CONSTRAINED CENTRALIZED ASSIGNMENT: A FIELD STUDY OF THE UEFA CHAMPIONS LEAGUE. 1. Introduction

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1 CONSTRAINED CENTRALIZED ASSIGNMENT: A FIELD STUDY OF THE UEFA CHAMPIONS LEAGUE PRELIMINARY DRAFT MARTA BOCZO AND ALISTAIR J. WILSON Abstract. We analyze the tournament matching procedures in the UEFA Champions League. Focusing in on the Round of 16 match, we show how the competition's association restrictions distort the resulting match process, even for unconstrained teams. Empirically, we map out who are the winners and losers from the current procedure and how the constraints aect later stages of the competition. From a market design point of view, we show the following: (i) Given the constraints there are no mechanisms for the application that oer substantial improvement. (ii) However, the constraints can be slacked a little to produce substantially less distortion. (iii) The procedure used represents a simple, proven way of publicly randomizing a constrained assignment. 1. Introduction The Union of European Football Association's (UEFA) Champions League (UCL) has been one of the most successful pan-european ventures over the past half-century, and certainly one with the most enthusiasm from the general public. This tournament brings together football clubs from across the continent (and beyond) that normally play within their own country-level associations. Selection into the competition is limited to the very top clubs from each nation, though countries with deeper footballing history are allotted more places. A number of initial rounds whittle the number of participating teams down to 32. Next, 16 clubs advance to a knockout stage that determines a nal European champion. Outside of the World Cup nals, the UCL nal games are one of the most-watched global sporting Date: June, Boczo«: Dept. of Economics, University of Pittsburgh; mjb249@pitt.edu; Wilson: Dept. of Economics, University of Pittsburgh; alistair@pitt.edu. 1

2 events, even eclipsing the viewership of the National Football League's Superbowl in the United States. Because the UCL has been put under a magnifying glass by both the general public and partisan-nature of football fans, UEFA has a clear interest in creating fair rules for the tournament. In particular, the o-the-eld elements such as the tournament assignment rules need to be fair and meritocratic, and publicly seen to be so. In many situations, this would be a simple problem to solve, where teams could be matched to one another through an urn draw without replacement. However, the tournament's round of 16 (R16) assignment problem is complicated due to a number of constraints on the possible matchings. First, teams are seeded into two groups determined on merit, group winners and group runnersup, where each R16 pair must be between a group winner and a group runner-up. Second, in order to increase the variation in the tournament match-ups, teams that played one another in the group stage are proscribed from matching again in the R16. Finally, teams from the same nation cannot play one another. In reaction to the need for a constrained assignment mechanism with a publicly veriable draw, UEFA developed a novel mechanism for the assignment of teams to one another. While respecting the above constraints on the possible aggregate matchings, the UEFA mechanism assembles each team-to-team match through a dynamic public draw of balls from an urn. While there is obviously more to learn from the specics, UEFA's approach to solving this problem oers a eld-proven solution. Our paper studies the precise UEFA assignment mechanism. Our initial contributions are theoretical; We formalize and generalize the draw procedure in the UCL R16. Next, we shift to thinking about the specics of the UEFA solution. We show that the exclusion constraints on same-nation matches create substantial spillovers to the likelihoods of other pairings. In particular, we demonstrate that even though two teams A and B might have the same prior performance in the tournament, and neither is explicitly constrained from a match with team c, there can be large dierences in the likelihoods of the Ac and Bc matchups. A natural question then is whether other assignment mechanisms exist that would 2

3 provide a more equal treatment of equals. To tackle this question, we make use of a result from a recent AER paper (Budish et al., 2013) that generalizes the Birkhovon-Neumann decomposition to constrained assignments. As the constraints imposed on the R16 matching satisfy a biheiracrchy condition, any expected assignment satisfying the constraints can be implemented as a lottery over pure assignments that satisfy the constraint. Having relaxed the problem, we go on to show that only marginal improvements over the UEFA procedure are possible. Our paper demonstrates that the UEFA draw has two nice properties: (i) it allows for a simple-to-follow and fully veriable public randomization over a constrained assignment; and (ii) it produces expected assignments that are close to an optimal outcome (from a fairness point of view). As such, generalizations of the mechanism might be useful in any market design application with constraints where openness in the randomizations is paramount. For example, the mechanism could be used to publicly and fairly assign students to a timetable of classes or donorrecipient pairs to a kidney exchange chain. While our results suggest a fairly minimal scope for improvements given the constraints, a separate question is the extent to which fairer outcomes are possible when relaxing the constraints. In particular, we show that providing a little slackness in the association constraint can go a long way towards reducing the distortions. Interestingly, this can be practically obtained with a small adjustment to the current matching procedure, allowing for at most one association match in the R16. The paper is structured as follows: In Section 2, we discuss the background of the tournament and the constraint. In Section 3, we formalize the mechanism and some of its properties through a number of propositions. In Section 4, we numerically examine the tournament match through the past 15 years. Finally, in Section 5, we conclude. 2. Application Background The UCL is the most prestigious football club competition in Europe. Its importance within Europe is similar to that of the Superbowl in the United States, though with stronger 3

