Condorcet method

From Electowiki
Jump to navigation Jump to search
Wikipedia has an article on:

Any election method conforming to the Condorcet criterion - that is, one which always elects the pairwise champion if such exists - is known as a Condorcet method. The name comes from the 18th century mathematician and philosopher Marquis de Condorcet, although the method was previously described by Ramon Llull in the 13th century.

At present, the synonymous phrase "Instant Round Robin Voting" (IRRV) is being coined to leverage the public's greater familiarity with Instant Runoff Voting (IRV). This phrase is currently being used in a legislative effort to implement a Condorcet variant (CSSD) in the state of Washington.

Condorcet is sometimes used to indicate the family of Condorcet methods as a whole.

Simple explanation[edit | edit source]

If one candidate is preferred by more voters than all other candidates (when compared one-on-one), that candidate is the Condorcet Winner. This can be determined through use of ranked or rated ballots. On rare occasions, there is no Condorcet winner. In that case it is necessary to use some tiebreaking procedure.

Casting ballots[edit | edit source]

Each voter fills out a ranked ballot or rated ballot. The voter can include less than all candidates under consideration. Usually when a candidate is not listed on the voter's ballot they are considered less preferred than listed candidates, and ranked accordingly. However, some variations allow a "no opinion" default option where no for- or against- preference is counted for that candidate. Write-ins are possible, but are somewhat more difficult to implement for automatic counting than in other election methods. This is a counting issue, but results in the frequent omission of the write-in option in ballot software.

Counting ballots[edit | edit source]


If a candidate is preferred over all other candidates, that candidate is the Condorcet candidate (Condorcet winner). However, a Condorcet candidate may not exist, due to a fundamental paradox: It is possible for the electorate to prefer A over B, B over C, and C over A simultaneously. This is called a majority rule cycle or Condorcet cycle, and it must be resolved by some other mechanism.

Counting with matrices[edit | edit source]


A 0 0 1
B 1 1 1
C 1 0 1
D 0 0 0





Key terms in ambiguity resolution[edit | edit source]


The following are key terms when discussing ambiguity resolution methods:


Different ambiguity resolution methods[edit | edit source]


Examples of Condorcet methods include:

  • Black chooses the Condorcet winner when it exists and otherwise the Borda winner. It is named after Duncan Black.
  • Baldwin Computes the Borda count for all candidates, then iteratively deletes the candidate with the lowest count.
  • Smith/IRV is instant-runoff voting with the candidates restricted to the Smith set.
  • Copeland selects the candidate that wins the most pairwise matchups. Note that if there is no Condorcet winner, Copeland will often still result in a tie.
  • Llull-Approval Voting- Elects the member of the Schwartz set with the greatest number of approvals
  • Minimax (also called Simpson) chooses the candidate whose worst pairwise defeat is less bad than that of all other candidates.1
  • Smith/Minimax restricts the Minimax algorithm to the Smith set.1
  • Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) with variations such as Maximize Affirmed Majorities (MAM) and Maximum Majority Voting (MMV)1
  • Schulze with several reformulations/variations, including Schwartz Sequential Dropping (SSD) and Cloneproof Schwartz Sequential Dropping (CSSD)1
  • Approval-Condorcet Hybrids, such as Definite Majority Choice, use an Approval Cutoff to augment the Condorcet pair wise array. Many believe that such a method would make a good first-round public proposal.
  • Smith//Score chooses the candidate with the highest summed or average score in the Smith Set. Condorcet//Score chooses the Score winner when no Condorcet winner exists. (These can only be done with rated ballots, or with ranked ballots modified to include approval thresholds).


Ranked Pairs and Schulze are procedurally in some sense opposite approaches:


The text below describes (variants of) these methods in more detail.





Schulze method[edit | edit source]

The Schulze method resolves votes as follows:



Related terms[edit | edit source]

Other terms related to the Condorcet method are:

  • Condorcet loser: the candidate who is less preferred than every other candidate in a pair wise matchup.
  • weak Condorcet winner: a candidate who beats or ties with every other candidate in a pair wise matchup. There can be more than one weak Condorcet winner.
  • weak Condorcet loser: a candidate who is defeated by or ties with every other candidate in a pair wise matchup. Similarly, there can be more than one weak Condorcet loser.

An example[edit | edit source]


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


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%


Connection to cardinal methods[edit | edit source]


It's possible to modify Score to be more like a traditional Condorcet method by allowing voters to write the scores they would give to every possible pair of candidates in a Score runoff, and then using a Condorcet method to process this, treating a score of, say, A5 B3 (where the max score is 5) as 0.4 votes for A>B. As this would be utterly infeasible with just a few candidates running however, one way to accomplish most of the same objective is to allow voters to mark on their ballots that they want their vote strategically optimized, meaning that if their cardinally expressed preferences are A5 B3 Z2, instead of having their vote considered as B3 Z2 in an B vs. Z runoff, it would be considered as B5 Z0 (if the max score is 5), which is functionally equivalent to the Plurality voting runoffs that are used for the traditional Condorcet winner definition. This strategic optimization can be done fractionally to allow a voter to customize how much optimization they want to be done with their scores in each runoff. It is also possible for voters to indicate an approval threshold, meaning that for all approved candidates, no strategic optimization is applied to pairwise matchups between them, but all pairwise matchups between approved and disapproved candidates are strategically optimized. With this modification, if all voters use strategic optimization, Score becomes a traditional Condorcet method (which will need a cycle resolution method to be applied at times), but if no voters strategically optimize, it remains Score (which never needs cycle resolution methods to be applied).

Note that the above schemes can make Score fail the logical property that a voter's strength of preference between any pair of candidates must equal the sum of the strengths of preference between all sequential pairs of candidates in a beatpath from the first candidate of the pair to the second. The failure of this property seems to be the major reason traditional Condorcet methods can have Condorcet cycles and one major reason for why they fail certain properties such as Favorite Betrayal and Independence of Irrelevant Alternatives.

Demonstrating pairwise counting[edit | edit source]


Use of Condorcet voting[edit | edit source]


  1. The Debian project uses the Schulze method.
  2. The Software in the Public Interest (SPI) project uses the Schulze method.
  3. The Gentoo Linux project uses the Schulze method.
  4. The UserLinux project uses the Schulze method.
  5. The Free State Project used a Condorcet method for choosing its target state
  6. The voting procedure for the uk.* hierarchy of Usenet
  7. Five-Second Crossword Competition

External links[edit | edit source]