Schulze method

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Cloneproof Schwartz Sequential Dropping (CSSD) is a voting system developed by Markus Schulze that selects a single winner using votes that express preferences. CSSD can also be used to create a sorted list of winners. CSSD is also known as "Schwartz Sequential Dropping", "Beatpath Method", "Beatpath Winner", "Path Voting", "Path Winner", and "Schulze Method".

If there is a candidate who is preferred over the other candidates, when compared in turn with each of the others, CSSD guarantees that that candidate will win. Because of this property, CSSD is (by definition) a Condorcet method. Note that this is different from some other preference voting systems such as Borda and Instant-runoff voting, which do not make this guarantee.

The Schwartz Set

The definition of a Schwartz set, as used in CSSD, 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.


The voters cast their ballots by ranking the candidates according to their preferences, just like for any other Condorcet election.

CSSD uses Condorcet pairwise matchups between the candidates and a winner is chosen in each of the matchups.

From there, CSSD 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. Go to 1.

An Example

The Situation

Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county):

  • Memphis (Shelby County): 826,330
  • Nashville (Davidson County): 510,784
  • Chattanooga (Hamilton County): 285,536
  • Knoxville (Knox County): 335,749

Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows:

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

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

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

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

The results would be tabulated as follows:

Pairwise Election Results
Memphis Nashville Chattanooga Knoxville
BMemphis[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/A68%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
  • [NP] indicates voters who expressed no preference between either candidate

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.


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, and C.

  • A > B 72%
  • B > C 68%
  • C > A 52%

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

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

  • A > B 72% (as C has been removed)

Therefore, A is the winner.

(It may be more accessible to phrase that as "drop the weakest win", though purists may complain.)


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. Using Instant-runoff voting in this example would result in Knoxville winning, even though more people preferred Nashville over Knoxville.

Satisfied Criteria

CSSD satisfies the following criteria:

  1. Monotonicity criterion
  2. Pareto criterion
  3. Condorcet criterion
  4. Smith criterion (aka Generalized Condorcet criterion)
  5. local independence from irrelevant alternatives
  6. Schwartz criterion
  7. Strategy-Free criterion
  8. Generalized Strategy-Free criterion
  9. Strong Defensive Strategy criterion
  10. Weak Defensive Strategy criterion
  11. Summability criterion
  12. Independence of clones

CSSD violates the following criteria:

  1. Participation criterion
  2. Consistency criterion
  3. invulnerability to compromising
  4. invulnerability to burying

It isn't known whether CSSD satisfies the following criteria:

  1. Favorite Betrayal criterion

Use of CSSD

CSSD is not currently used in government elections. However, it is starting to receive support in some public organizations. Organizations which currently use CSSD are:

  1. the Debian project (See here and here!)
  2. the Software in the Public Interest (SPI) project (See here!)
  3. the UserLinux project
  4. the Park Alumni Society (PAS) (See here!)
  5. the Blitzed project (See here!)
  6. the Leader of the Free World (LFW) project (See here!)
  7. the L'Expérience Démocratique (DemExp) project (See section 14.6 of this paper!)
  8. the Glasnost / Easter Eggs / libre-entreprise project
  9. the Green Mountain Mutual Aid (GMMA) project
  10. the Kingman Hall (See here and here!)
  11. the Johns Hopkins Animation Club (JHAC)
  12. the Haifa Linux Club (Haifux) (See here!)
  13. the Free Software Club of Kirksville (FSCK) (See here!)
  14. the College of Science Students' Advisory Council (COSSAC)
  15. the NationStates Wiki (NSwiki) project (See here!)
  16. the Sparta Wiki project

External Resources

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