A paradox of tournament seeding - L aszl o Csat
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A paradox of tournament seeding
László Csató∗
Institute for Computer Science and Control (SZTAKI)
Laboratory on Engineering and Management Intelligence, Research Group of Operations
Research and Decision Systems
arXiv:2011.11277v2 [physics.soc-ph] 15 Dec 2020
Corvinus University of Budapest (BCE)
Department of Operations Research and Actuarial Sciences
Budapest, Hungary
16th December 2020
Große Beispiele sind die besten Lehrmeister, aber freilich ist es schlimm, wenn sich eine
Wolke von theoretischen Vorurteilen dazwischenlegt, denn auch das Sonnenlicht bricht und
färbt sich in Wolken. Solche Vorurteile, die sich in mancher Zeit wie ein Miasma bilden
und verbreiten, zu zerstören, ist eine dringende Pflicht der Theorie, denn was menschlicher
Verstand fälschlich erzeugt, kann auch bloßer Verstand wieder vernichten.1
(Carl von Clausewitz: Vom Kriege)
Abstract
A mathematical model of seeding is analysed for sports tournaments where the
qualification is based on round-robin contests. The conditions of strategyproofness
are found to be quite restrictive: if each team takes its own coefficient (a measure of
its past performance), only one or all of them should qualify from every round-robin
contest. Thus the standard draw system creates incentives for tanking in order to
be assigned to a stronger pot as each team prefers to qualify with teams having a
lower coefficient. Major soccer competitions are shown to suffer from this weakness.
Strategyproofness can be guaranteed by giving to each team the highest coefficient
of all teams that are ranked lower in its round-robin contest. The proposal is
illustrated by the 2020/21 UEFA Champions League.
Keywords: OR in sports; incentive compatibility; mechanism design; seeding; soccer
∗
Corresponding author. E-mail: laszlo.csato@sztaki.hu
1
“Great examples are the best teachers, but it is certainly a misfortune if a cloud of theoret-
ical prejudices comes between, for even the sunbeam is refracted and tinted by the clouds. To des-
troy such prejudices, which many a time rise and spread themselves like a miasma, is an imperative
duty of theory, for the misbegotten offspring of human reason can also be in turn destroyed by pure
reason.” (Source: Carl von Clausewitz: On War, Book 4, Chapter 11 [The Battle—Continuation:
The Use of the Battle], translated by Colonel James John Graham, London, N. Trübner, 1873.
http://clausewitz.com/readings/OnWar1873/TOC.htm)MSC class: 62F07, 91B14
JEL classification number: C44, D71, Z20
21 Introduction
According to a survey of scientific research on the laws and rules of sporting competitions,
there are few fields of science with higher importance for the general public (Wright, 2014).
Sporting contests represent a major branch of the entertainment industry (Szymanski,
2003), illustrated by the fact that about 50% of the global population watched the final
of the 2010 FIFA World Cup (Palacios-Huerta, 2014). Soccer, probably the most popular
sport around the world, seems to have a powerful influence on everyday life: the success of
the national team can reduce violence in some countries (Depetris-Chauvin et al., 2020).
Academic scholars clearly have something to offer for sports governing bodies by
improving the design of tournaments, especially when poorly drafted rules create per-
verse incentives (Preston and Szymanski, 2003), which is not as rare as one might think
(Kendall and Lenten, 2017). This paper uncovers that the standard draw system for tour-
naments with round-robin qualifications suffers from similar shortcomings: a team may
be tempted to tank, namely, to deliberately lose a game in order to be assigned to a
higher-ranked seeding pot, which reduces the expected strength of future opponents.
The problem is rooted in seeding, used to increase the discriminatory power of a
tournament. This is usually based on a ranking that takes into account the performance of
the teams not only during the qualification itself. The teams are placed in pots according
to the ranking, and each group gets one team from each pot. Therefore, every team has an
incentive to qualify with lower-ranked teams, which can be achieved by reordering them in
a round-robin competition through strategic losses. Even though a situation susceptible
to manipulation occurs with a low probability, the rules at least may punish a team for
its better achievements, which is unfair and violates the spirit of the game.
The FIFA World Cup, the most prestigious soccer tournament and the most widely
viewed and followed sporting event in the world, is shown to exhibit this paradox. Further
examples include the UEFA European Championship, the primary soccer competition for
European national teams, as well as the UEFA Champions League—the most prestigi-
ous club competition in European football with its final being the most-watched annual
sporting event around the world (BBC, 2010)—and UEFA Europa League, the second-tier
European club competition.
Let us see two motivating examples.
Example 1.1. Assume the following hypothetical modifications to real-world results in
the 2018 FIFA World Cup qualification:
• Wales vs. the Republic of Ireland was 1-1 (instead of 0-1) on 9 October 2017 in
UEFA Group D. Consequently, Wales would have been 18 and the Republic of
Ireland would have been 17 points in the group, thus Wales would have advanced
to the UEFA Second Round. There Wales would have been in Pot 1 instead of
Denmark, therefore the tie Wales vs. Denmark would have been possible (in fact,
Denmark played against the Republic of Ireland). Suppose that Wales qualified
for the World Cup instead of Denmark.
• The first leg Sweden vs. Italy was 1-1 (instead of 1-0) on 10 November 2017 in
the UEFA Second Round, hence Italy qualified for the World Cup.
In the draw for the 2018 FIFA World Cup, the composition of the pots was determined
by each national team’s October 2017 FIFA World Ranking. The only exception was
the automatic assignment of the hosts—Russia—to Pot 1 with the seven highest-ranked
3Table 1: Pot composition in the hypothetical 2018 FIFA World Cup
Pot 1 Pot 2 Pot 3 Pot 4
Russia (65) Spain (8) Uruguay (17) Serbia (38)
Germany (1) Peru (10) Iceland (21) Nigeria (41)
Brazil (2) Switzerland (11) Costa Rica (22) Australia (43)
Portugal (3) England (12) Sweden (25) Japan (44)
Argentina (4) Colombia (13) Tunisia (28) Morocco (48)
Begium (5) Wales (14) Egypt (30) Panama (49)
Poland (6) Italy (15) Senegal (32) South Korea (62)
France (7) Mexico (16) Iran (34) Saudi Arabia (63)
The pots are determined by each national team’s October 2017 FIFA World Ranking, see the numbers
in parenthesis.
Russia is the top seed as hosts.
