Schulze method

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 unranked candidates and that this voter is indifferent between all unranked 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.

Candidate A pairwise beats candidate B indirectly if either

• 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

• there is a path from candidate A to candidate B and no path from candidate B to candidate A.

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] . 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 matrix of pairwise defeats looks as follows:

	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 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.

The strongest paths are:

	... 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 A ...
from B ...	B-(25)-A		B-(33)-D-(28)-C	B-(33)-D	B-(33)-D-(28)-C-(24)-E	from B ...
from C ...	C-(29)-B-(25)-A	C-(29)-B		C-(29)-B-(33)-D	C-(24)-E	from C ...
from D ...	D-(28)-C-(29)-B-(25)-A	D-(28)-C-(29)-B	D-(28)-C		D-(28)-C-(24)-E	from D ...
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		from E ...
	... to A	... to B	... to C	... to D	... to E

The strengths of 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

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 matrix of pairwise defeats looks as follows:

	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 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.

The strongest paths are:

	... to A	... to B	... to C	... to D
from A ...		A-(20)-C-(21)-B	A-(20)-C	A-(20)-C-(17)-D	from A ...
from B ...	B-(19)-A		B-(19)-A-(20)-C	B-(19)-A-(20)-C-(17)-D	from B ...
from C ...	C-(21)-B-(19)-A	C-(21)-B		C-(17)-D	from C ...
from D ...	D-(18)-B-(19)-A	D-(18)-B	D-(18)-B-(19)-A-(20)-C		from D ...
	... to A	... to B	... to C	... to D

The strengths of 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

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 matrix of pairwise defeats looks as follows:

	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 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.

The strongest paths are:

	... 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 A ...
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 B ...
from C ...	C-(19)-A	C-(19)-A-(18)-B		C-(19)-A-(21)-D	C-(19)-A-(21)-E	from C ...
from D ...	D-(20)-C-(19)-A	D-(20)-C-(19)-A-(18)-B	D-(20)-C		D-(30)-E	from D ...
from E ...	E-(20)-C-(19)-A	E-(20)-C-(19)-A-(18)-B	E-(20)-C	E-(20)-C-(19)-A-(21)-D		from E ...
	... to A	... to B	... to C	... to D	... to E

The strengths of 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

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 matrix of pairwise defeats looks as follows:

	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 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.

The strongest paths are:

	... to A	... to B	... to C	... to D
from A ...		A-(5)-B	A-(5)-C	A-(5)-C-(5)-D	from A ...
from B ...	B-(5)-D-(6)-A		B-(7)-C	B-(5)-D	from B ...
from C ...	C-(5)-D-(6)-A	C-(5)-D-(6)-A-(5)-B		C-(5)-D	from C ...
from D ...	D-(6)-A	D-(6)-A-(5)-B	D-(6)-A-(5)-C		from D ...
	... to A	... to B	... to C	... to D

The strengths of 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

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.

for i : = 1 to C
begin
   for j : = 1 to C
   begin
      if ( i ≠ j ) then
      begin
         if ( d[i,j] > d[j,i] ) then
         begin
            p[i,j] : = d[i,j]
         end
         else
         begin
            p[i,j] : = 0
         end
      end
   end
end

for i : = 1 to C
begin
   for j : = 1 to C
   begin
      if ( i ≠ j ) then
      begin
         for k : = 1 to C
         begin
            if ( i ≠ k ) then
            begin   
               if ( j ≠ k ) then
               begin
                  p[j,k] : = max ( p[j,k], min ( p[j,i], p[i,k] ) )
               end
            end
         end
      end
   end
end

The Schwartz set heuristic

Procedure

The Schwartz set heuristic for the Schulze method is an iterative heuristic.

At each stage, we proceed as follows:

1. For each pair of undropped candidates X and Y: If there is a directed path of undropped links from candidate X to candidate Y, then we write "X → Y"; otherwise we write "not X → Y".
2. For each pair of undropped candidates V and W: If "V → W" and "not W → V", then candidate W is dropped and all links, that start or end in candidate W, are dropped.
3. The weakest undropped link is dropped. If several undropped links tie as weakest, all of them are dropped.

The procedure ends when all links have been dropped. The winners are the undropped candidates.

