Journal of Sports Analytics xx (2021) x–xx 1 DOI 10.3233/JSA-200527 IOS Press

1 “Choose your opponent”: A new knockout † 2 design for hybrid

4 ∗ 3 Julien Guyon 5 Department of Mathematics, Columbia University Courant Institute of Mathematical Sciences, New York Uni- versity

6 Abstract. We present a new, simple knockout format for sports tournaments, that we call “Choose Your Opponent”, where the 7 teams that have performed best during a preliminary group stage can choose their opponents during the subsequent knockout 8 stage. The main benefit of this format is that it essentially solves a recently identified incentive compatibility problem when 9 more than one teams from a group advance to the knockout stage, by effectively canceling the risk of tanking. This new 10 design also makes the group stage more exciting, by giving teams a strong incentive to perform at their best level, and more 11 fair, by limiting the risk of collusion and making sure that the best group winners are fairly rewarded in the knockout round. 12 The choosing procedure would add a new, exciting strategic component to the competition. Advancing teams would choose 13 their opponent during new, much anticipated TV shows which would attract a lot of media attention. We illustrate how this 14 new format would work for the round of 16 of the UEFA Champions League, the most popular soccer club competition in 15 the world.

16 Keywords: Sports, design, fairness in sports, win incentive, collusion, tanking, strategy

17 1. Introduction stage. Since group rank is supposed to reflect team 28 abilities, the bracket gives, on paper, an incentive to 29 18 Many sports tournaments consist of a group stage advance with the best group rank. Probably the most 30 19 played as a round-robin, followed by a single- famous example of a competition using such a for- 31 1 20 elimination knockout stage. Often, the best k teams mat is the FIFA World Cup (soccer). The knockout 32 21 in each group advance to the knockout stage, and bracket of the 2018 FIFA World Cup is shown in 33 22 the knockout stage follows a predetermined bracket Figure 1; the 8 groups are labeled A to H; C1 denotes 34 23 route that depends only on group labels (e.g., A, B, C, the winner of Group C, F2 denotes the runner-up 35 24 etc.) and group ranks (group winner, group runner- of Group F, etc. Some variations exist; for instance, 36 25 up, etc.). Usually, for the teams that advance, a better when the number of groups is not a power of 2, the 37 26 group rank means facing an opponent with a worse best k teams in each group advance to the knockout 38 27 group rank during the first round of the knockout stage, as well as the best of the teams ranked k + 1in 39 their group (compared across groups), so as to popu- 40 † N A first version of this document, containing the main idea and late a bracket with 2 teams. For example, the 1986, 41 applications to the round of 16 of the UEFA Champions League and 1990, 1994 FIFA World Cups, as well as the UEFA 42 to maximizing the number of home games during the UEFA Euro n = 2020, was communicated to UEFA in August 2016. A preprint Euros since 2016, use a format with 6 groups of 43 (Guyon, 2019a) was posted online in November 2019. The author 4, where the best k = 2 teams in each group advance 44 N also published an article on this new knockout design in the French to the round of 16 (2 = 16), together with the best 45 newspaper Le Monde in December 2019 (Guyon, 2019d), and in N 2 − kn = 4 third-placed teams; see (Guyon, 2016) 46 Four Four Two in December 2020 (Guyon, 2020a). for details. 47 1Throughout the article, we use the word “teams” to denote the participants of the tournament, but of courseUncorrected they could also Compared Author to a pure round-robin Proof tournament 48 be individual players. involving all teams, this hybrid format made of a 49 ∗ Corresponding author: Julien Guyon, Department of Mathe- group stage followed by a knockout stage has more 50 matics, Columbia University Courant Institute of Mathematical outcome uncertainty, but it has much less matches, 51 Sciences, New York University. E-mail: [email protected], [email protected]. so it can fit in a given calendar frame. It also has a 52

ISSN 2215-020X © 2021 – The authors. Published by IOS Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (CC BY-NC 4.0). 2 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments

