Champions League: the most likely opponents for PSG and Lyon in the round of 16

Champions League: the most likely opponents for PSG and Lyon in the round of 16

December 13, 2018 0 By admin

The player of the Red Star of Belgrade Lorenzo Ebicilio grappling with Neymar, December 11, in Belgrade.
The player of the Red Star of Belgrade Lorenzo Ebicilio grappling with Neymar, December 11, in Belgrade. FRANCK FIFE / AFP

Despite very moderate performances in the group stage of the Champions League (4 wins only in 12 games), Paris-Saint-Germain and Olympique Lyonnais managed, in extremis, to qualify for the knockout stages of the European competition. All qualified clubs are now known, and the draw for the round of 16 will take place Monday, December 17 at noon in Nyon (Switzerland), at UEFA headquarters.

As every year, so here comes the time for clubs and fans to study the list of potential opponents and play to imagine the worst and the best draw possible. It is also the moment to indulge in a small exercise of probabilities, more subtle than it seems at first glance.

The three basic rules of the draw

  1. a group winner is opposed to a group second.
  2. two clubs from the same group can not be drawn against each other.
  3. two clubs from the same country can not compete.

The eight group winners are: Dortmund, Barcelona, ​​PSG, Porto, Bayern Munich, Manchester City, Real Madrid, Juventus.

The eight corresponding group seconds are: Atlético Madrid, Tottenham, Liverpool, Schalke 04, Ajax Amsterdam, Lyon, AS Roma, Manchester United.

A first good news for PSG

The Paris club will avoid Barcelona and Real Madrid, his tormentors in the last two seasons, because they also finished first in their group.

Potential opponents of PSG are all second in group, except Liverpool (which appeared in his qualifying group) and Lyon.

You might think that Paris has a six-chance chance of hitting each of its six potential opponents … but it's a bit more complicated than that.

Compare for example two possible opponents of the PSG: Atlético Madrid and Ajax Amsterdam. Ajax has seven potential opponents: all the group winners, except Bayern, which he faced in groups.

Atlético, he has only five possible opponents, since he can meet Dortmund, also from Group A, Barcelona and Real Madrid.

This mechanically increases the probability that the PSG falls on Atlético.

The operating mode of the draw complicates the simulations a little

We could therefore think that this is a simple enumeration exercise: we list all the eligible results of the draw (there are 3,694 this year), we count, for example, in how many cases the PSG meets Atlético (696 out of 3,694), and we deduce that the PSG has 696/3 694 = 18.8% chance of falling on Atlético. And that Manchester City has 1 068/3 694 = 28.9% chance of falling on the Griezmann club.

It is almost true … but the operating mode of the draw comes to complicate a little more the thing.

This counting reasoning would be correct if the draw was to draw a random ball out of 3,694, each ball representing an eligible round of 16 game. That's not how UEFA does it.

The second group are placed in an urn, and each time a second group is drawn, an algorithm provides the list of his possible opponents among the group winners still available, then a club is drawn in the draw. this list.
The algorithm is not as simple as one might think since it must prevent future impasse.

Atlético Madrid or Shalke 04, Dortmund, Barcelona or Real

I simulated 2 million draws, following the official procedure of UEFA, to obtain the true probabilities (to 0,1% close).

Bad news for the PSG: his most likely opponent is also his opponent a priori the most dangerous, Atlético Madrid, but with a probability of only 18.5%. Schalke 04 follows with 18%. The least likely opponent of PSG is Ajax Amsterdam (13.3% chance).

As for Lyon, its most likely opponent is Borussia Dortmund (17.5%), closely followed by Barcelona (17.3%) and Real Madrid (17.1%), while its opponent the least likely is also his opponent a priori the most accessible, Porto (14.6%).

The most likely eighth final is Manchester City-Atletico (28.2%), followed by Manchester City-Schalke 04 (27.3%).

From the quarterfinals, the draw becomes full, without constraints, so much more readable … but much less funny in the eyes of the mathematician!

Julien Guyon (Polytechnician, PhD in Applied Mathematics, Associate Professor in the Departments of Mathematics at Columbia University and NYU)

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