Teams written in italics qualified only in the hypothetical but feasible scenario of Example 1.1.
Uruguay (17)—the top team of Pot 3—would have been drawn from Pot 2 due to losing against
Paraguay (34) in the South American section of the 2018 FIFA World Cup qualification since either
Paraguay (34) or New Zealand (122) would have qualified for the World Cup instead of Peru (10).
The national teams affected by this modification are written in bold.
qualified teams. Hence Uruguay (17th in the relevant FIFA ranking) would have been
drawn from Pot 3 as among the best 16 teams only Chile would have not qualified in the
above scenario (Wales was the 14th and Italy was the 15th in the October 2017 FIFA
World Ranking).
The allocation of the teams in the above scenario is given in Table 1. Consider what
would have happened if the result of the match Paraguay (34) vs. Uruguay, played on
5 September 2017 in the South American qualifier, would have been 2-1 instead of 1-2.
Then Uruguay would have remained the second and Paraguay would have been the fifth.
Paraguay would have played against New Zealand (122) in the OFC–CONMEBOL qualification play-off,
thus Peru (10) could not have qualified for the World Cup. Therefore, Uruguay would
have been drawn from the stronger Pot 2 instead of Pot 3 due to its loss against Paraguay.
This probably means a substantial advantage: in the 2018 FIFA World Cup, seven and
two teams advanced to the knockout stage from Pots 2 and 3, respectively.
Example 1.1 contains a small sloppiness since it is not checked what would have
happened with the October 2017 FIFA World Ranking if the result of Paraguay vs. Ur-
uguay would have changed. However, this does not affect the potential case of incentive
incompatibility.
Example 1.2. In the draw for the 2020/21 UEFA Europa League group stage, the com-
position of the pots was determined by the 2020 UEFA club coefficients, available at
https://kassiesa.net/uefa/data/method5/trank2020.html. Assume the following
hypothetical modifications to real-world results in the play-off round of the qualifying phase
with the first favourite team advancing to the group stage in place of the second unseeded
underdog (the club coefficients are given in parenthesis):
• Viktoria Plzeň (34.0) against Hapoel Be’er Sheva (14.0);
• Basel (58.5) against CSKA Sofia (4.0);
• Sporting CP (50.0) against LASK (14.0);
4Table 2: Pot composition in the hypothetical 2020/21 UEFA Europa League
Pot 1 Pot 2 Pot 3 Pot 4
Arsenal (91.0) Gent (39.5) Leicester City (22.0) 1899 Hoffenheim (14.956)
Tottenham Hotspur (85.0) PSV Eindhoven (37.0) PAOK (21.0) CFR Cluj (12.5)
Roma (80.0) VfL Wolfsburg (36.0) Qarabağ (21.0) Zorya Luhansk (12.5)
Napoli (77.0) Celtic (34.0) Standard Liège (20.5) Nice (11.849)
Benfica (70.0) Viktoria Plzeň (34.0) Real Sociedad (20.476) Lille (11.849)
Bayer Leverkusen (61.0) Dinamo Zagreb (33.5) Granada (20.456) Dundalk (8.5)
Basel (58.5) Sparta Prague (30.5) Milan (19.0) Slovan Liberec (8.0)
Villarreal (56.0) Slavia Prague (27.5) AZ Alkmaar (18.5) Antwerp (7.58)
Sporting CP (50.0) Ludogorets Razgrad Feyenoord (17.0) Lech Poznaṅ (7.0)
(26.0)
CSKA Moscow (44.0) Young Boys (25.5) Maccabi Tel Aviv (16.5) Sivasspor (6.72)
Copenhagen (42.0) Crvena Zvezda (22.75) Rangers (16.25) Wolfsberger AC (6.585)
Braga (41.0) Rapid Wien (22.0) Molde (15.0) Omonia (5.35)
The pots are determined by the 2020 UEFA club coefficients, shown in parenthesis.
Teams written in italics qualified only in the hypothetical but feasible scenario of Example 1.2.
Leicester City (22.0)—the top team of Pot 3—could have been drawn from Pot 2 due to losing against Wolverhampton
(18.092) in the 2019/20 English Premier League since the latter could have qualified for the UEFA Europa League
group stage instead of Tottenham Hotspur (85.0). The clubs affected by this modification are written in bold.
• Copenhagen (42.0) against Rijeka (11.0);
• VfL Wolfsburg (36.0) against AEK Athens (16.5).
There were 48 teams in the group stage, Leicester City (22.0) was the 20th highest-ranked
because Rapid Wien (22.0) had the same 2020 UEFA club coefficient but the tie-breaking
criterion—coefficient in the next most recent season in which they are not equal (UEFA,
2020c, Annex D.8)—favoured the latter club. Due to the above changes, five teams having
a higher coefficient than Leicester City would have qualified instead of five teams having
a lower coefficient. Hence, Leicester City would have been only the 25th highest-ranked,
that is, the best club in Pot 3 because each of the four pot contained 12 clubs.
In addition, suppose that Leicester defeated Norwich City at home by 2-1 (instead of
0-0) in the 2019/20 English Premier League. Then Leicester City would have remained
the fifth at the end of the season with 64 points.
The allocation of the clubs in the above scenario is given in Table 2. Consider what
would have happened if the outcome of the match Wolverhampton Wanderers (18.092) vs.
Leicester City, played on 14 February 2020 in the 2019/20 English Premier League, would
have been 1-0 instead of 0-0. Leicester would have remained the fifth with 62 points, while
Wolverhampton would have been the sixth with 61 points rather than Tottenham Hotspur
(85.0), which scored 59 points. Consequently, Wolverhampton would have played in the
Europa League qualification from the second qualifying round, and it could have qualified
for the group stage in the place of Tottenham. Then Leicester would have been drawn
from the stronger Pot 2 due to its loss against Wolverhampton. This probably means an
advantage, although in the 20202/21 Europa League, six teams advanced to the knockout
stage from both Pots 2 and 3, respectively.
A reasonable general mechanism is provided to avoid the above problem of poor in-
centives and achieve strategyproofness. Hopefully, the Sveriges Riksbank Prize in Eco-
nomic Sciences in Memory of Alfred Nobel 2020—awarded for theoretical discoveries that
5have benefited sellers, buyers, and taxpayers around the world by proposing new auc-
tion formats—will persuade sports administrators to consider our findings in the design
of future rules. A recent decision by the ten members of the South American Football
Confederation (CONMEBOL) indicates that organisers are increasingly open to proposals
from the academic community: they unanimously approved a better schedule for their
qualifying competition to the FIFA World Cup (Alarcón et al., 2017; Durán et al., 2017).
