The Schulze method is a voting 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), Beatpath Method, Beatpath Winner, Path Voting, and Path Winner. The Elections and Parties Series Democracy Representative democracy History of democracy Referenda Liberal democracy Representation Voting Voting systems Ideology Elections Elections by country Elections by calendar Politics Politics by country Political campaigns Political science Political philosophy Related topics Political parties Parties by country Parties by name Parties by ideology Ideologies... Jump to: navigation, search 1997 is a common year starting on Wednesday of the Gregorian calendar. ...

If there is a candidate who is preferred pairwise over the other candidates, when compared in turn with each of the others, the Schulze method guarantees that that candidate will win. Because of this property, the Schulze method is (by definition) a Condorcet method. Note that this is different from some other preference voting systems such as Borda and Instant-runoff voting, which do not satisfy the Condorcet criterion; in these systems the Condorcet winner may lose. Jump to: navigation, search Any election method conforming to the Condorcet criterion is known as a Condorcet method. ... The Borda count is a voting system used for single-winner elections in which each voter rank-orders the candidates. ... When the Single Transferable Vote voting system is applied to a single-winner election it is sometimes called instant-runoff voting (IRV), as it is much like holding a series of runoff elections in which the lowest polling candidate is eliminated in each round until someone receives majority vote. ... The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ...

Many different heuristics for the Schulze method have been proposed. The most important heuristics are the path heuristic and the Schwartz heuristic.

The path heuristic

Each ballot contains a complete list of all candidates. Each voter ranks these candidates in order of preference. The individual voter may give the same preference to more than one candidate and he may keep candidates unranked. When a given voter does not rank all candidates, then it is presumed that this voter strictly prefers all ranked candidates to all not ranked candidates and that this voter is indifferent between all not ranked candidates.

Procedure

Suppose d[V,W] is the number of voters who strictly prefer candidate V to candidate W.

A path from candidate X to candidate Y of strength z is an ordered set of candidates C(1),...,C(n) with the following four properties:

1. C(1) is identical to X.
2. C(n) is identical to Y.
3. For i = 1,...,(n-1): d[C(i),C(i+1)] > d[C(i+1),C(i)].
4. For i = 1,...,(n-1): d[C(i),C(i+1)] ≥ z.

If there is a p such that there is a path from candidate A to candidate B of strength p and no path from candidate B to candidate A of strength p, then candidate A disqualifies candidate B.

Candidate D is a potential winner if and only if there is no candidate E such that candidate E disqualifies candidate D.

Examples

A path from candidate X to candidate Y is an ordered set of candidates C(1),...,C(n) with the following three properties:

1. C(1) is identical to X.
2. C(n) is identical to Y.
3. For i = 1,...,(n-1): d[C(i),C(i+1)] > d[C(i+1),C(i)].

The strength of the path C(1),...,C(n) is min { d[C(i),C(i+1)] | i = 1,...,(n-1) }.

In other words: The strength of a path is the strength of its weakest link.

p[A,B] : = max { min { d[C(i),C(i+1)] | i = 1,...,(n-1) } | C(1),...,C(n) is a path from candidate A to candidate B }.

In other words: p[A,B] is the strength of the strongest path from candidate A to candidate B.

Then the Schulze method can be described as follows: Candidate A is a potential winner if and only if p[A,B] ≥ p[B,A] for every other candidate B.

Example 1

Example (45 voters; 5 candidates):

5 ACBED

5 ADECB

8 BEDAC

3 CABED

7 CAEBD

2 CBADE

7 DCEBA

8 EBADC

The critical defeats of the strongest paths are underlined.

Candidate E is a potential winner, because p[E,X] ≥ p[X,E] for every other candidate X.

Example 2

Example (30 voters; 4 candidates):

5 ACBD

2 ACDB

3 ADCB

4 BACD

3 CBDA

3 CDBA

1 DACB

5 DBAC

4 DCBA

The critical defeats of the strongest paths are underlined.

Candidate D is a potential winner, because p[D,X] ≥ p[X,D] for every other candidate X.

Example 3

Example (30 voters; 5 candidates):

3 ABDEC

5 ADEBC

1 ADECB

2 BADEC

2 BDECA

4 CABDE

6 CBADE

2 DBECA

5 DECAB

The critical defeats of the strongest paths are underlined.

Candidate B is a potential winner, because p[B,X] ≥ p[X,B] for every other candidate X.

Example 4

Example (9 voters; 4 candidates):

3 ABCD

2 DABC

2 DBCA

2 CBDA

The critical defeats of the strongest paths are underlined.

Candidate B and candidate D are potential winners, because p[B,X] ≥ p[X,B] for every other candidate X and p[D,Y] ≥ p[Y,D] for every other candidate Y.

