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The method Condorcet (or vote Condorcet ) is a Voting system in which the single winner is that, if there exists, which compared in turn with all the other candidates, would each time prove to be the preferred candidate.

Nothing guarantees the presence of a candidate satisfying this criterion. Thus, any voting system based on the comparative method of Condorcet must envisage a means of solving the votes for which this ideal candidate does not exist.

This method owes its name with the marquis de Condorcet, mathematician and philosopher French of the 18th century, although the method was already known of Raymond Lulle (1299).

Motivation

In its Essai on the application of the analysis to the probability of the decisions returned to the plurality of the voices , Condorcet highlights the fact that the vote with plurality can not represent the desires of the voters very well.

Example

Let us consider an assembly of 60 voters having the choice between three proposals has , B and C . The preferences are distributed thus (by noting > B has, the fact that has is preferred with B ):

23 voters prefer: has > C > B

19 voters prefer: B > C > has

16 voters prefer: C > B > has

2 voters prefer: C > has > B

In a pluralist procedure of vote, has carries it with 23 vote, on B with 19 vote and on C with 18 from where has > B > C .

In the majority comparisons per pairs, one obtains:

35 prefers B > has against 25 for has > B

41 prefer C > B against 19 for B > C

37 prefer C > has against 23 for has > C

What leads to the majority preference C > B > has , exactly opposite with the pluralist choice.

He then proposes his own method while admitting that the very heavy organization that she implies does not make it very realistic for important elections. She can, according to him, to only be associated with presorting with the candidates to limit the number of it. It highlights moreover the existence of situations where its own method does not make it possible to undoubtedly choose the good candidate. It is what is called the Paradoxe of Condorcet. There exist several methods to reduce the conflicts generated in these situations.

Course of the vote

Each voter classifies the candidates by order preferably.

Deduct votes

The counting of the votes consists with to simulate the whole of the possible duels : for each pair of candidates, one determines the number of voters having voted for one or the other while checking, on each ballot paper, how one was classified compared to the other. Thus for each duel, there is a victorious candidate. In the majority of the cases, there is a single candidate who gains all his duels : it is about the winner of the poll. The following section described what occurs in the rare cases where no candidate does not gain all its duels.

Resolution of the conflicts

It happens that no candidate is elected following the calculation of the votes. Condorcet had noticed an important internal contradiction in this method, called the Paradoxe of Condorcet: in an election, as soon as the number of the voters is higher than two, has can be preferred with B, even preferred to him with C, itself preferred with A. Several methods are usable to solve this conflict of circularity.

one uses the Méthode Bordered, it acts then of the Méthode Black
one uses the Méthode Schulze
one sets up a group of head: a candidate belongs to the group of head if, at the time of the calculation of the votes, he managed to beat all the other candidates, except those of the group of head. Once this group of head defined, one proceeds to a choice in this group of head
1. by using a alternative Vote
2. by choosing the candidate who lost the least in his worse confrontation
3. by progressively eliminating those whose defeat was weakest
one seeks the candidate who gained the most confrontation (but there are likely to be ex-æquos)
one directly chooses that which lost the least during its worse confrontation without determining of group of head
one classifies the pairs according to the greatest differences observed, one builds a directed graph then (which gains against which?) while starting with the pair having the strongest difference and while going down in the scale by eliminating the pairs which could lead to a loop (paradox of Condorcet). The final graph makes it possible to define one gaining. It is the Méthode Condorcet with arrangement of the pairs per decreasing order (Ranked pars)

The choice of the method must be fixed before the vote, this choice having consequences on the determination of the winner.

Examples

Group head and less bad defeat

At the time of an election, 3 candidates (X, Y and Z) find themselves in the group of head . The results of the votes for these 3 candidates are:

41 voters voted for 1st =X; 2e=Y; 3e=Z
33 voters voted for 1st =Y; 2e=Z; 3e=X
22 voters voted for 1st =Z; 2e=X; 3e=Y

When one carries out the comparisons per pairs, X: against Y = 41+22-33 = +30 (thus X gains by 30 votes) against Z = 41-33-22 = -14 (thus X loses by 14 votes) Y: against X = -30 against Z = 52 Z: against X = 14 against Y = -52

X thus gains, since its worse result (- 14) is better than those of Y and Z (respectively -30 and -52)

Classification of the pairs and graph

Let us take again the preceding example and add a fourth candidate T. Imagine that

41 voters voted for 1st =X; 2e=Y; 3e=T; 4th = Z
33 voters voted for 1st =Y; 2e=Z; 3e=T; 4th = X
22 voters voted for 1st =Z; 2e=X; 3e=T; 4th = Y

The classification of the pairs gives

Y gains against Z and Y gains against T (52)
X gains against Y (30)
X gains against T, Z gains against X, Z gains against T (14)

The constitution of the graph is done in the order: Y > Z and Y > T, then X > Y (not of cycle), then one eliminates Z > X which would constitute a loop, one preserves X > T and Z > T. gaining It is then X because X > Y > Z > T.

The Condorcet method compared with the alternative vote

The alternative Vote is another voting system per classification used mainly in Australia. It happens that the alternative vote does not give the same result as the method of Condorcet.

However, if there exists one gaining of Condorcet without conflict, i.e. a candidate placed better than all his adversaries, and if it is supposed that the votes were sincere and nonstrategic, confrontation between gaining of Condorcet and gaining it of the alternative vote at the time of a simple majority poll will turn obviously to the advantage of gaining of Condorcet. But is this always the case if conflicts should have been solved? …

Use of the vote of Condorcet

This method is not currently used in national elections. It however starts to be used in certain public organizations. Among them, one can trouver :

Of the projects in the field of the Free software (Debian, Software in the Public Interest, Gentoo, UserLinux) which use the Méthode Schulze.
the “Free State Project”
procedure of vote for the hierarchy the U.K.. * of Usenet
Five-Second Crossword Competition
the network free-company

External resources

the democratic experiment
Condorcet' S Method
Ranked Pars
On-line computer of results by method Condorcet

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