Schulze method.html



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.

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 candidate will win. Because of this property, the Schulze method is (by definition) a Condorcet method.

Currently, the Schulze method is the most wide-spread Condorcet method (list). The Schulze method is used by several organizations including Wikimedia, Debian, Gentoo, and Software in the Public Interest.

Many different heuristics for computing the Schulze method have been proposed. The most important heuristics are the path heuristic and the Schwartz set heuristic that are described below. All heuristics find the same winner and only differ in the details of the computational procedure to determine this winner.

The path heuristic

Example Schulze method ballot

Under the Schulze method (as well as under other preferential single-winner election methods), each ballot contains a complete list of all candidates and the individual voter ranks this list in order of preference. Under a common ballot layout (as shown in the image to the right), ascending numbers are used, whereby the voter places a '1' beside the most preferred candidate, a '2' beside the second-most preferred, and so forth. 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.

The basic idea of the path heuristic for the Schulze method is the concept of indirect defeats, the so-called paths.

If candidate C(1) pairwise beats candidate C(2), candidate C(2) pairwise beats candidate C(3), candidate C(3) pairwise beats candidate C(4), ..., and C(n-1) pairwise beats candidate C(n), then we say that there is a path from candidate C(1) to candidate C(n). The strength of the path C(1),...,C(n) is the weakest pairwise defeat in this sequence.

In other words:

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

• A path is a sequence of candidates C(1),...,C(n) with d[C(i),C(i+1)] > d[C(i+1),C(i)] for all i = 1,...,(n-1).

• The strength of the path C(1),...,C(n) is the minimum of all d[C(i),C(i+1)] for i = 1,...,(n-1).

The strength of the strongest path from candidate A to candidate B is the maximum of the strengths of all paths from candidate A to candidate B.

If (a) the strength of the strongest path from candidate A to candidate B is stronger than the strength of the strongest path from candidate B to candidate A or (b) there is a path from candidate A to candidate B and no path from candidate B to candidate A, then candidate A pairwise beats candidate B indirectly.

Indirect defeats are transitive. That means: If candidate A pairwise beats candidate B indirectly and candidate B pairwise beats candidate C indirectly, then also candidate A pairwise beats candidate C indirectly. Therefore, no tie-breaker is needed for indirect defeats.

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 p is a sequence of candidates C(1),...,C(n) with the following properties:

1. C(1) = X and C(n) = Y.
2. For all i = 1,...,(n-1): d[C(i),C(i+1)] > d[C(i+1),C(i)].
3. For all i = 1,...,(n-1): d[C(i),C(i+1)] ≥ p.

p[A,B], the strength of the strongest path from candidate A to candidate B, is the maximum value such that there is a path from candidate A to candidate B of that strength. If there is no path from candidate A to candidate B at all, then p[A,B] : = 0.

Candidate D is better than candidate E if and only if p[D,E] > p[E,D].

Candidate D is a potential winner if and only if p[D,E] ≥ p[E,D] for every other candidate E.

Remark

It is possible to prove that p[X,Y] > p[Y,X] and p[Y,Z] > p[Z,Y] together imply p[X,Z] > p[Z,X] ¹. Therefore, it is guaranteed (1) that the above definition of "better" really defines a transitive relation and (2) that there is always at least one candidate D with p[D,E] ≥ p[E,D] for every other candidate E.

Examples

Example 1

Example (45 voters; 5 candidates):

5 ACBED (that is, 5 voters have order of preference: A > C > B > E > D)

5 ADECB

8 BEDAC

3 CABED

7 CAEBD

2 CBADE

7 DCEBA

8 EBADC

The graph of pairwise defeats looks as follows:

The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table lists the strongest path from candidate X to candidate Y. 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.

As 25 = p[E,A] > p[A,E] = 24, candidate E is better than candidate A.

As 28 = p[E,B] > p[B,E] = 24, candidate E is better than candidate B.

As 28 = p[E,C] > p[C,E] = 24, candidate E is better than candidate C.

As 31 = p[E,D] > p[D,E] = 24, candidate E is better than candidate D.

As 28 = p[A,B] > p[B,A] = 25, candidate A is better than candidate B.

As 28 = p[A,C] > p[C,A] = 25, candidate A is better than candidate C.

As 30 = p[A,D] > p[D,A] = 25, candidate A is better than candidate D.

As 29 = p[C,B] > p[B,C] = 28, candidate C is better than candidate B.

As 29 = p[C,D] > p[D,C] = 28, candidate C is better than candidate D.

As 33 = p[B,D] > p[D,B] = 28, candidate B is better than candidate D.

