The Condorcet paradox

Since we are in an intense election period , it is essential to me today to talk about the Condorcet paradox. This is a finding in the eighteenth century by Nicolas - mathematician -philosopher Marquis de Condorcet , who observed that in some situations, regardless of the voting method one chooses , it is impossible to appoint a indisputable winner. This may seem surprising , but as we shall see, the Condorcet paradox is far from a theoretical situation . However you will see, it does not mean completely unable to imagine a fair democratic election .

An example of the paradox Imagine that an election will have three candidates : Alain Beatrice and Claude . Suppose that about 40 % of the population prefers to Alain Beatrice, Beatrice but prefers to Claude . For these 40 % of the population , so we Alain > Beatrice > Claude .
 

Now assume that 35% of people we have Beatrice > Claude > Alain , and 20% for the remaining Claude > Alain > Beatrice. We will note it like this :

 Group 1 (40 %): A> B > C 
 Group 2 (35 %): B > C > A 
 Group 3 ( 25%): C > A> B

Where is the paradox? It is that regardless of the voting method used to determine the winner , there will always be a majority of the population that will be ready to change it for another . No winner is indisputable ! Imagine that the winner is Beatrice. Then the groups 1 and 3 ( which weigh 65 % of the population between them ) would agree to replace Béatrice Alain , since both of them prefer A to B. And you see that all cases are similar : if Alain who is elected , then the groups 2 and 3 (60% of the population) prefer to have Claude in its place. In short, it is inextricable : there can be no undisputed winner. And you see that it in no way depends on the electoral system , just the respective preferences of each other .

The Condorcet winner 
 Fortunately, all situations are paradoxical! There are cases where one escapes the paradox. This happens when a candidate would win a duel against any of the others. So if he is elected, there is no possibility for a majority of the population wants to replace it with another. There is therefore an indisputable winner was then called the Condorcet winner. Let's make a small retrospective. In the 2007 presidential election in France, there was clearly a Condorcet winner: François Bayrou. Indeed according to the time of polls, it would beat Nicolas Sarkozy in the second round, but Ségolène Royal! When there is a Condorcet winner, we should be thankful because it means that escapes the paradox. And yet you see it on the example of 2007, a Condorcet winner is not necessarily the winner in a classic election! It is also very rarely the case with the typical modes of ballots in force. A voting system that allows fail to elect the Condorcet winner (if any) is called Condorcet method. Here is one very simple: organize duels between all possible candidates. If someone wins all his duels, he is the Condorcet winner, and if no one wins all his duels, it is in the case "paradoxical" (and wrong!). Obviously if you have 15 candidates, we must organize a hundred duels, I doubt that people accept this type of election where you have to give 100 times its opinion, such as "Do you prefer Philippe Poutou and Nathalie Arthaud? ". Good fast way to do this is to ask everyone to classify the 10 candidates in order of preference. That would be perfectly playable.