4 global viewership. The tournament is played annually between September and May between the top-performing teams from 57 national associations across Europe (and beyond), and features the best players in the sport. In the 2017 season, 6.8 million spectators attended the UCL matches (from the rst qualifying round onward) and more than 65,000 attended the nal game alone. In addition to the in-person audience, the tournament has vast media exposure. The UCL nal game has become one of the most watched annual sporting events in the world outside of the World Cup. The 2015 nal had an estimated global reach of 400 million viewers across 200 countries and a projected live audience of 180 million. 1 The UCL (initially called the European Champion Clubs' Cup) was founded in 1955 as a knockout tournament solely between national champions. Over the years the format of the competition has changed, and the tournament in its present form is open to multiple entrants (at most ve) from the same national association. The last major change to the competition's format took place in Hence, in this paper, we focus on the UCL seasons from 2004 onward and consequently, restrict our attention to the 15 most-recent years. 2 The main UCL tournament consists of a group and a knockout stages, similar in format to the World Cup, but played in parallel to the European club season. In the group stage, 32 teams are divided in eight groups of four. 3 Beginning in September each team plays the other three in its group twice (once at home, once away). At the end of the group stage in December the worst-two teams in each group are eliminated, and the group winner and the group runner-up advance. 4 1 Similar numbers are also true for social media. In the 2017 season, the UCL ocial Facebook page became the worlds' most followed for a sports competition having 63 million fans that triggered 98 million interactions over the same year nal. 2 For more details regarding the format changes, see Table 2 in the Appendix sides automatically qualify while ten more qualify through an extended pre-tournament qualication stage. The titleholder of the competition (or of the sister Europa League tournament if the titleholder qualies through another route) automatically qualies along with 21 automatic qualiers. The 21 are: the top three clubs from the Spanish, German, and English associations; top two from the Italian, Portuguese, and French leagues; champions from Russia, Ukraine, Belgium, Netherlands, Turkey, and Switzerland. See Regulations of the UEFA Champions League Cycle ( Download/Regulations/uefaorg/Regulations/02/46/71/38/ _DOWNLOAD.pdf ). 4 The group match assigns the 32 teams through four seeded pots of eight. Prior to the 2015 season, teams were seeded based on their UEFA club coecients with the titleholder being placed in Pot 1 automatically. Starting from the 2016 season, the titleholder and the champions of the top seven associations based on their UEFA country coecients are being placed in Pot 1. The remaining teams are seeded to Pots 2-4 based on 4

5 The rest of the knockout stages (except for the nal) follows a two-legged format, where each team plays one leg at home. Teams that score more goals over the two legs advance to the next round, whereas the remaining teams are immediately eliminated from the tournament. 5 The knockout stage begins with the R16, followed by four quarter-nals, two semi-nals, followed by the nal game in May. UEFA generates large revenues from its three agship club competitions, which are the UCL (the competition we focus on), UEFA Europa League, and the UEFA Super Cup. The 2017 gross commercial revenue is estimated at 2.35 billion euros. From this amount, UEFA oers teams participating in the UCL substantial monetary rewards. 6 Each club that makes it to the group stage receives 12.7 million euros, while for an R16 appearance, each team receives a further 6 million euros (group winners an additional 1.5 million euros). Beyond the R16, a quarternal appearance garners a further 6.5 million euros, a seminal appearance 7.5 million euros, and reaching the nal game provides an additional 11 million euros (with the winner receiving an additional bonus of 4.5 million euros). In this paper, we analyze the UCL R16 constrained assignment problem. The R16 requires UEFA to assemble a perfect matching of eight team pairs. While there are many possible pairings, the assignment must satisfy the following constraints: (i) each pairing must be between a group winner and a group runner-up (bipartite constraints); (ii) teams that played one another in the prior group stage cannot be matched (group constraints); and (iii) teams from the same national association cannot play one another (association constraints). 7,8 their UEFA club coecients. The eight groups are formed by making sequential draws from the four pots with a restriction that teams from the same association cannot be drawn against each other, enforced in a similar way to the R16 match we detail in the paper. 5 Technically, the scoring rule is lexicographic over total goals, and goals away from home. A draw on both results in extra time, and then a penalty shootout until a winner is determined. 6 The rst-order beneciaries are the clubs participating in the tournament proper (the group stage onward); the remaining UEFA revenue is distributed among the second- and third-order beneciaries, clubs participating in the qualifying rounds and non-participating clubs, respectively. The solidarity payments made to the latter teams are distributed via their national associations and allocated, for the most part, to their youth training programs. 7 The quarter- and seminal draws are free from seeding and domestic country protection, and are conducted by standard draws from an urn without replacement. 8 Between 2004 and 2018, 17 national associations participated in the R16. Five associations (England, Spain, France, Germany, and Italy) were represented in all 15 seasons. These national associations, which we refer to as the Top Five had a total of at least 11 teams in the R16 each season. Top-Five associations 5