tanking irrelevant in practice, since in all realistic 94 situations tanking can only decrease a team’s proba- 95 bility of achieving its goal, whether it be winning the 96 tournament or reaching a particular round. 97 Moreover, advancing teams would pick their oppo- 98 nent during new, much anticipated TV shows which 99 would attract a lot of media attention; the picking 100 strategies of the teams would certainly be highly 101 debated by the media and among fans. 102 Fig. 1. Bracket of the knockout stage of the 2018 FIFA World Cup. The “Choose Your Opponent” knockout format 103 appears to have been used in the Austrian 104 53 much smaller proportion of noncompetitive matches. league since the 2012–2013 postseason “Playoff 105 54 Compared to a pure knockout tournament, this Pick” (Austrian website; Matisz, 106 55 hybrid format has less outcome uncertainty, and it 2020). The top three teams in the league can choose 107 56 gives all teams a minimum number of matches (at their opponent from the playoff spots 5 to 8. For 108 57 least two) and a chance to advance even if they lose instance, in 2016–17, the first-placed Vienna Capi- 109 58 their first game. tals chose the fifth-placed HC TWK Innsbruck and 110 59 Those traditional designs suffer from well-known the second-placed EC Red Bull Salzburg picked 111 60 and well-documented flaws. Fairness issues include the seventh-placed Graz 99ers (Wikipedia, 2016– 112 61 the risk of collusion, the risk of tanking/shirking, the 17 Austrian Hockey League season). In 2017–18, 113 62 possible absence of win incentive in the last group also finished in first place and again 114 63 games, and group winners being poorly rewarded in picked HC TWK Innsbruck, this time six-placed 115 64 the knockout stage despite a great performance in the (Wikipedia, 2017–18 Austrian Hockey League sea- 116 65 group stage. Those flaws are discussed in detail in son). In 2019–20, they finished in third place and 117 66 Section 2. preferred the fifth-placed EC Graz 99ers to the 118 67 In this article, in order to eliminate these flaws, seventh-placed Black Wings Linz (Wikipedia, 2019– 119 68 we suggest a new knockout format. In this new for- 20 Austrian Hockey League season). 120 69 mat, that we call “Choose Your Opponent”, the teams The “Choose YourOpponent” design has also been 121 70 that have performed best during the group stage can used in the Southern Professional Hockey League 122 71 choose their opponent during the subsequent knock- in the U.S. under the name “Challenge Round” 123 72 out stage. The main benefit of this format is that it in 2017–18 and 2018–19 (Southern Professional 124 2 73 makes the group stage more exciting, by giving teams Hockey League website). The English rugby league 125 74 a strong incentive to perform at their best level. It also applied it under the name “Club Call” in 2009– 126 75 makes the group stage more fair, by limiting the risk 14 for semifinals only (Wikipedia, Super League 127 76 of collusion, effectively canceling the risk of tank- play-offs). The format has also recently been used 128 77 ing, and making sure that the best group winners are in bridge (Canadian Bridge Federation; The United 129 78 fairly rewarded for their group stage performance and States Bridge Federation), chess (Chess.com web- 130 79 do not meet strong opponents in the first knockout site), sailing (2015 World Match Racing Tour), and in 131 80 round. the multiplayer online battle arena video game Dota 132 81 In particular, one of the remarkable benefits of this 2 (The International Dota 2 Championships). 133 82 design is that it practically solves the problem iden- However, it seems that it has never been used in the 134 83 tified by Vong (2017), who has shown that in the most popular sports tournaments. It could be used in 135 84 classical design of hybrid tournaments, i.e., when many major tournaments, including: 136 85 bracket routes are predetermined based on group 2 86 labels and group ranks, in order to exclude the risk Note that between 2010 and 2013 the Finnish ice hockey league made a surprising use of “Choose Your Opponent”, allow- 87 of tanking, it is both necessary and sufficient to allow ing the lowest-ranked teams at some point during the 14-team 88 only the top-ranked team in each groupUncorrected to advance. regular season toAuthor pick their opponent amongProof the best-ranked teams 89 That is crucially important because regulations that for an additional two games in the regular season (International 90 allow only one qualifier from each group may not Hockey Wiki website; Wikipedia, 2010–11 SM-liiga season). The 14th-placed team at the time was the first picking team, followed by 91 be desirable, as they tend to produce many stake- the 13th-placed team, etc. The restricted choice version was used: 92 less games. Though the “Choose Your Opponent” the picking teams could only choose opponents ranked between 93 design cannot fully cancel the risk of tanking, it makes the first and seventh place. J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments 3