Therefore, the Schulze ranking is E > A > C > B > D.

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 graph of pairwise defeats looks as follows:

The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table lists the strongest path from candidate X to candidate Y. 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.

As 18 = p[D,A] > p[A,D] = 17, candidate D is better than candidate A.

As 18 = p[D,B] > p[B,D] = 17, candidate D is better than candidate B.

As 18 = p[D,C] > p[C,D] = 17, candidate D is better than candidate C.

As 20 = p[A,B] > p[B,A] = 19, candidate A is better than candidate B.

As 20 = p[A,C] > p[C,A] = 19, candidate A is better than candidate C.

As 21 = p[C,B] > p[B,C] = 19, candidate C is better than candidate B.

Therefore, the Schulze ranking is D > A > C > B.

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 graph of pairwise defeats looks as follows:

The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table lists the strongest path from candidate X to candidate Y. 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.

As 19 = p[B,A] > p[A,B] = 18, candidate B is better than candidate A.

As 19 = p[B,C] > p[C,B] = 18, candidate B is better than candidate C.

As 19 = p[B,D] > p[D,B] = 18, candidate B is better than candidate D.

As 19 = p[B,E] > p[E,B] = 18, candidate B is better than candidate E.

As 20 = p[A,C] > p[C,A] = 19, candidate A is better than candidate C.

As 21 = p[A,D] > p[D,A] = 19, candidate A is better than candidate D.

As 21 = p[A,E] > p[E,A] = 19, candidate A is better than candidate E.

As 20 = p[D,C] > p[C,D] = 19, candidate D is better than candidate C.

As 30 = p[D,E] > p[E,D] = 19, candidate D is better than candidate E.

As 20 = p[E,C] > p[C,E] = 19, candidate E is better than candidate C.

Therefore, the Schulze ranking is B > A > D > E > C.

Example 4

Example (9 voters; 4 candidates):

3 ABCD

2 DABC

2 DBCA

2 CBDA

The graph of pairwise defeats looks as follows:

The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table lists the strongest path from candidate X to candidate Y. 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.

As 7 = p[B,C] > p[C,B] = 5, candidate B is better than candidate C.

As 6 = p[D,A] > p[A,D] = 5, candidate D is better than candidate A.

Possible Schulze rankings are B > C > D > A, B > D > A > C, B > D > C > A, D > A > B > C, D > B > A > C, and D > B > C > A.

Implementation

Suppose C is the number of candidates. Then the strengths of the strongest paths can be calculated with the Floyd–Warshall algorithm. The following Pascal-like pseudocode illustrates the determination of such a path.

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

Output: p[i,j] is the strength of the strongest path from candidate i to candidate j.

 1 for i : = 1 to C
 2 begin
 3    for j : = 1 to C
 4    begin
 5       if ( i ≠ j ) then
 6       begin
 7          if ( d[i,j] > d[j,i] ) then
 8          begin
 9             p[i,j] : = d[i,j]
10          end
11          else
12          begin
13             p[i,j] : = 0
14          end
15       end
16    end
17 end
18
19 for i : = 1 to C
20 begin
21    for j : = 1 to C
22    begin
23       if ( i ≠ j ) then
24       begin
25          for k : = 1 to C
26          begin
27             if ( i ≠ k ) then
28             begin   
29                if ( j ≠ k ) then
30                begin
31                   p[j,k] : = max { p[j,k]; min { p[j,i]; p[i,k] } }
32                end
33             end
34          end
35       end
36    end
37 end

The Schwartz set heuristic

The Schwartz set

The definition of a Schwartz set, as used in the Schulze method, is as follows:

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.

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. If several defeats tie as weakest, drop all of them. Go to 1.

An example

The situation

Imagine that Tennessee is having an election 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 these four cities, and that everyone wants to live as near the capital as possible.

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:

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

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, C, and D.

• A > B 68%
• C > A 52%
• A > D 62%
• B > C 72%
• B > D 84%
• C > D 91%

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

• A > B 68%
• B > C 72%
• C > A 52%

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

• A > B 68%
• B > C 72%

The new Schwartz set is now A, as it is unbeaten by anyone outside its set. With A in a Schwartz set by itself, it is now the winner.

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. Instant-runoff voting in this example would select Knoxville, even though more people preferred Nashville than Knoxville.

