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Articles with example pseudocode Debian Electoral systems Monotonic Condorcet methods Single-winner electoral systems The Schulze method (), also known as the beatpath method, is a single winner ranked-choice voting rule developed by Markus Schulze. The Schulze method is a Condorcet completion method, which means it will elect a
majority-preferred candidate A Condorcet winner (, ) is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condo ...
if one exists. In other words, if most people rank ''A'' above ''B'', ''A'' will defeat ''B'' (whenever this is possible). Schulze's method breaks cyclic ties by using indirect victories. The idea is that if
Alice Alice may refer to: * Alice (name), most often a feminine given name, but also used as a surname Literature * Alice (''Alice's Adventures in Wonderland''), a character in books by Lewis Carroll * ''Alice'' series, children's and teen books by ...
beats Bob, and Bob beats Charlie, then Alice (indirectly) beats Charlie; this kind of indirect win is called a "beatpath". For
proportional representation Proportional representation (PR) refers to any electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to political divisions (Political party, political parties) amon ...
, a
single transferable vote The single transferable vote (STV) or proportional-ranked choice voting (P-RCV) is a multi-winner electoral system in which each voter casts a single vote in the form of a ranked ballot. Voters have the option to rank candidates, and their vot ...
(STV) variant known as
Schulze STV Schulze STV is a proposed multi-winner ranked voting system designed to achieve proportional representation. It was invented by Markus Schulze, who developed the Schulze method for resolving ties using a Condorcet method. Schulze STV is similar ...
also exists. The Schulze method is used by several organizations including
Debian Debian () is a free and open-source software, free and open source Linux distribution, developed by the Debian Project, which was established by Ian Murdock in August 1993. Debian is one of the oldest operating systems based on the Linux kerne ...
,
Ubuntu Ubuntu ( ) is a Linux distribution based on Debian and composed primarily of free and open-source software. Developed by the British company Canonical (company), Canonical and a community of contributors under a Meritocracy, meritocratic gover ...
, Gentoo,
Pirate Party Pirate Party is a label adopted by various Political party, political parties worldwide that share a set of values and policies focused on Civil and political rights, civil rights in the digital age. The fundamental principles of Pirate Partie ...
political parties and
many others Many (/ˈmɛni/) may refer to: * grammatically plural in number *an English quantifier used with count nouns indicating a large but indefinite number of; at any rate, more than a few ;Place names * Many, Moselle, a commune of the Moselle depart ...
. It was also used by
Wikimedia The Wikimedia Foundation, Inc. (WMF) is an American 501(c)(3) nonprofit organization headquartered in San Francisco, California, and registered there as a charitable foundation. It is the host of Wikipedia, the eighth most visited website ...
prior to their adoption of
score voting Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approva ...
.


Description of the method

Schulze's method uses
ranked ballots Ranked voting is any voting system that uses voters' Ordinal utility, rankings of candidates to choose a single winner or multiple winners. More formally, a ranked vote system depends only on voters' total order, order of preference of the cand ...
with equal ratings allowed. There are two common (equivalent) descriptions of Schulze's method.


Beatpath explanation

The idea behind Schulze's method is that if
Alice Alice may refer to: * Alice (name), most often a feminine given name, but also used as a surname Literature * Alice (''Alice's Adventures in Wonderland''), a character in books by Lewis Carroll * ''Alice'' series, children's and teen books by ...
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. As a result, the Schulze method is a
Condorcet method A Condorcet method (; ) is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the ...
, providing a full extension of the
majority rule In social choice theory, the majority rule (MR) is a social choice rule which says that, when comparing two options (such as bills or candidates), the option preferred by more than half of the voters (a ''majority'') should win. In political ...
to any set of ballots.


Iterative description

The Schulze winner can also be constructed iteratively, using a defeat-dropping method: # Draw a directed graph with all the candidates as nodes; label the edges with the number of votes supporting the winner. # If there is more than one candidate left: #* Check if any candidates are tied (and if so, break the ties by
random ballot A random ballot or random dictatorship is a randomized electoral system where the election is decided on the basis of a single randomly-selected ballot. A closely-related variant is called random serial (or sequential) dictatorship, which repeats ...
). #* Eliminate all candidates outside the majority-preferred set. #* Delete the edge closest to being tied. The winner is the only candidate left at the end of the procedure.