Choosing an effective procedure could certainly improve societal and economic outcomes
by circumventing the suboptimal characteristics of the current seeding policy.
Our roadmap is as follows. Section 2 presents an overview of related studies. A
mathematical model is given in Section 3. Section 4 examines the strategyproofness
of some major soccer tournaments. A strategyproof seeding mechanism is discussed in
Section 5. Finally, Section 6 concludes.
2 Literature review
Our topic is connected to at least three different fields of scientific research.
Firstly, a number of works aim to found the best seeding in knockout tournaments
through an axiomatic approach: a seeding is said to be desirable when it satisfies certain
criteria. Hwang (1982) calls a tournament monotone if the probability of winning is in-
creasing in team skill, and proves that this property may be violated by the traditional
method of seeding but reseeding after each round guarantees monotonicity. Schwenk
(2000) provides a randomisation procedure, which is free of perverse incentives and sat-
isfies two additional axioms of fairness. Vu and Shoham (2011) adapt two further fair-
ness properties, envy-freeness and order preservation, to derive some impossibility results.
Prince et al. (2013) introduce another set of conditions for the fairness of a knockout
bracket and show that up to 16 players all of them can be found within a reasonable
time. Karpov (2016) analyses the novel concept of equal gap seeding, which is uniquely
determined by a set of theoretical requirements and is the only one maximising the prob-
ability that the strongest participant is the winner, the strongest two participants are the
finalists, etc. Karpov (2018) extends this model to tournaments with more than two parti-
cipants in one match. Della Croce et al. (2020) suggest a fairness-based approach for the
allocation of unseeded players in tennis tournaments. Arlegi and Dimitrov (2020) specify
the class of seeding rules that let the structures satisfy equal treatment and monotonicity
in strengths.
Random knockout tournaments have also been investigated. Marchand (2002) com-
pares the winning probability of a top-seeded player for standard and random designs.
Adler et al. (2017) obtain upper and lower bounds for the winning probabilities in ran-
dom knockout tournaments. Kulhanek and Ponomarenko (2020) characterise the situ-
ations when the second-best player is more likely to win than to finish second in both the
random and seeded formats.
Further studies look for optimal seedings in knockout tournaments. Horen and Riezman
(1985) examine the draws of four-team tournaments with respect to various objective
functions such as maximising the winning probability for the top player or the expected
strength of the winner. Groh et al. (2012) analyse similar optimality criteria in a model
where the agents exert effort in order to win a match and advance to the next stage,
thus the winning probabilities are endogenous and depend on strategic choices. They
also identify the monotonic seedings. Dagaev and Suzdaltsev (2018) solve a discrete op-
timization problem to find the seeding that maximises the number of matches with high
6competitive intensity and high quality.
Finally, a whole line of research in the area of artificial intelligence focus on the
computational complexity of fixing a tournament, that is, detecting the optimal draw
for a particular player (Aronshtam et al., 2017; Aziz et al., 2018; Eidelstein et al., 2019;
Russell and Walsh, 2009; Stanton and Williams, 2013; Walsh, 2011).
Secondly, seeding in real-world tournaments has been widely scrutinised in the literat-
ure. Jones (1990) and Rathgeber and Rathgeber (2007) explain the consequences of the
flawed draw procedure used in the 1990 and 2006 FIFA World Cups, respectively. Accord-
ing to Monks and Husch (2009), assignment to the top pot increased the probability of
playing in the quarterfinals by 26% points in the World Cups organised between 1982 and
2006. It has been shown that Italy would have gained a place in the first pot in the 2014
FIFA World Cup by avoiding to play two (optional) friendly matches in 2013 because of
the poor construction of the FIFA World Ranking used for seeding (Lasek et al., 2016,
p. 1356). Cea et al. (2020) enumerate several weaknesses in the group assignments for the
2014 World Cup. Guyon (2015b) summarises these mistakes and proposes a better draw
procedure, which produces eight balanced and geographically diverse groups, is fair to all
teams, and gives equally likely outcomes. The solution presented by Laliena and López
(2019) for the same problem outperforms the system of Guyon (2015b). Interestingly,
the seeding rule has been changed for the 2018 FIFA World Cup, and it addresses some
deficiencies identified in Guyon (2015b).
Compared to this impressive list of papers, seeding in the UEFA Champions League
and UEFA Europa League has received less attention. Klößner and Becker (2013) and
Boczoń and Wilson (2018) analyse the UEFA Champions League Round of 16 draw.
Guyon (2019b) suggests a new format for the knockout stage of the Champions League.
UEFA has investigated how seedings translate into results in the Champions League
(UEFA, 2016). Although the overall winning ratio of seeded clubs is around 75% in
the knockout round of European cups (FootballSeeding.com, 2019), Engist et al. (2021)
find no evidence from a regression-discontinuity approach that seeding itself contributes
positively to success in the UEFA Champions League and UEFA Europa League.
Several simulation studies attempt to quantify the role of seeding. McGarry and Schutz
(1997) highlight the importance of an accurate seeding policy in knockout tournaments.
Scarf and Yusof (2011) consider the effect of seeding rules on tournament outcome un-
certainty, while taking account of competitive balance, illustrated using the FIFA World
Cup. Corona et al. (2019) and Dagaev and Rudyak (2019) evaluate the effects of a seeding
reform in the UEFA Champions League group stage.
The draw of major soccer tournaments has been discussed in the mainstream me-
dia, too (Guyon, 2017, 2019a,c,d). In particular, Guyon (2015a) proposes an alternative
seeding policy for the UEFA Champions League group stage.
The third area of related literature considers incentive (in)compatibility. Tourna-
ments with a draft system designed to equalise the strengths of all teams at the start
of the next season tend to provide an incentive to lose towards the end of the season
(Taylor and Trogdon, 2002; Price et al., 2010). Fornwagner (2019) presents field evidence
that teams have a concrete losing strategy in the NHL (National Hockey League), beyond
the possible effects of lower motivation or disappointment. On the contrary, tanking is
not prevalent in the AFL (Australian Football League), probably due to the relatively low
benefits (Borland et al., 2009; Chandrakumaran, 2020). Naturally, there are many ideas
to guarantee incentive compatibility by cleverly devised mechanisms (Banchio and Munro,
2020; Gold, 2010, 2011; Lenten, 2016; Lenten et al., 2018; Kazachkov and Vardi, 2020).
