Condorcet's method

Condorcet is partly an electoral system, and partly a way of thinking about preference electoral systems that elect one candidate. Condorcet's method derives its name from its inventor, the 18th century mathematician/philosopher Marquis de Condorcet.

The Condorcet winner is the candidate who, when compared in turn with each of the other candidates, is preferred to them. It is not guaranteed that there will be any candidate to whom this applies, so any Condorcet electoral system must have a way of resolving such results.

Table of contents 1 Voting 2 Counting The Votes 3 An Example 4 Resolving Disputes 5 Condorcet compared to Instant Runoff 1 Use of Condorcet voting 1.1 External Resources

Voting

Each voter ranks all candidates in order of preference.

Counting The Votes

For each pair of candidates, it is determined how many voters preferred each candidate by counting whether they were higher-ranked on the ballot. If any candidate is preferred to all other candidates, they are declared the winner.

An Example

The easiest way to visualize how a Condorcet election would work is to 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 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%	0%	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. Howver, Nashville won every simulated pairwise election outright. Note that using Instant Runoff Voting in this same example would result in Knoxville winning.

Resolving Disputes

If there is no initial winner, then there is a situation which involves a voting paradox, and the winner must be determined some other way. There are numerous ways of doing this:

• Marquis de Condorcet described his preferred method, which is to choose the candidate who has the smallest defeat to any other candidate. How Condorcet intended to measure this is unclear; some proponents measure the margin of defeat, while others measure winning votes.
• One can define the 'top cycle' to mean that candidates are said to be in the 'top cycle' if each of them will beat all candidates outside the top cycle in pairwise competition but not all the candidates inside the top cycle. The top cycle is known as the "Smith set". 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/Condorcet" picks the candidate from the Smith set who has incurred the smallest defeat.
• One can also have a winner can be chosen by having an Instant Runoff (IRV) election between the Smith set candidates. This system would be called "Smith/IRV".
• In a variant called "Copeland's method", the winner is the candidate that wins the most pairwise matchups.

Condorcet compared to Instant Runoff

There are reasonable arguments to regard the Condorcet criterion, when fulfilled, as the best test of who should win: if there is a Condorcet winner, then a system for selecting one winner ought to select the Condorcet winner. On this 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.

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 Free State Project.

External Resources

• Condorcet's Method (http://www.eskimo.com/~robla/politics/condorcet)
• electionmethods.org (http://electionmethods.org/)
• Ranked Pairs (http://condorcet.org/rp)
• Accurate Democracy (http://accurate-democracy.com/)