preservation) and investigate several impossibility 172 results. Groh et al. (2012) aim to find optimal seed- 173 ings in elimination tournaments for three different 174 criteria: (1) maximization of total tournament effort; 175 (2) maximization of the probability of a final among 176 the two top ranked teams; (3) maximization of the 177 win probability for the top player. They also identify 178 the seedings ensuring that higher-ranked players have 179 a higher winning probability. Pauly (2014) develops 180 a mathematical model of strategic manipulation in 181 complex sports competition formats. Karpov (2016, 182 2018) develops an axiomatic theory of knockout tour- 183 Fig. 2. Ideal bracket, when teams have been ranked from 1 (the naments, gives axiomatic justification for various 184 N best group winner) to 2 after the group stage is over. Here N = 4. seedings methods, suggests two new seeding meth- 185 ods (equal gap seeding and increasing competitive 186 intensity seeding), and provides many useful refer- 187 ences. Dagaev and Suzdaltsev (2018) study optimal 188 137 • Soccer: FIFA World Cup, UEFA European ways to seed a knockout tournament in order to 189 138 Championship, Copa America, African Cup maximize the overall spectator interest in it, assum- 190 139 of Nations, UEFA Champions League, UEFA ing that neutral spectators tend to prefer to watch 191 140 Europa League, of European qualifiers close and intense matches and matches that involve 192 141 to the FIFA World Cup, etc. strong teams. As it has already been mentioned, 193 142 • Football (NFL), basketball, baseball, ice Vong (2017) has shown that in the classical design 194 143 hockey: playoffs of the regular domestic season. of hybrid tournaments, in order to exclude the risk 195 144 • Handball, basketball, volleyball: World Cup, of tanking, it is necessary and sufficient that only 196 145 continental championships, European club the top-ranked team in each group advances. Arlegi 197 146 competitions. and Dimitrov (2020) study the impact of two basic 198 147 • Rugby: World Cup, European club competi- principles of fairness on the structure of elimination- 199 148 tions, playoffs of the regular domestic season. type competitions: that stronger players should have 200 149 • Tennis: ATP Finals. a larger chance of winning than weaker players, and 201 150 • Cricket: the ICC Cricket World Cup. that equally strong players should have the same 202 151 In this article we will illustrate how this new format chances of being the final winner. They find that 203 152 would work for the round of 16 of the UEFA Champi- a new competition structure that they call antler 204 153 ons League, the most popular soccer club competition plays a referential role in the design of fair knockout 205 154 in the world. brackets. 206 155 There exists a vast literature on knockout tour- Our new format is radically different from the ones 207 156 nament design. Several theoretical studies have studied in those papers, as in our approach teams are 208 157 attempted to find a best seeding and formalize dif- not placed in a bracket using a predefined mecha- 209 158 ferent goals of seeding. Hwang (1982) defines a nism. Rather, the teams that have performed best so 210 159 bracket to be monotone if the probability of win- far during the tournament decide against who they 211 160 ning the tournament increases with team skill. He will play in the next knockout round, while the teams 212 161 shows that the canonical knockout bracket (see that are chosen have no say. This new feature prac- 213 162 Figure 2) is not necessarily monotone and suggests an tically solves the win incentive issues raised in the 214 163 adaptive reseeding method that he proves to be mono- abovementioned papers. 215 164 tone. Schwenk (2000) suggests three axioms for fair Numerous statistical and simulation studies aim 216 165 seeding (delayed confrontation, sincerity rewarded, at comparing tournament designs. Glickman (2008) 217 166 favoritism minimized). He proves thatUncorrected the canonical assumes only Author partial information Proof about competitors’ 218 167 knockout bracket fails to satisfy the second axiom relative rankings and develops Bayesian locally- 219 168 and suggests a variant that satisfies all three seeding optimal design of adaptive knockout tournaments to 220 169 axioms, in which subgroups of teams are randomly maximize the probability that the best team advances 221 170 shuffled. Vu and Shoham (2011) introduce two alter- to the next round. Recently, Glickman and Hen- 222 171 native criteria for fairness (envy-freeness and order nessy (2016) have extended this approach in order 223 4 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments

224 to identify the overall best team in fixed knock- South American Qualifiers, which overcomes 275 225 out tournament brackets. Other utility functions are the main drawbacks of the previous policy 276 226 also considered. Scarf, Yusof, and (2009) and has unanimously been approved by all 277 227 propose tournament metrics that can be used to CONMEBOL (South American Football Con- 278 228 measure the success of a sporting tournament, and federation) members to be used in the qualifier 279 229 describe how these metrics may be evaluated for a for the 2018 FIFA World Cup; see also (Alarcon´ 280 230 particular tournament design. This allows them to et al., 2017). 281 231 compare competing designs, such as round-robin, The remainder of this article is structured as fol- 282 232 pure knockout and hybrids of these designs. They use lows. Section 2 discusses the main flaws of the 283 233 the UEFA Champions League (soccer) to illustrate traditional hybrid designs. In Section 3, we describe 284 234 their methodology, while Scarf and Yusof (2011) use the “Choose Your Opponent” knockout format and 285 235 the FIFA World Cup Finals (soccer). Csato´ (2021a) investigate its main benefits and drawbacks. Section 286 236 compares four hybrid designs for the World Men’s 4 investigates how this new format would work for the 287 237 Handball Championships. Other relevant literature round of 16 of the UEFA Champions League. Finally, 288 238 includes Adler et al. (2017), Annis and Wu (2006), Section 5 concludes. 289 239 Appleton (1995), Baumann, Matheson, and Howe 240 (2010), Csato´ (2021b), Dagaev and Rudyak (2019), 241 Edwards (1998), Goossens, Belien,¨ and Spieksma 2. Flaws of the traditional hybrid tournament 290 242 (2012), Lasek and Gagolewski (2018), Marchand design 291 243 (2002), McGarry and Schutz (1997), and Ross and 244 Ghamami (2008). Fair draws of round-robin groups The hybrid tournament design, though it is widely 292 245 are investigated in (Cea, 2020; Guyon, 2014a, 2015a; used, has serious fairness issues: 293 246 Laliena and Lopez,´ 2019). Another recent line of 247 research is concerned with fixing a knockout tour- • Risk of collusion: When k ≥ 2, two teams may 294 248 nament (Aziz et al. 2014; Stanton and Vassilevska be tempted to collude if they play against each 295 249 Williams, 2011; Vassilevska Williams, 2010; Vu, Alt- other during the last match day of the group 296 250 man, and Shoham, 2009). stage, and if a given result satisfies both of 297 251 In a sense, the “Choose Your Opponent” policy is a them, whatever the result of the other group 298 252 “reversed” draft allocation mechanism as the former matches. For instance, one team may be sat- 299 253 prefers the best teams and the latter prefers the worst isfied by securing the first position in the group, 300 254 teams. Draft allocation has a very large literature, while the other team may be satisfied by secur- 301 255 see, e.g., Banchio and Munro (2021), Fornwagner ing qualification to the knockout stage, even in 302 256 (2019), Kazachkov and Vardi (2020), Lenten (2016), second position. Collusion does not have to be 303 257 Lenten, Smith, and Boys (2018), Price et al. (2010), explicit, it may be tacit and simply take the form 304 258 and Taylor and Trogdon (2002). of two teams refusing to attack each other when 305 259 Note that academic research has already led to the current score of the match satisfies both of 306 260 changing tournament designs in soccer. Let us men- them. 307 261 tion three examples: One of the most famous examples of collusion 308 in sports is the “disgrace of Gijon”´ (see, e.g., 309 262 • Inspired by Guyon (2014a, 2015a), the draw for (Kendall and Lenten, 2017)). It refers to the 310 263 the 2018 FIFA World Cup—unlike the previous match between West and who 311 264 editions of this championship—has determined refused to attack each other during 80 minutes, 312 265 all pots by positions in the FIFA World Rank- satisfied by the 1-0 Germany win that let both 313 266 ing in order to increase balancedness across the teams advance to the second round of the 1982 314 267 groups while still satisfying geographic con- FIFA World Cup at the expense of Algeria, who 315 268 straints; had played their last group game the day before. 316 • 269 UEFA has followed the proposalUncorrected of Guyon To prevent Author this to happen Proof again, FIFA, soccer’s 317 270 (2016) to reduce the biases in the design of the world governing body, decided that all teams in 318 271 knockout bracket of the 2020 UEFA European a given group would play their last group match 319 272 Championship; at the same time. 320 273 • Duran´ et al. (2017) have constructed an alter- However, playing the last two group games 321 274 native schedule for the FIFA World Cup at the exact same time does not fully prevent 322 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments 5