7Baumann et al. (2010) reveal an anomaly in the National Collegiate Athletic Associ-
ation (NCAA) men’s basketball tournament, which violates the intended incentive struc-
ture of seeded tournaments. Morris and Bokhari (2012) assess the statistical evidence in
support of this oddity. Zimmerman et al. (2020) examine whether perceived anomalies
truly are anomalous and if so, what is responsible for them. According to Pauly (2014)
and Vong (2017), round-robin tournaments where more players advance to the next stage
from each group give rise to strategic manipulation. Krumer et al. (2020) analyse the sub-
game perfect equilibrium of round-robin all-pay tournaments with four symmetric players
and two identical prizes. They show that one of the winners in the first round maximises
its expected payoff by losing in the second round, which can be avoided by making the
schedule of the tournament contingent on the outcomes in the previous rounds.
In a sense, the situation can be even more serious: sometimes an advantage can be
gained not only in expected terms, and a team might be strictly better off by losing. Tour-
nament systems, consisting of multiple round-robin and knockout tournaments with non-
cumulative prizes, usually suffer from this shortcoming (Dagaev and Sonin, 2018). Such
an incentive has appeared in the qualification for the UEFA Champions League between
2015 and 2018 (Csató, 2019b). The comparison of teams playing in separate round-robin
groups often leads to similar perverse incentives (Csató, 2018, 2020a), including the pos-
sibility that both teams are interested in avoiding winning (Csató, 2020c).
The current work is not the first paper on the connection between seeding rules and
incentive incompatibility. However, while Csató (2020b) has already revealed a violation
of strategy-proofness in the UEFA Champions League group stage draw, that deficiency
has emerged only in the 2015/16 season and is caused by a misaligned policy of filling
vacant slots. Here we show a more general result, that is, seeding systems based on
exogenous measures of teams’ strengths are generically incentive incompatible—but can
be made strategyproof in a straightforward way.
3 The theoretical model
Let (N, M) be a round-robin competition where N is the set of competitors and the array
M = [mij ] contains the result of the game between competitors i, j ∈ N with competitor
i playing at home. mij = mji if it is a single round-robin tournament. Note that mij is
not a number—therefore, M is not a matrix—since it should represent all features of the
match that can be relevant for ranking the set of competitors N: minimally, the number of
goals scored by competitors i and j but maybe other match statistics such as the number
of yellow cards or the outcome of a penalty shootout. For example, the ranking criteria in
the qualifying group stage of the 2020 UEFA European Championship is given in UEFA
(2018, Article 15).
Let R ⊆ N × N be a total, reflexive and transitive binary relation (preorder) on the
set of competitors N, namely, iRi for each i ∈ N, iRj and jRk implies iRk for each
i, j, k ∈ N, and iRj or jRi for each i, j ∈ N, i 6= j.
Let M be a total preorder on the set of competitors N associated with the round-
robin competition (N, M), which depends on M. Ranking ≻M denotes the asymmetric
part M . For the sake of simplicity, this contains a slight abuse of notation because
a strict ranking of the competitors can be sometimes obtained only by knowing their
identity or the outcome of drawing of lots that are not contained in array M.
The competitors from the round-robin competition (N, M) can qualify for a champi-
onship where the set of participants is denoted C. In particular:
8• The competitors ranked between p and q qualify directly for the championship
(p ≤ q + 1 with p > q meaning that no competitor can qualify directly), that is,
p − 1 ≤ |j ∈ N : j ≻M i| ≤ q − 1 implies i ∈ C for each i ∈ N.
• The competitors ranked between q + 1 and r qualify for the play-offs of the
championship (q ≤ r with q + 1 > r meaning that no competitor can qualify for
the play-offs), namely, q ≤ |j ∈ N : j ≻M i| ≤ r − 1 implies that i ∈ C has a
positive probability for each i ∈ N, which is smaller than one and monotonically
decreases with the number of competitors ranked higher by ≻M than i ∈ N.
• Otherwise, if |j ∈ N : j ≻M i| ≤ p − 2 or r ≤ |j ∈ N : j ≻M i|, competitor i ∈ N
is said to be eliminated.
The play-offs are not detailed in the model.
Each competitor i ∈ N has a coefficient c(i) ∈ R and a coefficient in the championship
C(i) ∈ R if i ∈ C, too. According to the usual rule in sports (see the next section),
C(i) = c(i) for each i ∈ N ∩ C, that is, if a competitor qualifies for the championship, then
it is considered with its own coefficient there.
The championship consists of k groups with |C| = kℓ participants that are ranked by
their coefficients C(i). The first k are placed in pot 1, the next k are placed in pot 2, and
so on, until the k participants with the lowest coefficients are placed in pot ℓ. Then each
group gets one participant from each pot. The model can be easily extended to the case
when |C| is not divisible by k.
The participants of the championship want to play in a group where the average
coefficient is lower. In other words, it is a common belief that the coefficients are positively
correlated with true abilities. The coefficients used in practice are constructed along this
line. For instance, the FIFA World Ranking and the UEFA coefficients for national teams,
countries, and clubs alike award more points for wins than for draws and more points
for draws than for losses, thus better achievements in the past translate into a higher
coefficient.
A tournament consists of a championship and at least one round-robin competition.
Consider a round-robin competition (N, M) of a tournament. Tanking of competitor
i ∈ N against competitor j ∈ N results in the round-robin group (N, M′ ) where M and
M′ are identical except for m′ij ≪ mij , that is, the result of the game between competitors
i and j becomes less favourable for i. Ranking ≻M is assumed to be monotonic, hence
competitor i cannot be ranked higher than another competitor due to the tanking, namely,
[h ≻M i] =⇒ [h ≻M′ i] for each h ∈ N, while competitor j cannot be ranked lower than
another competitor due to the tanking, namely, [j ≻M h] =⇒ [j ≻M′ h] for each h ∈ N.
