Schulze method

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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.

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. 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.

Contents

[edit] The path heuristic

Example Schulze method ballot
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.

[edit] 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. p is the minimum of all d[C(i),C(i+1)] for i = 1,...,(n-1).

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.

[edit] Implementation

Suppose C is the number of candidates. Then the strengths of the strongest paths can be calculated with the Floyd–Warshall algorithm.

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

Output: Candidate i is a potential winner if and only if "winner[i] = true".

 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
38
39 for i : = 1 to C
40 begin
41    winner[i] : = true
42 end
43
44 for i : = 1 to C
45 begin
46    for j : = 1 to C
47    begin
48       if ( i ≠ j ) then
49       begin
50          if ( p[j,i] > p[i,j] ) then
51          begin
52             winner[i] : = false
53          end
54       end
55    end
56 end

[edit] Examples

[edit] 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
d[*,A] d[*,B] d[*,C] d[*,D] d[*,E]
d[A,*] 20 26 30 22
d[B,*] 25 16 33 18
d[C,*] 19 29 17 24
d[D,*] 15 12 28 14
d[E,*] 23 27 21 31
The matrix of pairwise defeats looks as follows:

The critical defeats of the strongest paths are underlined.

... to A ... to B ... to C ... to D ... to E
from A ... A-(30)-D-(28)-C-(29)-B A-(30)-D-(28)-C A-(30)-D A-(30)-D-(28)-C-(24)-E
from B ... B-(25)-A B-(33)-D-(28)-C B-(33)-D B-(33)-D-(28)-C-(24)-E
from C ... C-(29)-B-(25)-A C-(29)-B C-(29)-B-(33)-D C-(24)-E
from D ... D-(28)-C-(29)-B-(25)-A D-(28)-C-(29)-B D-(28)-C D-(28)-C-(24)-E
from E ... E-(31)-D-(28)-C-(29)-B-(25)-A E-(31)-D-(28)-C-(29)-B E-(31)-D-(28)-C E-(31)-D
The strongest paths are:
p[*,A] p[*,B] p[*,C] p[*,D] p[*,E]
p[A,*] 28 28 30 24
p[B,*] 25 28 33 24
p[C,*] 25 29 29 24
p[D,*] 25 28 28 24
p[E,*] 25 28 28 31
The strengths of the strongest paths are:

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

[edit] 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
d[*,A] d[*,B] d[*,C] d[*,D]
d[A,*] 11 20 14
d[B,*] 19 9 12
d[C,*] 10 21 17
d[D,*] 16 18 13
The matrix of pairwise defeats looks as follows:

The critical defeats of the strongest paths are underlined.

... to A ... to B ... to C ... to D
from A ... A-(20)-C-(21)-B A-(20)-C A-(20)-C-(17)-D
from B ... B-(19)-A B-(19)-A-(20)-C B-(19)-A-(20)-C-(17)-D
from C ... C-(21)-B-(19)-A C-(21)-B C-(17)-D
from D ... D-(18)-B-(19)-A D-(18)-B D-(18)-B-(19)-A-(20)-C
The strongest paths are:
p[*,A] p[*,B] p[*,C] p[*,D]
p[A,*] 20 20 17
p[B,*] 19 19 17
p[C,*] 19 21 17
p[D,*] 18 18 18
The strengths of the strongest paths are:

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

[edit] 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
d[*,A] d[*,B] d[*,C] d[*,D] d[*,E]
d[A,*] 18 11 21 21
d[B,*] 12 14 17 19
d[C,*] 19 16 10 10
d[D,*] 9 13 20 30
d[E,*] 9 11 20 0
The matrix of pairwise defeats looks as follows:

The critical defeats of the strongest paths are underlined.

... to A ... to B ... to C ... to D ... to E
from A ... A-(18)-B A-(21)-D-(20)-C A-(21)-D A-(21)-E
from B ... B-(19)-E-(20)-C-(19)-A B-(19)-E-(20)-C B-(19)-E-(20)-C-(19)-A-(21)-D B-(19)-E
from C ... C-(19)-A C-(19)-A-(18)-B C-(19)-A-(21)-D C-(19)-A-(21)-E
from D ... D-(20)-C-(19)-A D-(20)-C-(19)-A-(18)-B D-(20)-C D-(30)-E
from E ... E-(20)-C-(19)-A E-(20)-C-(19)-A-(18)-B E-(20)-C E-(20)-C-(19)-A-(21)-D
The strongest paths are:
p[*,A] p[*,B] p[*,C] p[*,D] p[*,E]
p[A,*] 18 20 21 21
p[B,*] 19 19 19 19
p[C,*] 19 18 19 19
p[D,*] 19 18 20 30
p[E,*] 19 18 20 19
The strengths of the strongest paths are:

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

[edit] Example 4

Example (9 voters; 4 candidates):

3 ABCD
2 DABC
2 DBCA
2 CBDA
d[*,A] d[*,B] d[*,C] d[*,D]
d[A,*] 5 5 3
d[B,*] 4 7 5
d[C,*] 4 2 5
d[D,*] 6 4 4
The matrix of pairwise defeats looks as follows:

The critical defeats of the strongest paths are underlined.

