The Voting series:
This series is part of the Politics and the Election series It has been suggested that ballot be merged into this article or section. ... Politics is a process by which decisions are made within groups. ... An election is a decision making process where people vote for preferred political candidates or parties to act as representatives in government. ...
Voting Voting system Ballot Vote counting systems Approval voting Block voting Borda count Condorcet method Coombs' method Copeland's method Cumulative voting D'Hondt method Droop quota Election threshold Hamilton method Highest averages method Instant-runoff voting Largest remainder method Party list Plurality voting Preferential voting Proportional approval voting Range voting Sainte-Laguë method Schulze method Single transferable vote Spoilt vote
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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. It has been suggested that ballot be merged into this article or section. ... A voting system is a means of choosing between a number of options, based on the input of a number of voters. ... A ballot is a device used to record choices made by voters. ... There exist various methods through which the ballots cast at an election may be counted, prior to applying a voting system to obtain one or more winners. ... On an approval ballot, the voter can vote for any number of candidates. ... Bloc voting (or block voting) (also called Plurality-at-large) refers to a class of voting systems which can be used to elect several representatives from a single constituency. ... The Borda count is a single winner election method in which voters rank candidates in order of preference. ... Any election method conforming to the Condorcet criterion is known as a Condorcet method. ... The Coombs method, created by Clyde Coombs, is a voting system used for single-winner elections in which each voter rank-orders the candidates. ... Copelands method is a Condorcet method in which the winner is determined by finding the candidate with the most pairwise victories. ... A points method ballot design like this one is the most common for governmental elections using cumulative voting. ... The DHondt method is a highest averages method for allocating seats in party-list proportional representation. ... The Droop quota is the quota most commonly used in elections held under the Single Transferable Vote (STV) system. ... In party-list proportional representation systems, an election threshold is a clause that stipulates that a party must receive a minimum percentage of votes, either nationally or within a particular district, to get any seats in the parliament. ... The Hamilton method is a version of the largest remainder method for allocating seats proportionally for representative assemblies with party list voting systems. ... The highest averages method is one way of allocating seats proportionally for representative assemblies with party list voting systems. ... Instant-runoff voting (IRV) (also known as the Alternative Vote and by several other names) is an electoral system used for single winner elections in which voters rank candidates in order of preference. ... The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems. ... Party lists are used in elections to legislatures which use Party-list proportional representation or additional member proportional representation to designate a partys nominees in the at-large portion of the vote. ... 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. ... A preferential voting system (or preference voting system) is an electoral system in which voters rank a list of candidates or group of candidates in order of preference. ... Proportional approval voting (PAV) is a theoretical voting system for multiple-winner elections, in which each voter can vote for as many or as few candidates as the voter chooses. ... Range voting (also called ratings summation, or average voting, or cardinal ratings, or 0-99 voting, or the score system or point system) is a voting system used for single-seat elections. ... The Sainte-Laguë method of the highest average (sometimes identified with Websters method or divisor method with standard rounding) is one way of allocating seats proportionally for representative assemblies with party list voting systems. ... This STV ballot for the Australian Senate illustrates group voting tickets. ... In voting, a ballot paper is considered to be spoilt, void, or null if it is regarded by the election authorities to contain irregularities during vote counting, and hence cannot be recorded as a valid vote. ... A voting system is a means of choosing between a number of options, based on the input of a number of voters. ... 1997 (MCMXCVII) was 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. Any election method conforming to the Condorcet criterion is known as a Condorcet method. ...

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

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Schulze method

Image File history File links Commons-logo. ... Wikimedia Commons logo by Reid Beels The Wikimedia Commons (also called Commons or Wikicommons) is a repository of free content images, sound and other multimedia files. ...