Definition 1. A round-robin competition (N, M) of a tournament is said to be manip-
ulable upwards if there exist three competitors i, j, k ∈ N such that the following conditions
hold:
• (N, M′ ) is a round-robin competition achieved by tanking of competitor i against
competitor j;
• competitor i is not worse off with respect to qualification for the champion-
ship due to the tanking: p − 1 ≤ |h ∈ N : h ≻M i| ≤ q − 1 and p − 1 ≤
|h ∈ N : h ≻M′ i| ≤ q−1 (it remains directly qualified), or q ≤ |h ∈ N : h ≻M i| =
|h ∈ N : h ≻M′ i| ≤ r − 1 (it qualifies for the play-offs with preserving its rank);
9• competitor j is eliminated due to the tanking: p − 1 ≤ |h ∈ N : h ≻M j| ≤ r − 1
and |h ∈ N : h ≻M′ j| < p − 1;
• competitor k qualifies directly or for the play-offs due to the tanking: |h ∈ N : h ≻M k| <
p − 1 and p − 1 ≤ |h ∈ N : h ≻M′ k| ≤ r − 1;
• C(j) > C(i) > C(k).
Upwards manipulation means that competitor i “reorders” one higher-ranked compet-
itor as k ≻M i and k ≻M j but j ≻M′ k ≻M′ i to face better prospects in the draw of the
championship.
Proposition 1. A round-robin competition (N, M) is not manipulable upwards if one of
the following conditions hold:
• p ≥ r;
• p = 1;
• g ≻A h implies C(g, A) ≥ C(h, A) for each g, h ∈ N and (N, A);
• C(g) = max{c(h) : g ≻M h} for each g, h ∈ N.
Proof. If p ≥ q, then p−1 ≤ |h ∈ N : h ≻M i| ≤ r−1 and p−1 ≤ |h ∈ N : h ≻M j| ≤ r−1
contradict i 6= j.
If p = 1, then competitor j cannot be eliminated due to the tanking of competitor i.
If g ≻A h implies C(g, A) ≥ C(h, A), then k ≻M′ i means that C(k, M′ ) ≥ C(i, M′ ).
If C(g) = max{c(h) : g ≻M h}, then C(k) ≥ C(i) even if c(i) > c(k) since k ≻M′ i.
Definition 2. A tournament is up-strategyproof if neither of its round-robin competitions
is manipulable upwards.
Definition 3. A round-robin competition (N, M) of a tournament is said to be ma-
nipulable downwards if there exist three competitors i, j, k ∈ N such that the following
conditions hold:
• (N, M′ ) is a round-robin competition achieved by tanking of competitor i against
competitor j;
• competitor i is not worse off with respect to qualification for the champion-
ship due to the tanking: p − 1 ≤ |h ∈ N : h ≻M i| ≤ q − 1 and p − 1 ≤
|h ∈ N : h ≻M′ i| ≤ q−1 (it remains directly qualified), or q ≤ |h ∈ N : h ≻M i| =
|h ∈ N : h ≻M′ i| ≤ r − 1 (it qualifies for the play-offs with preserving its rank);
• competitor j is better off with respect to qualification for the championship due
to the tanking: q ≤ |h ∈ N : h ≻M j| and p − 1 ≤ |h ∈ N : h ≻M′ j| ≤ q − 1
(direct qualification instead of qualification for the play-offs or being eliminated),
or q ≤ |h ∈ N : h ≻M′ j| ≤ r − 1 and |h ∈ N : h ≻M j| > |h ∈ N : h ≻M′ j|
(qualification for the play-offs with a better rank);
• competitor k is worse off with respect to qualification for the championship due
to the tanking: p − 1 ≤ |h ∈ N : h ≻M k| ≤ q − 1 and q ≤ |h ∈ N : h ≻M′ k|
(qualification for the play-offs or being eliminated instead of direct qualification),
or q ≤ |h ∈ N : h ≻M k| ≤ r − 1 and |h ∈ N : h ≻M k| < |h ∈ N : h ≻M′ k|
(qualification for the play-offs with a worse rank or being eliminated instead of
qualification for the play-offs);
10• C(k) > C(i) > C(j).
Downwards manipulation means that competitor i “reorders” one lower-ranked com-
petitor as i ≻M k ≻M j but i ≻M′ k and j ≻M′ k to face better prospects in the draw of
the championship.
Proposition 2. A round-robin competition (N, M) is not manipulable downwards if
• p ≥ r;
• q = |N|;
• g ≻A h implies C(g, A) ≥ C(h, A) for each g, h ∈ N and (N, A);
• C(g) = max{c(h) : g ≻M h}.
Proof. If p ≥ r, then p−1 ≤ |h ∈ N : h ≻M′ i| ≤ r−1 and p−1 ≤ |h ∈ N : h ≻M′ j| ≤ r−1
contradict i 6= j.
If q = |N|, then competitor k cannot be worse off with respect to qualification due to the
tanking of competitor i against competitor j.
If g ≻A h implies C(g, A) ≥ C(h, A), then i ≻M k means that C(i, M) ≥ C(k, M).
If C(g) = max{c(h) : g ≻M h}, then C(i) ≥ C(k) even if c(i) < c(k) since i ≻M k.
Definition 4. A tournament is down-strategyproof if neither of its round-robin competi-
tions is manipulable downwards.
Now consider how up- and down-strategyproofness can be achieved simultaneously by
combining the first two conditions of Propositions 1 and 2, that is, without reconsidering
the usual draw procedure of the championship:
• p ≥ r means that at most one competitor qualifies directly for the championship
and no competitor advances to the play-offs;
• p ≥ r ≥ q = |N| means that at most the last competitor qualifies directly for the
championship and no competitor advances to the play-offs;
• p = 1 ≥ r means that at most the top competitor qualifies directly for the
championship and no competitor advances to the play-offs;
• p = 1 and q = |N| mean that all competitors qualify directly for the champion-
ship;
Usually, none of the above policies is desirable. Then strategyproofness can be guaranteed
only by modifying the definition of coefficients.
4 The strategyproofness of real-world tournaments
This empirical section examines some real-world sports tournaments in the view of Defin-
itions 2 and 4.
Example 4.1. 2018 FIFA World Cup
The draw for the group stage was based on the October 2017 FIFA World Ranking. The
host Russia was automatically assigned to Pot 1. Disregarding the inter-confederation play-offs,
the 2018 FIFA World Cup qualification was organised separately in the six FIFA confed-
erations:
11• The Asian (AFC) section was not down-strategyproof because the top two teams
qualified directly from each of the two groups with six teams in the third round.
The manipulation is analogous to Example 1.1.
