Analysing the restricted assignment problem of the group draw in sports tournaments - arXiv

Analysing the restricted assignment problem of
 the group draw in sports tournaments
 László Csató∗
 Institute for Computer Science and Control (SZTAKI)
 Eötvös Loránd Research Network (ELKH)
 Laboratory on Engineering and Management Intelligence
arXiv:2103.11353v1 [physics.soc-ph] 21 Mar 2021

 Research Group of Operations Research and Decision Systems

 Corvinus University of Budapest (BCE)
 Department of Operations Research and Actuarial Sciences

 Budapest, Hungary

 23rd March 2021

 Die Unvollkommenheit der menschlichen Einsicht, die Scheu vor einem üblen Ausgang,
 die Zufälle, von welchen die Entwicklung der Handlung berührt wird, machen, daß von
 allen durch die Umstände gebotenen Handlungen immer eine Menge nicht zur Ausführung
 kommen.1

 (Carl von Clausewitz: Vom Kriege)

 Abstract
 Many sports tournaments contain a group stage where the allocation of teams
 is subject to some constraints. The standard draw procedure extracts the teams
 from pots sequentially and places them in the first available group in alphabetical
 order such that at least one assignment of the teams still to be drawn remains
 acceptable. We show how this mechanism is connected to generating permutations
 and provide a backtracking algorithm to find the solution for any given sequence.
 The consequences of draw restrictions are investigated through the case study of the
 European Qualifiers for the 2022 FIFA World Cup. We quantify the departure of its
 draw procedure from even distribution and propose two alternative approaches to
 increase the excitement of the draw.
 Keywords: assignment problem; backtracking algorithm; football; mechanism design;
 permutation
 MSC class: 05A05, 68U20, 68W40, 91B14
 JEL classification number: C44, C63, Z20
 ∗
 E-mail: laszlo.csato@sztaki.hu
 1
 “The imperfection of human insight, the fear of evil results, accidents which derange the development
 of designs in their execution, are causes through which many of the transactions enjoined by circumstances
 are never realised in the execution.” (Source: Carl von Clausewitz: On War, Book 6, Chapter 30 [Defence
 of a theatre of war (continued) When no decision is sought for]. Translated by Colonel James John
 Graham, London, N. Trübner, 1873. http://clausewitz.com/readings/OnWar1873/TOC.htm)

1 Introduction
Mechanism design usually focuses on theoretical properties like efficiency, fairness, and
incentive compatibility (Abdulkadiroğlu and Sönmez, 2003; Roth et al., 2004; Csató,
2021). On the other hand, institutions—like governing bodies in major sports—often
emphasise simplicity and transparency, which calls for a comprehensive review of how these
procedures that exist in the real world perform with respect to the above requirements.
 The present paper offers such an analysis of a mechanism used to solve a complex
assignment problem. Several sports tournaments are organised with a group stage where
the teams are assigned to groups subject to some rules. This is implemented by a draw
system that satisfies the established criteria.
 In particular, we analyse the draw procedure of the Union of European Football
Associations (UEFA), applied for various competitions of national teams such as the UEFA
Nations League (UEFA, 2020b), the UEFA Euro qualifying (UEFA, 2018), or the European
Qualifiers for the FIFA World Cup (UEFA, 2020a). The mechanism works as follows to
generate a sense of excitement and to ensure transparency. First, the teams are divided
into seeding pots based on an exogenous ranking. For each pot, balls representing the
teams are placed in a bawl and drawn randomly. The teams are assigned to the groups in
alphabetical order, i.e. the first team drawn from each pot is allocated to the first group,
the second team to the second group, and so forth. However, there are draw constraints
to provide an assignment “that is fair for the participating teams, fulfils the expectations
of commercial partners and ensures with a high degree of probability that the fixture can
take place as scheduled” (UEFA, 2020a). Consequently, if a draw restriction applies or is
anticipated to apply, the actual team is allotted to the first available group as indicated
by a computer program in order to avoid any dead end, a situation when the teams still
to be drawn cannot be assigned to the remaining empty slots.
 This procedure is not so simple as intuition suggests.

