# who are the most likely opponents for PSG and Lyon

Like last year, despite very weak performances – apart from Paris Saint-Germain – in the group stage of the Football Champions League (2 victories only in 12 games for Lille and Lyon), France managed in extremis to qualify two representatives in the eighth in the final, PSG and Olympique Lyonnais.

For the first time in the history of the Champions League, the 16 clubs qualified all come from the five major European championships: England and Spain qualified 4 representatives each, Germany and Italy 3, and France 2. No Portuguese, Russian, Ukrainian, Dutch, Greek or Belgian representative this year, signs that the elite continues to tighten.

The draw for the round of 16 will take place on Monday 16 December at noon in Nyon at UEFA headquarters. Like every year, now is the time for clubs and fans to study the list of potential opponents and try to imagine the worst and the best draw. Now is also the time to indulge in a little exercise in probabilities, more subtle than it seems at first glance.

The three rules of the draw are simple:

1. A group winner is opposed to a group second.
2. Two clubs from the same group cannot be drawn against each other.
3. Two clubs from the same country cannot compete.

The eight group winners are PSG, Bayern Munich, Manchester City, Juventus, Liverpool, Barcelona, ​​Leipzig and Valencia. The corresponding eight group runners-up are Real Madrid, Tottenham, Atalanta Bergamo, Atletico Madrid, Naples, Dortmund, Lyon and Chelsea.

PSG's possible opponents are all second in the group, except Real, who were with Paris in Group A, and Lyon. You would think that Paris has a one in six chance of falling on each of its six potential opponents … but it's a little more complicated than that.

For example, compare two possible PSG opponents: Atletico Madrid and Naples. Naples has six potential opponents: all the group winners, except Liverpool, whom it faced in pools, and Juventus. Atletico has only five possible opponents, since he cannot meet Juventus, also from Group D, Barcelona and Valencia.

This mechanically increases the probability that PSG will fall on Atletico. The probability of a PSG-Atletico cannot be equal to 1/6 (seen from PSG) and 1/5 (seen from Atletico). How to find the true probability?

## An algorithm not as simple as you might think

You would think that this is a simple counting exercise: we list all the admissible results of the draw (there are 2002 this year), we count, for example, in how many cases the PSG meets the Atletico (359 cases in 2002), and we deduce that PSG has 359/2002 = 17.9% chance of falling on Atletico. It’s almost true… but the mode of operation of the draw complicates the matter a little more.

This reasoning for counting would be correct if the draw consisted in drawing a ball among 2002, each ball representing a table of possible round of 16. Of course, 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 known as "backtracking" provides a list of their possible opponents from among the group winners still available, then a club is drawn from this list. The algorithm is not as simple as you might think since it must consider all possible scenarios for the rest of the draw to prevent any future deadlock.

I simulated 1 million draws, following the official UEFA procedure, to get the real odds (to the nearest 0.1%). Bad news for PSG: its most likely opponent is also its a priori most dangerous opponent (with Tottenham), Atletico Madrid, with a probability of 18.4% (and not 17.9%). Chelsea follows with 18.2%.

The two least likely opponents of PSG are Naples and Bergamo, the weakest opponent a priori (14.6% chance).

As for Lyon, its most likely opponent is, on paper, the most affordable, Valencia (17.7%), followed closely by Barcelona (17.6%), while its least likely opponent is Bayern Munich (14.3%).

This year again, the most likely round of 16 is Barcelona-Chelsea (23.3%). Surprisingly, Valence is Lyon’s most likely opponent, but Lyon is Valence’s least likely opponent. This is the beauty of mathematics!

## What if Chelsea had finished first in their group?

Note that the probabilities would have been radically different if Chelsea had finished ahead of Valence in group H. Tottenham would then have had a more than 32% chance of meeting Barcelona, ​​Lyon more than 22% chance of falling on the Catalans, and PSG more 21% chance of hitting Tottenham.

However, it didn't take long for this to happen, with Valencia and Chelsea both having 11 points in Group H and only being decided based on their face-to-face matches.

Besides, if UEFA had used the classic criteria (goal difference, number of goals scored) to decide between Valencia and Chelsea, it would have been Chelsea who would have finished first and the round of 16 would probably have had a completely different face.

It is remarkable and very surprising that the result of a single face-to-face (or the arbitrary choice of a rule to decide between two teams) can have such an impact on the rest of the competition.

Julien Guyon is a mathematician and football fan. A quantitative analyst, he is also an associate professor in the Department of Mathematics at Columbia University and the Courant Institute of Mathematical Sciences at New York University. His works are available on his web page: http://cermics.enpc.fr/~guyon/. Other probability tables are available on his Twitter account: @ julienguyon1977. More details are available here: https://www.fourfourtwo.com/features/champions-league-last-16-draw-probabilities-liverpool-chelsea-tottenham-man-city-real-madrid-barcelona