<<

Analysing the restricted assignment problem of the group draw in sports

L´aszl´oCsat´o∗

Institute for Computer Science and Control (SZTAKI) E¨otv¨osLor´andResearch Network (ELKH) Laboratory on Engineering and Management Intelligence Research Group of Operations Research and Decision Systems

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

Budapest,

23rd March 2021

Die Unvollkommenheit der menschlichen Einsicht, die Scheu vor einem ublen¨ Ausgang, die Zufalle¨ , von welchen die Entwicklung der Handlung beruhrt¨ wird, machen, daß von allen durch die Umstande¨ gebotenen Handlungen immer eine Menge nicht zur Ausfuhrung¨ 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 ofits draw procedure from even distribution and propose two alternative approaches to increase the excitement of the draw. Keywords: assignment problem; backtracking algorithm; ; mechanism design; arXiv:2103.11353v1 [physics.soc-ph] 21 Mar 2021 permutation MSC class: 05A05, 68U20, 68W40, 91B14 JEL classification number: C44, C63, Z20

∗ E-mail: [email protected] 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, , N. Tr¨ubner,1873. http://clausewitz.com/readings/OnWar1873/TOC.htm) 1 Introduction

Mechanism design usually focuses on theoretical properties like efficiency, fairness, and incentive compatibility (Abdulkadiro˘gluand S¨onmez, 2003; Roth et al., 2004; Csat´o, 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 from Pot 1, Ukraine from Pot 2, Iceland from Pot 3, and Serbia from Pot 4. The draw continues with Pot 5. First, is drawn and allotted to Group A. Second, Cyprus is drawn and assigned to Group B. Third, —a country without any draw constraints—is drawn and placed in. . . Group H, the first available group according to the computer.

Example1 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 (Section 4.3);

• We propose two alternative approaches for solving the group assignment problem to increase uncertainty during the draw (Section5).

Group allocation is an extensively discussed topic in the mainstream media. Several articles published in famous dailies such as Le Monde and 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 ) 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 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 , were not (Rathgeber and Rathgeber, 2007). Guyon(2015) identifies severe shortcomings of the procedure used for the 2014FIFA 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 L´opez(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 givea 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 (henceforth football) club competition around the world. Kl¨oßnerand 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, Boczo´nand 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 Example1.

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 푛 = 4 teams 푇 1–푇 4 drawn sequentially to be assigned to groups A–D. The order of permutations implied by the draw mechanism is shown in Figure1. Note that team 푇 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 푇 1 or teams 푇 2–푇 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 푇 1–푇 4 in Example2, respectively. Figure 1: The sequence of permutations according to the UEFA draw procedure, 푛 = 4

∙ If team 푇 1 cannot be placed in group A and team 푇 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: 푇 2, 푇 1, 푇 4, 푇 3.

∙ If teams 푇 2–푇 4 cannot be placed in group C and team 푇 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: 푇 3, 푇 2, 푇 1, 푇 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 Figure1.

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 Example1, 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 Figure2. 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 inthe 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 푖 of pot 푗 be placed in group 푘?

No Yes

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

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

Yes Yes No No

SOLUTION 푖 = 푖 + 1 There exists 푘 = [group of team (푖 − 1)] + 1 is found START NO allocation All teams ℓ ≥ 푖 are removed from their groups 푖 = 푖 − 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 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 11 21 (1) 20 12 22 Hungary 30 13 Poland 23 40 Portugal 14 24 Czech Republic 50 15 Austria 25 Norway 60 Italy 16 Ukraine 26 Northern Ireland 70 Croatia 17 Serbia 27 Iceland 80 18 Turkey 28 90 Germany 19 Slovakia 29 Greece 10 20 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 (3) 53 Liechtenstein 34 44 (1) 54 (2) 35 45 Estonia (2) 55 San Marino 36 46 (3) 37 47 (5) 38 (1) 48 Lithuania (2) 39 Georgia 49 (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 Table1. 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, onefrom 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 (Csat´o, 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 ofgroups. Therefore, it is sufficient to consider the permutations of the (maximal 10) teamsina 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 inthe appropriate order and checking them sequentially is an inefficient approach. In Example1, 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 assignthe first ten teams to the groups: the four Nations League finalists are placed inGroupsA–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. Table1 presents the maximal number of such groups in parenthesis. The first dead end may occur in Pot3

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 arefour 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 Example1 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 thereexist (10!)5 × 5! ≈ 7.55 × 1034 different permutations of the teams. These probabilities areas 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 Figures3 and4. The theoretical value is not given because of the complex interactions of draw constraints.

