This document discusses various methods for tabulating ranked choice voting results based on the ideas of the Marquis de Condorcet. It summarizes several leading Condorcet methods - Schulze, Ranked Pairs, Kemeny-Young, Copeland, and Minimax - and compares their properties and whether they satisfy the Condorcet and monotonicity criteria. The Schulze method is recommended as the most popular and well-established Condorcet method used by organizations like Wikipedia and Debian.

1. A look around the various methods of
Condorcet
Introduction
These methods are modern mathematical algorithms, sometimes heavy, to extend and complement the original methods of the
Marquis de Condorcet, without ever results do contradict.
Specifically, these methods can overcome shortages such as the Condorcet paradox (https://en.wikipedia.org/wiki/Voting_paradox). A
special case where no winner or loser can not normally be determined. These methods are also used to represent a complete ranking
of the election.
Each has its biases, its advantages and disadvantages, and may require more computing power or less. Meet certain anti-
manipulation advanced criteria like "independence of clone", "Smith criterion" or "local independence".
If you do not have specific needs, we recommend the use of Schulze Winning method.
Specifications
Above all, a small table summarizing the characteristics of the main methods of advanced votes, that they meet or fail the Condorcet
criterions.
Monotonic
(http://en.wikipedia.org/wiki/Monotonicity_criterion)
Condorcet
(http://en.wikipedia.org/wiki/Condorcet_criterion)
Schulze Winning (http://en.wikipedia.org/wiki/Schulze_method) Yes Yes
Schulze Margin (http://en.wikipedia.org/wiki/Schulze_method) Yes Yes
Ranked Pairs (http://en.wikipedia.org/wiki/Ranked_Pairs) Yes Yes
Kemeny-Young
(http://en.wikipedia.org/wiki/Kemeny%E2%80%93Young_method)
Yes Yes
Copeland (http://en.wikipedia.org/wiki/Copeland%27s_method) Yes Yes
MiniMax Winning (http://en.wikipedia.org/wiki/Minimax_Condorcet) Yes Yes
MiniMax Margin (http://en.wikipedia.org/wiki/Minimax_Condorcet) Yes Yes
MiniMax-Opposition
(http://en.wikipedia.org/wiki/Minimax_Condorcet)
Yes No
Dodgson (http://en.wikipedia.org/wiki/Dodgson%27s_method) No Yes
Nanson (http://en.wikipedia.org/wiki/Nanson%27s_method) No Yes
Baldwin
(http://en.wikipedia.org/wiki/Nanson%27s_method#Baldwin_method)
No Yes
Instant-runoff voting (http://en.wikipedia.org/wiki/Instant-
runoff_voting)
No No
Borda (http://en.wikipedia.org/wiki/Borda_count) Yes No
Bucklin (http://en.wikipedia.org/wiki/Bucklin_voting) Yes No
Coombs (http://en.wikipedia.org/wiki/Coombs%27_method) No No
Plurality (http://en.wikipedia.org/wiki/Plurality_voting_system) Yes No
Anti-plurality (http://en.wikipedia.org/wiki/Anti-plurality_voting) Yes No
Contingent voting (http://en.wikipedia.org/wiki/Contingent_vote) No No
Sri Lankan contingent voting
(http://en.wikipedia.org/wiki/Sri_Lankan_contingent_vote)
No No
Supplementary voting
(http://en.wikipedia.org/wiki/Supplementary_vote)
No No
Source : Wikipedia EN → (https://en.wikipedia.org/wiki/Schulze_method#Comparison_table)
Supported method on Condorcet-Vote.org
Schulze method also know as: Schwartz Sequential Dropping (SSD), Cloneproof Schwartz Sequential Dropping (CSSD), the
Beatpath Method...
Ranked Pairs (exprimental implementation at this date)
Kemeny-Young method also know as: Kemeny rule, VoteFair popularity ranking, maximum likelihood method and the median
relation.
(Cause of performance consideration, works only for election with less than 7 candidates)
Copeland method
MiniMax Family methods also know as: Simpson-Kramer method, Successive reversal method
◦ MiniMax Winning
◦ MiniMax Margin