Satisfied and failed criteria

Satisfied criteria

The Schulze method satisfies the following criteria:

• Unrestricted domain
• Non-imposition (a.k.a. citizen sovereignty)
• Non-dictatorship
• Pareto criterion ²
• Monotonicity criterion ³
• Majority criterion
• Majority loser criterion
• Condorcet criterion
• Condorcet loser criterion
• Schwartz criterion
• Smith criterion
• Independence of Smith-dominated alternatives
• Mutual majority criterion
• Independence of clones ⁴
• Reversal symmetry ⁵
• Mono-append
• Mono-add-plump
• Resolvability criterion ⁶
• Polynomial runtime ⁷

If winning votes is used as the definition of defeat strength, it also satisfies:

• Woodall's plurality criterion ⁸
• Woodall's CDTT criterion ⁸

If margins as defeat strength is used, it also satisfies:

• Symmetric-completion

Failed criteria

The Schulze method violates the following criteria:

• All criteria that are incompatible with the Condorcet criterion (e.g. independence of irrelevant alternatives, participation ⁹, consistency, invulnerability to compromising, invulnerability to burying, later-no-harm)

Independence of irrelevant alternatives

The Schulze method fails independence of irrelevant alternatives. However, the method adheres to a less strict property that is sometimes called independence of Smith-dominated 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.

Comparison with other preferential single-winner election methods

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

This is the main difference between the Schulze method and the Ranked Pairs method:

Suppose the MinMax score of a set X of candidates is the strength of the strongest pairwise win of a candidate A ∉ X against a candidate B ∈ X. Then the Schulze method, but not the Ranked Pairs method, guarantees that the winner is always a candidate of the set with minimum MinMax score ¹⁰. So, in some sense, the Schulze method minimizes the strongest pairwise win that has to be overturned when determining the winner.

History of the Schulze method

The Schulze method was developed by Markus Schulze in 1997. The first times that the Schulze method was discussed in a public mailing list were in 1998 ¹¹ and in 2000 ¹². In the following years, the Schulze method has been adopted e.g. by Software in the Public Interest (2003) ¹³, Debian (2003) ¹⁴, Gentoo (2005), TopCoder (2005), Sender Policy Framework (2005), and the French Wikipedia section (2005). The first books on the Schulze method were written by Tideman (2006) and by Stahl and Johnson (2007). In the then following years, the Schulze method has been adopted e.g. by Wikimedia (2008) and KDE (2008).

Use of the Schulze method

sample ballot for Wikimedia's Board of Trustees elections

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:

• Annodex Association ¹⁵
• Blitzed ¹⁶
• BoardGameGeek ¹⁷
• Codex Alpe Adria ¹⁸
• County Highpointers ¹⁹
• Debian ^{14 20}
• EnMasse Forums
• EuroBillTracker ²¹
• Fair Trade Northwest ²²
• Free Hardware Foundation of Italy ²³
• Free Software Foundation Europe (FSFE) ²⁴
• Free Software Foundation Latin America (FSFLA) ²⁵
• Gentoo Foundation ²⁶
• GNU Privacy Guard (GnuPG) ²⁷
• KDE e.V. ²⁸
• Kingman Hall ²⁹
• Kumoricon ³⁰
• League of Professional System Administrators (LOPSA) ³¹
• Libre-Entreprise ³²
• Lumiera/Cinelerra ³³
• Mathematical Knowledge Management Interest Group (MKM-IG) ³⁴
• Music Television (MTV) ³⁵
• North Shore Cyclists (NSC) ³⁶
• OpenCouchSurfing ³⁷
• Pitcher Plant of the Month
• Pittsburgh Ultimate ³⁸
• RPMrepo ³⁹
• Sender Policy Framework (SPF) ⁴⁰
• Software in the Public Interest (SPI) ¹³
• Students for Free Culture ⁴¹
• TopCoder ⁴²
• WikIAC ⁴³
• Wikimedia Foundation ⁴⁴
• Wikipedia in French ⁴⁵
• Wikipedia in Hungarian ⁴⁶
• Wikipedia in Spanish ⁴⁷

Wikimedia Foundation, 2008

In June 2008, the Wikimedia Foundation used the Schulze method for the election to its Board of Trustees ⁴⁴: One vacant seat had to be filled. There were 15 candidates, about 26,000 eligible voters, and 3,019 valid ballots.

As Chen was a clear Condorcet winner, he won the vacant seat. However, there was a tie for sixth to ninth position between Heiskanen, Postlethwaite, Smith, and Saintonge. Heiskanen beat Postlethwaite; Postlethwaite beat Smith; Smith beat Saintonge; Saintonge beat Heiskanen.

Each figure represents the number of voters who ranked the candidate at the left better than the candidate at the top. A figure in green represents a victory in that pairwise comparison by the candidate at the left. A figure in red represents a defeat in that pairwise comparison by the candidate at the left.

Notes