According to Figure3, restrictions affecting the teams still to be drawn cause the most severe problem in Pot 5, where they proliferate as Table1 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 Example1, 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 teamsin 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 inthesame 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 Figure4 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 푇 1–푇 3 but 푇 2 cannot be placed in group B. Then 푇 2 should play in group A and group C with an equal chance of 50%, respectively. However, the UEFA draw procedure assigns 푇 2 to group A with a probability of 1/3 (when 푇 2 is drawn first) and to group C with a probability of 2/3 (when 푇 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 Section2 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. Table2 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 (allthefive countries from Pot 6 are present together with Andorra in the second part of Table2), 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 UEFAdraw 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 Kl¨oßnerand 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 푇 1–푇 4 drawn in this order, while 푇 3 cannot be assigned to group C. After team 푇 1 is placed in group A and 푇 2 in group B, uncertainty entirely disappears since 푇 3 should play in group D and 푇 4 in group C. The problem of Example3 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 givento 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 Example4: • Mechanism A assigns 푇 1 to group A, 푇 2 to group C (otherwise, only one feasible allocation remains), 푇 3 to group B, and 푇 4 to group D. • Mechanism B assigns 푇 1 to group C (since 푇 3 cannot be placed here), 푇 2 to group A, 푇 3 to group B, and 푇 4 to group D. Note that after allocating the first team 푇 1, six assignments satisfy all restrictions under Mechanism B but only four under Mechanism A as 푇 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 willbe the topic of future research.

Acknowledgements

This paper could not have been written without my father (also called L´aszl´oCsat´o), who coded the simulations in Python. We are grateful to Julien Guyon for inspiration. Lajos R´onyai 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.

References

Abdulkadiro˘glu,A. and S¨onmez,T. (2003). School choice: A mechanism design approach. American Economic Review, 93(3):729–747. Aisch, G. and Leonhardt, D. (2014). Mexico, the World Cup’s Luckiest Country. The New York Times. 5 June. https://www.nytimes.com/2014/06/06/upshot/mexicos- run-of-world-cup-luck-has-continued.html. Arndt, J. (2010). Matters Computational: Ideas, Algorithms, Source Code. Springer, Heidelberg. Boczo´n,M. and Wilson, A. J. (2018). Goals, constraints, and public assignment: A field study of the UEFA Champions League. Technical Report 18/016, Univer- sity of Pittsburgh, Kenneth P. Dietrich School of Arts and Sciences, Department of Economics. https://www.econ.pitt.edu/sites/default/files/working_papers/ Working%20Paper.18.16.pdf.

13 Cea, S., Dur´an,G., Guajardo, M., Saur´e,D., Siebert, J., and Zamorano, G. (2020). An analytics approach to the FIFA ranking procedure and the World Cup final draw. Annals of Operations Research, 286(1-2):119–146.

Csat´o,L. (2020). Two issues of the UEFA Euro 2020 qualifying play-offs. International Journal of Sport Policy and Politics, 12(3):471–484.

Csat´o,L. (2021). Tournament Design: How Operations Research Can Improve Sports Rules. Palgrave Pivots in Sports Economics. Palgrave Macmillan, Cham, Switzerland.

Depetris-Chauvin, E., Durante, R., and Campante, F. (2020). Building nations through shared experiences: Evidence from African football. American Economic Review, 110(5):1572–1602.

FIFA (2017). FIFA Council: FIFA Council confirms contributions for FIFA World Cup participants. 27 October. http://www.fifa.com/about- fifa/news/y=2017/m=10/news=fifa-council-confirms-contributions-for- fifa-world-cup-participants-2917806.html.

Guyon, J. (2014). A Better Way to Rank Soccer Teams in a Fairer World Cup. The New York Times. 13 June. https://www.nytimes.com/2014/06/14/upshot/a- better-way-to-rank-soccer-teams-in-a-fairer-world-cup.html.

Guyon, J. (2015). Rethinking the FIFA World CupTM final draw. Journal of Quantitative Analysis in Sports, 11(3):169–182.

Guyon, J. (2017a). Barcelona x Chelsea ´eo confronto com mais chances de acontecer na Champions; veja outras probabilidades. El Pa´ıs Brasil. 11 December. https: //brasil.elpais.com/brasil/2017/12/07/deportes/1512643772_372491.html.

Guyon, J. (2017b). Losowanie nie do ko´ncalosowe. analityk policzy l, na kogo trafi Polska. Przeglad Sportowy. 1 December. https://www.przegladsportowy.pl/pilka- nozna/mundial-2018/losowanie-ms-2018-na-kogo-trafi-polska-analityk-to- wyliczyl/10qxhsg.

Guyon, J. (2017c). Por qu´eel Barcelona tiene un 41,3% de probabilidades de emparejarse con el Chelsea en octavos. El Pa´ıs. 7 December. https://elpais.com/deportes/2017/ 12/07/actualidad/1512643772_372491.html.

Guyon, J. (2017d). Por qu´eespa˜natiene el doble de probabilidades de estar con Argentina o Brasil en el grupo del Mundial. El Pa´ıs. 30 November. https://elpais.com/deportes/ 2017/11/30/actualidad/1512033514_352769.html.

