Schulze method

TheSchulze method(/ˈʃʊltsə/) is asingle winnerranked-choice voting ruledeveloped by Markus Schulze. It is also known as thebeatpath method.The Schulze method is aCondorcet method,which means it will elect a majority-choice candidate if one exists; in other words, if most people rank A above B, A will defeat B (whenever this is possible).

Schulze's method is based on the idea of breakingcyclic tiesby using indirect victories. The idea is that ifAlicebeats Bob, and Bob beats Charlie, then Alice (indirectly) beats Charlie; this kind of indirect win is called abeatpath.

Forproportional representation,asingle transferable vote(STV) variant known asSchulze STValso exists. The Schulze method is used by several organizations includingDebian,Ubuntu,Gentoo,Pirate Partypolitical parties andmany others.It was also used byWikimediaprior to their adoption ofscore voting.

Description of the method

A sample ballot asking voters to order candidates by preference

Schulze's method usesranked ballotswith equal ratings allowed. There are two common (equivalent) descriptions of Schulze's method.

Beatpath explanation

The idea behind Schulze's method is that ifAlicedefeats Bob, and Bob beats Charlie, then Alice "indirectly" defeats Charlie; this kind of indirect win is called a 'beatpath'.

Every beatpath is assigned a particularstrength.The strength of a single-step beatpath from Alice to Bob is just the number of voters who rank Alice over Bob. For a longer beatpath, consisting of multiple "beats", the strength of a beatpath is as strong as its weakest link (i.e. the beat with the smallest number of winning votes).

We say Alice has a "beatpath-win" over Bob if her strongest beatpath to Bob is stronger than all of Bob's beatpaths to Alice. The winner is the candidate who has a beatpath-win over every other candidate.

Markus Schulze proved that this definition of a beatpath-win istransitive;in other words, if Alice has a beatpath-win over Bob, and Bob has a beatpath-win over Charlie, Alice has a beatpath-win over Charlie.As a result, the Schulze method is aCondorcet method,providing a full extension of themajority ruleto any set of ballots.

Iterative description

The Schulze winner can also be constructed iteratively, using a defeat-dropping method:

1. Draw adirected graphwith all the candidates as nodes; label the edges with the number of votes supporting the winner.
2. If there is more than one candidate left:
   • Check if any candidates are tied (and if so, break the ties byrandom ballot).
   • Eliminate all candidates outside themajority-preferred set.
   • Delete the edge closest to being tied.

The winner is the only candidate left at the end of the procedure.

Example

In the following example 45 voters rank 5 candidates.

Number of voters	Order of preference
5	ACBED
5	ADECB
8	BEDAC
3	CABED
7	CAEBD
2	CBADE
7	DCEBA
8	EBADC

The pairwise preferences have to be computed first. For example, when comparingAandBpairwise, there are5+5+3+7=20voters who preferAtoB,and8+2+7+8=25voters who preferBtoA.So$d[A,B]=20$and$d[B,A]=25$.The full set of pairwise preferences is:

Directed graphlabeled with pairwise preferences d[*, *]

Matrix of pairwise preferences

	$d[*,A]$	$d[*,B]$	$d[*,C]$	$d[*,D]$	$d[*,E]$
$d[A,*]$		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 cells for d[X, Y] have a light green background if d[X, Y] > d[Y, X], otherwise the background is light red. There is no undisputed winner by only looking at the pairwise differences here.

Now the strongest paths have to be identified. To help visualize the strongest paths, the set of pairwise preferences is depicted in the diagram on the right in the form of adirected graph.An arrow from the node representing a candidate X to the one representing a candidate Y is labelled with d[X, Y]. To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d[X, Y] > d[Y, X] (i.e. the table cells with light green background), omitting the one in the opposite direction (the table cells with light red background).

One example of computing the strongest path strength is p[B, D] = 33: the strongest path from B to D is the direct path (B, D) which has strength 33. But when computing p[A, C], the strongest path from A to C is not the direct path (A, C) of strength 26, rather the strongest path is the indirect path (A, D, C) which has strength min(30, 28) = 28. Thestrengthof a path is the strength of its weakest link.

For each pair of candidates X and Y, the following table shows the strongest path from candidate X to candidate Y in red, with the weakest link underlined.