Example

In the following example 45 voters rank 5 candidates. The pairwise preferences have to be computed first. For example, when comparing ' and ' pairwise, there are voters who prefer ' to ', and voters who prefer ' to '. So d
, B The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
= 20 and d
, A The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
= 25. The full set of pairwise preferences is: The cells for d
, Y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
have a light green background if d
, Y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
> d
, X The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
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
, Y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d
, Y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
> d
, X The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
(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
, D The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
= 33: the strongest path from B to D is the direct path (B, D) which has strength 33. But when computing p
, C The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
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 ' and ', since (28 =) p ,B> p ,A(= 25), for the Schulze method candidate ' is ''better'' than candidate '. Another example is that (31 =) p ,D> p ,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 ' wins. In other words, ' wins since p ,X\ge p ,E/math> 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 In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The widest path problem is also known as the maxim ...
. One simple way to compute the strengths, therefore, is a variant of the
Floyd–Warshall algorithm In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest paths in a directed weighted graph with po ...
. The following
pseudocode In computer science, pseudocode is a description of the steps in an algorithm using a mix of conventions of programming languages (like assignment operator, conditional operator, loop) with informal, usually self-explanatory, notation of actio ...
illustrates the algorithm. # Input: d ,j the number of voters who prefer candidate i to candidate j. # Output: p ,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 ,j> d ,ithen p ,j:= d ,j else p ,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 ,k:= max (p ,k min (p ,i p ,k) This algorithm is efficient and has
running time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
O(''C''3) 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
, B The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
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 ''d''s are defined, the Schulze ranking has no cycles, and assuming the ''d''s 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 A random ballot or random dictatorship is a randomized electoral system where the election is decided on the basis of a single randomly-selected ballot. A closely-related variant is called random serial (or sequential) dictatorship, which repeats ...
. 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. # 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. # If there is a candidate with no red on their line, they win. # 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. # Cut away every part of the table outside the box. # 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: *
Monotonicity criterion Electoral system criteria In social choice, the negative response, perversity, or additional support paradox is a pathological behavior of some voting rules where a candidate loses as a result of having too much support (or wins because of in ...
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.
* Majority criterion *
Majority loser criterion The majority loser criterion is a criterion to evaluate single-winner voting systems. The criterion states that if a majority of voters give a candidate no support, i.e. do not list that candidate on their ballot, that candidate must lose (unles ...
*
Condorcet criterion A Condorcet winner (, ) is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condo ...
*
Condorcet loser criterion In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion. A voting system complying wi ...
*
Smith criterion The Smith set, sometimes called the top-cycle or Condorcet winning set, generalizes the idea of a Condorcet winner to cases where no such winner exists. It does so by allowing cycles of candidates to be treated jointly, as if they were a single ...
*
Independence of Smith-dominated alternatives Independence of Smith-dominated alternatives (ISDA, also known as Smith- IIA) is a voting system criterion which says that the winner of an election should not be affected by candidates who are not in the Smith set. Another way of defining ISDA i ...
*
Mutual majority criterion The mutual majority criterion is a criterion for evaluating electoral systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion. This criterion requires that whenever a majority of voters prefe ...
* Independence of clones *
Reversal symmetry The reversal symmetry criterion is a voting system criterion which says that if every voter's opinions on each of the candidates is perfectly reversed (i.e. they rank candidates in order from worst to best), the outcome of the election should b ...
* Mono-appendDouglas R. Woodall
Properties of Preferential Election Rules
''Voting Matters'', issue 3, pages 8–15, December 1994
* Mono-add-plump *
Resolvability criterion A voting system is called decisive, resolvable, or resolute if it ensures a low probability of tied elections. There are two different criterion that formalize this. * In Nicolaus Tideman's version of the criterion, adding one extra vote (with no ...
* Polynomial runtime * prudence * MinMax sets * Woodall's plurality criterion if winning votes are used for d ,Y* Symmetric-completion if
margins Margin may refer to: Physical or graphical edges *Margin (typography), the white space that surrounds the content of a page *Continental margin, the zone of the ocean floor that separates the thin oceanic crust from thick continental crust *Leaf ...
are used for d ,Y