Example 1. Consider the European Qualifiers for the 2022 FIFA World Cup. Assume
that Pots 1–4 are already emptied and Group H consists of Portugal from Pot 1, Ukraine
from Pot 2, Iceland from Pot 3, and Serbia from Pot 4. The draw continues with Pot 5.
First, Armenia is drawn and allotted to Group A. Second, Cyprus is drawn and assigned
to Group B. Third, Andorra—a country without any draw constraints—is drawn and
placed in. . . Group H, the first available group according to the computer.

 Example 1 uncovers that the number of options available to a team depends not only
on its own attributes but also on the characteristics of the remaining teams. As we will
see, Andorra can be allocated only to Group H, otherwise, no feasible assignment exists.
 The mechanism described above is used generally to draw groups in the presence of
some constraints. Nonetheless, UEFA does provide neither an exact algorithm to determine
the group allocation for a given random order of the teams, nor an analysis on the effects of
the particular conditions. Our work aims to fill this research gap. The main contributions
can be summarised as follows:

 • We highlight how the restricted group assignment problem is linked to generating
 all permutations of a sequence (Section 3.1);

 • We present a backtracking algorithm to produce the group allocation for teams
 drawn randomly, which also finds the first available group for the team drawn
 (Section 3.2);

 2

• We reveal the implications of the draw constraints in the case of the European
 Qualifiers for the 2022 FIFA World Cup (Section 4.2);

 • We quantify the departure of the UEFA draw procedure from the “evenly distrib-
 uted” system in this particular tournament (Section 4.3);

 • We propose two alternative approaches for solving the group assignment problem
 to increase uncertainty during the draw (Section 5).

 Group allocation is an extensively discussed topic in the mainstream media. Several
articles published in famous dailies such as Le Monde and The New York Times illustrate
the significant public interest in the FIFA World Cup draw (Aisch and Leonhardt, 2014;
Guyon, 2014, 2017b,d,e,f; McMahon, 2013), as well as in the UEFA Champions League
group round draw (Guyon, 2020a,c) and the Champions League knockout stage draw
(Guyon, 2017a,c, 2020b,d). Thus a better understanding of these draw procedures and their
consequences is relevant not only for the academic community but for sports administrators
and football fans around the world.

2 Literature review
Several scientific works focus on the FIFA World Cup draw. Before the 2018 edition, the
host nation and the strongest teams were assigned to different groups, while the remaining
teams were drawn randomly with maximising geographic separation: countries from the
same continent (except for Europe) could not have played in the same group and at most
two European teams could have been in the same group.
 For the 1990 FIFA World Cup, Jones (1990) shows that the draw was not mathematically
fair. For example, West Germany would be up against a South American team with a
probability of 4/5 instead of 1/2—as it should have been—due to the incorrect consideration
of the constraints. Similarly, the host Germany was likely to play in a difficult group in the
2006 edition, but other seeded teams, such as Italy, were not (Rathgeber and Rathgeber,
2007).
 Guyon (2015) identifies severe shortcomings of the procedure used for the 2014 FIFA
World Cup draw: imbalance (the eight groups are at different competitive levels), unfairness
(certain teams have a greater chance to end up in a tough group), and uneven distribution
(the feasible allocations are not equally likely). The paper also discusses alternative
proposals to retain the practicalities of the draw but improve its outcome.
 Laliena and López (2019) develop two evenly distributed designs for the group round
draw with geographical restrictions that produce groups having similar (or equal) compet-
itive levels.
 Cea et al. (2020) analyse the deficiencies of the 2014 FIFA World Cup draw and give a
mixed integer linear programming model to create groups. The suggested method takes
into account draw restrictions and aims to balance “quality” across the groups.
 Other studies deal with the UEFA Champions League, the most prestigious association
football (henceforth football) club competition around the world. Klößner and Becker
(2013) investigate the procedure to determine the matches in the round of 16, where eight
group winners should be paired with eight runners-up. There are 8! = 40,320 possible
outcomes depending on the order of runners-up, but clubs from the same group or country
cannot face each other, and the group constraint reduces the number of feasible solutions
to 14,833. The draw system is proved to inherently imply different probabilities for certain

