This is rare, even in games of chance. Monday, December 13, in Nyon (Switzerland), the Union of European Football Associations (UEFA) missed its draw for the knockout stages of the Champions League. After carrying out a full draw, the European body had to cancel it and organize a second a little later in the afternoon. Julien Guyon, mathematician and football fan, decipher what happened.
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Page Contents
How is the draw going?
The draw consists of matching eight group runners-up with eight group winners. But not just any old way: UEFA prohibits two teams from the same group or belonging to the same federation. This year there are exactly 4,781 knockout tables that meet all these constraints.
Since it is not really practical to draw one of these 4,781 results directly, and in order to maintain the ceremonial and dramatic character of the draw, UEFA has opted for a sequential procedure. A ballot box containing the eight group runners-up is emptied sequentially, and each time a group runner-up is drawn, an algorithm provides the list of eligible opponents from among the remaining group winners; these opponents are then placed in another ballot box, and one of them is drawn.
Determining the list of eligible opponents is less straightforward than it looks. In addition, the sequential procedure modifies the probabilities of the selection: the 4,781 possible outcomes are no longer equiprobable.
Here are the odds of the different pairings this year:
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What happened on Monday and from when was the printout skewed?
Let us go back to the chronological thread of events. Benfica is the first runner-up to be drawn. He can play against all group winners, except Bayern, whom he met in pools. It is Real Madrid which is drawn.
So far so good. Each draw of a ball changes the probabilities; this is what they were at that moment. Note how likely Chelsea-Lille becomes from that point on.
Villarreal is the second club drawn from the second-in-group ballot box. As soon as Villarreal is drawn, the odds change for all teams; in particular, Villarreal now has exactly a one in six chance of meeting each of their six possible opponents (all group winners except Real and Manchester United, who were in Villarreal’s squad):
This is where the first error arises: while the software used by UEFA, provided by an external provider, appears frozen (on the screen at least, the list of possible Villarreal opponents does not appear), it At the same time, it seems to indicate to Giorgio Marchetti, UEFA deputy general secretary and master of ceremonies in the draw, that Villarreal can play against all group winners except Real Madrid. Manchester United is therefore added to the ballot box of possible opponents. Mr. Marchetti seems to immediately notice the error.
There, UEFA is unlucky: there was only one chance in seven that United would be drawn, and it was! Mr. Marchetti reacts correctly: he rejects this draw, and asks that another team be drawn, among the six (real) eligible opponents of Villarreal. Manchester City are drawn.
Note that adding Manchester United (and even Real!) To the ballot box and rejecting them if they are drawn does not change the odds of the draw: Villarreal does have a one in six chance of falling against each of his eligible opponents. It has a name in probability theory: the “rejection method”.
The first error is therefore benign: it is a little embarrassing, but in no way modifies the fairness and the probabilities of the drawing, which, at this moment, are the following :
Atlético de Madrid are the third runner-up to be drawn. At this point everything is still going well from the point of view of the fairness of the draw. Atlético have a 20% chance of falling against each of their five possible opponents: Ajax Amsterdam, Bayern Munich, Manchester United, Lille and Juventus:
This is where the second, much more serious error comes in. The software removes Manchester United from the list of possible opponents of Atlético, and adds Liverpool, yet from the same group as the Madrilenians. Adding Liverpool is benign: he would have been rejected if he had been drawn, without impacting the odds of the draw. But not including Manchester United distorts the draw: it changes the odds, and Atlético is deprived of a possible match against Manchester United, in poor form this year. Instead, the Colchoneros inherit powerful Bayern Munich.
The draw continues normally, giving rise in particular to a confrontation between the two enemy brothers, Leo Messi and Cristiano Ronaldo (PSG-Manchester United). But the leaders of Atlético lodge a complaint with UEFA, which quickly decides to redo the entire draw, arguing that irregularities may have occurred from the start of the draw. However, we have seen that this is not the case: it was only after the draw for Atlético that the draw was biased.
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Where should the draw have been resumed?
Many believe the draw should have been restarted after Benfica-Real was drawn. But in this case, why not start over just after the Villarreal draw, which is also perfectly valid, and which changes the odds of the remaining draw?
It can, in fact, be argued that since the draw was in no way biased by the first error, UEFA should have restarted the draw after the Villarreal-Manchester City draw, and even after the Atlético draw. This is what UEFA should have done, to avoid other possible claims – from Real Madrid in particular, very cold with UEFA since the abortive launch of the Super League, and aggrieved by the decision to cancel the first draw entirely.
We can even push the reasoning further: the successive drawing of all runners-up having been done without irregularity, it could (should?) Have been kept too, since it influences the probabilities of the drawing.
The two French clubs finally fell against their most likely opponents (PSG-Real Madrid and Chelsea-Lille), and Lille even “fell” on Chelsea during the two draws!
Julien Guyon is a mathematician and football fan. A quantitative analyst, he is also an associate professor in the mathematics department at Columbia University and at the Courant Institute of Mathematical Sciences at New York University. His work is available on his page http://cermics.enpc.fr/~guyon/ and on his Twitter account @ julienguyon1977.