6 While the bipartite and group constraints impose a symmetric restriction over equivalent teams, the association constraints create a strong asymmetry that aects the entire matching. 9 If it was just matching two equal-sized groups together, the problem would be simple, and two urns (one for group winners, one for runners-up) could be used to assemble the matching through a sequential draw without replacement. However, with the exclusion constraints, a draw cannot be conducted in this piecewise for two reasons. The rst is direct and is easy to address; The partners to be matched with need to remove potential partners excluded by the constraints. The second is indirect and is a more-complicated combinatoric inference. Creating some matches to a seemingly valid partner can force an excluded match at a later point in the draw, and as such must be excluded. As an example, consider three group winners A, B and C, that each have to be matched to a group runner-up d, e or f. Suppose that the matches Ad and Be are excluded. An initial draw from the group-winners pot selects team A. The partner draw needs to exclude d directly, and we suppose f is selected to create the match Af. At the next round, C is chosen from the group winners pot. Although C has no directly excluded partners, it cannot match with d, as otherwise B will have no partners at the next stage. As such, if Af were selected at the rst stage, the unique complete matching under the constraints would have to be (Af, Ce, Bd). Although this logic is easy to follow when matching three teams to three teams, with eight teams on each side and more constraints, the combinatorics become involved. In response to the constraints, UEFA developed a dynamic draw procedure to determine the R16 tournament matching as follows: (i) eight balls representing the group winners are placed in the rst urn and one ball is drawn without replacement; (ii) the computer determines which runners-up can match with the drawn group winner given the current and previous draws and the constraints; (iii) balls representing the possible runners-up are placed into the second represented an average of between 1.5 (France) and 3.4 (England) teams in the R16. The presence of multiple teams from the same nation led to between 2 and 10 association restrictions in the R16 draw (the average year had 5.5). With the exception of the 2009 season, all association restrictions were generated by one of the Top-Five associations, with the majority being driven by Spanish and English teams. See Tables 3 and 4 in the Appendix for numbers supporting the above discussion. 9 In the presence of just the bipartite and group constraints, each group winner would have an expected chance of matching to a runner-up from another group given by 1 7, and vice versa for runners-up. 6

7 urn and one ball is drawn without replacement; (iv) a pair of the two drawn teams is added to the aggregate R16 matching. A unique feature of the UEFA draw procedure is its simplicity and openness. Representatives from each of the R16 teams attend the draw ceremony in person. Moreover, the event draws substantial attentions from the media and general public. The 20-minute draw ceremony is streamed live by UEFA over the Internet. Moreover, many national media companies broadcast the event. Examining the viewership gures for the Internet stream, we nd that the most recent UCL R16 draw was watched by 813,000 viewers on UEFA.tv alone. 10 While this gure may be small relative to game-day TV audiences for the UCL games, it is a huge audience when we think about centralized assignments. Note that in the absence of association constraints the tournament has 14,833 possible R16 matchings, where each association constraint signicantly reduces the number of valid assignments. Across the 15 seasons we examine, the number of valid assignments ranged from 2,988 to 9,200. The relationship between the number of possible matchings and the number of association constraints is illustrated in Figure 2.1. Note that the number of assignments is not purely a function of the number of association restrictions, as it depends on the precise constellation of constraints. 11 Nevertheless, the relationship in question can be fairly accurately approximated by a linear function, with a decrease of approximately 1,400 matchings for each association constraint Theory for the Current Matching Procedure In what follows, we describe the dynamic mechanism used by UEFA to draw an assignment under the constraints. First, we characterize the eective lottery over matchings that the mechanism induces. Next, we show that the dynamic mechanism is distinct from the most obvious static implementations, as well as from a number of dynamic variants. Finally, we 10 Online audiences have increased for the draw over time, in the 2017 season the R16 draw was watched by 618,000 viewers, in the 2016 season by 415,000, and in the 2015 season by 262, See Table 5 in the Appendix for the exact constellation of constraints in seasons In Figure B.1 in the Appendix, we show that the number of valid assignments each year is time invariant using a simulation of the group stage. 7

8 Figure 2.1. Relationship between possible assignments and number of association constraints. show that the constraints used by UEFA allow us to make use of the main result in Budish et al. (2013). Thus, in order to analyze the extent to which the UEFA mechanism might be improved upon, we abstract from the precise features of the mechanism and instead, focus on what is feasible in the reduced form Description of the Dynamic Mechanism. Let R (a set of group runners-up) and W (a set of group winners) be two sets that need to be matched together. The aim of the mechanism is to construct a perfect matching between the winners and runners-up. The dynamic process used by the UEFA is part of a family of mechanisms ψ : 2 V V, where V denotes the set of all possible perfect matchings between W and R. Given a set of admissible matchings Γ V, the mechanism selects a realized matching ψ(γ) Γ according to the following procedure: Algorithm (W to R dynamic mechanism). Given an input set of admissible matchings Γ V, the algorithm denes the selected matching ψ(γ) in K = W steps. Initialization: set W 0 = W, and Γ 0 = Γ. 8