323 collusion. - (0-0 on June 26, Denmark- at UEFA Euro 2004 is 375 324 2018 during the 2018 FIFA World Cup) is a another example of a tacit collusion situation 376 325 recent example of tacit collusion in this con- (Kendall and Lenten, 2017): a 2-2 would 377 326 text: both teams knew that a draw would let qualify both teams at the expense of , what- 378 327 them both advance to the knockout stage what- ever the result of the game between Italy and 379 328 ever the result of the other game in the group, . The game indeed ended as a 2-2 draw, 380 329 Australia-Peru. They did little effort to attack raising complaints from the Italian team and 381 330 each other, which resulted in a boring game fans, even though Sweden and Denmark seemed 382 331 and the only goalless match of the 2018 World to attack each other without restraint and try to 383 332 Cup. The crowd made its displeasure known, win the game. 384 333 as well as football fans around the world on The pre-2017 examples above are studied in 385 334 social media (The Sun, 2018). Denmark’s man- Kendall and Lenten (2017) where the authors 386 335 ager Age˚ Hareide said after the game: “We just also provide other examples of tacit collusion 387 336 needed one point, we were up against one of in sports, and more generally examples where 388 337 the best teams in the world at counterattacks, the rules of sports have led to unforeseen and/or 389 338 so we would have been stupid to open up a lot unwanted consequences. Guyon (2020d) exam- 390 339 of space. We stood back and got the result we ines the risk of collusion in groups of 3, a format 391 340 needed, it was a 0-0 and we’re very pleased with that FIFA considers using for the 48 team group 392 341 that” (, 2018). stage of the 2026 World Cup. He explains how 393 342 The last 10 minutes of the 2018 World Cup to build the match schedule so as to minimize 394 343 match between Japan and Poland is another the risk of collusion, and also quantifies how the 395 344 example of tacit collusion. Poland, already risk of collusion depends on competitive bal- 396 345 eliminated, was leading 1-0 and happy to leave ance and the point system. He also describes 397 346 the tournament on a win. When Senegal, which alternate formats for a 48 team World Cup that 398 347 was playing at the same time against Colombia, would eliminate or strongly decrease the risk 399 348 conceded a goal at the 74th minute, Japan was of collusion. Stronka (2020) attempts to quan- 400 349 perfectly even with Senegal in the group rank- tify the probability of match-fixing in groups of 401 350 ing on points, goal difference, and goals scored. four teams under various match schedules and 402 351 The next tie-breaker was fair-play. With two pair matching procedures. Schedules to reduce 403 352 yellow cards less than Senegal during the group match-fixing opportunities have been recently 404 353 stage, Japan would advance at the expense of studied in (Chater et al. 2021). 405 354 Senegal. Even though Senegal could still score, • Risk of tanking/shirking: When k ≥ 2 and 406 355 Japan was happy with a 0-1 loss. Japan and the knockout bracket has predetermined routes 407 356 Poland then suddenly stopped to attack each (like in Figure 1), some teams may even be 408 357 other, in scenes reminiscent of the disgrace of tempted to tank, i.e., to intentionally lose their 409 358 Gijon´ (BBC, 2018). last group stage match. (Tanking is also known 410 359 Another recent example of tacit collusion in as shirking.) This could for instance happen in 411 360 soccer was the last minutes of the Peru- the example of Figure 1 (k = 2) if Group B 412 361 Colombia World Cup qualifying game (1-1) teams play their last group game after Group 413 362 in October 2017. The current score saw both A has completed all its matches, and a team 414 363 sides progress in World Cup qualifying. With considered as one of the best in the world has 415 364 results elsewhere going in their favor, Colom- finished second in Group A, due to bad luck 416 365 bia knew a draw would see them finish in or poor seeding (see Section 4.4) for instance. 417 366 fourth place and qualify automatically, while Imagine that two Group B teams have already 418 367 Peru would advance to a playoff against New secured their qualification to the knockout stage 419 368 Zealand after finishing fifth. Both teams then and play against each other in the last group 420 369 stopped to attack each other. TheUncorrected Independent game. Author Both teams may then Proof be tempted to lose 421 370 (2017) reported that, following allegations of this game in order to finish second in Group B 422 371 match-fixing, Radamel Falcao, the Colombian and thus avoid the runner-up of Group A. 423 372 striker and , admitted that he discussed Tanking famously happened in the badminton 424 373 playing for a draw with his opponents, as could tournament of the 2012 Olympics in . 425 374 be seen on TV. On August 1, 2012, four teams were ejected 426 6 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments

427 from the competition for “not using one’s best studies on incentive incompatibility or lack of 476 428 efforts to win a match” and “conducting oneself strategy-proofness, i.e., the possibility that a 477 429 in a manner that is clearly abusive or detrimental team is strictly better off with a weaker per- 478 430 to the sport” following round-robin matches the formance, include (Csato,´ 2018, 2019, 2020b; 479 431 previous evening, during which the teams were Dagaev and Sonin, 2018; Vong, 2017); see also 480 432 accused of trying to lose in order to manipu- the references therein. 481 3 433 late the knockout bracket. The decision was • Absence of win incentive: Some teams may 482 434 highly debated. Indeed it can be argued that the have secured the first place of their group before 483 435 bad design of the tournament could be blamed their last group game. For instance, this was 484 436 more than the teams: while the teams have not the case of Brazil and Nigeria during the 1998 485 437 been performing their best effort to win the FIFA World Cup, and of Italy during the UEFA 486 438 game at hand, they have been in fact doing Euro 2016. This happens more frequently in the 487 439 their best to win the tournament (Badminton UEFA Champions League and UEFA Europa 488 440 at the 2012 Summer Olympics). Moreover, it is League than in the FIFA World Cup or UEFA 489 441 a common practice in every competitive sport Euro because the former are played in a home- 490 442 to save forces in earlier matches in order to per- away round-robin format while the latter use 491 443 form better in later stages of a tournament. The a single round-robin format. When the knock- 492 444 badminton tournament of the 2012 Olympics is out bracket has predetermined routes (like in 493 445 discussed in (Kendall and Lenten, 2017, Section Figure 1), teams that have already secured the 494 446 3.3.1). It is also used as a motivating example first place of their group have no incentive at 495 447 by Pauly (2014). all to win their last group match and often send 496 448 The game between and at the their B team to play this game, distorting the 497 449 2018 FIFA World Cup is another good illustra- results of their group and the fairness of the 498 450 tion of the risk of tanking. England and Belgium competition. In the three examples mentioned 499 451 were playing against each other in the last above, Brazil, Nigeria, and Italy all lost their 500 452 round of the group stage, and the winner could last group matches against Norway, Paraguay, 501 453 face a more difficult path to the final (Brazil and the Republic of , respectively, after 502 454 in quarterfinals and France in semifinals) than securing the first place of their group after only 503 455 the loser (England lost against Belgium and two games. 504 456 faced Sweden and Croatia instead). In the case Teams may also just need a draw in their last 505 457 of France-Denmark, it was not clear either group game to secure the first place, and pre- 506 458 whether winning the group was much of an fer securing a draw rather than taking risks and 507 459 advantage: the group winner could possibly face trying to win the game by playing attacking soc- 508 460 twice World Cup winners Argentina, while the cer. This usually results in more defensive, less 509 461 runner-up would very likely face Croatia (and attractive play. 510 462 all this indeed happened). Note that teams might have at least some finan- 511 463 Tanking also famously happened during the cial incentives to win as many matches as 512 464 soccer game Thailand vs Indonesia at the 1998 possible in the group stage. This is the case 513 465 AFF championship (Kendall and Lenten, 2017, for instance in UEFA club competitions, where 514 466 Section 3.9.2), in which Indonesian defender the revenue distribution system incentivizes win 515 467 Mursyid Effendi deliberately scored an own even for teams that have no sporting win incen- 516 468 goal during injury time to secure a 2-3 loss that tive, see for instance (Total Sportal website; 517 469 would allow Indonesia to play against Singa- UEFA website, 2019/20 UEFA club competi- 518 470 pore in semifinals rather than against Vietnam, tions revenue distribution system). 519 471 the hosts of the tournament. Csato´ (2020e) • Group winners may be poorly rewarded: 520 472 investigates an interesting example of tanking When group winners are paired with runners- 521 473 in soccer, where in a given matchUncorrected the two teams up during Author the first round ofProof the knockout stage 522 474 must not win in order to advance to the next (FIFA World Cup, UEFA Champions League, 523 475 stage, due to a bad tournament design. Other Cup, rugby European cups, ATP 524 Finals, etc.), they may face runners-up that they 525 3It seems that the teams wanted to avoid one half of the bracket would not have picked if given the choice. 526 (Badminton at the 2012 Summer Olympics). They can feel poorly and unfairly rewarded 527 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments 7