Strongest paths

To From	A	B	C	D	E
A	—	A-(30)-D-(28)-C-(29)-B	A-(30)-D-(28)-C	A-(30)-D	A-(30)-D-(28)-C-(24)-E	A
B	B-(25)-A	—	B-(33)-D-(28)-C	B-(33)-D	B-(33)-D-(28)-C-(24)-E	B
C	C-(29)-B-(25)-A	C-(29)-B	—	C-(29)-B-(33)-D	C-(24)-E	C
D	D-(28)-C-(29)-B-(25)-A	D-(28)-C-(29)-B	D-(28)-C	—	D-(28)-C-(24)-E	D
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	—	E
	A	B	C	D	E	From To

Strengths of the strongest paths

	$p[*,A]$	$p[*,B]$	$p[*,C]$	$p[*,D]$	$p[*,E]$
$p[A,*]$		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

Now the output of the Schulze method can be determined. For example, when comparingAandB, since$(28=)p[A,B]>p[B,A](=25)$,for the Schulze method candidateAisbetterthan candidateB.Another example is that$(31=)p[E,D]>p[D,E](=24)$,so candidate E isbetterthan candidate D. Continuing in this way, the result is that the Schulze ranking is$E>A>C>B>D$,andEwins. In other words,Ewins since$p[E,X]\geq p[X,E]$for every other candidate X.

Implementation

The only difficult step in implementing the Schulze method is computing the strongest path strengths. However, this is a well-known problem in graph theory sometimes called thewidest path problem.One simple way to compute the strengths, therefore, is a variant of theFloyd–Warshall algorithm.The followingpseudocodeillustrates the algorithm.

# Input: d[i,j], the number of voters who prefer candidate i to candidate j.
# Output: p[i,j], the strength of the strongest path from candidate i to candidate j.

for i from 1 to C
for j from 1 to C
if i ≠ j then
if d[i,j] > d[j,i] then
p[i,j]:= d[i,j]
else
p[i,j]:= 0

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

This algorithm isefficientand hasrunning timeO(C³)whereCis the number of candidates.

Ties and alternative implementations

When allowing users to have ties in their preferences, the outcome of the Schulze method naturally depends on how these ties are interpreted in defining d[*,*]. Two natural choices are that d[A, B] represents either the number of voters who strictly prefer A to B (A>B), or themarginof (voters with A>B) minus (voters with B>A). But no matter how theds are defined, the Schulze ranking has no cycles, and assuming theds are unique it has no ties.

Although ties in the Schulze ranking are unlikely, they are possible. Schulze's original paper recommended breaking ties byrandom ballot.

There is another alternative way todemonstratethe winner of the Schulze method. This method is equivalent to the others described here, but the presentation is optimized for the significance of steps beingvisually apparentas a human goes through it, not for computation.

1. Make the results table, called the "matrix of pairwise preferences", such as used above in the example. Then, every positive number is a pairwise win for the candidate on that row (and marked green), ties are zeroes, and losses are negative (marked red). Order the candidates by how long they last in elimination.
2. If there is a candidate with no red on their line, they win.
3. Otherwise, draw a square box around the Schwartz set in the upper left corner. It can be described as the minimal "winner's circle" of candidates who do not lose to anyone outside the circle. Note that to the right of the box there is no red, which means it is a winner's circle, and note that within the box there is no reordering possible that would produce a smaller winner's circle.
4. Cut away every part of the table outside the box.
5. If there is still no candidate with no red on their line, something needs to be compromised on; every candidate lost some race, and the loss we tolerate the best is the one where the loser obtained the most votes. So, take the red cell with the highest number (if going by margins, the least negative), make it green—or any color other than red—and go back step 2.

Here is a margins table made from the above example. Note the change of order used for demonstration purposes.

Initial results table

	E	A	C	B	D
E		1	−3	9	17
A	−1		7	−5	15
C	3	−7		13	−11
B	−9	5	−13		21
D	−17	−15	11	−21

The first drop (A's loss to E by 1 vote) does not help shrink the Schwartz set.

