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 1 The path heuristic 1.1 Procedure 1.2 Implementation 1.3 Examples 1.3.1 Example 1 1.3.2 Example 2 1.3.3 Example 3 1.3.4 Example 4 2 The Schwartz set heuristic 2.1 The Schwartz set 2.2 Procedure 2.3 An example 2.3.1 The situation 2.3.2 Pairwise winners 2.3.3 Dropping 2.3.4 Ambiguity resolution example 2.3.5 Summary 3 Satisfied and failed criteria 3.1 Satisfied criteria 3.2 Failed criteria 3.2.1 Independence of irrelevant alternatives 4 Comparison with other preferential single-winner election methods 5 History of the Schulze method 6 Use of the Schulze method 6.1 Wikimedia Foundation 7 References 8 External links 8.1 General 8.2 Tutorials 8.3 Advocacy 8.4 Research papers 8.5 Books 8.6 Software 8.7 Legislative projects

[edit] The path heuristic

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. 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.

[edit] 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: 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

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 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 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 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.

[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

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 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 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 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.

[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

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 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 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 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.

[edit] 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 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 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 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.

[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^[clarify] defeats) among the candidates of that set. Go to 1.

[edit] An example

[edit] 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:

42% of voters (close to Memphis)	26% of voters (close to Nashville)	15% of voters (close to Chattanooga)	17% of voters (close to Knoxville)
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville 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:

• Universality
• Non-imposition (a.k.a. citizen sovereignty)
• Non-dictatorship
• Pareto criterion ^[1]
• Monotonicity criterion ^[2]
• Majority criterion
• Condorcet criterion (a.k.a. Condorcet winner criterion)
• Condorcet loser criterion
• Smith criterion
• Schwartz criterion
• Local independence of irrelevant alternatives
• Mutual majority criterion
• Independence of clones ^[3]
• Reversal symmetry ^[4]
• Mono-append
• Mono-add-plump
• Resolvability criterion ^[5]
• Polynomial runtime ^[6]

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

• Woodall's plurality criterion ^[7]
• Woodall's CDTT criterion ^[7]

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

• Symmetric-completion

[edit] 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 ^[8], 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	Participation, Consistency
Schulze	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No
Ranked Pairs	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No
Kemeny-Young	Yes	Yes	Yes	Yes	Yes	No	Yes	No	No
MiniMax	Yes	Yes	No	Yes	No	No	No	Yes	No
Nanson	No	Yes	Yes	Yes	Yes	No	Yes	Yes	No
Baldwin	No	Yes	Yes	Yes	Yes	No	No	Yes	No
Instant-runoff voting	No	No	Yes	Yes	Yes	Yes	No	Yes	No
Coombs	No	No	Yes	Yes	Yes	No	No	Yes	No
Contingent voting	No	No	Yes	Yes	No	No	No	Yes	No
Sri Lankan contingent voting	No	No	No	Yes	No	No	No	Yes	No
Supplementary voting	No	No	No	Yes	No	No	No	Yes	No
Borda	Yes	No	Yes	No	No	No	Yes	Yes	Yes
Bucklin	Yes	No	No	Yes	Yes	No	No	Yes	No
Plurality	Yes	No	No	Yes	No	No	No	Yes	Yes
Anti-plurality	Yes	No	No	No	No	No	No	Yes	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 times that the Schulze method was discussed in a public mailing list were in 1998 ^[9] and in 2000 ^[10]. In the following years, the Schulze method has been adopted e.g. by Software in the Public Interest (2003) ^[11], Debian (2003) ^[12], 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

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 Alma Mater Society of the University of British Columbia (AMS) ^[13] and the Langara Students' Union (LSU) ^[14] are using the Schulze method for their voter-funded media system.
• Annodex Association ^[15]
• Blitzed ^[16]
• BoardGameGeek ^[17]
• Codex Alpe Adria ^[18]
• County Highpointers ^[19]
• Debian ^{[12] [20]}
• EnMasse Forums
• EuroBillTracker ^[21]
• Fair Trade Northwest (see article XI section 2 of their bylaws)
• Free Software Foundation Latin America (FSFLA) ^[22]
• Gentoo Foundation ^[23]
• GNU Privacy Guard (GnuPG) ^[24]
• Kingman Hall ^[25]
• Kumoricon ^[26]
• 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.) ^[27]
• Metalab ^[28]
• Music Television (MTV) ^[29]
• North Shore Cyclists (NSC) ^[30]
• OpenCouchSurfing ^[31]
• Pittsburgh Ultimate ^[32]
• RPMrepo ^[33]
• Sender Policy Framework (SPF) ^[34]
• Software in the Public Interest (SPI) ^[11]
• Students for Free Culture ^[35]
• TopCoder ^[36]
• Wikimedia Foundation ^[37]
• Wikipedia in French (The Schulze method is one of three methods recommended for decision-making.) (see here)
• Wikipedia in Hungarian (see here and here)
• Wikipedia in Spanish (see here)

[edit] Wikimedia Foundation

As of July 2008, the largest Schulze poll was the election to Wikimedia's Board of Trustees in June 2008 ^[37]: 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.

[edit] References

[edit] External links

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

[edit] 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

[edit] Tutorials

• Schulze-Methode (German) by the University of Stuttgart

[edit] 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

[edit] Research papers

• A Continuous Rating Method for Preferential Voting by Rosa Camps, Xavier Mora, and Laia Saumell
• Voting Systems by Paul E. Johnson
• 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 (German) by Martin Wilke
• Approaches to Constructing a Stratified Merged Knowledge Base by Anbu Yue, Weiru Liu, and Anthony Hunter

[edit] Books

• Understanding Modern Mathematics by Saul Stahl and Paul E. Johnson (ISBN 0-7637-3401-2)
• Collective Decisions and Voting: The Potential for Public Choice [3] by Nicolaus Tideman (ISBN 0-7546-4717-X)

[edit] 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
• Java implementation of the Schulze method by Thomas Hirsch
• Electowidget by Rob Lanphier
• Haskell Condorcet Module by Evan Martin
• Condorcet Internet Voting Service (CIVS) by Andrew Myers
• BetterPolls.com by Brian Olson
• OpenSTV by Jeffrey O'Neill

[edit] Legislative projects

• Arizonans for Condorcet Ranked Voting [4] [5] [6] [7] [8] [9]