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A look around the various methods of Condorcet

Introduction

These methods are modern mathematical algorithms, sometimes computationally heavy, that extend and complement the original methods of the Marquis de Condorcet, without ever contradicting the results.

Specifically, these methods can overcome shortcomings such as the Condorcet paradox, a special case where no winner or loser can 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 or less computing power. They meet certain advanced anti-manipulation criteria like "independence of clones", "Smith criterion", or "local independence".

If you do not have specific needs, we recommend the use of Schulze Winning method.

Specifications

First, a small table summarizing the characteristics of the main advanced voting methods, and whether they meet or fail the Condorcet criteria.


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

Supported methods on Condorcet-Vote.org

Condorcet Methods Overview

Schulze

Summary

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 collaborative organizations such as Wikipedia, Debian, KDE, Pirate Party, Free Software Foundation Europe, OpenStack...

Documentation by Martin Schulze himself :

Variants

  1. Schulze Winning (Recommended 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 meets the criterion of resolvability; the possibility of a draw is then already very low and increasingly unlikely when the number of voters exceeds the number of candidates.

Ranked Pairs

Summary

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

Note

Our own implementation of this method is experimental.
The results look good, but it does not handle contingency equalities, which are frequent in small votes; such disputes are then resolved arbitrarily. Paradoxically, our implementation is more reliable on large elections than small ones.

Kemeny-Young

Summary

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 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 possible final classification. It is therefore dependent on the number of candidates (and not specifically the number of voters). Thus, if there are five candidates, there are 120 possibilities to calculate; for six there are 720; and for ten there are 3,628,800 possible solutions to compute and store!
For this reason, the results of this method will only be provided here for elections 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.

Copeland

Summary

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.

The Copeland method is very fast and simple. But this method fails the criterion of resolvability, so there are often ties in the results. Due to its ease of understanding, this method has been proposed and is still used in some local elections by universal suffrage around the world.

MiniMax

Summary

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!