Failed criteria

Since the Schulze method satisfies the Condorcet criterion, it automatically fails the following criteria: * Participation *
Consistency In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
* Invulnerability to burying * Later-no-harm Likewise, since the Schulze method is not a
dictatorship A dictatorship is an autocratic form of government which is characterized by a leader, or a group of leaders, who hold governmental powers with few to no Limited government, limitations. Politics in a dictatorship are controlled by a dictator, ...
and is a
ranked voting Ranked voting is any voting system that uses voters' Ordinal utility, rankings of candidates to choose a single winner or multiple winners. More formally, a ranked vote system depends only on voters' total order, order of preference of the cand ...
system (not rated),
Arrow's Theorem Arrow's impossibility theorem is a key result in social choice theory showing that no Ordinal utility, ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Kenneth Arrow, Arrow showed no ...
implies it fails
independence of irrelevant alternatives Independence of irrelevant alternatives (IIA) is an axiom of decision theory which codifies the intuition that a choice between A and B (which are both related) should not depend on the quality of a third, unrelated outcome C. There are several dif ...
, meaning it can be vulnerable to the
spoiler effect In social choice theory and politics, a spoiler effect happens when a losing candidate affects the results of an election simply by participating. Voting rules that are not affected by spoilers are said to be spoilerproof. The frequency and se ...
in some rare circumstances. The Schulze method also fails
Peyton Young Hobart Peyton Young (born March 9, 1945) is an American game theorist and economist known for his contributions to evolutionary game theory and its application to the study of institutional and technological change, as well as the theory of lea ...
's criterion of
Local Independence of Irrelevant Alternatives In social choice theory and politics, a spoiler effect happens when a losing candidate affects the results of an election simply by participating. Voting rules that are not affected by spoilers are said to be spoilerproof. The frequency and se ...
.


Comparison table

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


Difference from ranked pairs

Ranked pairs is another
Condorcet method A Condorcet method (; ) is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the ...
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 Minimax (sometimes Minmax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, combinatorial game theory, statistics, and philosophy for ''minimizing'' the possible loss function, loss for a Worst-case scenari ...
. Say that the
minimax Minimax (sometimes Minmax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, combinatorial game theory, statistics, and philosophy for ''minimizing'' the possible loss function, loss for a Worst-case scenari ...
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. 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. 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 Social organisms, including human(s), live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, and whether the exchange is voluntary or not. Etymology The word "social" derives fro ...
''.


Usage


Government

The Schulze method is used by the city of Silla, Spain for all referendums. It is also used by the cities of
Turin Turin ( , ; ; , then ) is a city and an important business and cultural centre in northern Italy. It is the capital city of Piedmont and of the Metropolitan City of Turin, and was the first Italian capital from 1861 to 1865. The city is main ...
and
San Donà di Piave San Donà di Piave (; ) is a city and ''comune'' in the Metropolitan City of Venice, Veneto, northern Italy. It is one of the historical main towns of the ''Eastern Veneto'' territory, although it was totally reconstructed in the early 1920s aft ...
in Italy and by the
London Borough of Southwark The London Borough of Southwark ( ) in South London forms part of Inner London and is connected by bridges across the River Thames to the City of London and the London Borough of Tower Hamlets. It was created in 1965 when three smaller council ...
through their use of the WeGovNow platform, which in turn uses the
LiquidFeedback LiquidFeedback is free software for political opinion formation and decision making. The software incorporates insights from social choice theory in order to aggregate opinions more effectively. Description LiquidFeedback helps groups (such as ...
decision tool.