• The African (CAF) section was down-strategyproof due to Proposition 2 because
only the top team qualified directly from each of the five groups with four teams
in the third round, hence p = q = r = 1.
• The North, Central American and Caribbean (CONCACAF) section was not down-
strategyproof because the top three teams qualified directly from the group of
six teams in the fifth round. The manipulation is analogous to Example 1.1.
• The South American (CONMEBOL) section was not down-strategyproof as it
has already been presented in Example 1.1.
• The Oceanian (OFC) section was down-strategyproof because only the top team
qualified for the fourth round from each of the two groups with three teams in the
third round, hence p = q = r = 1 (if advancing to the fourth round is interpreted
as direct qualification in the theoretical model).
• The European (UEFA) section was not down-strategyproof because the top team
qualified directly and the runner-up qualified for the play-offs from the nine
groups of six teams in the first round. The manipulation is analogous to Ex-
ample 1.1: the group winner is interested in providing the second position for a
team with a lower FIFA rating.
In each confederation, the qualification satisfied up-strategyproofness due to Proposition 1
as p = 1.
The 2018 FIFA World Cup qualification is not covered entirely by the model of Sec-
tion 3 since the October 2017 FIFA World Ranking, underlying the draw of the final group
stage, can also be affected by tanking: c(i) is not fixed and can somewhat decrease when
(N, M) is changed to (N, M′ ). Nonetheless, while its consideration minimally narrows the
opportunity to tank, it does not guarantee incentive compatibility.
Example 4.2. 2016 UEFA European Championship
The draw for the group stage was based on the UEFA national team coefficients at the
end of the qualifying group stage. The host France was automatically assigned to Group
A, Pot 1 contained the titleholder Spain and the four highest-ranked teams. Further
pots were formed according to the standard procedure. The qualification contained one
group of five and eight groups of six teams such that the top two teams qualified directly.
Consequently, the tournament was not down-strategyproof in any groups but it was up-
strategyproof.
Again, it has been neglected that tanking marginally affects the team coefficients.
Example 4.3. 2020 UEFA European Championship
The draw for the group stage was based on the results of the qualification, which con-
tained five groups of five and five groups of six teams. In particular, the group winners
were assigned to the 1–10 positions, the runners-up to the 11–20 positions, and the four
play-off winners were placed in the last pot of six teams. Hence both up- and down-
strategyproofness hold due to the third condition of Propositions 1 and 2.
12But the matches played against sixth-placed teams were not considered in this ranking,
which created another form of incentive incompatibility, see Csató (2020a, Remark 3.3).
Example 4.4. 2020/21 UEFA Champions League
The draw for the group stage was based on the 2020 UEFA club coefficients except for Pot
1, which contained the UEFA Champions League and UEFA Europa League titleholders
together with the champions of the top six national associations. If one or both titleholders
were one of the champions of the top six associations, the champion(s) of the next highest-
ranked association(s) were assigned to Pot 1.
Because of the strange definition of Pot 1, the tournament was not up-strategyproof
despite that p = 1 for each country. As an illustration, consider the sixth-ranked Russian
league where p = 1, q = 2, and r = 3. Since the third-placed club (Krasnodar) had a
coefficient (35.5) between the coefficients of the champion (Zenit Saint Petersburg; 64.0)
and the runner-up (Lokomotiv Moscow; 33.0), it was favoured by the outcome of the
Russian championship: if Lokomotiv would have been the champion in place of Zenit,
then Krasnodar would have had a worse position in the seeding. Therefore, Krasnodar
was interested in the winning of Zenit, possibly through tanking. This situation might
have occurred in the top eight UEFA national associations. In the 2020/21 season, these
were Spain, England, Italy, Germany, France, Russia, Portugal, and Belgium according
to the 2019 UEFA country coefficients.
The tournament was neither down-strategyproof due to the national associations with
q ≥ 2, that is, where the runner-up qualified directly. In the 2020/21 season, these
were the six highest-ranked, Spain, England, Italy, Germany, France, and Russia. For
example, consider the German league with p = 1, q = 4, and r = 4. The third-placed RB
Leipzig (49.0) benefited from Borussia Mönchengladbach (26.0) being the fourth rather
than Bayer Leverkusen (61.0). Analogously, down-strategyproofness did not hold for the
national associations with r ≥ 2 (the runner-up qualified at least for the play-offs) as
the champion could have preserved its rank even after tanking. These were the next
nine countries, Portugal, Belgium, Ukraine, Turkey, the Netherlands, Austria, the Czech
Republic, Greece, and Croatia in the 2020/21 season. Note that the Portuguese champion
was not necessarily placed in Pot 1.
Example 4.5. 2020/21 UEFA Europa League
The draw for the group stage was based on the 2020 UEFA club coefficients. However,
the Europa League cannot fit into our theoretical model since the teams participating in
the Champions League qualification may end up in the Europa League. Nonetheless, this
could not have occurred in the 11 highest-ranked countries where the champion qualified
for the Champions League group stage (including the 11th ranked Netherlands with its
domestic titleholder filling the vacancy created by the Champions League titleholder).
Therefore, the Europa League was not up-strategyproof. As an illustration, start
from Example 1.2. The sixth-placed Tottenham (club coefficient: 85.0), which qualified
for the play-offs, was clearly better off with Leicester (22.0) finishing fifth instead of the
third-placed Manchester United (100.0).
The violation of down-strategyproofness has already been shown in Example 1.2, at
least for the associations with r ≥ p + 2, namely, all countries except for Liechtenstein—
which does not organise a domestic league—and the five lowest-ranked countries (Gibral-
tar, Northern Ireland, Kosovo, Andorra, San Marino): the best club qualifying for the
Europa League only was interested in giving the next position to a club having a lower
coefficient.
13The situation is even more serious as the cup winners participate in the Europa League
qualification and their position is not threatened by any tanking strategy pursued in the
domestic league. This creates perverse incentives in each country except for Liechtenstein.
5 Policy implications
The following section presents a general procedure to guarantee strategyproofness on the
basis of our theoretical model. Some alternative ideas are also outlined shortly.
As Propositions 1 and 2 show, both up- and down-strategyproofness hold if the coeffi-
cients are calculated in all round-robin tournaments such that C(g) = max{c(h) : g ≻M h}
for each g, h ∈ N. In other words, team g is seeded in the championship according to
the maximum of individual coefficients c(h) of all teams ranked lower in its round-robin
competition. This is a reasonable rule: if team i finishes ahead of team j in the league,
why is it judged worse for the draw of the championship? The proposal can be called
strategyproof seeding.