2. Schulze Schulze Method on Wikipedia → (https://en.wikipedia.org/wiki/Schulze_method)
Ranked Pairs Ranked Pairs on Wikipedia → (http://en.wikipedia.org/wiki/Ranked_pairs)
Kemeny-Young Kemeny-Young on Wikipedia → (http://en.wikipedia.org/wiki/Kemeny%E2%80%93Young_method)
Copeland Copeland on Wikipedia → (http://en.wikipedia.org/wiki/Copeland%27s_method)
◦ MiniMax Opposition This is not a Condorcet method, because it fails occasionally its criteria!
And of course, naturals winner & loser from original Marquis de Condorcet method if there s not paradox of Condorcet
(https://en.wikipedia.org/wiki/Voting_paradox)
Condorcet Methods Overview
Resume
If for a pairwise contest X either beats or ties Y, then we say that X has a path to Y, with a strength equal to the number of voters
ranking X over Y.
If X has a path to Y of strength m, and Y has a path to Z of strength n, then we say that X has a path to Z equal to the minimum of m
and n.
Of all the paths from X to Y, a maximum path strength can be found. If the maximum path strength from X to Y is greater than the
maximum path strength from Y to X, then Y cannot win. The winner is the candidate that does not lose any such maximum path
strength comparisons.
The Schulze method is undoubtedly the most popular advanced Condorcet method. It is commonly used in organizations to
collaborative vocation as Wikipedia, Debian, KDE, Pirate Party, Free Software Foundation Europe, OpenStack...
Documentation by Martin Schulze himself :
http://m-schulze.9mail.de (http://m-schulze.9mail.de/)
Variants
1. Schulze Winning (Recommanded by M.Schulze)
2. Schulze Margin
3. Schulze Ratio
Note
Our implementation is simple and safe. It does not include the complex and heavy possible supplements (STV. ..) for advanced tie-
breaking. Schulze meet the criterion of resolvability, the possibility of a draw as then already very low and increasingly unlikely when
the number of voters exceeds the number of candidates.
Resume
Ranked Pairs finds a complete ranking. pairwise victories are processed starting from the greatest margin, and working down. These
victories are locked, which means that the final ranking will agree with this pairwise decision. If a victory is processed that is
incompatible with the previously locked victories, it is skipped. Once all victories are processed, a complete ranking is left.
Documentation
Independence of Clones (https://www.condorcet.vote/view/DOCS/IndependenceofClones.pdf)
Complete Independence of Clones (https://www.condorcet.vote/view/DOCS/CompleteIndependenceofClones.pdf)
Note
Our own implementation of this method is actually strange and experimental.
The results look good, but do not take care of contingencies equalities, yet frequent votes on small, sharp then these disputes
arbitrarily. Paradoxically, our implementation is more reliable on large elections as small.
Resume
Each possible complete ranking of the candidates is given a "distance" score. For each pair of candidates, find the number of ballots
that order them the the opposite way as the given ranking. The distance is the sum across all such pairs. The ranking with the least
distance wins.
Note
This method, particularly heavy, simply involves a series of calculations for each final classification possible. It is therefore dependent
on the number of candidates in the presence (and not specifically the number of voters). Thus, if five candidates are 120 possibilities
to calculate, six are 720 and ten are 3 628 800 possible solutions to compute and store!
For this reason, the results of this method will here provides for election comprising at most 5 candidates.
Also, this method although excellent, tends not to reach a solution in the case of a very small number of voters (less than the number
of candidates). This will be indicated in bold, and an arbitrary classification (but realistic) is provided for reference only.

3. MiniMax MiniMax on Wikipedia → (http://en.wikipedia.org/wiki/Minimax_Condorcet)
Resume
Each alternative's Copeland score is calculated by subtracting the number of alternatives that pairwise beat it from the number that it
beats. The alternatives with the highest Copeland score win.
Copeland method is very fast and simple. But this method failed to the criterion of resolvability, so there is often tie on result. Of its
ease of understanding, this method has been proposed and is still some local elections by universal suffrage around the world.
Resume
Minimax selects as the winner the candidate whose greatest pairwise defeat is smaller than the greatest pairwise defeat of any other
candidate.
Variants
When it is permitted to rank candidates equally, or to not rank all the candidates, three interpretations of the rule are possible. When
voters must rank all the candidates, all three variants are equivalent.
1. MiniMax Winning
2. MiniMax Margin
3. MiniMax Opposition This is not a Condorcet method, because it fails occasionally its criteria!
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