528 for winning their group. This happens quite would be ranked from 1 to n, compared across groups 575 529 often during the draw of the round of 16 of based on sporting criteria which depend on the sport 576 530 the UEFA Champions League (see Section 4). in consideration. For example, in soccer, the criteria 577 531 In this case tanking is not an option, as the could be in order: number of points; goal difference; 578 532 bracket is randomly drawn and does not have number of goals scored; fair-play score (based on the 579 533 predetermined routes. A similar scenario could number of yellow and red cards received), etc. Then 580 534 have occurred in the playoffs of the UEFA Euro the n runners-up would be ranked from n + 1to2n, 581 535 2020 qualifying tournament as a group winner compared across groups based on the same criteria, 582 536 in the UEFA Nations League might have faced and so on. We call the resulting ranking the “global 583 N 537 stronger opponents than a non-group winner ranking 1–2 ”. For instance, UEFA uses this mecha- 584 538 from the same league despite its better perfor- nism for its Nations League to build a global ranking 585 539 mance in the UEFA Nations League, see (Csato,´ of the 55 member associations from the rankings of 586 540 2020c). the different groups of the different leagues; this over- 587 541 Note that at UEFA European Championships, all ranking was used in the draw of the groups of the 588 542 since 2016, some group winners face third- Euro 2020 qualifiers (Wikipedia, UEFA Euro 2020 589 543 placed teams while other play runners-up, qualifying). UEFA also used the suggested global 590 544 depending only on their group label (see a full ranking mechanism in the seeding of the Euro 2020 591 545 study of the corresponding so-called “group (Wikipedia, UEFA Euro 2020 qualifying). 592 546 advantage” in (Guyon, 2016)). Remark 1. Note that when there is one unique group 593 in the group stage (n = 1), like in the NBA confer- 594

547 3. The “Choose Your Opponent” knockout ences for instance, the global ranking is trivially built 595 548 design from the group rankings. 596

549 3.1. Description of the format Remark 2. Currently, in the NFL and MLB, the teams 597 that advance to the postseason playoffs are the divi- 598

550 In classical hybrid tournament designs, the teams sion winners, as well as “wild cards”, i.e., teams with 599 551 that advance to the knockout round either the best records that did not win their division. The 600 NHL uses a similar format, with wild cards advancing 601 552 • follow a predetermined bracket route entirely on top of the best teams in each division. A similar 602 553 decided by their group label and group rank, as rule could be used to produce the global ranking, with 603 554 illustrated for example in Figure 1; or division winners being ranked above wild cards. 604 555 • follow the predetermined “ideal” bracket At this point, one natural knockout format would 605 556 entirely decided by their rank during the regular consist of placing teams in the “ideal” bracket shown 606 557 season, like during the NBA playoffs (see Fig- in Figure 2 in the case where N = 4 (see (Guyon, 607 558 ure 2 for an illustration of the 16-team “ideal” 2016) and the references therein). The ideal bracket 608 559 bracket); or is a perfectly balanced bracket, where the ranks (from 609 560 • are automatically reseeded in the “ideal” N N 1to2 ) of the two opponents sum to 2 + 1 in each 610 561 bracket, like during the NFL Conference Cham- N−1 of the 2 matches of the first knockout round, and 611 562 pionships, where the strongest remaining teams then, assuming the best ranked team always advances 612 563 face the weakest remaining teams; or N−k+1 to the next round, sum to 2 + 1inthekth 613 564 • are randomly drawn against opponents, like knockout round. It ensures that teams 1 to 8 cannot 614 565 during the draws of the knockout rounds of meet before the quarterfinals, in which case the quar- 615 566 the UEFA Champions League and the UEFA terfinals are 1-8, 2-7, 3-6, and 4-5; that teams 1, 2, 616 567 Europa League. 3, and 4 cannot meet before the semifinals, in which 617