9 Step-k: (for k = 1 to K) (i) Choose w k through a uniform draw over W k 1 ; (ii) Choose r k through a uniform draw over admissible partners R k = {r R V Γ k s.t. w k r V }; (iii) Set W k = W k 1 \{w k } and Γ k = {V Γ k 1 w k r k V }. Finalization: After K steps Γ K has a unique element given by {w 1 r 1,..., w K r K }, which we set as the realization of ψ(γ). The particular mechanism used by the UEFA is dened by a set of match constraints H W R, where H = H A H G is given by a set of association-level proscriptions H A and a set of group-level proscriptions H G. The admissible matchings for the UEFA mechanism is given by Γ H := {V V V H = }, where the draw chooses a realized matching ψ(γ H ). At each step k of the mechanism, the probability for the uniform draw from W is given by 1 K k+1. For any sequence of matches V = (w 1r 1,..., w K r K ) the probability of selecting r k at step k is given by 1 H k (V ), where H k (V ) := { r R V Γ s.t. wk r V and k j=1 w j r j V, } is the number of possible partners for w k at step k, given the prior matches. Dening the set of possible permutations of a matching V as P(V ), we get the following characterization of the mechanism: Proposition 1. Under the mechanism ψ(γ) the probability of any matching V Γ is given by Pr {V } = 1 K 1 K! V P(V ) k=1 H k, where P(V ) is the set of possible permutations of V (V ) and H k (V ) is the number of match partners at step-k. Proof. Each matching V Γ is reached through one of K! possible permutations of the elements of V, where each permutation has a positive probability of selection under the mechanism. The articulated mechanism produces a random sequence in (W R) K, so we 9

10 can write the probability of V as Pr {V } = V P(V ) Pr {V = ψ(w, R, H)}, where V (W R) K is a K 2 vector. The calculation can be reformulated as a series of conditional probabilities using the chain rule given the current draw at step k(i) or k(ii) as Pr {V } = V P(V ) k=1 K ( Pr{V k,1 I k 1 (V )} Pr{V k,2 I k 2 (V )} ), where I k j (V ) represents the selected elements of V from step (1, 1) to step (k, j). By uniformity of the draws at each step, we have Pr{V k,1 I k a (V )} = 1 K k+1 state, while Pr{V k,2 I k 2 (V )} = 1 H k (V ). regardless of the current The proposal shows that the induced probability distribution over Γ can be calculated in (K! Γ )-steps. The cardinality of Γ is always less than K! when there are exclusion constraints in H, and can be substantially smaller with many constraints. However, computation of G generally involves between K! and (K!) 2 steps, and can be taxing even for K = 8 in our applications. Though the process is combinatorically involved, the UEFA mechanism has three useful features when dealing with a public draw. First, all randomizations are uniform draws, which can be accomplished with an urn. Second, the number of realizations from each draw is less than n = 8 at all points, and so the draw is manageable and easy to follow, particularly, as each realization is naturally labeled as an agent within the draw (a team). Finally, the draw is fully veriable in that the non-obvious parts of the draw conducted by the computer (the combinatoric calculation on the valid partners) can be checked at post. Consequently, as long as the urn draws are fair, it is not possible to cheat within the mechanism, as any deviation from the prescribed matching set along the path of play is observable and checkable by an outside observer. This, therefore, inoculates the one opaque part of the randomization from corruption. 10

11 Given the characterization in Proposition 1, one question is the extent to which this mechanism is a more-convoluted version of a simpler one. We now attempt to show that there is no much hope for such an equivalence. Dene two matching mechanisms as being equivalent if they produce the same distribution over V, and distinct if they dier. Proposition 2 (Characteristics). The dynamic W-to-R mechanism is distinct from: (i) The mechanism that uniformly draws over Γ. (ii) The dynamic mechanism that uniformly draws admissible pairs. (iii) The R-to-W mechanism. Proof. See Appendix. The rst two parts of the Proposition 2 are essentially negative results, indicating that our environment is not equivalent to simpler uniform settings, while the third part speaks to the mechanism's asymmetry. While the main takeaway from Proposition 2 is negative, there is some positive spin: the three parts of the proposition set out three constructive design channels that might be used to aect the nal probabilistic assignment mechanisms. Generalizing the dynamic mechanism to settings such as the school choice, the ability to vary outcomes by choosing to draw students rst and schools second or vice versa, might be leveraged to obtain more-desirable outcomes. While these theoretical results demonstrate that the three alternative mechanisms are not equivalent, we later show that in our particular setting, there is little dierence between them. Indeed, while Proposition 2 points to the potential for error when calculating the likelihood of various matches, the simple uniform selection mechanism over Γ, where each 1 V Γ has a selection probability of, is much more tractable, where analytic calculation Γ becomes practically impossible Generally, the standings of the UCL groups are determined two days before the draw is carried out. Analytic calculation of the exact matching probabilities for the draw would take approximately two months on a powerful desktop. In contrast, probabilities for the uniform draw can be calculated in seconds. 11

