A note on the UEFA Champions League Round of 16 draw L aszl o Csat o

2025-04-30
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A note on the UEFA Champions League
Round of 16 draw
L´aszl´o Csat´o∗
Institute for Computer Science and Control (SZTAKI)
E¨otv¨os Lor´and Research Network (ELKH)
Laboratory on Engineering and Management Intelligence
Research Group of Operations Research and Decision Systems
Corvinus University of Budapest (BCE)
Institute of Operations and Decision Sciences
Department of Operations Research and Actuarial Sciences
Budapest, Hungary
7th February 2023
Abstract
A paper published in a leading journal of management science has recently analysed
the mechanism used for the UEFA Champions League Round of 16 draw. Since the
authors have not been familiar with the related sports analytics literature, this note
attempts to summarise what has already been done, and what can still be done on
the issue of group draw in sports tournaments.
Keywords: constrained assignment; draw mechanism; fairness; transparency; UEFA
Champions League
MSC class: 90-10, 91B14
JEL classification number: C44, C63, Z20
1 Introduction
An article published in Management Science (Boczo´n and Wilson,2022) on 2 September
2022 studies the example UEFA Champions League Round of 16 draw to call the attention
of economists and market designers that transparency is a first-order concern in complex
assignment problems used in practice. According to the main result, the mechanism
adopted by the Union of European Football Associations (UEFA) is close to a constrained-
best solution, and the organiser does not lose too much in terms of fairness by preferring
transparency.
However, Boczo´n and Wilson (2022) fail to discuss the related sports analytics literature,
which has resulted in some weaknesses:
∗E-mail: laszlo.csato@sztaki.hu
1
arXiv:2210.15555v4 [physics.soc-ph] 6 Feb 2023
•
The UEFA Champions League Round of 16 draw is presented as a unique field
solution to a constrained assignment problem in sports. However, the same
question emerges in the group draw of several competitions. Crucially, the
F´ed´eration Internationale de Football Association (FIFA) has adopted the credible
and transparent solution of the UEFA for the 2018 FIFA World Cup draw (Guyon,
2018) as suggested by Guyon (2014).
•
Previous results on the UEFA Champions League Round of 16 draw (Kiesl,2013;
Kl¨oßner and Becker,2013) and the FIFA World Cup draw (Guyon,2015) have
been ignored.
•
The paper does not mention and evaluate the entirely fair (evenly distributed) al-
gorithms proposed for the constrained assignment problem (Guyon,2015;Kl¨oßner
and Becker,2013).
Inspired by these issues, our note endeavours to summarise all research connected to
the UEFA Champions League Round of 16 draw. Hopefully, we can also inspire future
research on this topic.
2 Is the UEFA Champions League Round of 16 draw
a unique field setting?
Boczo´n and Wilson (2022) present the procedure used for the UEFA Champions League
Round of 16 draw as a unique field solution to an assignment problem in sports. Here,
the eight group runners-up should be matched to the eight group winners such that both
the association and the group constraints are satisfied: group winners cannot play against
the runner-up from the same group, or a runner-up from the same country. This setting
is equivalent to a group draw with two pots, the first containing the runners-up and
the second containing the group winners, where the allocation is required to meet the
association and group constraints.
Then the randomisation procedure used for the UEFA Champions League Round of 16
draw is equivalent to the following mechanism:
•
Placing the runners-up in the eight groups according to the order in which they
are drawn from their urn;
•Labelling the groups in alphabetical order;
•
Placing the group winners in the first available group in alphabetical order such
that any dead end—a situation when the remaining group winners cannot be
assigned with satisfying all draw constraints—is avoided.
This interpretation reveals that the bipartite constraint (each pairing must be between a
group winner and the runner-up) is essentially not a constraint because the draw mechanism
would be entirely fair, that is, evenly distributed in the absence of the other restrictions.
Furthermore, the same randomisation procedure is used in the FIBA Basketball World Cup
(FIBA,2019), the FIFA World Cup draw (FIFA,2017,2022), the draw of the European
Qualifiers for the FIFA World Cup (UEFA,2020a), the UEFA Euro qualifying draw (UEFA,
2018,2022), and the UEFA Nations League draw (UEFA,2020b,2021). However, the
draw rules of these competitions are more complicated than the Champions League Round
2
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时间:2025-04-30
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