# Abstract

The UEFA Champions League Round of 16 is characterized by restrictions that prevent teams from the same preliminary group and the same nations from matches against each other. Together with the draw procedure currently employed by UEFA, this leads to odd probabilities: in 2012/2013, there were more outcomes of the draw with German Schalke 04 facing Ukrainian Shakhtar Donetsk than there were results where they were matched with Galatasaray Istanbul. In contrast, the probability of Schalke being drawn against Galatasaray exceeded that of playing Shakhtar. We show that this strange effect is due to the group restriction and the mechanism used by UEFA for the draw. Additionally, we provide procedures with which UEFA could produce adequate probabilities for the draw.

## 7 Appendix

### 7.1 Proof of Theorem 1

**Recapitulation of Theorem 1 ***For every constellation of nationalities of the group winners and runners-up, the probability of any subset of draw results can be written as*

*with m a natural number.*

*Proof.* The runners-up are drawn sequentially from a bowl, so the probability for a given sequence of runners-up is

where *m*_{i} denotes the number of possible opponents of the *i*-th runner-up. Obviously, all *m*_{i} are less or equal to 7 due to the group restriction, and thus the only prime numbers appearing in the denominator *m*_{1} · ·· · *m*_{g} are 2, 3, 5, and 7. Because *m*_{i}<7 for all *i*>2 (there are at most 6 group winners left after two matches have been drawn) and *m*_{i}<5 for all *i*>4 (there are at most 4 group winners left after four pairings have been determined), *m*_{1} · ·· · *m*_{g} can contain 7 at most twice and 5 at most four times. A similar argument shows that the prime number 3 (which may appear for *m*_{i}=6=3·2 or *m*_{i}=3) cannot enter the denominator more often than six times. The most difficult case is the prime number 2 which enters once if *m*_{i}=2 or *m*_{i}=6=3·2 and twice if *m*_{i}=4=2·2. So the maximal multiplicity of 2 is given when *m*_{1}=*m*_{5}=4 and *m*_{6}=*m*_{7}=2 (it always holds that *m*_{8}=1) in which case it is 12. □

### 7.2 Details of Algorithm for Probabilities of UEFA draw

For calculating the probabilities of the draw results, we build on the following fact: as the drawing from the bowl with the runners-up is not affected at all by the drawing of the group winners, we have 8!=40320 possibilities for the order in which the runners-up are drawn, and each of these orders happens with probability 1/40320. Conditional on the order in which the runners-up are drawn, the probability of a certain draw result is just

where*m*

_{i}as in the proof of Theorem 1 denotes the number of possible opponents of the

*i*-th runner-up under this draw result. The probability of any draw result is therefore simply 1/40320 times the sum of the conditional probabilities of this draw result over all 40320 possible orders in which the runners-up may be drawn. The problem of calculating the probabilities therefore essentially boils down to calculating

*m*

_{1}through

*m*

_{g}for all draw results and all orders of the runners-up, which fortunately can be done recursively.

### 7.3 Additional Tables and Figures

Bayern | Inter | Benfica | Real | Chelsea | Arsenal | Apoel | Barca | |
---|---|---|---|---|---|---|---|---|

Napoli | – | – | –0.420 | –0.420 | 0.742 | –0.420 | –0.420 | 0.898 |

CSKA | 1.129 | – | –0.518 | –0.518 | –0.740 | –0.518 | –0.518 | 1.407 |

Basel | –0.330 | 0.318 | – | 0.119 | 0.086 | 0.119 | 0.119 | –0.498 |

Lyon | –0.330 | 0.318 | 0.119 | – | 0.086 | 0.119 | 0.119 | –0.498 |

Bayer | – | –1.055 | 0.542 | 0.542 | – | 0.542 | 0.542 | –0.546 |

Marseille | –0.330 | 0.318 | 0.119 | 0.119 | 0.086 | – | 0.119 | –0.498 |

Zenit | –0.330 | 0.318 | 0.119 | 0.119 | 0.086 | 0.119 | – | –0.498 |

Milan | 0.091 | – | 0.076 | 0.076 | –0.405 | 0.076 | 0.076 | – |

Tottenham | Schalke | ManU | Barca | Bayern | Chelsea | Real | Shakhtar | |
---|---|---|---|---|---|---|---|---|