 3

assignments, which are translated into more than ten thousand Euros in expected revenue
due to the substantial amount of prize money. Finally, the authors propose a better suited
mechanism for the draw.
 Analogously, Boczoń and Wilson (2018) examine the matching problem in the knockout
phase of this tournament. The number of valid assignments is found to be ranged
from 2,988 (2008/09 season) through 6,304 (2010/11) to 9,200 (2005/06), determined
by the same-nation exclusion that varies across the years. It is analysed how the UEFA
procedure affects expected assignments and addresses the normative question of whether
a fairer randomisation mechanism exists. They conclude that the current design comes
quantitatively close to a constrained best in fairness terms. Guyon (2019) presents a new
tournament format where the teams performing best during a preliminary group round
can choose their opponents in the subsequent knockout stage. The proposal is illustrated
with the round of 16 of the Champions League.
 To summarise, the previous academic literature of constrained matching mechanisms
for sports tournaments mostly discusses either the FIFA World Cup draw or the UEFA
Champions League knockout phase draw. Both problems are simpler than the one discussed
here. The World Cup draw does not require backtracking as the group skipping policy
could not lead to impossibility (Jones, 1990; Guyon, 2015). Even though dead ends should
be avoided in the knockout stage of the Champions League, only 16 teams need to be
paired, thus the number of feasible solutions remains tractable and the complexity of
backtracking is more limited compared to Example 1.

3 Algorithmic background
This section studies the question of how we can obtain the solution, i.e. the group allocation
provided by the UEFA procedure.

3.1 Connection to the generation of permutations
A permutation of an already ordered set is a rearrangement of its elements. Here the
initial order of the teams is determined by the random draw. In the absence of restrictions,
the teams can be assigned to the groups sequentially. Otherwise, one should find the
permutation that corresponds to the allocation implied by the UEFA rule. This can be
achieved by checking all permutations of the initial order in an appropriate sequence such
that the first permutation satisfying all constraints solves the group assignment problem.
 According to the UEFA draw procedure (UEFA, 2020a): “when a draw condition
applies or is anticipated to apply, the team drawn is allocated to the first available group
in alphabetical order”. In other words, the team drawn is assigned to the first empty slot
except if all permutations of the remaining teams, including the one drawn presently,
violate at least one restriction.

Example 2. Assume that there are = 4 teams 1– 4 drawn sequentially to be assigned
to groups A–D. The order of permutations implied by the draw mechanism is shown in
Figure 1. Note that team 1 is assigned to group A in the first six permutations as it
can be placed in another group only if either group A is unavailable for team 1 or teams
 2– 4 cannot be allocated to groups B–D. The draw system is illustrated by two case
studies:

 4

Team assignment: the first 12 permutations
 Group
 1 2 3 4 5 6 7 8 9 10 11 12

 A
 B
 C
 D
 Team assignment: the last 12 permutations
 Group
 13 14 15 16 17 18 19 20 21 22 23 24

 A
 B
 C
 D
 The symbols , , , represent teams 1– 4 in Example 2, respectively.

Figure 1: The sequence of permutations according to the UEFA draw procedure, = 4

 ∙ If team 1 cannot be placed in group A and team 3 cannot be placed in group
 C, then the first six permutations are unacceptable due to the first constraint,
 and permutation 7 is skipped because of the second condition. The solution is
 permutation 8: 2, 1, 4, 3.

 ∙ If teams 2– 4 cannot be placed in group C and team 2 cannot be placed
 in group A, then the first 12 permutations are unacceptable due to the first
 constraint, and the next two are skipped because of the second condition. The
 solution is permutation 15: 3, 2, 1, 4.

 Generating all permutations of a given sequence of values in a specific order is a
well-known problem in computer science (Sedgewick, 1977). The classic algorithm of
lexicographic ordering goes back to Narayan.a Pan.d.ita, an Indian mathematician from
the 14th century (Knuth, 2005). The ordering corresponding to the UEFA mechanism is
called representation via swaps (Arndt, 2010) and has been presented first in Myrvold and
Ruskey (2001); Arndt (2010, Figure 10.1-E) contains the same order of permutations as
Figure 1.

