Analysing the restricted assignment problem of the group draw in sports tournaments 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 Budapest (BCE) Department of Operations Research and Actuarial Sciences Budapest, Hungary 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; football; 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, London, 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 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. 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 tournament (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 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 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 association football (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.