12 3.2. Reduced-Form Approach. Above we characterize the mechanism employed by UEFA in the R16 match. While our paper goes on to discuss how the constraints in the mechanism aect the expected outcomes, we also wish to examine the extent to which alternative mechanisms might exist. To aid us in that endeavor we make use of the core result in Budish et al. (2013) to relax the problem to one of nding expected assignments. This allows us to focus on understanding what outcomes are feasible given the constraints, without worrying about the precise details of the mechanism, nor dealing with mixed-integer problems in our subsequent optimizations. Given W = {w 1,..., w K } and R = {r 1,..., r K }, we can represent any perfect matching V as a K K rook-matrix A with generic element a ij = 1 {w i r j V }. Proposition 3 (Implementability). Every expected assignment matrix A satisfying the group and association constraints is implementable as a mechanism that produces pure assignments. Proof. The group and assignment constraints, H A and H G respectively, place exclusion restrictions on singleton elements for the matrix (a ij = 0 if w i r j H A H G ). The only other constraints on A are that each row and column sum to exactly one. We can therefore organize the constraints into two distinct sets: (i) every singleton exclusion and the K 1 row constraints; and (ii) the K 1 column constraints. 14 As such the constrained problem satises the biheiracrchy denition for Budish et al. (2013, Theorem 1). An easy-to-show corollary of this result is that any expected assignment matrix can be implemented by randomizing over the input parameter to the W-to-R mechanism, the set of admissible matchings Γ. Corollary 1. Any expected assignment matrix satisfying the group and association constraints is implementable by randomizing over a nite collection of dynamic W-to-R mechanisms {ψ (Γ j )} J j=1 where each Γ j Γ H. 14 The quotas for each element are therefore a min and max of zero for the excluded singletons; a min and max of zero and one, respectively for the non-excluded singletons; a min and max of one for the rows; and a min and max of one for the columns. 12

13 Table 1. Matching probabilities for the 2018 season. FC Basel 1893 FC Bayern Munchen Chelsea FC Juventus Sevilla FC FC Shakhtar Donetsk FC Porto Real Madrid CF Manchester United FC Paris Saint-Germain AS Roma FC Barcelona Liverpool FC Manchester City FC Besiktas JK Tottenham Hotspur FC * Probablities are derived from a simulation (N = 10 6 ) of the UEFA draw procedure. Proof. Set J = Γ H < K! and for each entry V j Γ H set Γ j = {V j }. By Proposition 5, there exists a probability p j of selecting each admissible matching V j that leads to any implementable expected assignment matrix. The result follows from setting Pr {ψ (Γ j )} = p j. While Proposition 5 gives us a way to check whether better mechanisms might be found, Corollary 1 provides us with a directed tool to construct better outcomes while maintaining at least some of the desirable properties. 4. Constraint Effects on the Champions League 4.1. Current UEFA mechanism. In this section, we rst provide some examples of how the mechanism aects expected assignments, and how spillovers between the constraints aect otherwise equal teams. After motivating our main assessment criteria, we then go on to examine the extent to which better mechanisms might be available. After showing that substantially better mechanisms do not exist, in our nal sub-section we turn to the extent to which gains can me made by slightly relaxing the association constraints. We start with an example from the UCL R16 draw in the 2018 season. The expected assignment matrix for the draw is given in Table 1. Each row represents a group winner and each column a group runner-up. In row i and column j, we provide the probability (calculated by simulating the algorithm 10 6 times) that the R16 matching contains the (ij) pair. 15 The constraints in the expected assignment matrix are as follows: First, along the diagonal, the probabilities of each match are zero, reecting the eight group constraints. Second, seven 15 Given the simulation size, 95-percent condence intervals for each coecient are contained in the ball of radius around each number (see Proposition 4 in the Appendix). 13

14 same-nation matches are excluded (England: Chelsea with Manchester United, Liverpool FC, Manchester City, and Tottenham; Italy: Roma with Juventus; and Spain: Barcelona with Sevilla and Real Madrid). Finally, each row and column must sum to exactly one, as each represents the marginal match distribution for the respective team. Note that the above constraints interact with each other and consequently, even though the matrix is of dimension 8 8, there are only 34 degrees of freedom. Despite having a uniform draw from an urn at each point, the likelihoods of two teams playing each other are not uniform. As an example, consider Paris Saint-Germain (PSG), the only French team in the R16 in the 2018 season (Table 1, row 2). PSG had seven potential partners, but with substantial variation in the likelihoods of the dierent matchups. Specically, the probability of PSG playing Chelsea was almost three times larger than that of playing Basel, Shaktar, or Porto. One approach to assess the distortions caused by the constrains is to investigate the differences between the realized and uniform distributions. In this paper, however, we focus on a dierent concept, equal treatment of equals. In the following example, consider Besiktas and Barcelona (see Table 1, rows 4 and 7, respectively). These two clubs were both group winners and each had Chelsea as a potential match partner. However, because of the exclusion restriction between Barcelona and Sevilla, there was a substantially higher chance (an odds ratio of 1.41) of a Barcelona-Chelsea match than that of Besiktas-Chelsea. Formalizing our notion of equal treatment of equals, for any pair of teams in the same pot (i, j), where both have the option to match to another team k, we think of the distance p ik p jk as being costly to the designer. Dening the non-excluded matches as M = {ik i W, k R, ik H }, our overall measure of distortion assembles all possible pairs in the overall assignment with an intersecting team Υ = {(ik, jk) ik, jk M } {(ki, kj) ki, kj M }. 14