The Schwartz heuristic

The Schwartz set

The definition of a Schwartz set, as used in the Schulze method, is as follows: The Schwartz set is a term used in regard to voting systems. ...

1. An unbeaten set is a set of candidates of whom none is beaten by anyone outside that set.
2. An innermost unbeaten set is an unbeaten set that doesn't contain a smaller unbeaten set.
3. The Schwartz set is the set of candidates who are in innermost unbeaten sets.

Procedure

The voters cast their ballots by ranking the candidates according to their preferences, just like for any other Condorcet election.

The Schulze method uses Condorcet pairwise matchups between the candidates and a winner is chosen in each of the matchups. Jump to: navigation, search Any election method conforming to the Condorcet criterion is known as a Condorcet method. ...

From there, the Schulze method operates as follows to select a winner (or create a ranked list):

1. Calculate the Schwartz set based only on undropped defeats.
2. If there are no defeats among the members of that set then they (plural in the case of a tie) win and the count ends.
3. Otherwise, drop the weakest defeat among the candidates of that set. Go to 1.

An example

The situation

Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county): State nickname: Volunteer State Other U.S. States Capital Nashville Largest city Memphis Governor Phil Bredesen (D) Senators Bill Frist (R) Lamar Alexander (R) Official languages English Area 109,247 km² (36th)  - Land 106,846 km²  - Water 2,400 km² (2. ...

• Memphis (Shelby County): 826,330
• Nashville (Davidson County): 510,784
• Chattanooga (Hamilton County): 285,536
• Knoxville (Knox County): 335,749

Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows: Census. ...

The results would be tabulated as follows:

• [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption
• [NP] indicates voters who expressed no preference between either candidate

Pairwise winners

First, list every pair, and determine the winner:

Note that absolute counts of votes can be used, or percentages of the total number of votes; it makes no difference.

Dropping

Next we start with our list of cities and their matchup wins/defeats

• Nashville 3-0
• Chattanooga 2-1
• Knoxville 1-2
• Memphis 0-3

Technically, the Schwartz set is simply Nashville as it beat all others 3 to 0.

Therefore, Nashville is the winner.

Ambiguity resolution example

Let's say there was an ambiguity. For a simple situation involving candidates A, B, and C.

• A > B 72%
• B > C 68%
• C > A 52%

In this situation the Schwartz set is A, B, and C as they all beat someone.

Schulze then says to drop the weakest defeat, so we drop C > A and are left with

• A > B 72% (as C has been removed)

Therefore, A is the winner.

(It may be more accessible to phrase that as "drop the weakest win", though purists may complain.)

Summary

In the (first) example election, the winner is Nashville. This would be true for any Condorcet method. Using the first-past-the-post system and some other systems, Memphis would have won the election by having the most people, even though Nashville won every simulated pairwise election outright. Nashville would also have been the winner in a Borda count. Using Instant-runoff voting in this example would result in Knoxville winning, even though more people preferred Nashville over Knoxville. Jump to: navigation, search Any election method conforming to the Condorcet criterion is known as a Condorcet method. ... The first-past-the-post electoral system is a voting system for single-member districts, variously called first-past-the-post (FPTP or FPP), winner-take-all, plurality voting, or relative majority. ... The Borda count is a voting system used for single-winner elections in which each voter rank-orders the candidates. ... When the Single Transferable Vote voting system is applied to a single-winner election it is sometimes called instant-runoff voting (IRV), as it is much like holding a series of runoff elections in which the lowest polling candidate is eliminated in each round until someone receives majority vote. ...

Satisfied and failed criteria

Satisfied criteria

If winning votes is used as the definition of defeat strength, the Schulze method satisfies the following criteria: Jump to: navigation, search Any election method conforming to the Condorcet criterion is known as a Condorcet method. ...

1. Majority criterion
2. Monotonicity criterion (a.k.a. mono-raise)
3. Condorcet criterion (a.k.a. Condorcet winner criterion)
4. Condorcet loser criterion
5. Independence of clones (See clones.)
6. Pareto criterion
7. Mutual majority criterion
8. Local independence from irrelevant alternatives (see below)
9. Weak defensive strategy criterion
10. Strong defensive strategy criterion
11. Summability criterion
12. non-dictatorial
13. non-imposition (a.k.a. citizen sovereignty)
14. Woodall's plurality criterion
15. Smith criterion (a.k.a. Generalized Condorcet criterion)
16. Schwartz criterion
17. Strategy-Free criterion
18. Generalized Strategy-Free criterion
19. Woodall's CDTT criterion
20. Minimal Defense criterion
21. Resolvability criterion
22. Reversal symmetry
23. mono-append
24. mono-add-plump

