Condorcet method

The Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. A Condorcet winner will not always exist (see Condorcet's paradox), which means that an alternative winner must be chosen.

Any voting system which chooses the Condorcet winner when it exists is known as a Condorcet method, after its deviser, the 18th century mathematician and philosopher Marquis de Condorcet, although it appears that the method was already thought up by Ramon Llull in the 13th century. [1]

Condorcet is partly a class of voting systems, and partly a school of thought regarding single-winner preference electoral systems.

Other similar terms 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.

Table of contents

1 Procedures

1.1 Resolving Disputes

2 An Example
3 Condorcet compared to Instant Runoff
4 Use of Condorcet voting
5 External Resources

Procedures

Each voter ranks the candidates in order of preference. For each pair of candidates, it is determined how many voters prefer one candidate to the other by counting how many times it is higher ranked on the ballot (any candidates not placed on the ballot at all is considered to be less preferred by all those that are). If one candidate is preferred to every other candidate, it is declared the winner.

Resolving Disputes

If there is no initial winner, then the winner must be determined in some other way. There are numerous ways of doing this:

One method chooses the candidate whose strongest pair wise defeat is the weakest of all candidates. This is also known as Minimax. However, one must still choose how to measure the strength of each defeat; some use the margin of defeat (the difference between votes for and votes against), while others use winning votes (the votes favoring the defeat in question).
In a variant called "Copeland's method", the winner is the candidate that wins the most pairwise matchups. Assuming no Condorcet winner, Copeland will often still result in a tie.
An alternative winner can be chosen by having an Instant Runoff (IRV) election between the candidates. This system would be called "Condorcet/IRV".
Candidates are said to be in the 'top cycle' (or 'Smith set') if each of them will beat all candidates outside the top cycle in pairwise competition but not all the candidates inside the top cycle. Though this doesn't constitute a tie-breaker by itself, it is a useful filter for other methods. For example, a method often called "Smith/Minimax" picks the candidate from the Smith set who has incurred the smallest biggest defeat. In the same way we can construct other methods, like "Smith/IRV".
Instead of the Smith set, one can also restrict the winner(s) to the Schwartz set, but in practice this would rarely cause a different result, especially in public elections.

The fact that Condorcet himself already spearheaded the debate of which particular Condorcet method to promote, has made the term "Condorcet's method" ambiguous.

An Example

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):

Image:CondorcetTennesee.png

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 Tennesee 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
A
Memphis Nashville Chattanooga Knoxville

B Memphis [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/A 68% 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

In this election, the winner is Nashville. Using other systems, Memphis may have won the election by having the most people. However, Nashville won every simulated pairwise election outright. Note that using Instant Runoff Voting in this same example would result in Knoxville winning.

Condorcet compared to Instant Runoff

There are reasonable arguments to regard the Condorcet criterion as a requirement of a voting system: if there is a Condorcet winner, then it should be selected by the system as the (sole) winner. From this point of view, Instant Runoff is not as good as the Condorcet scheme, because there are circumstances in which it will fail to pick the Condorcet winner. On the other hand, the Condorcet winner could be a candidate with very weak core support, raising questions about that winner's legitimacy.

Assuming that preferences are sincerely expressed and remain consistent, in a head-to-head election with only two candidates, the Condorcet winner of an election would always beat the Instant Runoff winner from the same set of ballots, provided of course that the Condorcet Winner exists and differs from the Instant Runoff winner.

Use of Condorcet voting

Condorcet voting is not currently used in government elections. However, it is starting to receive support in some public organizations, such as the Debian project and the Free State Project. It also used in the voting procedure for the uk.* hierarchy of Usenet.