3.2 Finding the allocation for a given draw order
The description of the European Qualifiers for the 2022 FIFA World Cup draw procedure
(UEFA, 2020a) does not provide an algorithm to find the implied group assignment.
Nonetheless, according to Example 1, this is a non-trivial task. In addition, researchers,
journalists, or football fans may be interested in simulating similar draw systems.
 The scheme of an appropriate computer program is provided in Figure 2. It is based
on backtracking: if the remaining teams cannot be assigned to the empty group slots
in any order such that all restrictions are satisfied, then the last team is placed in the
next available group in alphabetical order. This process is repeated until the solution is
obtained or the existence of a feasible allocation is excluded.

 5

START

 Can team of pot 
 be placed in group ?

 No
 Yes

 Team of pot is
 Is smaller than
 allotted to group 
 the number of groups
 = +1
 available for pot ?
 =1
 Yes No

 Is smaller than the = +1
 Is = 1?
 number of teams ? START

 Yes No Yes
 No

 SOLUTION = +1 There exists = [group of team ( − 1)] + 1
 is found START NO allocation All teams ℓ ≥ are
 removed from their groups
 = −1
 START

 Figure 2: Backtracking algorithm for the restricted group assignment problem

 For readers following operations research in sports, backtracking can be familiar from
the problem of scheduling round robin tournaments, where an unlucky assignment of
games to slots can result in a schedule that could not be completed (Rosa and Wallis,
1982; Schaerf, 1999). Backtracking is also widely used to solve puzzles such as the eight
queens puzzle, crosswords, or Sudoku. The name stems from the American mathematician
D. H. Lehmer.

4 An analysis of the European Qualifiers for the 2022
 FIFA World Cup group draw
The FIFA World Cup attracts millions of fans, the final of the 2010 edition has been watched
by about half of the humans who were alive on its time (Palacios-Huerta, 2014). Sports has
a huge influence on society: in sub-Saharan Africa, national football team success improves
attitudes toward other ethnicities and reduces interethnic violence (Depetris-Chauvin
et al., 2020). Participation in the FIFA World Cup yields substantial economic benefits on
its own as each team received at least 9.5 million USD in the 2018 edition (FIFA, 2017).

 6

Table 1: Seeding pots in the European Qualifiers for the 2022 FIFA World Cup

 Pot 1 Pot 2 Pot 3
 10 Belgium 11 Switzerland 21 Russia (1)
 20 France 12 Wales 22 Hungary
 30 England 13 Poland 23 Republic of Ireland
 40 Portugal 14 Sweden 24 Czech Republic
 50 Spain 15 Austria 25 Norway
 60 Italy 16 Ukraine 26 Northern Ireland
 70 Croatia 17 Serbia 27 Iceland
 80 Denmark 18 Turkey 28 Scotland
 90 Germany 19 Slovakia 29 Greece
 10 Netherlands 20 Romania 30 Finland

 Pot 4 Pot 5 Pot 6
 31 Bosnia and Herzegovina 41 Armenia (1) 51 Malta (1)
 32 Slovenia 42 Cyprus (1) 52 Moldova
 33 Montenegro 43 Faroe Islands (3) 53 Liechtenstein
 34 North Macedonia 44 Azerbaijan (1) 54 Gibraltar (2)
 35 Albania 45 Estonia (2) 55 San Marino
 36 Bulgaria 46 Kosovo (3)
 37 Israel 47 Kazakhstan (5)
 38 Belarus (1) 48 Lithuania (2)
 39 Georgia 49 Latvia (2)
 40 Luxembourg 50 Andorra
 The number before the member association indicates its rank according to the November 2020 FIFA
 Rankings.
 Numbers in parenthesis show the maximal number of groups that can be unavailable for the team,
 except for countries in Pot 1. Zeros are not displayed.