Nonetheless, Section 4 demonstrates that the real-world is more complex than the
theoretical model of Section 3, hence it should be seen how the above policy could be
realised in practice.
Example 5.1. The implementation of strategyproof seeding for real-world tournaments:
• 2018 FIFA World Cup: The teams qualifying directly from a round-robin group
obtain the best FIFA rating of all lower-ranked teams in their groups. The teams
qualifying through play-offs (at most one from the AFC, CONCACAF, CONME-
BOL, and OFC; four from the UEFA) obtain either the highest FIFA rating of
all lower-ranked teams in their groups (e.g. Australia, the third-placed team of
Group B in the third round of the AFC qualification, receives the maximal FIFA
ratings of Australia, United Arab Emirates (4th), Iraq (5th), Thailand (6th))
or even the FIFA rating of a team defeated in the play-offs (e.g. Australia won
against Syria in the fourth round of the AFC qualification and against Honduras
in the intercontinental play-offs). The rule of using the maximum can be ap-
plied iteratively, that is, each team may take the highest coefficient of the teams
overtaken in the previous stages of the qualification.
• 2016 UEFA European Championship: The teams qualifying directly from a round-
robin group obtain the best UEFA national team coefficient of all lower-ranked
teams in their groups (e.g. Austria is considered with 31,345, the coefficient of
Russia, instead of its own coefficient of 30,932 since Austria won Group G ahead
of Russia). The teams qualifying through the play-offs obtain either the highest
UEFA national team coefficient of all lower-ranked teams in their groups or even
the UEFA national team coefficient of a team defeated in the play-offs (e.g. the
Republic of Ireland is considered with 30,367, the coefficient of its opponent in
the play-offs—Bosnia and Herzegovina—instead of its own coefficient of 26,902).
Nonetheless, the latter policy seems to be unfair because Poland is the runner-up
in Group D ahead of the third-placed Republic of Ireland but has a coefficient
28,306. If Poland can also inherit the coefficient of Bosnia and Herzegovina, this
may be disadvantageous for Group A, where Turkey qualified directly as the best
third-placed team.
14• UEFA Champions League: All participants of the group stage take the highest
UEFA club coefficient of the teams ranked lower in their domestic league.
• UEFA Europe League: All participants of the group stage are considered with
the highest UEFA club coefficient of the teams ranked lower in their domestic
league.
However, the presence of cup winners—possibly finishing at the bottom of their
national championships—substantially complicates the situation since they do
not fit into the theoretical model of Section 3. Unfortunately, providing them
with the highest UEFA club coefficient of the teams ranked lower than the best
club of the country that goes to the UEFA Europa League does not guarantee
strategyproofness: the cup winners remain interested in placing the club with the
highest coefficient in their domestic league lower. The only remaining solution
is to give the highest UEFA club coefficient of the national championship to the
cup winner.
Although this policy can be justified as the cup winner, directly or at least indir-
ectly, has defeated even the champion in the cup, and the vacancy created if the
Europa League titleholder qualifies for the Europa League through its domestic
championship is filled by the cup winners (UEFA, 2020c), it might strongly prefer
the cup winners over other Europa League participants. Therefore, any mono-
tonic transformation, for instance, one-half or one-third of the highest UEFA
club coefficient in the domestic league can be taken for the cup winner as a lower
bound of its coefficient.
Table 3 applies strategyproof seeding for the 2020/21 Champions League group stage.
Despite that the club coefficient of 15 teams—including the 11 lowest-ranked—improve
due to our proposal, it has only a moderated effect on the composition of pots: one German
and two French teams benefit at the expense of four teams from the Netherlands, Austria,
Greece, and Russia. The amendment favours the highest-ranked associations, where some
clubs emerging without a European record (remember the unlikely triumph of Leicester
City in the 2015/16 English Premier League (BBC, 2016)) can “obtain” the records of
clubs with considerable achievements in the European cups. Consequently, strategyproof
seeding may be advantageous for the long-run competitive balance in the top leagues by
contributing to the success of underdogs in the European cups. In addition, it probably
better reflects the true abilities of the teams because it is more difficult to perform better
in a round-robin league than in the Champions League or Europa League: playing more
matches clearly reduces the role of luck in sports tournaments (McGarry and Schutz, 1997;
Scarf et al., 2009; Lasek and Gagolewski, 2018; Csató, 2019a).
In fact, from the 2018/19 season onwards, UEFA club coefficients are determined either
as the sum of all points won in the previous five years or as the association coefficient over
the same period, whichever is the higher (Kassies, 2020; UEFA, 2020a). This rule was not
effective in the 2020/21 Champions League but the lower bound applied in the case of some
Spanish, German, or French teams in the 2020/21 Europa League. A somewhat similar
mechanism is used in the UEFA Champions League and Europa League qualification,
too, when the subsequent round is drawn before the identity of the teams is known: “If,
for any reason, any of the participants in such rounds are not known at the time of the
draw, the coefficient of the club with the higher coefficient of the two clubs involved in an
undecided tie is used for the purposes of the draw.” (UEFA, 2020b, Article 13.03).