Political parties

Schulze was adopted by the Pirate Party of Sweden (2009),See: * * * and the
Pirate Party of Germany The Pirate Party Germany (), commonly known as Pirates (), is a list of political parties in Germany, political party in Germany founded in September 2006 at c-base. It states general agreement with the Swedish Pirate Party (Sweden), Piratpartie ...
(2010).11 of the 16 regional sections and the federal section of the
Pirate Party of Germany The Pirate Party Germany (), commonly known as Pirates (), is a list of political parties in Germany, political party in Germany founded in September 2006 at c-base. It states general agreement with the Swedish Pirate Party (Sweden), Piratpartie ...
are usin
LiquidFeedback
for unbinding internal opinion polls. In 2010/2011, the Pirate Parties of
Neukölln Neukölln (), officially abbreviated Neuk, is one of the twelve boroughs of Berlin. It is located south-east of Berlin's center and stretches from the inner city southward to the border with Brandenburg, encompassing the eponymous quarter of Neu ...

link
,
Mitte Mitte () is the first and most central borough of Berlin. The borough consists of six sub-entities: Mitte proper, Gesundbrunnen, Hansaviertel, Moabit, Tiergarten and Wedding. It is one of the two boroughs (the other being Friedrichshain-Kreuz ...

link
,
Steglitz-Zehlendorf Steglitz-Zehlendorf () is the sixth Boroughs of Berlin, borough of Berlin, formed in Berlin's 2001 administrative reform by merging the former boroughs of Steglitz and Zehlendorf, Berlin, Zehlendorf. Home to the Free University of Berlin, the Be ...

link
,
Lichtenberg Lichtenberg may refer to: Places * Lichtenberg, Austria * Lichtenberg, Bas-Rhin, France * Lichtenberg, Bavaria, Germany * Lichtenberg, Berlin, Germany * Lichtenberg, Mittelsachsen, Saxony, Germany * Lichtenberg (Lausitz), Saxony, Germany * Lichte ...

link
, and
Tempelhof-Schöneberg Tempelhof-Schöneberg () is the seventh borough of Berlin, formed in 2001 by merging the former boroughs of Tempelhof and Schöneberg. Situated in the south of the city it shares borders with the boroughs of Mitte and Friedrichshain-Kreuzberg in ...