 Therefore, it is important to design fairly all details of the 2022 FIFA World Cup
qualification. One of them is the draw procedure of the European Qualifiers. This
competition contains two rounds, the first being the group stage played from March to
November 2021. 55 national teams have entered the contest where they are divided into
10 groups. The group winners qualify directly for the 2022 FIFA World Cup, and the
runners-up advance to the play-offs.

4.1 The draw conditions
The draw system aims to produce groups that are balanced in terms of strength. For this
purpose, the teams are allocated to six seeding pots on the basis of the November 2020
FIFA Rankings as shown in Table 1. They have to be divided into Groups A–E consisting
of five teams, one from Pots 1–5 each, and Groups F–J consisting of six teams, one from
Pots 1–6 each. Four types of draw constraints apply (UEFA, 2020a):

 • Competition-related reasons: the four participants of the UEFA Nations League
 Finals 2021 (Belgium, France, Italy, Spain; all in Pot 1) should be drawn into
 Groups A–E.

 7

• Prohibited team clashes: based on the UEFA Executive Committee decisions,
 certain teams cannot be drawn into the same group for political reasons (Ar-
 menia/Azerbaijan, Gibraltar/Spain, Kosovo/Bosnia-Herzegovina, Kosovo/Serbia,
 Kosovo/Russia, Russia/Ukraine).

 • Winter venue restrictions: ten countries are identified with a risk of severe winter
 conditions; a maximum of two such countries can be drawn into the same group
 (Belarus, Estonia, Faroe Islands, Finland, Iceland, Latvia, Lithuania, Norway,
 Russia, Ukraine). Since the Faroe Islands and Iceland have the highest risk, these
 teams cannot play in the same group.

 • Excessive travel restrictions: twenty country pairs are identified with excessive
 travel relations; a maximum of one such pair can be drawn into the same group
 (Kazakhstan – Andorra, England, France, Faroe Islands, Gibraltar, Iceland,
 Malta, Northern Ireland, Portugal, Republic of Ireland, Scotland, Spain, Wales;
 Azerbaijan – Gibraltar, Iceland, Portugal; Iceland – Armenia, Cyprus, Georgia,
 Israel).

Some conditions are automatically guaranteed by the allocation of seeding pots. For
instance, Spain has to play in a smaller group of five teams (competition-related reasons),
while Gibraltar from Pot 6 should be in a larger group of six, thus the prohibited clash
Gibraltar/Spain is not an effective constraint.
 The group stage draw was held on 7 December 2020 as a virtual event. Its video
is available at https://www.fifa.com/worldcup/news/uefa-preliminary-draw-for-
qatar-2022-to-be-streamed-live.

4.2 The effects of restrictions
The draw conditions are worth a thorough analysis for at least two reasons: (1) they might
be logically inconsistent (Csató, 2020); and (2) the probability of events that threaten the
transparency of the draw can be determined.
 The number of permutations is prohibitively large due to the 55 participants but the
problem can be simplified as there are 10 groups and no constraints involve a set of groups.
Therefore, it is sufficient to consider the permutations of the (maximal 10) teams in a
given pot, and modify the allocation of the teams drawn from the previous pot(s) only if
no permutation satisfies the restrictions. Nonetheless, generating all permutations in the
appropriate order and checking them sequentially is an inefficient approach. In Example 1,
Andorra should be assigned to Group H instead of the still empty Group C, meaning that
more than 5 × 7! = 25,200 permutations should be rejected due to the violation of at least
one constraint (seven teams come after Andorra in Pot 5 and no solution exists if Andorra
is assigned to Groups C–G).
 The condition of competition-related reasons is different from the other constraints
since it affects only Pot 1 where the draw starts. Hence it is straightforward to assign the
first ten teams to the groups: the four Nations League finalists are placed in Groups A–E
according to their random order together with the team drawn first from the remaining six,
which is allotted to the first empty slot in Groups A–E. This can probably be understood
by any viewer.
 Draw restrictions can make a group unavailable for certain teams. Table 1 presents
the maximal number of such groups in parenthesis. The first dead end may occur in Pot 3