568 Instead, in the “Choose Your Opponent” knockout case the semifinals are 1-4 and 2-3; and that teams 618 1 and 2 cannot meet before the final. This bracket 619 569 format, the teams that have performedUncorrected best during the Author Proof 570 preliminary group stage can choose their opponent is not only perfectly balanced, it is also free of any 620 571 during the subsequent knockout stage. group advantage (see (Guyon, 2016)) and, compared 621 N 572 First, the 2 teams that qualified for the knockout to brackets built only on group labels and group ranks 622 N 573 stage would be ranked from 1 to 2 , based on their (like the one in Figure 1), it also increases win incen- 623 574 group stage results. For instance, the n group winners tive during the group stage, as the more a team wins 624 8 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments

625 during the group stage, the weaker their opponents in − Team 2, if not already picked, chooses their 676 626 the knockout stage will be, at least on paper, and the opponent among the remaining teams. 677 627 higher the probability is that it goes far in the tour- − Team 3, if not already picked, chooses their 678 628 nament. Guyon (2016) has suggested two ways of opponent among the remaining teams. 679 629 modifying the “ideal” bracket to make sure that two − And so on, until all matchups are formed. 680 630 teams from the same group cannot meet again before • Restricted choice: The restricted choice imple- 681 N−1 631 the semifinals. mentation would protect Teams {2,...,2 } 682 632 However, it often happens that teams perceived as (from the better half of the advancing teams), 683 633 strong get quite poor results in the group stage and which could not be chosen: 684 634 advance to the knockout round with a poor global − Team 1 would choose their opponent among 685 N−1 N 635 ranking, maybe due to bad luck, or a bad seeding Teams {2 + 1,...,2 } (the weaker half 686 636 (see Section 4.4), or because their physical training of the advancing teams). 687 637 aims at peak performance during the knockout round − Team 2 would choose their opponent among 688 638 and players have suffered too much fatigue during the the remaining teams in the weaker half of the 689 639 group stage. In such a case, a team advancing into the advancing teams. 690 640 “ideal” knockout bracket with a good rank may end up − Team 3 would choose their opponent among 691 641 facing an opponent that it perceives as stronger than the remaining teams in the weaker half of the 692 642 other, better ranked opponents, and may feel unfairly advancing teams. 693 643 rewarded for its good performance in the group stage. − And so on, until all matchups are formed. 694 644 For example, if UEFA had adopted the “ideal” bracket For example, with the restricted choice imple- 695 645 for the UEFA Euro 2016, Germany would have been mentation, Germany would not have been 696 646 ranked 2nd after the group stage and therefore would allowed to pick (ranked 6th out of 697 647 have had to face , ranked 15th, a team gen- 16) in the round of 16 of the UEFA Euro 698 648 erally considered as a soccer powerhouse (and, as 2016; Hungary would have been protected by 699 649 a matter of fact, the future winner of the competi- their good performance during the group stage. 700 650 tion), in the round of 16. Germany may have preferred Other restrictions can easily be accomodated. 701 651 to play, say, against Ireland, , , or For instance, in the UEFA Champions League, 702 652 Hungary, teams that finished the group stage with a teams from the same country or group cannot 703 653 better rank than Portugal but are considered weaker play against each other during the round of 16 704 4 654 by many specialists (see (Guyon, 2016)). There exist (see Section 4). Also, if the tournament orga- 705 655 numerous examples of teams unfairly rewarded for nizer wants to avoid repeated matchups year 706 656 winning their group and having to face strong oppo- after year, it may add the constraint that, if possi- 707 657 nents in the first knockout round: Brazil having to ble, a team cannot pick a team that it has selected 708 658 face Argentina in the round of 16 of the 1990 FIFA in the previous edition (or p previous editions) 709 659 World Cup; Arsenal and Napoli having to face Bay- of the tournament. The tournament organizer 710 660 ern and Real Madrid in the round of 16 of may also add a constraint to prevent two teams 711 661 the 2016–17 UEFA Champions League; Tottenham qualifying from the same city to play at home 712 662 and Saint-Germain, the best two group winners on the same day, as this may overload the infras- 713 663 in the 2017–18 group stage of the UEFA Champions tructure. 714 664 League, having to face Juventus and Real Madrid in 665 the round of 16, etc. For subsequent rounds, the same choice procedure 715 666 The “Choose Your Opponent” knockout format would be applied: 716 667 precisely aims to avoid these situations by allowing • 717 668 the teams that have performed best during the group Frozen rankings:Inthefrozen ranking imple- 718 669 stage to choose their opponent during the final knock- mentation, the team rankings would not be 719 670 out stage among the pool of opponents that are still updated after each round of the knockout stage; teams would be ranked based on their group 720 671 available. Uncorrected Author Proof stage performance during the whole knockout 721