Inter | – | –1.55 | 0.49 | –0.60 | –1.55 | 1.99 | –0.60 | 1.23 |

Lyon | 0.29 | – | –0.95 | 0.84 | –0.87 | 0.29 | 0.84 | –0.48 |

Valencia | –1.17 | 1.40 | – | – | 1.40 | –1.17 | – | 0.28 |

Kobenhavn | –0.41 | –0.79 | 0.44 | – | –0.79 | –0.41 | 2.61 | –0.97 |

Roma | 0.29 | –0.87 | –0.95 | 0.84 | – | 0.29 | 0.84 | –0.48 |

Marseille | 1.99 | –1.55 | 0.49 | –0.60 | –1.55 | – | –0.60 | 1.23 |

Milan | –0.41 | –0.79 | 0.44 | 2.61 | –0.79 | –0.41 | – | –0.97 |

Arsenal | – | 1.95 | – | –1.55 | 1.95 | – | –1.55 | – |

Girond. | ManU | Real | Chelsea | Firenze | Barca | Sevilla | Arsenal | |
---|---|---|---|---|---|---|---|---|

Bayern | 0 | 1192 | 1236 | 1192 | 1854 | 1236 | 1192 | 1192 |

CSKA | 1395 | 0 | 1203 | 1161 | 1810 | 1203 | 1161 | 1161 |

Milan | 1757 | 1459 | 0 | 1459 | 0 | 1501 | 1459 | 1459 |

Porto | 1395 | 1161 | 1203 | 0 | 1810 | 1203 | 1161 | 1161 |

Lyon | 0 | 1501 | 1545 | 1501 | 0 | 1545 | 1501 | 1501 |

Inter | 1757 | 1459 | 1501 | 1459 | 0 | 0 | 1459 | 1459 |

VfB | 1395 | 1161 | 1203 | 1161 | 1810 | 1203 | 0 | 1161 |

Olymp. | 1395 | 1161 | 1203 | 1161 | 1810 | 1203 | 1161 | 0 |

Girond. | ManU | Real | Chelsea | Firenze | Barca | Sevilla | Arsenal | |
---|---|---|---|---|---|---|---|---|

Bayern | 0.00 | 13.12 | 13.48 | 13.12 | 20.57 | 13.48 | 13.12 | 13.12 |

CSKA | 15.33 | 0.00 | 13.18 | 12.82 | 19.86 | 13.18 | 12.82 | 12.82 |

Milan | 19.34 | 16.00 | 0.00 | 16.00 | 0.00 | 16.68 | 16.00 | 16.00 |

Porto | 15.33 | 12.82 | 13.18 | 0.00 | 19.86 | 13.18 | 12.82 | 12.82 |

Lyon | 0.00 | 16.43 | 17.13 | 16.43 | 0.00 | 17.13 | 16.43 | 16.43 |

Inter | 19.34 | 16.00 | 16.68 | 16.00 | 0.00 | 0.00 | 16.00 | 16.00 |

VfB | 15.33 | 12.82 | 13.18 | 12.82 | 19.86 | 13.18 | 0.00 | 12.82 |

Olymp. | 15.33 | 12.82 | 13.18 | 12.82 | 19.86 | 13.18 | 12.82 | 0.00 |

Girond. | ManU | Real | Chelsea | Firenze | Barca | Sevilla | Arsenal | |
---|---|---|---|---|---|---|---|---|