 8

because of the prohibited clash between Russia and Ukraine. The winter venue constraint
can be effective in Pot 4. The case of Pot 5 becomes the most complex as there are four
countries with an issue of winter venue, four affected by excessive travel restrictions, and
one involved in prohibited clashes. Finally, excessive travel conditions may exclude two
teams from Pot 6.
 The biggest threat to simplicity and transparency is when the team drawn is not
assigned to the first available group in alphabetical order. Recall Example 1 where
Andorra is placed in Group H because (1) this group contains Ukraine and Iceland, thus
no more country with a risk of severe winter conditions (Estonia, Faroe Islands, Latvia,
Lithuania) can be drawn here; (2) this group contains Serbia, thus Kosovo cannot be
drawn here due to prohibited team clashes; and (3) this group contains Portugal and
Iceland, two teams involved in excessive travel relations together with both Azerbaijan
and Kazakhstan. Consequently, among the eight remaining teams in Pot 5, only Andorra
can be allotted to Group H.
 Even though UEFA (2020a) emphasises that “the number of options available to a team
depends not only on the team’s own attributes (for example, “winter venue”) and those of
the teams already drawn, but also on the attributes of the other teams still to be drawn”,
explaining such a situation poses a serious challenge, especially as the fans of Andorra are
not much interested in the constraints that affect other teams. The integrity of the draw
mechanism may be harmed if the stakeholders cannot be immediately persuaded about
the necessity of a similar assignment to another group.
 The probability that the team drawn is not placed in the first available group is
quantified by simulating the official draw procedure 10 million times since there exist
(10!)5 × 5! ≈ 7.55 × 1034 different permutations of the teams. These probabilities are as
follows.

 • Pot 1: 0 because competition-related reasons are disregarded.

 • Pot 2: 0 as no draw conditions can be effective.

 • Pot 3: varies between 1.1 × 10−3 and 1.13 × 10−3 for any team drawn ninth except
 for Russia. The theoretical value can also be computed. Group H should contain
 Ukraine and Russia had to be the last team drawn from Pot 3. This has a chance
 of 0.01 being shared equally among the nine other nations of Pot 3, which results
 in 1.111 × 10−3 . The discrepancy is due to the stochastic nature of the simulation.

 • Pot 4: varies between 3.61 × 10−4 and 3.76 × 10−4 for any team drawn ninth
 except for Belarus. The theoretical value can be computed, too. Group H should
 contain Ukraine from Pot 2 and either Finland, or Iceland, or Norway from Pot
 3. In addition, Belarus has to be the last team drawn from Pot 4. As Russia
 from Pot 3 cannot be in Group H, this has a chance of 3.333 × 10−3 being shared
 equally among the nine other countries of Pot 4, which leads to 3.7 × 10−4 . Again,
 the bias is caused by the stochastic nature of the simulation.

 • Pots 5 and 6: see Figures 3 and 4. The theoretical value is not given because of
 the complex interactions of draw constraints.

 According to Figure 3, restrictions affecting the teams still to be drawn cause the most
severe problem in Pot 5, where they proliferate as Table 1 uncovers. For instance, Andorra
(50) could not be assigned to the first empty slot with a probability of almost 2.5%, which

