Paradox of Condorcet

The Paradoxe of Condorcet is actually, plus a thorny question concerned with the Decision theory, or more one dilemma in Démocratie, that a pure logical paradox.

Nicolas de Condorcet

In 1785, Nicolas of ''' Condorcet ''' published one of its principal work: the Test on the application of the analysis to the probability of the decisions returned to the plurality of the voices . In this work, it explores the paradox of Condorcet , which it describes like the possible Intransitivité of the majority: among the same electorate, and at the time of the same election, it is possible that a majority prefers has with B, that another majority prefers B with C, and that a third majority prefers C with A. the decisions taken with a popular majority by this way of voting would be thus incoherent compared to those which would take a rational individual. Condorcet specifies itself, in its work, how to raise its paradox.

Examples

Example 1 preferences

Let us consider a system preferably majority with 3 criteria. Objects are judged on 3 criteria and one prefers an object with another since 2 criteria are better.

Preferably let us consider the 3 following objects in a system growing (the highest note is the best):

$A (1, 3,2)$

B (2, 1,3)

C (3, 2,1)

with the final one:

$B$ is preferred with $A$ because better on criteria 1 and 3.
$C$ is preferred with $B$ because better on criteria 1 and 2.
$A$ is preferred with $C$ because better on criteria 2 and 3.

$B$ is thus preferred with $A$ which is him even preferred with $C$ which is him even preferred with $B$ .

Example 2 the vote

Let us consider for example 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 > B > C

17 voters prefer: B > C > has

2 voters prefer: B > has > C

10 voters prefer: C > has > B

8 voters prefer: C > B > HAS

In the majority comparisons per pairs, one obtains:

33 prefers has > B against 27 for B > has

42 prefer B > C against 18 for C > B

35 prefer C > has against 25 for has > C

What leads to contradiction interns has > B > C > has .

In a case like this one, Condorcet proposes to eliminate the least powerful winner (here has because has > B gains the weakest score) and to make a duel between B and C which will be gained by B. But other solutions are possible (see Méthode Condorcet#Résolution of the conflicts).

The French presidential election of 1974 is sometimes quoted like example of the paradox of Condorcet: François Mitterrand, Valery Giscard d'Estaing and Jacques Chaban-Delmas had obtained respectively with the first turn 43,2%,32,6% and 15,1% of the votes. To the second turn, it is Giscard d'Estaing, however arrived in second position at the time of the first turn, which is elected with 50,81% of the voices.

Polemics

Contrary to a widespread opinion (inter alia by Robert Badinter in its biography of Condorcet), this paradox blames only the coherence of certain voting systems and not that of the democracy itself .

It is necessary to await the Théorème of impossibility of Arrow at the 20th century which will affirm that the problem is quite inherent in the democracy, on the basis of assumption reasonable, and obviously discussed taking into account the range of the problem.

In its test, Condorcet also exposes the method of Condorcet, a method conceived to simulate elections per pairs of candidates. It states however that matters of time practical of the examination return the method that it considers difficult to realize, in any case at its time. It had many discussions with Jean-Charles of Bordered, at the time which they compared their respective methods. This Condorcet method is used nowadays in Data mining.

External bonds

Illustration of the paradox of Condorcet in the surveys of the presidential election Frenchwoman 2007
Software DemExp which tries to propose a solution of participative democracy, based on the contribution of new technologies - which make possible of the votes of Condorcet to large scales.

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