Barcelona and Atlético, most likely opponents for PSG

Kylian Mbappé, after the Champions League final lost by PSG against Bayern Munich, on August 23, in Lisbon.

Barely a month after the final won (1-0) by Bayern Munich against Paris-Saint-Germain, the Champions League (C1) is back. The European Cup football season kicks off tonight with the draw for the C1 group stage, which this year will last for barely a month and a half, from late October to early December.

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The eight groups of four teams will be formed from four pots of eight teams. Each group is made up of a team drawn from each pot. Since 2015, pot 1 has been made up of the defending champion and championship winners from the countries best ranked on the UEFA index. Since 2018, the winner of the Europa League (Seville, this year again) has also been placed in pot 1. Pots 2 to 4, meanwhile, are always built according to the UEFA coefficient of the clubs, in descending order.

PSG are in pot 1, as the winner of Ligue 1. Olympique de Marseille and Stade Rennais, which are making their debut in the competition, are in pot 4 and can fear very difficult draws .

Champions League hats

Hat 1:

Bayern Munich – Sevilla FC – Real Madrid – Liverpool – Juventus Turin – Paris-SG – Zenith Saint-Petersburg – FC Porto

Hat 2:

FC Barcelona – Atlético de Madrid – Manchester City – Manchester United – Chakhtior Donetsk – Borussia Dortmund – Chelsea – Ajax Amsterdam

Hat 3:

Dynamo Kiev – RB Salzburg – RB Leipzig – Inter Milan – Olympiakos Piraeus – Lazio Rome – FC Krasnodar – Atalanta Bergamo

Hat 4:

Lokomotiv Moscow – Olympique de Marseille – Club Bruges – Borussia Mönchengladbach – Istanbul Basaksehir – Midtjyland – Rennes – Ferencvaros

A complex procedure

Groups are not formed completely at random. A UEFA rule well known to football fans prevents two teams from the same country being drawn in the same group. This strongly impacts the probabilities of the draw. For example, Liverpool can only meet five teams from Pot 2: Barcelona, ​​Atlético de Madrid, Chakhtior Donetsk, Borussia Dortmund and Ajax Amsterdam. As a result, Ajax, even if they can fall against any of the eight teams in Pot 1, are much more likely to fall against Liverpool than against PSG, Juventus Turin or Porto.

You might think Liverpool have a 20% chance of falling against each of their five potential opponents, but the exact odds are much more complicated than it looks. The probability of a Liverpool-Ajax cannot be at the same time equal to 20%, “sight” of Liverpool, and 12.5%, “sight” of Amsterdam. The true probabilities take into account the draw procedure followed by UEFA, which is relatively complex.

For the group stage, an additional element complicates the problem: UEFA uses a “TV pairings” in order to maximize TV audiences. For example, for France, PSG and OM are paired, which means that if the PSG is drawn in one half of the groups (ABCD or EFGH), the OM will automatically be placed in the other half, so that these two clubs, which are the ones that generate the most TV audiences in France, will never play the same evening.

In particular, during the draw, OM fans will know very quickly, as soon as pot 1 has been emptied, which are the four groups in which they can fall (ABCD or EFGH, depending on the half where PSG will have been placed). Nothing of the sort, however, for Rennes supporters: for them, all options will remain open.

The impact of TV pairings

TV pairings further complicate the calculation of probabilities. They have a fairly complex and significant impact, often underestimated, on the probabilities of the draw.

To estimate these probabilities, at least for pots 1 and 2, I simulated 90,000 draws by precisely following the procedure adopted by UEFA. The most likely clashes are Liverpool-Barcelona and Liverpool-Atlético (around 21.5%); the least likely are Ajax against PSG, Juventus or Porto (around 8%). PSG’s most likely opponents are Barcelona and Atlético Madrid (around 15.5%), far ahead of Chakhtior Donetsk (9%) and Ajax (8%). The probabilities are more homogeneous for OM and Rennes.

The slight fluctuations that can be observed on the probability matrix are due to sampling error – the fact that only a finite number of draws are simulated. In particular, when it comes to the probabilities of the clashes between the teams in pots 1 and 2, PSG, Juventus and Porto are perfectly interchangeable, and therefore in fact share exactly the same probabilities. The same goes for Sevilla and Real Madrid, Barcelona and Atlético, Manchester City and Chelsea – Manchester United is not perfectly interchangeable with City and Chelsea because of TV pairings.

In order to measure the impact of TV pairings on the probabilities, I also simulated draws by removing the rule of TV pairing. We can see that the latter indeed have a significant impact, up to +/– 2%. In particular, they increase the likelihood of PSG falling on Barcelona or Atlético de Madrid by around 1%, and decrease the likelihood of PSG meeting Donetsk by 2.5%.

Finally, let’s note a funny curiosity: if Dortmund is Liverpool’s least likely opponent, Liverpool is also Dortmund’s most likely opponent! And there is nothing contradictory here. Ah, the beauty of mathematics!

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Julien Guyon

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.

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