15 Figure 4.1. Mean absolute distance between the two equally treated teams match-up chances as a function of the number of association constraints. Our main assessment metric is the mean absolute distance between the two equally treated teams match-up chances: Q (A) = 1 Υ (ik,jk) Υ p ik p jk. For the 2018 season, Table 1 produces a mean absolute dierence of While we provide the full expected assignment Ât in the Appendix for the 2004 through 2017 tournaments, in Figure 4.1 we summarize the average deviation Q(Ât) across these years. Above, we show that the constraints lead to dierential treatment of otherwise equally treated teams in terms of the likelihood of matching. We now turn to quantifying how the constraints aect a key economic outcome of the tournament: the prize money earned by each team. To do this we design a method for simulating the entire tournament across a counterfactual matching procedure, where the association constraints are not present. Specically, we estimate a well-used structural model from the sports economics literature: a bivariate Poisson (Maher, 1982; Dixon and Coles, 1997, see). For each team in our tournament, we 15

16 Figure 4.2. Predicted versus actual winnings in the 2018 tournament (given realized R16 draw). estimate an attacking and a defensive parameter via maximum likelihood (as well as a global constant and home-stadium eect parameters) using scoreline data from the trailing three years. Armed with this structural model, we are able to both simulate the entire tournament (the R16 draw, the games, later draws, etc.) and change the matching constraints. Our main outcome metric is the expected prize-money assigned to each team (where we x the prize amounts to the 2017 tournament for consistency). Figure 4.2 illustrates the strong relationship between the predicted and actual tournament winnings in the 2018 competition (xing the R16 draw to the realized one). 16 Looking across all 15 seasons, the expected prize money is strongly correlated with the realized prizes (given the teams' performance: ρ = 0.639). From a linear regression on 240 team-year observations (16 teams across 15 seasons), we nd that the simulated prizes are highly predictive of the 16 Here we exclude data from games that came after the 2018 R16 draw, so the gure only uses out-ofsample information. In general though we also use the contemporaneous year games in estimating the model parameters as our core exercise is not attempting to predict tournament outcomes. 16

17 actual prizes (p = 0.000). 17 While these exercises help us validate the bivariate Poisson model, its true utility lies in its ability to examine more structured counterfactuals. In what follows, we quantify the importance of the R16 match for any given team, and measure the distortions caused by the association constraints. We run two simulations of the UEFA draw mechanism that dier in the underlying set of exclusion restrictions. In the rst simulation, we impose the association and group constraints, whereas in the second simulation, we maintain the group restrictions but allow same-associations matches. In each simulation, we rst draw J = 1, 000 matchings ( { } Vj t J ) for the R16 for each j=1 tournament year (xing the R16 teams). For each draw V t j, we simulate all remaining games in the tournament S = 1, 000 times (the R16 home/away games, quarter and semi-nals home/away games, and the nal game on neutral soil), and calculate the average earnings π t ij for each team i in each year t. First, we examine the extent to which the R16 match aects realized prize outcomes. For each team-year observation (it), we calculate the range of prizes as we vary the draw V t j, in particular the dierence between the 10th and 90th percentiles of the empirical distribution for { πij} t J. In Figure 4.3 we graph the prize range against the average expected prize j=1 across all draws π t i = 1 J J j=1 πt ij, where each point represents an R16 team in one year of the tournament. The gure makes clear that variation in the specic R16 draw mostly aects better teams. Clubs likely to be eliminated in the R16 (and so earning less than 20 million euros) have less than half a million euros response. However, for teams with greater aspirations, dierent realizations lead to more than a million euros swing in their expected earnings. We conclude that given the stakes involved, the draw is clearly of high importance to the clubs involved. Our rst simulation generates an expected prize for each team under the current matching procedure π t i through 10 6 simulations (1,000 draws and a 1,000 tournament simulations of each draw). In a similar manner, our second simulation generates an expected prize 17 Our model explains 41 percent of the variation in prize money, thought the model also does well predicting the extent of the variation in the actual prizes. For example the average prize standard deviation for the 16 teams in 2018 is 7.1 million euros, where the root mean squared error for that year is 8.4 million. 17

18 Figure 4.3. Unconditional expected prize versus expected prize range across draws. of π t i under a counterfactual draw mechanism, where we entirely remove the association exclusions (again through 10 6 total simulations). We dene the expected prize dierence as π t i := π t i π t i. Note that teams with a positive value for π t i are beneting from the association constraints, whereas those with a negative value are being disadvantaged. Across all 15 seasons of the modern tournament the eects of the association constraint have a standard deviation of 0.3 million euros (they are mean-zero by construction) and range between a 0.8 million euros cost (Barcelona in 2007) and a million euros subsidy (Liverpool in 2009, Real Madrid in 2017). In order to validate the above eects as being driven by the association constraints, we explain the variation in teams outcomes that is orthogonal to their ability. While the estimated Poisson attack and defense parameters provide proxies for ability, we dene an easier to parse index. Using the Poisson model, we calculate each team's average chance of winning (on a neutral ground) against each of the other 15 teams participating in that year's 18