Example

Example (30 voters; 4 candidates):

3 ACDB

9 BACD

8 CDAB

5 DABC

5 DBCA

The matrix of pairwise defeats looks as follows:

	d[*,A]	d[*,B]	d[*,C]	d[*,D]
d[A,*]		16	17	12
d[B,*]	14		19	9
d[C,*]	13	11		20
d[D,*]	18	21	10

The graph of pairwise defeats looks as follows:

We get: "A → B", "A → C", "A → D", "B → A", "B → C", "B → D", "C → A", "C → B", "C → D", "D → A", "D → B", and "D → C".

As there is no pair of candidates V and W with "V → W" and "not W → V", no candidate can be dropped.

The weakest undropped link (A:B=16:14) is dropped.

Thus, we get:

We get: "A → B", "A → C", "A → D", "B → A", "B → C", "B → D", "C → A", "C → B", "C → D", "D → A", "D → B", and "D → C".

As there is no pair of candidates V and W with "V → W" and "not W → V", no candidate can be dropped.

The weakest undropped link (A:C=17:13) is dropped.

Thus, we get:

We get: "not A → B", "not A → C", "not A → D", "B → A", "B → C", "B → D", "C → A", "C → B", "C → D", "D → A", "D → B", and "D → C".

As "B → A" and "not A → B", candidate A is dropped and all links, that start or end in candidate A, are dropped.

Thus, we get:

The weakest undropped link (B:C=19:11) is dropped.

Thus, we get:

We get: "not B → C", "not B → D", "C → B", "C → D", "D → B", and "not D → C".

As "C → B" and "not B → C", candidate B is dropped and all links, that start or end in candidate B, are dropped; as "C → D" and "not D → C", candidate D is dropped and all links, that start or end in candidate D, are dropped.

As candidate C is the unique undropped candidate, candidate C is the unique winner.

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
• Monotonicity criterion
• 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
• Reversal symmetry
• Mono-append
• Mono-add-plump
• Resolvability criterion
• Polynomial runtime

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

• Woodall's plurality criterion

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 , 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:

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

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 . 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 and in 2000 . In the following years, the Schulze method has been adopted e.g. by Software in the Public Interest (2003) , Debian (2003) , 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
• Blitzed
• BoardGameGeek
• Codex Alpe Adria
• College of Marine Science
• Computer Science Departmental Society for York University (HackSoc)
• County Highpointers
• Debian
• Digital Freedom in Education and Youth
• EnMasse Forums
• EuroBillTracker
• Fair Trade Northwest
• Free Hardware Foundation of Italy
• Free Software Foundation Europe (FSFE)
• Free Software Foundation Latin America (FSFLA)
• Gentoo Foundation
• GNU Privacy Guard (GnuPG)
• Graduate Student Organization at the State University of New York: Computer Science (GSOCS)
• Haskell
• Kanawha Valley Scrabble Club
• KDE e.V.
• Kingman Hall
• Kumoricon
• League of Professional System Administrators (LOPSA)
• Libre-Entreprise
• Lumiera/Cinelerra
• Mason Apollonic Society
• Mathematical Knowledge Management Interest Group (MKM-IG)
• Metalab
• Music Television (MTV)
• netznetz
• North Shore Cyclists (NSC)
• OpenCouchSurfing
• Park Alumni Society (PAS)
• Pitcher Plant of the Month
• Pittsburgh Ultimate
• RPMrepo
• Sender Policy Framework (SPF)
• Software in the Public Interest (SPI)
• Students for Free Culture
• TopCoder
• WikIAC
• Wikimedia Foundation
• Wikipedia in French
• Wikipedia in Hebrew
• Wikipedia in Hungarian
• Wikipedia in Russian
• Wikipedia in Spanish

Wikimedia Foundation, 2008

In June 2008, the Wikimedia Foundation used the Schulze method for the election to its Board of Trustees : 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.

elections to Wikimedia's Board of Trustees in 2008:

	TC	AB	SK	HC	AH	JH	RP	SS	RS	DR	CS	MB	KW	PW	GK
Ting Chen		1086	1044	1108	1135	1151	1245	1190	1182	1248	1263	1306	1344	1354	1421
Alex Bakharev	844		932	984	950	983	1052	1028	990	1054	1073	1109	1134	1173	1236
Samuel Klein	836	910		911	924	983	980	971	941	967	1019	1069	1099	1126	1183
Harel Cain	731	836	799		896	892	964	904	917	959	1007	1047	1075	1080	1160
Ad Huikeshoven	674	781	764	806		832	901	868	848	920	934	987	1022	1030	1115
Jussi-Ville Heiskanen	621	720	712	755	714		841	798	737	827	850	912	970	943	1057
Ryan Postlethwaite	674	702	726	756	772	770		755	797	741	804	837	880	921	1027
Steve Smith	650	694	654	712	729	750	744		778	734	796	840	876	884	1007
Ray Saintonge	629	703	641	727	714	745	769	738		789	812	848	879	899	987
Dan Rosenthal	595	654	609	660	691	724	707	699	711		721	780	844	858	960
Craig Spurrier	473	537	498	530	571	583	587	577	578	600		646	721	695	845
Matthew Bisanz	472	498	465	509	508	534	473	507	531	513	552		653	677	785
Kurt M. Weber	505	535	528	547	588	581	553	573	588	566	595	634		679	787
Paul Williams	380	420	410	435	439	464	426	466	470	471	429	521	566		754
Gregory Kohs	411	412	434	471	461	471	468	461	467	472	491	523	513	541

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

Schulze1: Markus Schulze, A New Monotonic, Clone-Independent, Reversal Symmetric, and Condorcet-Consistent Single-Winner Election Method

External links

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

General

• Proposed Statutory Rules for the Schulze Single-Winner Election Method by Markus Schulze
• A New Monotonic and Clone-Independent Single-Winner Election Method by Markus Schulze (mirrors: [1] [2])
• A New Monotonic, Clone-Independent, Reversal Symmetric, and Condorcet-Consistent Single-Winner Election Method by Markus Schulze
• Free Riding and Vote Management under Proportional Representation by the Single Transferable Vote by Markus Schulze
• Implementing the Schulze STV Method by Markus Schulze
• A New MMP Method by Markus Schulze
• A New MMP Method (Part 2) by Markus Schulze

Tutorials

• Schulze-Methode Template:De icon by the University of Stuttgart

Advocacy

• Election Methods Resource by Blake Cretney
• Voting Methods Survey by James Green-Armytage
• Descriptions of ranked-ballot voting methods by Rob LeGrand
• Accurate Democracy by Rob Loring
• 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

• A Continuous Rating Method for Preferential Voting by Rosa Camps, Xavier Mora, and Laia Saumell (mirror: [3])
• Voting Systems by Paul E. Johnson
• Test Run: Group Determination in Software Testing by James D. McCaffrey
• Distance from Consensus: a Theme and Variations by Tommi Meskanen and Hannu Nurmi
• Analyzing Political Disagreement by Tommi Meskanen and Hannu Nurmi
• Descriptions of voting systems by Warren D. Smith
• Election Systems by Peter A. Taylor
• Personalisierung der Verhältniswahl durch Varianten der Single Transferable Vote Template:De icon by Martin Wilke
• Objective Measures of Preferential Ballot Voting Systems by Barry Wright
• Approaches to Constructing a Stratified Merged Knowledge Base by Anbu Yue, Weiru Liu, and Anthony Hunter

Books

• Saul Stahl and Paul E. Johnson (2007), Understanding Modern Mathematics, Sudbury: Jones and Bartlett Publishers, ISBN 0-7637-3401-2
• Nicolaus Tideman (2006), Collective Decisions and Voting: The Potential for Public Choice, Burlington: Ashgate, ISBN 0-7546-4717-X

Software

• Voting Software Project by Blake Cretney
• Condorcet with Dual Dropping Perl Scripts by Mathew Goldstein
• Condorcet Voting Calculator by Eric Gorr
• Selectricity and RubyVote by Benjamin Mako Hill [4] [5]
• Java implementation of the Schulze method by Thomas Hirsch
• Electowidget by Rob Lanphier
• Votator.com by Louis Philippe Lessard
• Haskell Condorcet Module by Evan Martin
• Condorcet Internet Voting Service (CIVS) by Andrew Myers
• BetterPolls.com by Brian Olson
• OpenSTV by Jeffrey O'Neill

Legislative projects

• Arizonans for Condorcet Ranked Voting [6] [7] [8] [9]
• Schulze Method Phoenix