Bayern | 0.00 | 13.11 | 13.59 | 13.11 | 20.39 | 13.59 | 13.11 | 13.11 |

CSKA | 15.34 | 0.00 | 13.23 | 12.77 | 19.90 | 13.23 | 12.77 | 12.77 |

Milan | 19.32 | 16.04 | 0.00 | 16.04 | 0.00 | 16.51 | 16.04 | 16.04 |

Porto | 15.34 | 12.77 | 13.23 | 0.00 | 19.90 | 13.23 | 12.77 | 12.77 |

Lyon | 0.00 | 16.51 | 16.99 | 16.51 | 0.00 | 16.99 | 16.51 | 16.51 |

Inter | 19.32 | 16.04 | 16.51 | 16.04 | 0.00 | 0.00 | 16.04 | 16.04 |

VfB | 15.34 | 12.77 | 13.23 | 12.77 | 19.90 | 13.23 | 0.00 | 12.77 |

Olymp. | 15.34 | 12.77 | 13.23 | 12.77 | 19.90 | 13.23 | 12.77 | 0.00 |

Girond. | ManU | Real | Chelsea | Firenze | Barca | Sevilla | Arsenal | |
---|---|---|---|---|---|---|---|---|

Bayern | – | 0.073 | –0.796 | 0.073 | 0.873 | –0.796 | 0.073 | 0.073 |

CSKA | –0.057 | – | –0.392 | 0.410 | –0.224 | –0.392 | 0.410 | 0.410 |

Milan | 0.090 | –0.297 | – | –0.297 | – | 1.051 | –0.297 | –0.297 |

Porto | –0.057 | 0.410 | –0.392 | – | –0.224 | –0.392 | 0.410 | 0.410 |

Lyon | – | –0.431 | 0.838 | –0.431 | – | 0.838 | –0.431 | –0.431 |

Inter | 0.090 | –0.297 | 1.051 | –0.297 | – | – | –0.297 | –0.297 |

VfB | –0.057 | 0.410 | –0.392 | 0.410 | –0.224 | –0.392 | – | 0.410 |

Olymp. | –0.057 | 0.410 | –0.392 | 0.410 | –0.224 | –0.392 | 0.410 | – |

Roma | Panath. | Barca | Liverpool | ManU | Bayern | Porto | Juve | |
---|---|---|---|---|---|---|---|---|

Chelsea | – | 1.916 | –3.339 | – | – | 1.495 | 2.014 | 0.171 |

Inter | – | – | 2.383 | –0.911 | –0.911 | 0.793 | –1.656 | – |

Sporting | –1.497 | 0.049 | – | –0.115 | –0.115 | –0.328 | – | 2.123 |

Atletico | –0.907 | –1.251 | – | – | 2.027 | –1.387 | 0.258 | 0.332 |

Villarreal | –0.907 | –1.251 | – | 2.027 | – | –1.387 | 0.258 | 0.332 |

Lyon | –1.538 | –2.025 | 4.708 | –0.014 | –0.014 | – | –2.083 | –1.181 |

Arsenal | 3.623 | 0.128 | –1.592 | – | – | 0.322 | – | –1.665 |

Real | 0.351 | 1.785 | – | –0.998 | –0.998 | 0.201 | 0.522 | – |

^{1}See e.g., www.dailymail.co.uk/news/article-2251604/Champions-League-draw-produces-exact-pairings-dress-rehearsal–odds-staggering-2-000-000-1.html, http://redlondon.net/2012/12/19/picture-gunners-handed-tough-draw-in-uefa-rehearsal-draw/, www.dailymail.co.uk/sport/football/article-2251561/Champions-League-draw-5000-1-repeat-dress-rehearsal.html, uk.eurosport.yahoo.com/blogs/world-of-sport/champions-league-draw-exact-same-rehearsal-draw-115551048.html, http://sports.yahoo.com/blogs/soccer-dirty-tackle/champions-league-draw-matches-rehearsal-odds-better-might-191618943–sow.html, www.thesun.co.uk/sol/homepage/sport/football/4707472/Champions-League-draw-gives-same-results-as-dress-rehearsal.html, http://www.offthepost.info/blog/2012/12/champions-league-draw-was-exact-copy-of-rehearsal-no-one-agrees-on-the-odds-of-it-happening/, http://www.spiegel.de/sport/fussball/champions-league-auslosung-grafik-zeigt-gleiche-paarungen-a-874109.html.