19 Figure 4.4. Association constraint eects on expected prizes, controlling for team ability. tournament. Next, we linearly rescale the probabilities to run from 0 to 1 as an ability index for each year, α t i. 18 In Figure 4.4 we illustrate how the residuals from a regression of realized tournament prizes on our ability index vary with the association constraint eect π t i. 19 While there is still substantial noise (the constraints explain only 4.7 percent of the remaining variation), the model's estimated constraint eects are signicantly related to realized outcomes ( p = 0.001). Outside of realized prizes, a probit estimate examining whether a team makes the semi-nals of the competition, where we include both the ability index and association-constraint eect terms as explanatory variables, suggests that both are highly signicant. The marginal eects from this estimation indicate that higher ability increases the likelihood of making the semi-nals by 67.3 percent(p = 0.000), whereas a million euros predicted subsidy from the association constraints increases the chance of making the seminals by 19.7 percent (p = 0.004) For example, in the 2018 tournament the ability index runs from Shaktar Donetsk at 0, FC Basel at and Besiktas at 0.233, up to Real Madrid at 0.919, Liverpool at 0.999, and Barcelona at Our ability index coecient indicates that 17.8 million euros of the realized prizes can be explained by ability. 20 The eects on reaching later stages of the tournament are decreasing. The estimated marginal eects for reaching the quarter-nals indicate a 31.2 percentage increase given a million euros association-constraint subsidy (p = 0.001). However, the marginal eect for reaching the nal is just 9.0 percent and is only marginally signicant (p = 0.086). 19

20 The above all contributes evidence towards the following summary result: Result 1. Association constraints in the R16 generate substantial eects, (i) causing otherwise equally treated teams to have distinct chances of meeting with unconstrained teams; (ii) altering the expected tournament prizes by millions of euros; and (iii)signicantly aecting the chance of reaching later stages of the competition Near-Optimality of the Current Procedure. A natural question is whether a better matching mechanism exists. We start this section with a predominantly negative answer, which is: Result 2. While the UEFA mechanism is not optimal, given the constraints, it comes very close to the optimal mechanism when considering equal treatment of equals. In the following, we contextualize and provide evidence for the above result. To begin with, note that in Proposition 3, we show that an assignment-producing mechanism exists for every possible expected assignment that satises the constraints This substantially simplies the problem of nding better mechanisms as it allows us to look at the more-structured space of expected assignments. Our main argument for the above result is that the optimal expected assignment A t in each tournament year t is not substantially better than the expected assignment under the current mechanism  t. To generate the best feasible expected assignment we use our equal-treatment-of-equals objective function Q(A). The optimal feasible expected assignment is the solution to the following problem: A t = argmin Q(A), subject to the constraints: A ij H t : a ij = 0 if ij ij : 0 a ij 1 i : j a ij = 1, j a ji = 1 20

21 Figure 4.5. Mean absolute distance between the two equally treated teams match-up chances under the actual UEFA mechanism in comparison to the optimal mechanism, by season. We solve the optimization problem for all years, {A t } 2008 t=2004, and conclude that there is not much room for improvement. In Figure 4.5, we graph the equal treatment objective for optimal assignments against the objective for the actual assignments. As the gray 45-degree line represents no improvement, the gure illustrates that while there is some improvement possible across the tournament years, the gains are not large (a 10 percent improvement on average). On the ip-side of the coin, there are large potential costs from a switch: mechanisms that support the optimal expected assignment are complex in comparison to the current procedure. While Corollary 1 provides a channel through which the mechanism could be implementeda pre-stage draw where the organizers randomize the potential matchings even this would be cumbersome and engender some suspicion. Put against this tradeo, the ability to reduce the average match distortion among equals from 5 percent to 4.5 percent does not seem that alluring Outside of optimal, it is possible that some simple modication to the match procedure might push the outcomes in the direction of our optimal assignments. In the appendix, we show that the three distinct simple 21

22 The inability to improve much upon the expected assignment is not due to a limited scope in moving the expected assignments given the constraints. The univariate probability of any unconstrained team pairing can be moved fully between zero and one. Our equal treatment of equals objective can be made as high as by changing the expected assignment without violating the constraints. The conclusion instead is that the UEFA draw mechanism is close to a constrained-best. 22 While distinctly dierent constraints may violate this nding, the result provides some optimism that modications to this procedure could be used in constrained school choice settings were openness in the randomization is paramount Weakening the Constraints. One response to the previous result is to declare victory; accepting the distortions caused by the constraints because we cannot do better. In this section, we take a dierent tack and examine the extent to which gains can be made by weakening the association constraints, as they drive the distortions in the mechanisms. While fully weakening the constraints obviously removes all of the distortions, the interesting dimension is the extent to which we can respect the spirit of these exclusions while relaxing them. There are several ways in which this can be accomplished. In what follows, we focus solely on methods that weaken the constraint while still generating assignments within the current mechanism. Specically, we focus on the family of pure assignments that allow at most one association match within the resulting R16 matching. With a focus on pure assignments, we only need to modify the denition of the admissible set in the draw algorithm to Γ H := {V V V H 1}. As such, the modication retains the desirable features of the mechanism; the randomization is completely transparent (uniform urn draws) and the more-opaque combinatoric check is still fully veriable at all realizations. mechanisms set out in Proposition 2 are all nearly identical in their assignments to the actual mechanism (see Figures B.2-B.4 in the Appendix). 22 In the absence of any association constraints, the mechanism attains the lower bound for the objective, as the expected assignment for every match is one-in-seven. 23 For some example school choice applications see for instance Abdulkadiro lu et al. (2005); Pathak and Sönmez (2013); for theory see Abdulkadiro lu and Sönmez (2003); Ehlers et al. (2014). 22