Table 1 reinforces that the strategyproof seeding may result in more ties than the
15Table 3: Alternative rules for the draw of the UEFA Champions League group stage in the 2020/21 season
Club Country Position Coefficient Inherited from Pot allocation
Official Proposed Official Proposed Change
Bayern Munich CL TH (Germany 1st) 136 136 — 1 (1) 1 (1) —
Sevilla EL TH (Spain 4th) 102 102 — 1 (1) 1 (1) —
Real Madrid Spain 1st 134 134 — 1 (1) 1 (1) —
Liverpool England 1st 99 116 Manchester City (2nd) 1 (1) 1 (1) —
Juventus Italy 1st 117 117 — 1 (1) 1 (1) —
Paris Saint-Germain France 1st 113 113 — 1 (1) 1 (1) —
Zenit Saint Petersburg Russia 1st 64 64 — 1 (1) 1 (1) —
Porto Portugal 1st 75 75 — 1 (2) 1 (3) —
Barcelona Spain 2nd 128 128 — 2 (2) 2 (2) —
Atlético Madrid Spain 3rd 127 127 — 2 (2) 2 (2) —
Manchester City England 2nd 116 116 — 2 (2) 2 (2) —
Manchester United England 3rd 100 100 — 2 (2) 2 (2) —
Shakhtar Donetsk Ukraine 1st 85 85 — 2 (2) 2 (2) —
Borussia Dortmund Germany 2nd 85 85 — 2 (1) 2 (1) —
Chelsea England 4th 83 91 Arsenal (8th) 2 (2) 2 (2) —
➔
Ajax Netherlands 1st 69.5 69.5 — 2 (2) 3 (3)
Dynamo Kyiv Ukraine 2nd 55 55 — 3 (3) 3 (3) —
16
➔
Red Bull Salzburg Austria 1st 53.5 53.5 — 3 (3) 4 (4)
RB Leipzig Germany 3rd 49 61 Bayer Leverkusen (5th) 3 (3) 3 (3) —
Inter Milan Italy 2nd 44 80 Roma (5th) 3 (3) 3 (3) —
➔
Olympiacos Greece 1st 43 43 — 3 (3) 4 (4)
Lazio Italy 4th 41 80 Roma (5th) 3 (3) 3 (3) —
➔
Krasnodar Russia 3rd 35.5 44 CSKA Moscow (4th) 3 (3) 4 (4)
Atalanta Italy 3rd 33.5 80 Roma (5th) 3 (3) 3 (3) —
Lokomotiv Moscow Russia 2nd 33 44 CSKA Moscow (4th) 4 (4) 4 (4) —
➔
➔
Marseille France 2nd 31 83 Lyon (7th) 4 (4) 2 (2)
Club Brugge Belgium 1st 28.5 39.5 Gent (2nd) 4 (4) 4 (4) —
➔
Borussia Mönchengladbach Germany 4th 26 61 Bayer Leverkusen (5th) 4 (4) 3 (3)
Istanbul Başakşehir Turkey 1st 21.5 54 Beşiktaş (3rd) 4 (4) 4 (4) —
Midtjylland Denmark 1st 14.5 42 Copenhagen (2nd) 4 (4) 4 (4) —
➔
Rennes France 3rd 14 83 Lyon (7th) 4 (4) 3 (2)
Ferencváros Hungary 1st 9 10.5 Fehérvár (2nd) 4 (4) 4 (4) —
CL (EL) TH stands for the UEFA Champions League (Europa League) titleholder.
The column “Inherited from” shows the club of the domestic league whose UEFA club coefficient is taken over.
Proposed pot is the pot that contains the club if the current policy applies to Pot 1. Since this is incentive compatible (Csató, 2020b), the pot according to
the amendment suggested by Csató (2020b, Section 5) is reported in parenthesis for both the official and the strategyproof seeding.
The column “Change” shows the movements of clubs between the pots due to the strategyproof seeding if the current policy applies to Pot 1.current definition. Furthermore, if some teams inherit their coefficients from the same
lower-ranked team, then these remain identical, and the tie should be broken by drawing of
lots (UEFA, 2020c, Annex D.8). Although tie-breaking does not affect strategyproofness,
it is reasonable to prefer the teams ranked higher in the domestic championship. If there
are clubs from other associations with the same coefficient, an event that has a much
lower probability, they can be assigned arbitrarily in this equivalence class. Alternatively,
the original club coefficients can be used for tie-breaking.
The strategyproof mechanism has further favourable implications. For instance, UEFA
has modified the allocation policy of the clubs into pots in the Champions League from
the 2015/16 season, probably inspired by the previous year when Manchester City, the
English champion, was drawn from the second pot but Arsenal, the fourth-placed team in
England, was drawn from the first pot. The decision—intended to strengthen the position
of domestic titleholders (UEFA, 2015)—has considerable sporting effects (Corona et al.,
2019; Dagaev and Rudyak, 2019), especially because the poor way of filling vacancies leads
to incentive incompatibility (Csató, 2020b). The proposed seeding rule automatically
ensures that a national champion has at least the same UEFA club coefficient as any
team that is ranked lower in its domestic league.
Finally, other strategyproof seeding policies can be devised. One example is the sys-
tem of the 2020 UEFA European Championship (see Example 4.3), that is, the direct
ranking of all teams qualified for the group stage on the basis of their results in the qual-
ification. But that rule is not appropriate if the achievements in the qualification cannot
be compared: the qualification tournaments for the FIFA World Cup of the six confedera-
tions differ substantially, and the strength of clubs playing in the domestic championships
of UEFA associations varies greatly. Another solution might be to assign seeding posi-
tions not to the coefficients but to the type of qualification: for instance, to identify a
participant of the UEFA Champions League as the Spanish runner-up rather than by its
identity. Then these labels can be evaluated according to the records of the corresponding
teams (Guyon, 2015a). However, this scheme seems to be difficult to apply for the FIFA
World Cup or UEFA European Championship.
To summarise, the strategyproof seeding policy offers a general solution for incentive
compatibility. Other rules can also eliminate perverse incentives but they are unlikely to
be independent of the particular characteristics of the tournament.
6 Conclusions
The current work has analysed a mathematical model of seeding for sports tournaments
where the qualified teams are selected on the basis of round-robin contests. Several
championships are designed this way, including the most prestigious soccer tournaments
(FIFA World Cup, UEFA European Championship, UEFA Champions League). The
conditions of strategyproofness have turned out to be quite restrictive: if each competitor
is considered with its own coefficient (a measure of its past performance), only one or all
of them should qualify from every round-robin contest.
Similarly to the main theorem of Vong (2017) on the strategic manipulation problem
in multistage tournaments (i.e. the necessary and sufficient condition of incentive com-
patibility is to allow only the top-ranked player to qualify from each group), this finding
has the flavour of an impossibility result. However, we are able to conclude positively be-
cause strategyproofness can be achieved by giving to each qualified competitor the highest
coefficient of all competitors that are ranked lower in its round-robin competition for the
17seeding procedure.
The central message of our paper for decision makers is consonant with the motiva-
tion of Haugen and Krumer (2019)—tournament design should be included into the family
of traditional topics discussed by sports management. In particular, administrators are
strongly encouraged to follow our recommendation for future rules to prevent the occur-
rence of costly scandals.
Acknowledgements
We are indebted to the Wikipedia community for collecting and structuring valuable in-
formation on the sports tournaments discussed.
The research was supported by the MTA Premium Postdoctoral Research Program grant
PPD2019-9/2019.
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