First drop

	E	A	C	B	D
E		1	−3	9	17
A	−1		7	−5	15
C	3	−7		13	−11
B	−9	5	−13		21
D	−17	−15	11	−21

So we get straight to the second drop (E's loss to C by 3 votes), and that shows us the winner, E, with its clear row.

Second drop, final

	E	A	C	B	D
E		1	−3	9	17
A	−1		7	−5	15
C	3	−7		13	−11
B	−9	5	−13		21
D	−17	−15	11	−21

This method can also be used to calculate a result, if the table is remade in such a way that one can conveniently and reliably rearrange the order of the candidates on both the row and the column, with the same order used on both at all times.

Satisfied and failed criteria

Satisfied criteria

The Schulze method satisfies the following criteria:

• Monotonicity criterion
• Majority criterion
• Majority loser criterion
• Condorcet criterion
• Condorcet loser 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
• prudence
• MinMax sets
• Woodall's plurality criterionifwinning votesare used for d[X,Y]
• Symmetric-completionifmarginsare used for d[X,Y]

Failed criteria

Since the Schulze method satisfies the Condorcet criterion, it automatically fails the following criteria:

• Participation
• Consistency
• Invulnerability to burying
• Later-no-harm

Likewise, since the Schulze method is not a dictatorship and is aranked votingsystem (notrated),Arrow's Theoremimplies it fails:

• Independence of irrelevant alternatives

The Schulze method also fails

• Peyton Young's criterion ofLocal Independence of Irrelevant Alternatives.

Comparison table

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

Comparison of single-winner voting systems

Criterion Method	Majority	Majority loser	Mutual majority	Condorcet winner	Condorcet loser	Smith	Smith-IIA	IIA/LIIA	Clone­proof	Mono­tone	Participation	Later-no-harm	Later-no-help	No favorite betrayal	Ballot type
Anti-plurality	No	Yes	No	No	No	No	No	No	No	Yes	Yes	No	No	Yes	Single mark
Approval	Yes	No	No	No	No	No	No	Yes	Yes	Yes	Yes	No	Yes	Yes	Appr­ovals
Baldwin	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	No	Ran­king
Black	Yes	Yes	No	Yes	Yes	No	No	No	No	Yes	No	No	No	No	Ran­king
Borda	No	Yes	No	No	Yes	No	No	No	No	Yes	Yes	No	Yes	No	Ran­king
Bucklin	Yes	Yes	Yes	No	No	No	No	No	No	Yes	No	No	Yes	No	Ran­king
Coombs	Yes	Yes	Yes	No	Yes	No	No	No	No	No	No	No	No	Yes	Ran­king
Copeland	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	No	Yes	No	No	No	No	Ran­king
Dodgson	Yes	No	No	Yes	No	No	No	No	No	No	No	No	No	No	Ran­king
Highest median	Yes	Yes	No	No	No	No	No	Yes	Yes	Yes	No	No	Yes	Yes	Scores
Instant-runoff	Yes	Yes	Yes	No	Yes	No	No	No	Yes	No	No	Yes	Yes	No	Ran­king
Kemeny–Young	Yes	Yes	Yes	Yes	Yes	Yes	Yes	LIIA Only	No	Yes	No	No	No	No	Ran­king
Minimax	Yes	No	No	Yes	No	No	No	No	No	Yes	No	No	No	No	Ran­king
Nanson	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	No	Ran­king
Plurality	Yes	No	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	No	Single mark
Random ballot	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Single mark
Ranked pairs	Yes	Yes	Yes	Yes	Yes	Yes	Yes	LIIA Only	Yes	Yes	No	No	No	No	Ran­king
Runoff	Yes	Yes	No	No	Yes	No	No	No	No	No	No	Yes	Yes	No	Single mark
Schulze	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	No	No	No	No	Ran­king
Score	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	No	Yes	Yes	Scores
Sortition	No	No	No	No	No	No	No	Yes	No	Yes	Yes	Yes	Yes	Yes	None
STAR	No	Yes	No	No	Yes	No	No	No	No	Yes	No	No	No	No	Scores
Tideman alternative	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	No	No	No	No	No	Ran­king
Table Notes

The main difference between the Schulze method and theranked pairsmethod can be seen in this example:

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

On the other hand, Ranked Pairs minimizes the largest majority that has to be reversed to determine the order of finish, in the MinLexMax sense. In other words, when Ranked Pairs and the Schulze method produce different orders of finish, for the majorities on which the two orders of finish disagree, the Schulze order reverses a larger majority than the Ranked Pairs order.