23 Figure 4.6. mean absolute distance between the two equally treated teams match-up chances under the actual UEFA mechanism in comparison to the mechanism that allows at most one association match, by season. Our main nding is that: Result 3. Weakening the association constraints to allow for at most one association match in the R16 substantially weakens the distortions, while protecting associations from excessive early match-ups. Moreover, a secondary eect from weakening the association constraints reduces the number of same-nation games in the later stages of the tournament. We start by constructing analog results to those presented in Section 4.1. In Figure 4.6 we graph values of the equal-treatment-of equals objective for the UEFA mechanism with the weakened constraints against that for the current mechanism. While an optimal assignment under the constraints reduces the distortions by 10 percent, allowing for at most one association match decreases the distortions by more than 70 percent. Next, we generate the counterfactual dierence in expected prizes π i t. Specically, we calculate the expected tournament prizes under the new mechanism and compare them to 23

24 Figure 4.7. Reduction in the association eects. the expected prizes under a mechanism without association constraints. We illustrate the relationship between the two counterfactuals in Figure 4.8. Overall, allowing for a single association-match in the R16 reduces the prize distortions by 71 percent. Note that in Section 4.1, we show that the counterfactual dierence in expected prizes under the UEFA mechanism in comparison to a mechanism without association constraints is strongly related to actual tournament outcomes. Above, we focus on the benets from reducing the distortions caused by the association constraints. Importantly, however, there are also signicant costs associated with allowing for same-nation matches. As UEFA imposes the association constraints, there must be an underlying preference for the tournament to be primarily an international competition. Relaxing the association constraints leads to a same-association match at the R16 stage in approximately six out of ten tournaments. Moreover, in seasons with at least six association constraints, this ratio increases to seven-in-ten. 24

25 Figure 4.8. Changes in the number of international games when allowing for at most one R16 association match. On the other hand, imposing association constraints at earlier stages of the tournament make association match-ups in subsequent rounds more likely. Using our bivariate Poisson model and the at-most-one association match mechanism, we assess the predicted change in the number of same-association matches in later stages of the competition. We nd that for every same-association pairing generated in the R16, there is a reduction in the same nation games in later stages of the tournament (see Figure 4.8 that illustrates the two compensating eects in all 15 seasons between 2004 and 2018). 5. Conclusion We document a constrained-assignment problem under huge public interestand in addition where millions of euros are at stake in the draw outcomeswhere the randomization components need to be transparent. After documenting the particulars of our application (the UCL R16), we rst formalize the mechanism and then, show that a result from the related constrained matching literature facilitates the problem without loss of generality. In the main body of the paper, we dive into the particulars of our application. Using our theoretical model for the matching procedure and a commonly-used structural model for football match outcomes (a bivariate Possion model estimated with game data from the past 18 years), we quantify the substantial distortions caused by the constraints. Even though the 25

26 constraints on same-nation matches have signicant repercussions on the matching probabilities, we show that the existing UEFA solution is close to optimal. While alternative mechanisms can substantially shift the probabilities of particular match-ups, there is no clear way to do so in a mechanism that handles all equally treated teams equally. In the nal section of the paper, we quantify the eects from weakening the association constraint. In particular, we allow for at least seven out of eight R16 pairings to be international. This modication to the constraints reduces the distortions by approximately 70 percent, where we nd similar eect sizes both over R16 match probabilities and over expected tournament prizes. In years where the association constrains are numerous, relaxing the constraints leads to 0.7 same-association games in the R16 and moreover, it removes 0.07 same-association games in the quarter-nals, semi-nals and the nal. Provided that later stages of the tournament draw more attention from sponsors and the general public than either the group stage or the R16 weakening the association constraints might even be an unambiguous improvement. References Abdulkadiro lu, Atila and Tayfun Sönmez, School choice: A mechanism design approach, American Economic Review, 2003, 93 (3), , Joshua D Angrist, Yusuke Narita, and Parag A Pathak, Research design meets market design: Using centralized assignment for impact evaluation, Econometrica, 2017, 85 (5), , Parag A Pathak, and Alvin E Roth, The New York City High School Match, American Economic Review, 2005, 95 (2), Anderson, Ian, Combinatorial designs and tournaments, Vol. 6, Oxford University Press, Brown, Jennifer, Quitters never win: The (adverse) incentive eects of competing with superstars, Journal of Political Economy, 2011, 119 (5),