link
adopted the Schulze method for its primaries. Furthermore, the Pirate Party of
Berlin Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population withi ...
(in 2011)
link
and the Pirate Party of
Regensburg Regensburg (historically known in English as Ratisbon) is a city in eastern Bavaria, at the confluence of the rivers Danube, Naab and Regen (river), Regen, Danube's northernmost point. It is the capital of the Upper Palatinate subregion of the ...
(in 2012)
link
adopted this method for their primaries.
The
Boise, Idaho Boise ( ) is the List of capitals in the United States, capital and List of cities in Idaho, most populous city of the U.S. state of Idaho. As of the 2020 United States census, 2020 census, there were 235,685 people residing in the city. Loca ...
chapter of the
Democratic Socialists of America The Democratic Socialists of America (DSA) is a political organization in the United States and the country's largest Socialism, socialist organization. Sitting on the Left-wing politics, left-wing to Far-left politics, far-left of the politic ...
in February chose this method for their first special election held in March 2018.article IV section 3 of th
bylaws
/ref> *
Five Star Movement The Five Star Movement ( , M5S) is a political party in Italy, led by Giuseppe Conte. It was launched on 4 October 2009 by Beppe Grillo, a political activist and comedian, and Gianroberto Casaleggio, a web strategist. The party is primarily d ...
of
Campobasso Campobasso (, ; ) is a city and ''comune'' in southern Italy, the capital of the region of Molise and of the province of Campobasso. It is located in the high basin of the Biferno river, surrounded by Sannio and Matese mountains. Campobas ...
,
Fondi Fondi (; Southern Laziale: ''Fùnn'') is a city and ''comune'' in the province of Latina, Lazio, central Italy, halfway between Rome and Naples. As of 2017, the city had a population of 39,800. The city has experienced steady population growth si ...
,
Monte Compatri Monte Compatri () is a (municipality) in the Metropolitan City of Rome in the Italian region of Latium, located about southeast of Rome on the Alban Hills. It is one of the Castelli Romani. History Monte Compatri has been identified with the an ...
, Montemurlo,
Pescara Pescara (; ; ) is the capital city of the province of Pescara, in the Abruzzo Regions of Italy, region of Italy. It is the most populated city in Abruzzo, with 118,657 (January 1, 2023) residents (and approximately 350,000 including the surround ...
, and San Cesareo *
Pirate Parties Pirate Party is a label adopted by various political parties worldwide that share a set of values and policies focused on civil rights in the digital age. The fundamental principles of Pirate Parties include freedom of information, freedom of ...
of
Australia Australia, officially the Commonwealth of Australia, is a country comprising mainland Australia, the mainland of the Australia (continent), Australian continent, the island of Tasmania and list of islands of Australia, numerous smaller isl ...
,
Austria Austria, formally the Republic of Austria, is a landlocked country in Central Europe, lying in the Eastern Alps. It is a federation of nine Federal states of Austria, states, of which the capital Vienna is the List of largest cities in Aust ...
,
Belgium Belgium, officially the Kingdom of Belgium, is a country in Northwestern Europe. Situated in a coastal lowland region known as the Low Countries, it is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeas ...
,
Brazil Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...
,
Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
,
Iceland Iceland is a Nordic countries, Nordic island country between the Atlantic Ocean, North Atlantic and Arctic Oceans, on the Mid-Atlantic Ridge between North America and Europe. It is culturally and politically linked with Europe and is the regi ...
,
Italy Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...
,
the Netherlands , Terminology of the Low Countries, informally Holland, is a country in Northwestern Europe, with Caribbean Netherlands, overseas territories in the Caribbean. It is the largest of the four constituent countries of the Kingdom of the Nether ...
,
Sweden Sweden, formally the Kingdom of Sweden, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. It borders Norway to the west and north, and Finland to the east. At , Sweden is the largest Nordic count ...
,
Switzerland Switzerland, officially the Swiss Confederation, is a landlocked country located in west-central Europe. It is bordered by Italy to the south, France to the west, Germany to the north, and Austria and Liechtenstein to the east. Switzerland ...
, and
the United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...
* 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 Northwestern University (NU) is a Private university, private research university in Evanston, Illinois, United States. Established in 1851 to serve the historic Northwest Territory, it is the oldest University charter, chartered university in ...
* Associated Student Government at
University of Freiburg The University of Freiburg (colloquially ), officially the Albert Ludwig University of Freiburg (), is a public university, public research university located in Freiburg im Breisgau, Baden-Württemberg, Germany. The university was founded in 1 ...
* Associated Student Government at the Computer Sciences Department of the
University of Kaiserslautern-Landau The University of Kaiserslautern-Landau (German: ''Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau'', also known as RPTU) is a public research university in Kaiserslautern and Landau in der Pfalz, Germany. The university wa ...