The path heuristic

Each ballot contains a complete list of all candidates. Each voter ranks these candidates in order of preference. Voters may give the same preference to more than one candidate and 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 }.

p[A,B] : = 0 if there is no 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. 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 that the population of Tennessee, a state in the United States, is voting on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in one of these four cities, and that they would like the capital to be established as close to their city as possible. Image File history File links Tennessee_map_for_voting_example. ... Official language(s) English Capital Nashville Largest city Memphis Area  Ranked 36th  - Total 42,169 sq mi (109,247 km²)  - Width 120 miles (195 km)  - Length 440 miles (710 km)  - % water 2. ... In politics, a capital (also called capital city or political capital — although the latter phrase has an alternative meaning based on an alternative sense of capital) is the principal city or town associated with its government. ... In politics, an electorate is the group of people entitled to vote in an election. ...

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this: Flag Seal Nickname: The River City, The Bluff City, M-Town Location Location in Shelby County and the state of Tennessee Coordinates , Government Country State Counties United States Tennessee Shelby County Mayor W. W. Herenton (D) Geographical characteristics Area     City 294. ... Nickname: Music City Location in Davidson County and the state of Tennessee Coordinates: Country United States State Tennessee Counties Davidson County Founded: 1779 Incorporated: 1806 Mayor Bill Purcell (D) Area    - City 526. ... Nickname: The Marble City, K-Town, Big Orange Country, Knox Vegas Location Location within the U.S. State of Tennessee Coordinates , Government Cities in Tennessee Tennessee Mayor Bill Haslam (R) Geographical characteristics Area     City 254. ... Nickname: Scenic City (official), River City, Chatty, Chatt-Town, Chattavegas, The Nooga Location within the U.S. State of Tennessee Cities in Tennessee Tennessee Mayor Ron Littlefield Area    - City 370. ...

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 68%
• B > C 72%
• 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 68% (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. 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 single winner election method in which voters rank candidates in order of preference. ... Instant-runoff voting (IRV) (also known as the Alternative Vote and by several other names) is an electoral system used for single winner elections in which voters rank candidates in order of preference. ...

Satisfied and failed criteria

Satisfied criteria

The Schulze method satisfies the following criteria:

1. Non-imposition (a.k.a. citizen sovereignty)
2. Non-dictatorial
3. Pareto criterion
4. Monotonicity criterion (a.k.a. mono-raise)
5. Majority criterion
6. Condorcet criterion (a.k.a. Condorcet winner criterion)
7. Condorcet loser criterion
8. Smith criterion (a.k.a. Generalized Condorcet criterion)
9. Schwartz criterion
10. Local independence from irrelevant alternatives (see below)
11. Mutual majority criterion
12. Independence of clones (See clones)
13. Reversal symmetry
14. Mono-append
15. Mono-add-plump
16. Resolvability criterion

If winning votes is used as the definition of defeat strength, it also satisfies: In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ... AKA is an initialism for Also Known As. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ... Pareto efficiency, or Pareto optimality, is an important notion in economics with broad applications in game theory, engineering and the social sciences. ... A voting system is monotonic if it satisfies the monotonicity criterion, given below. ... AKA is an initialism for Also Known As. ... The majority criterion is a voting system criterion, used to objectively compare voting systems. ... 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. ... 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. ... Independence of irrelevant alternatives (IIA) is an axiom often adopted by social scientists as a basic condition of rationality. ... The mutual majority criterion is a criterion used to compare voting systems. ... 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. ... 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. ... 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. ... Any election method conforming to the Condorcet criterion is known as a Condorcet method. ...

1. Woodall's plurality criterion

If margins as defeat strength is used, it also satisfies: Plurality criterion is a voting system criterion devised by Douglas Woodall. ... Any election method conforming to the Condorcet criterion is known as a Condorcet method. ...

1. Symmetric-completion

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, later-no-harm, later-no-help)
2. The Schulze method doesn't guarantee that the winner is always chosen from the uncovered set.
3. Mono-remove-bottom
4. 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 (IIA) 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. ...