The majority criterion is a voting system criterion, used to objectively compare voting systems. ... A voting system is monotonic if it satisfies the monotonicity criterion, given below. ... AKA is an initialism for Also Known As. ... The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ... AKA is an initialism for Also Known As. ... The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ... Given a vote where voters rank options in order of preference, a Condorcet loser is an option that loses all of its pairwise comparisons. ... Strategic nomination is the manipulation of an election through its candidate set (compare this to tactical voting, where the manipulation comes from the voters). ... In the analysis of voting systems, a clone is an additional candidate who appears as attractive to each voter as an existing candidate. ... Pareto efficiency, or Pareto optimality, is a central theory in economics with broad applications in game theory, engineering and the social sciences. ... The mutual majority criterion is a criterion used to compare voting systems. ... Independence of irrelevant alternatives is an axiom often adopted by social scientists as a basic condition of rationality. ... Jump to: navigation, search The weak defensive strategy criterion is a voting system criterion devised by Mike Ossipoff. ... The introduction of this article does not provide enough context for readers unfamiliar with the subject. ... The summability criterion is a voting system criterion, used to objectively compare voting systems. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system meets all of a certain set of criteria when there are three or more choices. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system meets all of a certain set of criteria when there are three or more choices. ... AKA is an initialism for Also Known As. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system meets all of a certain set of criteria when there are three or more choices. ... Plurality criterion is a voting system criterion devised by Douglas Woodall. ... In voting systems, the Smith set is the smallest set of candidates in a particular election who, when paired off in pairwise elections, can beat all other candidates outside the set. ... AKA is an initialism for Also Known As. ... The Generalized Condorcet criterion or Smith criterion for a voting system is that it picks the winner from the Smith set, the smallest set of candidates such that every member of the set is preferred to every candidate not in the set. ... The Schwartz set is a term used in regard to voting systems. ... The strategy-free criterion is a voting system criterion devised by Mike Ossipoff for evaluating voting systems. ... Definitions A sincere vote is one with no falsified preferences or preferences left unspecified when the election method allows them to be specified (in addition to the preferences already specified). ... The Resolvability criterion pertains to Condorcet methods where at least in those cases in which there are no pairwise ties and no pairwise defeats of equal strength, the winner must be unique. ... Reversal symmetry is a voting method criterion that is stated as follows: If a candidate A is the unique winner, and the individual preferences of each voter are inverted, then candidate A must not be elected. ...

Failed criteria

The Schulze method violates the following criteria:

1. all criteria that are incompatible with the Condorcet criterion (e.g. independence from irrelevant alternatives, participation, consistency, invulnerability to compromising, invulnerability to burying, Favorite Betrayal criterion, later-no-harm,later-no-help)
2. The Schulze method doesn't guarantee that the winner is always chosen from the uncovered set.
3. Independence from Pareto-dominated alternatives (IPDA)
4. mono-remove-bottom
5. mono-add-top

The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ... Independence of irrelevant alternatives is an axiom often adopted by social scientists as a basic condition of rationality. ... Statement of Criterion Adding one or more ballots that vote X over Y should never change the winner from X to Y. Complying Methods Plurality voting, Approval voting, Cardinal Ratings, Borda count, and Woodalls DAC method all pass the Participation Criterion. ... A voting system is consistent if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same alternative being selected, an election of the entire electorate also selects that alternative. ... In voting systems, tactical voting (or strategic voting) occurs when a voter misrepresents his or her sincere preferences in order to gain a more favorable outcome. ... In voting systems, tactical voting (or strategic voting) occurs when a voter misrepresents his or her sincere preferences in order to gain a more favorable outcome. ... In social choice theory, the favorite betrayal criterion (or FBC) is a voting system criterion devised by Mike Ossipoff for evaluating voting systems. ...

Independence of irrelevant alternatives

The Schulze method fails independence from irrelevant alternatives. However, the method adheres to a less strict property is sometimes called local independence from irrelevant alternatives. It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the Smith set. Local IIA implies the Condorcet criterion. Independence of irrelevant alternatives is an axiom often adopted by social scientists as a basic condition of rationality. ... Independence of irrelevant alternatives is an axiom often adopted by social scientists as a basic condition of rationality. ... In voting systems, the Smith set is the smallest set of candidates in a particular election who, when paired off in pairwise elections, can beat all other candidates outside the set. ... Independence of irrelevant alternatives is an axiom often adopted by social scientists as a basic condition of rationality. ... The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ...

