743] (December 2009). Hungarian,See here and here. Russian,{{cite web|url=http://y.kalan.cc/schulze|trans-title=Result of Arbitration Committee Elections|script-title=ru:Девятнадцатые выборы арбитров, второй тур|work=kalan.cc|url-status=dead|archive-url=https://web.archive.org/web/20150222225428/http://y.kalan.cc/schulze|archive-date=2015-02-22|lang=ru}} and Persian.See [//fa.wikipedia.org/w/index.php?oldid=18462330 here]

{{div col end}}

In 2008, Camps et. al devised a method that, while ranking candidates in the same order of finish as Schulze, also provides ratings indicating the candidates' relative strength of victory.{{cite arXiv | last1=Camps | first1=Rosa | last2=Mora | first2=Xavier | last3=Saumell | first3=Laia | title=A continuous rating method for preferential voting | date=2008 | eprint=0810.2263 | class=math.OC}}

{{Commons}}

• {{cite arXiv |eprint=1804.02973 |last1=Schulze |first1=Markus |title=The Schulze Method of Voting |date=2018 |class=cs.GT }}
• [https://www.youtube.com/watch?v=_HVeN0GnnuA The Schulze Method] by Hubert Bray
• [https://gki.informatik.uni-freiburg.de/teaching/ss13/gametheory/spieltheorie.pdf Spieltheorie] {{in lang|de}} by Bernhard Nebel
• [http://accuratedemocracy.com/voting_rules.htm Accurate Democracy] by Rob Loring
• Christoph Börgers (2009), [https://books.google.com/books?id=dccBaphP1G4C&pg=PA37 Mathematics of Social Choice: Voting, Compensation, and Division], SIAM, {{ISBN|0-89871-695-0}}
• Nicolaus Tideman (2006), [https://books.google.com/books?id=RN5q_LuByUoC&pg=PA228 Collective Decisions and Voting: The Potential for Public Choice], Burlington: Ashgate, {{ISBN|0-7546-4717-X}}
• [http://www.public-software-group.org/preftools preftools] by the Public Software Group
• [http://apps.azsos.gov/election/2008/general/ballotmeasuretext/I-21-2008.pdf Arizonans for Condorcet Ranked Voting]
• [https://github.com/julien-boudry/Condorcet Condorcet PHP] Command line application and PHP library, supporting multiple Condorcet methods, including Schulze.
• [https://github.com/zephyr/schulze-voting Implementation in Java]
• [https://github.com/coorasse/schulze-vote Implementation in Ruby]
• [https://github.com/bradbeattie/python-vote-core Implementation in Python 2]
• [https://github.com/woctezuma/schulze-method Implementation in Python 3]

{{voting systems}}