^{2}^{3}Essentially, association is equivalent to nation. However, for instance, the United Kingdom takes part in European Football with several associations, such as England, Scotland, Wales, and Northern Ireland.

^{4}Typically, this is done by using some computer program. Such a program is availabe from the authors upon request. However, there are also elaborate mathematical methods to compute the number of admissible derangements. In particular, one may use bipartite graphs, adjacency matrices and their permanents, or one may use the technique of rook polynomials.

^{5}^{6}One should notice, however, that there was more than one test by UEFA, so actually one would have to consider the probability of the result of the actual draw happening in one of several test draws, given by 1–(1–1/5463)

^{n}*≈n*/5463, with*n*the (unknown) number of test draws.^{7}See http://www.uefa.com/uefachampionsleague/news/newsid=640504.html, where the procedure that is still used by UEFA is described for the 2007/2008 season.

^{8}These things can also be analyzed by using concepts from graph theory, in particular, the marriage theorem.

^{9}See R Core Team (2012).

^{10}The algorithm for calculating the probabilities is described in some detail in subsection 7.2 of the appendix.

^{11}^{12}^{13}Note that this is just a very simple way to arrive at reasonable strengths for illustration purposes. It would be much more difficult to determine the teams’ real strengths or, ultimately, the probabilities of advancing to the next round, in such a way that they come close to reality as measured by, say, the odds available from bookmakers.

^{14}^{15}^{16}^{17}Additionally, UEFA rules state that the group winners play the first leg abroad and the second leg at home, which is commonly thought to be an advantage. Critical on that, however, Eugster, Gertheiss, and Kaiser (2011).

^{18}One has to distinguish derangements of the following cycle types (sorted in ascending order of probability): four 2-cycles (abbreviated as (2222)), one 4-cycle and two 2-cycles (422), two 3-cycles and one 2-cycle (332), the forms (62), (53), (44), and (8) (one 8-cycle).

^{19}Notice that the probabilities of the UEFA procedure would change slightly if group winners were drawn first and only then matched with suitable runners-up. However, all the oddities described in this article would still be present. The same is true if at every point during the procedure, a random choice would be made whether to match a winner to a runner-up or vice versa.

^{20}Obviously, there are many ways for drawing one of 5463 objects at random, actually every decent statistics software offers this feature. However, UEFA most certainly is interested in a procedure that can be televised and is easily recognizable as being “fair.”

^{21}We will discuss matters for the 2012/2013 case, modifications to other numbers in other years being fairly obvious.

^{22}Recall that Schalke is not placed in Arsenal’s bowl because they joined the same preliminary group,

*B*, while English ManU is not eligible for English Arsenal.^{23}It would also be possible to stay with the current schedule in this case.

^{24}It is important to stick with the current matches in this case. If one were to draw until a change was possible, another “partial rejection sampling” would be created, leading again to unwanted probabilities.

^{25}Within the table, asterisks at the end of pairings indicate that these pairings are to be changed if possible, while asterisks in front of pairings indicate that these parings were (tried to be) changed in the previous step.

^{26}For the alternative discussed in footnote 23, speed of convergence to the uniform distribution would be slower, the corresponding eigenvalues being 0.33, 0.04, 0.11, 0.06, and 0.11. In any case, the transition matrix has only positive entries, showing that regardless of the initial stage, every result of the draw emerges with positive probability.

The authors would like to thank Ralph Friedmann, three anonymous referees, an anonymous associate editor and the journal editor for their helpful comments.

### References

Eugster, M. J. A., J. Gertheiss, and S. Kaiser. 2011. “Having the Second Leg at Home – Advantage in the UEFA Champions League Knockout Phase?” *Journal of Quantitative Analysis of Sports* (7): Article 6. Search in Google Scholar

R Core Team. 2012. *R: A Language and Environment for Statistical Computing*, R Foundation for Statistical Computing, Vienna, Austria, URL http://www.R-project.org/, ISBN 3-900051-07-0. Search in Google Scholar

**Published Online:**2013-08-14

**Published in Print:**2013-09-01

©2013 by Walter de Gruyter Berlin Boston