 9

Draw positions 2–5 Draw positions 6–9

 50 50
 49 49
 48 48
 47 47
 46 46
 45 45
 44 44
 43 43
 42 42
 41 41

 0 0.0005 0.001 0.0015 0.002 0 0.005 0.01 0.015 0.02

 Drawn 2nd Drawn 3rd Drawn 6th Drawn 7th
 Drawn 4th Drawn 5th Drawn 8th Drawn 9th

 Figure 3: Reassignments forced by draw restrictions for teams in Pot 5

 55

 54

 53

 52

 51
 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011

 Drawn 3rd Drawn 4th

 Figure 4: Reassignments forced by draw restrictions for teams in Pot 6

is explained by the lack of draw conditions for this team. It is followed by Armenia (41)
and Cyprus (42) that can be excluded from a group only because of a travel restriction
when Iceland from Pot 3 and either Israel or Georgia from Pot 4 play in the same group.
The unpleasant situation is the least frequent for the Faroe Islands (43) since, in addition
to the winter venue constraint, it cannot be placed in the group of Iceland.
 These necessary replacements are more difficult to justify if the number of remaining
teams in the pot is higher as shown by Example 1, which can be a proxy of complexity.
Backtracking may reach both Andorra and Kosovo (46) when they are drawn only second
from Pot 5: in almost four cases among a thousand, none of the eight teams still to be
drawn can be assigned to a particular group because it contains Ukraine from Pot 2 and
Iceland from Pot 3, as well as either Israel or Georgia from Pot 4. The rarest case—arising
in about one millionth of simulation runs—is when the Faroe Islands, drawn as 6th, could
not be assigned to the first available group. This occurs only if the remaining teams in
Pot 5 are Armenia, Cyprus, Azerbaijan (44), and Kazakhstan (47), furthermore, a pair of

 10

Table 2: The comparison of the UEFA mechanism to an evenly distributed random draw

Team 1 Team 2 Increase (%) Team 1 Team 2 Decrease (%)
Faroe Islands Gibraltar 3.97 Kosovo Iceland 6.08
Kosovo Gibraltar 3.96 Andorra Gibraltar 4.03
Kosovo Liechtenstein 3.79 Kosovo Belarus 3.89
Kosovo Malta 3.76 Andorra Malta 3.67
Kosovo San Marino 3.75 Andorra Moldova 3.60
Kosovo Moldova 3.70 Andorra San Marino 3.38
Faroe Islands Malta 3.66 Kazakhstan Belarus 3.24
Faroe Islands Liechtenstein 3.49 Andorra Liechtenstein 3.13
Andorra Iceland 3.46 Kosovo Ukraine 2.99
Faroe Islands Moldova 3.38 Armenia Gibraltar 2.63
Faroe Islands San Marino 3.29 Faroe Islands Italy 2.60
Kazakhstan Italy 3.16 Cyprus Gibraltar 2.56
Kazakhstan Belgium 3.07 Kazakhstan England 2.56
Belarus Russia 2.96 Cyprus Malta 2.53
Andorra Ukraine 2.81 Kazakhstan Portugal 2.43
 The first (last) three columns list the 15 highest (lowest) change in the probability to be placed in the same
 group caused by the UEFA draw procedure.
 Teams in Pot 6 are written in italics.

 countries suffering from excessive travel restrictions emerges in one of the last groups.
 Fortunately, draw conditions anticipated to apply require such a reassignment only with
 a probability of about 6.5% even in Pot 5. This event is still less frequent in the last pot as
 Figure 4 reveals. Here, the three unconstrained teams have about a 1% chance not to be
 assigned to the first available group, and this is less than 0.2% for Malta (51). Gibraltar
 (54) never needs to be reassigned due to a possible dead end because it is involved in all
 draw conditions concerning Pot 6.

 4.3 Quantifying the distortion of the UEFA draw procedure
 Another important requirement for any draw mechanism is fairness: whether the acceptable
 outcomes are evenly distributed, thus equally likely (Guyon, 2015). Jones (1990) warns
 that placing the team drawn in the first available group does not satisfy this axiom.

 Example 3. Assume that there are three groups A–C and three teams 1– 3 but 2
 cannot be placed in group B. Then 2 should play in group A and group C with an equal
 chance of 50%, respectively. However, the UEFA draw procedure assigns 2 to group A
 with a probability of 1/3 (when 2 is drawn first) and to group C with a probability of
 2/3 (when 2 is drawn second or third).

 Due to this bias, some teams might have a greater chance than others to end up in a
 tough group. Consequently, most papers discussed in Section 2 address the issue of even
 distribution. Analogously, we compare the UEFA mechanism to an ideal random choice
 among all valid group assignments. Again, both policies are simulated 10 million times,
 and the chance of playing in the same group is calculated for each pair of teams.
 Table 2 summarises the largest distortions. The implementation chosen by UEFA
 does not lead to substantial differences, only one bias (for the pair Iceland and Kosovo)