Organizations

It is used by the
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers (IEEE) is an American 501(c)(3) public charity professional organization for electrical engineering, electronics engineering, and other related disciplines. The IEEE has a corporate office ...
, by the
Association for Computing Machinery The Association for Computing Machinery (ACM) is a US-based international learned society for computing. It was founded in 1947 and is the world's largest scientific and educational computing society. The ACM is a non-profit professional membe ...
, and by
USENIX USENIX is an American 501(c)(3) nonprofit membership organization based in Berkeley, California and founded in 1975 that supports advanced computing systems, operating system (OS), and computer networking research. It organizes several confe ...
through their use of the HotCRP decision tool. Organizations which currently use the Schulze method include: * Annodex Association * (BVKJ) *
BoardGameGeek BoardGameGeek (BGG) is an online forum for board gaming hobbyists and a game database that holds reviews, images and videos for over 125,600 different tabletop games, including European-style board games, wargames, and card games. In addition t ...
* Cloud Foundry Foundation * County Highpointers * Dapr *
Debian Debian () is a free and open-source software, free and open source Linux distribution, developed by the Debian Project, which was established by Ian Murdock in August 1993. Debian is one of the oldest operating systems based on the Linux kerne ...
See:
Constitutional Amendment: Condorcet/Clone Proof SSD Voting Method
June 2003
Constitution for the Debian Project
appendix A6
Debian Voting Information
/ref> *
EuroBillTracker EuroBillTracker (EBT) is a website designed for tracking euro banknotes. It was inspired by the US currency bill tracking website Where's George? The aim is to record as many notes as possible to know details about their distribution and movement ...
*
European Democratic Education Community The European Democratic Education Community (EUDEC) is a European non-profit organisation that promotes democratic education as a sensible educational model for all democratic states. EUDEC aims to further democratic education in Europe. Founded i ...
(EUDEC) *
FFmpeg FFmpeg is a free and open-source software project consisting of a suite of libraries and programs for handling video, audio, and other multimedia files and streams. At its core is the command-line ffmpeg tool itself, designed for processing vide ...
* Free Geek * Free Hardware Foundation of Italy * Gentoo Foundation Project:Elections *
GNU Privacy Guard GNU Privacy Guard (GnuPG or GPG) is a free-software replacement for Symantec's cryptographic software suite PGP. The software is compliant with the now obsoleted , the IETF standards-track specification of OpenPGP. Modern versions of PGP are ...
(GnuPG) *
Haskell Haskell () is a general-purpose, statically typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research, and industrial applications, Haskell pioneered several programming language ...
* Homebrew *
Internet Corporation for Assigned Names and Numbers The Internet Corporation for Assigned Names and Numbers (ICANN ) is a global multistakeholder group and nonprofit organization headquartered in the United States responsible for coordinating the maintenance and procedures of several dat ...
(ICANN) (until 2023) * Kanawha Valley Scrabble Club * KDE e.V.section 3.4.1 of th
Rules of Procedures for Online Voting
/ref> *
Knight Foundation The John S. and James L. Knight Foundation, also known as the Knight Foundation, is an American non-profit foundation that provides grants for journalism, communities, and the arts. The organization was founded as the Knight Memorial Education ...
*
Kubernetes Kubernetes (), also known as K8s is an open-source software, open-source OS-level virtualization, container orchestration (computing), orchestration system for automating software deployment, scaling, and management. Originally designed by Googl ...
*
Kumoricon Kumoricon is an annual three-day anime convention held during October or November at the Oregon Convention Center in Portland, Oregon. The name of the convention comes from the Japanese language, Japanese word ''Kumori'' (曇り), meaning ''cloud ...
* League of Professional System Administrators (LOPSA) *
LiquidFeedback LiquidFeedback is free software for political opinion formation and decision making. The software incorporates insights from social choice theory in order to aggregate opinions more effectively. Description LiquidFeedback helps groups (such as ...
* Madisonium *
Metalab The Metalab is a hackerspace in Vienna's central Innere Stadt, first district. Founded in 2006, it is a meeting place of the Viennese tech community, hosting events from cultural festivals to user groups. It has played a catalyst role in the ...
*
MTV MTV (an initialism of Music Television) is an American cable television television channel, channel and the flagship property of the MTV Entertainment Group sub-division of the Paramount Media Networks division of Paramount Global. Launched on ...
* Neo *
Noisebridge Noisebridge is an anarchism, anarchistic maker and hackerspace located in San Francisco. It is inspired by the European hackerspaces Metalab in Vienna and c-base in Berlin. Noisebridge describes itself as "''a space for sharing, creation, coll ...
*
OpenEmbedded OpenEmbedded (OE) is a build automation framework and cross-compile environment used to create Linux distributions for embedded devices. The framework is developed by the OpenEmbedded community, which was formally established in 2003. OpenEmbed ...
*
Open Neural Network Exchange The Open Neural Network Exchange (ONNX) [] is an Open-source software, open-source artificial intelligence ecosystem of technology companies and research organizations that establish open standards for representing machine learning algorithms an ...
*
OpenStack OpenStack is a free, open standard cloud computing platform. It is mostly deployed as infrastructure-as-a-service (IaaS) in both public and private clouds where virtual servers and other resources are made available to users. The software pla ...
* OpenSwitch *
RLLMUK RLLMUK is an internet discussion forum that primarily contains discussion about video games, although many other topics of discussion have developed over the years. It was founded in 2003 by Rob Purnell, in response to the closure of the Edge di ...
*
Squeak Squeak is an object-oriented, class-based, and reflective programming language. It was derived from Smalltalk-80 by a group that included some of Smalltalk-80's original developers, initially at Apple Computer, then at Walt Disney Imaginee ...
* Students for Free Culture *
Sugar Labs Sugar Labs is a community-run software project whose mission is to produce, distribute, and support the use of Sugar (software), Sugar, an open source software, open source desktop environment and learning platform. Sugar Labs was initially estab ...
* Sverok *
TopCoder Topcoder (formerly TopCoder) is a crowdsourcing company with an open global community of designers, developers, data scientists, and competitive programmers. Topcoder pays community members for their work on the projects and sells community s ...
*
Ubuntu Ubuntu ( ) is a Linux distribution based on Debian and composed primarily of free and open-source software. Developed by the British company Canonical (company), Canonical and a community of contributors under a Meritocracy, meritocratic gover ...
*
Vidya Gaem Awards Vidya may refer to: * Vidya (philosophy), a concept in Hindu and Buddhist philosophy * ''Vidya'' (journal), a bimonthly journal published by the Triple Nine Society * ''Vidya'' (film), a 1948 Bollywood film * Vidya Academy of Science and Techn ...
*
Wikimedia The Wikimedia Foundation, Inc. (WMF) is an American 501(c)(3) nonprofit organization headquartered in San Francisco, California, and registered there as a charitable foundation. It is the host of Wikipedia, the eighth most visited website ...
(2008)See: * 2008 Board Elections, June 2008 * 2009 Board Elections, August 2009 * 2011 Board Elections, June 2011 *
Wikipedia Wikipedia is a free content, free Online content, online encyclopedia that is written and maintained by a community of volunteers, known as Wikipedians, through open collaboration and the wiki software MediaWiki. Founded by Jimmy Wales and La ...
in French,
Hebrew Hebrew (; ''ʿÎbrit'') is a Northwest Semitic languages, Northwest Semitic language within the Afroasiatic languages, Afroasiatic language family. A regional dialect of the Canaanite languages, it was natively spoken by the Israelites and ...
, Hungarian,
Russian Russian(s) may refer to: *Russians (), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *A citizen of Russia *Russian language, the most widely spoken of the Slavic languages *''The Russians'', a b ...
, and
Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...
.See /fa.wikipedia.org/w/index.php?oldid=18462330 here/ref>