... to A ... to B ... to C ... to D
from A ... A-(5)-B A-(5)-C A-(5)-C-(5)-D
from B ... B-(5)-D-(6)-A B-(7)-C B-(5)-D
from C ... C-(5)-D-(6)-A C-(5)-D-(6)-A-(5)-B C-(5)-D
from D ... D-(6)-A D-(6)-A-(5)-B D-(6)-A-(5)-C
The strongest paths are:
p[*,A] p[*,B] p[*,C] p[*,D]
p[A,*] 5 5 5
p[B,*] 5 7 5
p[C,*] 5 5 5
p[D,*] 6 5 5
The strengths of the strongest paths are:

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.

[edit] The Schwartz set heuristic

[edit] 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.

[edit] 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(s) (plural in the case of ex æquo defeats) among the candidates of that set. Go to 1.

[edit] An example

[edit] The situation

Tennessee's four cities are spread throughout the state

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:

42% of voters
(close to Memphis)		 26% of voters
(close to Nashville)		 15% of voters
(close to Chattanooga)		 17% of voters
(close to Knoxville)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

The results would be tabulated as follows:

Pairwise election results
A
Memphis Nashville Chattanooga Knoxville
B Memphis [A] 58%
[B] 42%
 [A] 58%
[B] 42%
 [A] 58%
[B] 42%
Nashville [A] 42%
[B] 58%
 [A] 32%
[B] 68%
 [A] 32%
[B] 68%
Chattanooga [A] 42%
[B] 58%
 [A] 68%
[B] 32%
 [A] 17%
[B] 83%
Knoxville [A] 42%
[B] 58%
 [A] 68%
[B] 32%
 [A] 83%
[B] 17%
Pairwise election results (won-lost-tied): 0-3-0 3-0-0 2-1-0 1-2-0
Votes against in worst pairwise defeat: 58% N/A 68% 83%
  • [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

[edit] Pairwise winners

First, list every pair, and determine the winner:

Pair Winner
Memphis (42%) vs. Nashville (58%) Nashville 58%
Memphis (42%) vs. Chattanooga (58%) Chattanooga 58%
Memphis (42%) vs. Knoxville (58%) Knoxville 58%
Nashville (68%) vs. Chattanooga (32%) Nashville 68%
Nashville (68%) vs. Knoxville (32%) Nashville 68%
Chattanooga (83%) vs. Knoxville (17%) Chattanooga: 83%

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

[edit] 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.

[edit] 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 margin, 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.

[edit] 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.

[edit] Satisfied and failed criteria

[edit] Satisfied criteria

The Schulze method satisfies the following criteria:

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

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

  1. Woodall's plurality criterion
  2. Woodall's CDTT criterion

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

  1. Symmetric-completion

[edit] Failed criteria

The Schulze method violates the following criteria:

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

[edit] 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 Local independence of 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.

[edit] Comparison with other preferential single-winner election methods

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

Monotonic Condorcet winner Condorcet loser Majority Mutual majority Clone independence Reversal symmetry Polynomial time
Schulze Yes Yes Yes Yes Yes Yes Yes Yes
Ranked Pairs Yes Yes Yes Yes Yes Yes Yes Yes
Kemeny-Young Yes Yes Yes Yes Yes No Yes No
MiniMax Yes Yes No Yes No No No Yes
Nanson No Yes Yes Yes Yes No Yes Yes
Baldwin No Yes Yes Yes Yes No No Yes
Instant-runoff voting No No Yes Yes Yes Yes No Yes
Coombs No No Yes Yes Yes No No Yes
Contingent voting No No Yes Yes No No No Yes
Sri Lankan contingent voting No No No Yes No No No Yes
Supplementary voting No No No Yes No No No Yes
Borda Yes No Yes No No No Yes Yes
Bucklin Yes No No Yes Yes No No Yes
Plurality Yes No No Yes No No No Yes
Anti-plurality Yes No No No No No No Yes

The difference between the Schulze method and the Ranked Pairs method is discussed in section 9 of this paper.

[edit] History of the Schulze method

The Schulze method was developed by Markus Schulze in 1997. The first time that the Schulze method was discussed in a public mailing list was in 1998 [1] [2] [3] [4] [5]. In the following years, the Schulze method has been adopted e.g. by Software in the Public Interest (2003), Debian (2003), UserLinux (2003), Gentoo (2005), TopCoder (2005), and Sender Policy Framework (2005). The first books on the Schulze method were written by Tideman (2006) and by Stahl and Johnson (2007).

[edit] Use of the Schulze method

sample ballot for Wikimedia's Board of Trustees elections
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:

[edit] External resources

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

[edit] General

[edit] Tutorials

[edit] Advocacy

[edit] Research papers

[edit] Books

[edit] Software

[edit] Legislative projects