Use of the Schulze method

The Schulze method is not currently used in government elections. However, it is starting to receive support in some public organizations. Organizations which currently use the Schulze method are:

• the Blitzed project (see [1])
• the CivicEvolution project (see [2] (PDF))
• the College of Science Students' Advisory Council (COSSAC)
• the Debian project (see [3] and [4])
  • Furthermore, the fact that the Schulze method is a part of Debian's voting software ("Debian Vote Engine", Devotee) means that it is the standard voting system in all Debian user groups (DUGs).
• the Democratic Experience (DemExp) project (see section 30.6 of this paper (PDF))
• the EuroBillTracker project (see [5] and [6])
• the Five-Second Crossword Competition (FSCC) (see [7])
• the Free Software Club of Kirksville (FSCK) (see [8])
• the Gentoo Linux project (see [9], [10], [11], and [12])
• the Glasnost / Easter Eggs / libre-entreprise project
• the Great Group Cruise (GGC) project (see [13], [14], [15], [16], and [17])
• the Haifa Linux Club (Haifux) (see [18])
• the Johns Hopkins Animation Club (JHAC)
• the Kingman Hall (see [19] and [20])
• the Leader of the Free World (LFW) / Open Elections / Simply Working Preferential Web Election (SWPWE) project (see [21] and [22])
• the Libertarian Party at Colorado State University (LPCSU) (see [23] (PDF))
• the Mathematical Knowledge Management Interest Group (MKM-IG) (see [24] The MKM-IG uses Condorcet with dual dropping. see [25] That means: The Schulze ranking and the ranked pairs ranking are calculated and the winner is the top-ranked candidate of that of these two rankings that has the better Kemeny score.)
• the NationStates Wiki (NSwiki) project (see [26])
• the Park Alumni Society (PAS) (see [27])
• the Sender Policy Framework (SPF) project (see [28] and [29])
• the Software in the Public Interest (SPI) project (see [30])
• the Suburban Otaku Squad (SOS) project (see [31])
• the UserLinux project (see [32] and [33])

Portable Document Format (PDF) is a file format developed by Adobe Systems for representing documents in a manner that is independent of the original application software, hardware, and operating system used to create those documents. ... Debian, organized by the Debian Project, is a widely used distribution of free software developed through the collaboration of volunteers from around the world. ... Debian, organized by the Debian Project, is a widely used distribution of free software developed through the collaboration of volunteers from around the world. ... Portable Document Format (PDF) is a file format developed by Adobe Systems for representing documents in a manner that is independent of the original application software, hardware, and operating system used to create those documents. ... Jump to: navigation, search This article is about the Linux distribution. ... Portable Document Format (PDF) is a file format developed by Adobe Systems for representing documents in a manner that is independent of the original application software, hardware, and operating system used to create those documents. ... Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) is a voting method that selects a single winner using votes that express preferences. ... Software in the Public Interest, Inc. ... UserLinux is a Linux distribution that aims to provide businesses with a freely available, high quality operating system accompanied by certifications, service, and support options designed to encourage productivity and security while reducing overall costs. ...

External resources

Note that these sources may refer to the Schulze method as CSSD, SSD, beatpath, path winner, etc.

Papers

• Election Methods Resource by Blake Cretney
• The Maximize Affirmed Majorities voting procedure (MAM) by Steve Eppley
• A Survey of Basic Voting Methods by James Green-Armytage
• Social Choice Under Incomplete, Cyclic Preferences (PDF) by Jobst Heitzig
• Voting Systems (PDF) by Paul E. Johnson
• Descriptions of ranked-ballot voting methods by Rob LeGrand
• Accurate Democracy by Rob Loring
• Single-Winner Methods by Mike Ossipoff
• A New Monotonic and Clone-Independent Single-Winner Election Method (mirrors: 1, 2) (PDF) by Markus Schulze
• Schulze-Methode by Markus Schulze
• Election Systems (PDF) by Peter A. Taylor
• Election Methods and Criteria by Kevin Venzke
• The Debian Voting System by Jochen Voss
• election-methods: a mailing list containing technical discussions about election methods

Portable Document Format (PDF) is a file format developed by Adobe Systems for representing documents in a manner that is independent of the original application software, hardware, and operating system used to create those documents. ... Portable Document Format (PDF) is a file format developed by Adobe Systems for representing documents in a manner that is independent of the original application software, hardware, and operating system used to create those documents. ... Portable Document Format (PDF) is a file format developed by Adobe Systems for representing documents in a manner that is independent of the original application software, hardware, and operating system used to create those documents. ... Wikipedia does not have an article with this exact name. ... Portable Document Format (PDF) is a file format developed by Adobe Systems for representing documents in a manner that is independent of the original application software, hardware, and operating system used to create those documents. ...

Software

• Voting Software Project by Blake Cretney
• Condorcet with Dual Dropping Perl Scripts by Mathew Goldstein
• Condorcet Voting Calculator by Eric Gorr
• A different way to vote by Anguo Ma
• Haskell Condorcet Module by Evan Martin
• Condorcet Internet Voting Service (CIVS) by Andrew Myers
• BetterPolls.com by Brian Olson