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 (IIA) is an axiom often adopted by social scientists as a basic condition of rationality. ... Independence of irrelevant alternatives (IIA) 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 (IIA) 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:

• Blitzed [1]
• CivicEvolution [2] (PDF)
• Codex Alpe Adria [3]
• Debian [4] [5] [6]
  • 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).
• Democratic Experience (DemExp) (see section 30.6 of this paper (PDF))
• EnMasse Forums [7]
• EuroBillTracker [8] [9]
• Five-Second Crossword Competition (FSCC) [10]
• Free Software Club of Kirksville (FSCK) [11]
• Free Software Foundation Latin America (FSFLA) [12] [13]
• Gentoo Foundation [14] [15] [16] [17] [18]
• Haifa Linux Club (Haifux) [19]
• Johns Hopkins Animation Club (JHAC) [20] [21]
• Kingman Hall [22] [23]
• Kumoricon [24]
• Leader of the Free World (LFW) / Open Elections / Simply Working Preferential Web Election (SWPWE) [25] [26]
• League of Professional System Administrators (LOPSA) (see article 8.3 of their bylaws)
• Libertarian Party at Colorado State University (LPCSU) [27] (PDF)
• Mathematical Knowledge Management Interest Group (MKM-IG) (The MKM-IG uses Condorcet with dual dropping. 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.) [28] [29] [30]
• NationStates Wiki (NSwiki) [31]
• Park Alumni Society (PAS) [32]
• Pittsburgh Ultimate [33]
• Sender Policy Framework (SPF) [34] [35]
• Software in the Public Interest (SPI) [36]
• TopCoder [37] [38] [39]
• UserLinux [40] [41]
• Wikipédia francophone (The Schulze method is one of three methods recommended for decision-making.) (see here)

Blitzed is an Internet Relay Chat (IRC) network founded on October 31st, 2000 in response to the collapse of WreckedNet. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... 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. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Eurobilltracker is a website designed for tracking Euro banknotes. ... Free Software Foundation Latin America (FSFLA) is the Latin American sister organisation of Free Software Foundation. ... Gentoo Linux is a Linux distribution named after the Gentoo Penguin. ... Kingman Hall is a student-run co-operative house for 50 residents, known as Kingmanites, in Berkeley, California, owned by the University Students Cooperative Association. ... Kumoricon is an anime convention in the Portland Metro Area. ... The League of Professional System Administrators, or LOPSA, is a non-profit organization. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... It has been suggested that Maximize Affirmed Majorities be merged into this article or section. ... In computing, Sender Policy Framework (SPF) is an extension to the Simple Mail Transfer Protocol (SMTP). ... Software in the Public Interest, Inc. ... A topcoder is a high-quality programmer or (especially if spelled TopCoder) one who competes in online programming contests at topcoder. ... 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.

General

• A New Monotonic and Clone-Independent Single-Winner Election Method (PDF) by Markus Schulze (mirrors: [42] [43])
• A New Monotonic, Clone-Independent, Reversal Symmetric, and Condorcet-Consistent Single-Winner Election Method (PDF) by Markus Schulze
• Free Riding and Vote Management under Proportional Representation by the Single Transferable Vote (PDF) by Markus Schulze
• Implementing the Schulze STV Method (PDF) by Markus Schulze

To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Advocacy

• 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
• Descriptions of ranked-ballot voting methods by Rob LeGrand
• Accurate Democracy by Rob Loring
• Single-Winner Methods by Mike Ossipoff
• Schulze beatpaths method by Warren D. Smith
• 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

Research papers

• Social Choice Under Incomplete, Cyclic Preferences (PDF) by Jobst Heitzig
• Voting Systems (PDF) by Paul E. Johnson
• Distance from Consensus: a Theme and Variations (PDF) by Tommi Meskanen and Hannu Nurmi
• Descriptions of voting systems (PDF) by Warren D. Smith
• Election Systems (PDF) by Peter A. Taylor

To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Books

• Collective Decisions and Voting: The Potential for Public Choice by Nicolaus Tideman

T. Nicolaus Tideman (born August 11, 1943 in Chicago, Illinois) is a Professor of Economics at Virginia Polytechnic Institute and State University. ...

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