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The Golden Goalkeeper

FIRST HALF

FIFA has decided to create a prize award to the best european goalkeepers of the first decade of the century. For this award, they've nominated these players as follows: Iker Casillas, Oliver Kahn, Petr Cech y Edwin van de Sar.

It will be held a gala in Monaco to give the Golden, Silver and Bronze Goalkeeper, respectively. The jury will consist of three prestigious goaldkeepers from other eras: Andoni Zubizarreta, Peter Shilton and Dino Zoff, who will give 4, 3, 2 and 1 point to the selected goalkeepers, depending on their preferences.

Iker Casillas is told some news that the majority of the jury's members prefer Oliver Kahn to Petr Cech; that most of them prefer Petr Cech to Van der Sar; and that most of the juries prefer Van der Sar to Casillas.

Kahn > Cech > Van der Sar > Casillas

With this info, Iker Casillas is thinking about either attending the gala, or training the next round of the Champions League, within 3 days, as he thinks he's not going to be at the podium of the Gala.

What do you think Casillas will do?

How do you think the podium of the Gala will stand?

Go to the second half to discover the solution

SECOND HALF

You've probably thought that the final ranking was:

1st Oliver Kahn - 2nd Petr Cech - 3rd Van der Sar - 4th Iker Casillas

And yes, that's one of the possible rankings that can be given, according to the known information. Although not the only, as we can see below.

In fact, Iker Casillas should attend the Gala, as the votes were as follows:

You can check that all what Iker Casillas was told about (most of the jury's members prefer Oliver Kahn to Petr Cech; Petr Cech to Van der Sar; and Van der Sar to Casillas) was absolutely certain, but the final result is very different. If we add the scores, we'll get the following ranking:

And this will be the resultant podium:

Curious, isn't it? This is what is called 'the electoral paradox'. And there are times when the most voted candidate, is not ultimately winning the election.

This paradox is also known as 'Arrow's paradox', in honor of Kenneth J. Arrow, Nobel laureate, who comes to show that it's impossible to find a perfect voting system. This occurs in not-transitive relationships, as the preference relation is.

Transitive relations, which are normally handled (to be 'greater than', 'smaller than', 'equal to', 'before than', 'faster than' ...) lead us to syllogisms like the following: if A is greater than B, and B is greater than C, A is greater than C.

However, there is another set of intransitive relations, as in the case of preferring something to anything else, in which the consequence can't be assumed: we might like it more to see a football game than to go to the movies, we can prefer going to the cinema instead of reading a book, and yet more to like reading a book than watching a football game. Hence, such paradoxes as in the Golden Goalkeeper can be produced.

As corollary from this topic we find Donald Gene Saari's theories. Saari, an expert in voting methods, showed that it's possible to cause, in any votation, whatever choice you want. That is, you can distort the popular will to make it coincide with one's will. So, you can always create formulas so that voters end up voting that what you want. Developing this example, if the organization of the Golden Goalkeeper knows about the preferences of individual jurors, it could make any of the goalkeeperes win the prize.

So, if they wanted Van der Sar winning the price, they could simply propose, for example, one first tie vote between Casillas and Van der Sar (two juries prefer Van der Sar to Casillas), then they'd make another tie vote between Kahn and Cech (2 prefer Kahn to Cech). And then, they'd make a final vote among the victors, Kahn and Van der Sar (1 for Kahn vs. 2 for Van de Sar), and they'd have forced Edwin to win the prize.

You can follow this example and try to make your preferred goalkeeper winning the Golden Goalkeeper. You'll see that it's always possible to find a sequence of votes that makes your favourite players be the winner.

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