 11

exceeds 5%. The highest values appear for the teams in Pot 5 where draw restrictions
proliferate, mainly concerning Andorra, the Faroe Islands, Kazakhstan, and Kosovo. The
most significant effect is assigning Andorra to a smaller group of five teams (all the five
countries from Pot 6 are present together with Andorra in the second part of Table 2),
while assigning the Faroe Islands and Kosovo to a larger group of six teams. They have a
probability of about 3.5%.
 At first glance, this might suggest that Andorra unfairly benefits from the UEFA draw
procedure since it is easier to obtain the first two positions against a smaller number of
opponents. However, teams in Pot 5 have no reasonable chance to progress, they can
mainly hope to score some goals and win occasionally. From this perspective, the Faroe
Islands and Kosovo are favoured by playing against a weaker team, which can result in
collecting more prize money and points in the FIFA ranking. Furthermore, Groups A–E
contain the four Nations League finalists that can be somewhat stronger than the average
in Pot 1.

5 Discussion
The current paper has addressed the restricted group assignment problem in sports
tournaments. The connection of UEFA implementation to permutation generation has
been discussed and a backtracking algorithm has been presented to find the valid allocation
implied by the rules. The draw of the European Qualifiers for the 2022 FIFA World Cup
has been investigated as a case study: we have examined what is the probability that a
team is not assigned to the first available group because of the attributes of the teams
still to be drawn, and how the official mechanism departs from even distribution.
 The draw mechanism is found to be somewhat biased but it remains close to the
principle of equal treatment. Its sporting effects are ambiguous and insignificant with
respect to qualification. Therefore, in contrast to the conclusion of Klößner and Becker
(2013), the UEFA mechanism does not lead to substantial financial differences here. The
chosen implementation seems to be a reasonable compromise until the draw constraints
do not exclude too many assignments.
 Previous studies have made several recommendations to create (more) evenly distributed
groups. However, these proposals usually use a fundamentally novel approach and/or are
less interesting for fans to watch, hence they are unlikely to be applied soon.
 Nonetheless, the UEFA policy has another shortcoming as it might lead to a determin-
istic assignment too early, which is detrimental to the excitement.
Example 4. Assume that there are four groups A–D and four teams 1– 4 drawn in
this order, while 3 cannot be assigned to group C. After team 1 is placed in group A
and 2 in group B, uncertainty entirely disappears since 3 should play in group D and
 4 in group C.
 The problem of Example 3 is caused by the principle that the team drawn is reassigned
from the first available group in alphabetical order only if no permutation of the remaining
teams satisfies the draw constraints. Therefore, two alternative policies are given to
increase uncertainty during the draw.
Definition 1. Mechanism A: If the team drawn can be allocated to a group such that at
least two feasible assignments of the remaining teams to the empty slots exist, then it is
placed in the first available group with this property in alphabetical order. Otherwise, the
team drawn is placed in the first available group in alphabetical order.

 12

Mechanism A retains at least two valid group assignments as long as possible.
Definition 2. Mechanism B: The team drawn is allocated to the first available group in
alphabetical order where the highest number of the remaining teams in its pot cannot
play.
 Mechanism B aims to maximise the number of acceptable assignments for the teams
still to be drawn.
Example 5. Consider the situation outlined in Example 4:
 • Mechanism A assigns 1 to group A, 2 to group C (otherwise, only one feasible
 allocation remains), 3 to group B, and 4 to group D.
 • Mechanism B assigns 1 to group C (since 3 cannot be placed here), 2 to
 group A, 3 to group B, and 4 to group D.
Note that after allocating the first team 1, six assignments satisfy all restrictions under
Mechanism B but only four under Mechanism A as 3 cannot be allotted to group C.
 In addition, mechanism B reduces the probability of a dead end situation by filling
first the groups with many draw constraints. The comparison of these procedures will be
the topic of future research.

Acknowledgements
This paper could not have been written without my father (also called László Csató), who
coded the simulations in Python.
We are grateful to Julien Guyon for inspiration.
Lajos Rónyai provided valuable comments and suggestions on an earlier draft.
We are indebted to the Wikipedia community for summarising important details of the
sports competitions discussed in the paper.
The research was supported by the MTA Premium Postdoctoral Research Program grant
PPD2019-9/2019.

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