History

The Schulze method was developed by Markus Schulze in 1997. It was first discussed in public mailing lists in 1997–1998and in 2000.

In 2011, Schulze published the method in the academic journalSocial Choice and Welfare.

Usage

Sample ballot forWikimedia's Board of Trusteeselections

Government

The Schulze method is used by the city ofSillafor all referendums.It is also used by the cities ofTurinandSan Donà di Piaveand by theLondon Borough of Southwarkthrough their use of the WeGovNow platform, which in turn uses theLiquidFeedbackdecision tool.

Political parties

Schulze was adopted by thePirate Party of Sweden(2009),and thePirate Party of Germany(2010).The newly formedBoise, Idaho,chapter of theDemocratic Socialists of Americain February chose this method for their first special election held in March 2018.

• Five Star MovementofCampobasso,Fondi,Monte Compatri,Montemurlo,Pescara,andSan Cesareo
• Pirate PartiesofAustralia,Austria,Belgium,Brazil,Germany,Iceland,Italy,the Netherlands,Sweden,Switzerland,andthe United States
• SustainableUnion
• Volt Europe

Student government and associations

• AEGEE – European Students' Forum
• Club der Ehemaligen der Deutschen SchülerAkademien e. V.
• Associated Student Government atÉcole normale supérieure de Paris
• Flemish Society of Engineering Students Leuven
• Graduate Student Organization at the State University of New York: Computer Science (GSOCS)
• Hillegass Parker House
• Kingman Hall
• Associated Students ofMinerva Schools at KGI
• Associated Student Government atNorthwestern University
• Associated Student Government atUniversity of Freiburg
• Associated Student Government at the Computer Sciences Department of theUniversity of Kaiserslautern

Organizations

It is used by theInstitute of Electrical and Electronics Engineers,by theAssociation for Computing Machinery,and byUSENIXthrough their use of the HotCRP decision tool.

Organizations which currently use the Schulze method include:

• Annodex Association
• Berufsverband der Kinder- und Jugendärzte(BVKJ)
• BoardGameGeek
• Cloud Foundry Foundation
• County Highpointers
• Dapr
• Debian
• EuroBillTracker
• European Democratic Education Community(EUDEC)
• FFmpeg
• Free Geek
• Free Hardware Foundation of Italy
• Gentoo Foundation
• GNU Privacy Guard(GnuPG)
• Haskell
• Homebrew
• Internet Corporation for Assigned Names and Numbers(ICANN) (until 2023)
• Kanawha Valley Scrabble Club
• KDE e.V.
• Knight Foundation
• Kubernetes
• Kumoricon
• League of Professional System Administrators(LOPSA)
• LiquidFeedback
• Madisonium
• Metalab
• MTV
• Neo
• Noisebridge
• OpenEmbedded
• Open Neural Network Exchange
• OpenStack
• OpenSwitch
• RLLMUK
• Squeak
• Students for Free Culture
• Sugar Labs
• Sverok
• TopCoder
• Ubuntu
• Vidya Gaem Awards
• Wikimedia(2008)
• WikipediainFrench,Hebrew,Hungarian,Russian,andPersian.

External links

• The Schulze Method of Votingby Markus Schulze
• The Schulze MethodbyHubert Bray
• Spieltheorie(in German)byBernhard Nebel
• Accurate Democracyby Rob Loring
• Christoph Börgers (2009),Mathematics of Social Choice: Voting, Compensation, and Division,SIAM,ISBN0-89871-695-0
• Nicolaus Tideman(2006),Collective Decisions and Voting: The Potential for Public Choice,Burlington: Ashgate,ISBN0-7546-4717-X
• preftoolsby the Public Software Group
• Arizonans for Condorcet Ranked Voting
• Condorcet PHPCommand line application andPHPlibrary,supporting multiple Condorcet methods, including Schulze.
• Implementation in Java
• Implementation in Ruby
• Implementation in Python 2
• Implementation in Python 3