Generalizations

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.


Notes


External links

*
The Schulze Method
by
Hubert Bray Hubert Lewis Bray is a mathematician and differential geometer. He is known for having proved the Riemannian Penrose inequality. He works as professor of mathematics and physics at Duke University. Early life and education He earned his B.A. a ...

Spieltheorie
by Bernhard Nebel
Accurate Democracy
by Rob Loring * Christoph Börgers (2009),
Mathematics of Social Choice: Voting, Compensation, and Division
', SIAM, *
Nicolaus Tideman Thorwald Nicolaus Tideman (, not ; born August 11, 1943, in Chicago, Illinois) is a Georgist economist and professor at Virginia Tech. He received his Bachelor of Arts in economics and mathematics from Reed College in 1965 and his PhD in economics ...
(2006),
Collective Decisions and Voting: The Potential for Public Choice
', Burlington: Ashgate,
preftools
by the Public Software Group
Arizonans for Condorcet Ranked Voting

Condorcet PHP
Command line application and
PHP PHP is a general-purpose scripting language geared towards web development. It was originally created by Danish-Canadian programmer Rasmus Lerdorf in 1993 and released in 1995. The PHP reference implementation is now produced by the PHP Group. ...
library A library is a collection of Book, books, and possibly other Document, materials and Media (communication), media, that is accessible for use by its members and members of allied institutions. Libraries provide physical (hard copies) or electron ...
, supporting multiple Condorcet methods, including Schulze.
Implementation in Java

Implementation in Ruby

Implementation in Python 2

Implementation in Python 3
{{voting systems