Guyon, J. (2017e). Sorteio da Copa: as chances de o Brasil pegar Espanha ou Inglaterra na fase de grupos. El Pa´ısBrasil. 1 December. https://brasil.elpais.com/brasil/ 2017/11/30/deportes/1512067618_495683.html.

Guyon, J. (2017f). Tirage au sort de la Coupe du monde : comment ¸ca marche et quelles probabilit´es pour la France. Le Monde. 30 November. https://www.lemonde.fr/mondial-2018/article/2017/11/30/tirage-au-sort- de-la-coupe-du-monde-comment-ca-marche-et-quelles-probabilites-pour-la- france_5222713_5193650.html.

14 Guyon, J. (2019). “Choose your opponent”: A new knockout format for sports tour- naments. Application to the Round of 16 of the UEFA Champions League and to maximizing the number of home games during the UEFA Euro 2020. Manuscript. DOI: 10.2139/ssrn.3488832.

Guyon, J. (2020a). Champions League group stage draw: Who are the most likely oppon- ents for Liverpool, Manchester City, Manchester United and Chelsea? FourFourTwo. 1 October. https://www.fourfourtwo.com/us/features/champions-league-group- stage-draw-who-are-the-most-likely-opponents-for-liverpool-manchester- city-manchester-united-and-chelsea.

Guyon, J. (2020b). Champions League knockout draw: Who are the most likely opponents for Liverpool, Manchester City and Chelsea? FourFourTwo. 11 December. https: //www.fourfourtwo.com/features/champions-league-knockout-draw-who-are- the-most-likely-opponents-for-liverpool-manchester-city-and-chelsea.

Guyon, J. (2020c). Ligue des champions : Barcelone et Atl´etico, adversaires les plus probables pour le PSG. Le Monde. 1 October. https://www.lemonde.fr/ football/article/2020/10/01/ligue-des-champions-barcelone-et-atletico- adversaires-les-plus-probables-pour-le-psg_6054370_1616938.html.

Guyon, J. (2020d). Ligue des champions : Borussia M’gladbach, adversaire le plus probable du PSG en huiti´eme de finale. Le Monde. 12 December. https://www.lemonde.fr/football/article/2020/12/12/ligue-des-champions- borussia-m-gladbach-adversaire-le-plus-probable-du-psg-en-huitieme-de- finale_6063134_1616938.html.

Jones, M. C. (1990). The World Cup draw’s flaws. The Mathematical Gazette, 74(470):335– 338.

Kl¨oßner,S. and Becker, M. (2013). Odd odds: The UEFA Champions League Round of 16 draw. Journal of Quantitative Analysis in Sports, 9(3):249–270.

Knuth, D. E. (2005). The Art of Computer Programming, Volume 4, Fascicle 2: Generating All Tuples and Permutations. Addison-Wesley.

Laliena, P. and L´opez, F. J. (2019). Fair draws for group rounds in sport tournaments. International Transactions in Operational Research, 26(2):439–457.

McMahon, B. (2013). Why the FIFA 2014 World Cup Finals will be unique and very unfair. Forbes Magazine. 1 December. https://www.forbes.com/sites/bobbymcmahon/ 2013/12/01/why-the-fifa-2014-world-cup-finals-will-be-unique-and-very- unfair/?sh=76e61ea62dab.

Myrvold, W. and Ruskey, F. (2001). Ranking and unranking permutations in linear time. Information Processing Letters, 79(6):281–284.

Palacios-Huerta, I. (2014). Beautiful Game Theory: How Soccer Can Help Economics. Princeton University Press, Princeton, New York.

Rathgeber, A. and Rathgeber, H. (2007). Why Germany was supposed to be drawn in the and why it escaped. Chance, 20(2):22–24.

15 Rosa, A. and Wallis, W. D. (1982). Premature sets of 1-factors or how not to schedule round robin tournaments. Discrete Applied Mathematics, 4(4):291–297.

Roth, A. E., S¨onmez,T., and Unver,¨ M. U. (2004). Kidney exchange. The Quarterly Journal of Economics, 119(2):457–488.

Schaerf, A. (1999). Scheduling sport tournaments using constraint logic programming. Constraints, 4(1):43–65.

Sedgewick, R. (1977). Permutation generation methods. ACM Computing Surveys (CSUR), 9(2):137–164.

UEFA (2018). UEFA EURO 2020 qualifying draw. 2 December. https://www.uefa.com/ european-qualifiers/news/newsid=2573388.html.

UEFA (2020a). FIFA World Cup 2022 Qualifying draw procedure. https: //www.uefa.com/MultimediaFiles/Download/competitions/WorldCup/02/64/ 22/19/2642219_DOWNLOAD.pdf.

UEFA (2020b). UEFA Nations League 2020/21 – league phase draw proced- ure. https://www.uefa.com/MultimediaFiles/Download/competitions/General/ 02/63/57/88/2635788_DOWNLOAD.pdf.

16