672 • Full choice: The full choice implementation of stage. 722 673 this format would work as follows: − 674 Team 1 in the global rankings chooses their 4 N Currently, Ukrainian and Russian teams are not allowed to 675 opponent among Teams {2,...,2 }. play against each other either. J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments 9

723 • Updated rankings:Intheupdated ranking • The opponents of other teams or a set of possible 768 724 implementation, the team rankings would be opponents for other teams. 769 725 updated after each round of the knockout stage. • Other features of its knockout bracket. For 770 726 For instance, teams qualified for the kth knock- example, the team ranked first might choose its 771 727 out round would be ranked from 1 (best ranked opponent not only in the Round of 16 but in 772 N−k+1 728 team) to 2 (lowest ranked team) based the quarterfinals, too, if it wins all of its group 773 729 on their results during the group stage and the matches. 774 730 previous knockout rounds, according to criteria 731 similar to those used to rank advancing teams Remark 4. (“Choose Your Opponent" in the draw 775 732 at the end of the group stage. For instance, in of the groups) It is also possible to use the “Choose 776 733 soccer, the criteria could be, in order: number Your Opponent" policy in the draw of the groups. For 777 734 of points; goal difference (per minute played); instance, assume a tournament with n groups and n 778 735 goals scored (per minute played); fair play (per seeded teams, ranked 1 to n according to past perfor- 779 736 minute played); etc. A win before extra-time mance. First, the unseeded teams are drawn into the 780 737 during a knockout round could be awarded with n groups; usually this is done using a system of pots 781 738 3 points, a win during extra-time with 2 points, with the objective of achieving a good group balance. 782 5 739 and a win after penalty shootout with 1 point. Then the n seeded teams would be allowed to pick 783 their group, in order. Team 1 would first choose one 784

n 785 740 Remark 3. (Alternatives for updated rankings) the of the groups, then Team 2 would choose one of n − 786 741 teams that advance to the postseaOne could also con- the 1 remaining groups, and so on. Here again, 787 742 sider an updated ranking version where the strengths restrictions can easily be accomodated. For example, 788 743 of the opponents that a team has eliminated in the past a seeded team may only be allowed to choose a group 789 744 knockout rounds are taken into account in its rank- in which all teams come from a different country. 790 745 ing. That could lead to interesting strategic debates: This procedure is a good way of rewarding the 791 746 should a team pick an easy opponent to maximize highest-ranked teams while ensuring group balance. 792 747 the chances that it advances to the next round, or It would actually ensure superior group balance com- 793 748 select a stronger opponent, which would improve its pared to drawing seeded teams into groups, as the 794 749 ranking in the following round if it advances? For higher-ranked seeded teams would probably choose 795 750 instance, should Team 1 pick the the lowest-ranked the weaker groups, while the lower-ranked seeded 796 751 advancing team, or select a higher-ranked team so teams would end up in stronger groups. 752 that it still has the first pick in the next round if it 753 advances? Taking the strength of the opponent into 3.2. Benefits and drawbacks 797 754 account would also reward more weaker teams for 755 eliminating stronger teams. For instance this could The main benefits of the “Choose Your Opponent” 798 756 be done using a dynamic Elo-like rating system. knockout format are the following: 799 757 Many variations of the “Choose Your Opponent” • Risk of tanking reduced to almost zero:In 800 758 policy can be considered. A team may be awarded by practice, teams would not be tempted to tank. It 801 759 the opportunity to choose not only its next opponent would always be better to score one more point, 802 760 but also, for instance: goal, etc., than not, because it can only give 803 you more options: more chances to pick your 804 761 • Whether they play the first leg of the next knock- opponent, and if you are already in a position 805 762 out round at home or away, when applicable. to pick one, a wider range of teams to choose 806 763 Several papers analyze whether playing the sec- from. For example, with such a system, the 2012 807 764 ond leg at home means a significant advantage Olympics badminton tournament scandal (see 808 765 or not (Amez et al., 2020; Eugster et al., 2011; Section 1) would have been avoided, and Eng- 809 766 Geenens and Cuddihy, 2018; Page and Page, land and Belgium would have exerted their best 810 767 2007). Uncorrected Author Proof effort to win their last group match at the 2018 811 FIFA World Cup. In particular, the “Choose 812 5For knockout rounds played over two legs, one at home and Your Opponent” format effectively solves the 813 one away, a team advancing thanks to the away goals rule may also be awarded less points than one advancing after scoring strictly problem of strategic manipulation in tourna- 814 more goals than its opponent over the two legs. ments identified by Vong (2017). 815 10 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments

816 The only (very unlikely but mathematically its opponent increase its ranking among group 868 817 possible) situations where a team might still winners and get a better picking position. 869 818 prefer to tank in the “Choose Your Oppo- • Added strategic component: The procedure of 870 819 nent” format is when, by tanking, a team thinks choosing opponents would bring a new, exciting 871 820 that it increases its probability of reaching its strategic component to the competition. It could 872 821 final objective, be it winning the tournament or give the tournament a whole new dynamics. The 873 822 reaching a particular round. For instance a team picking strategies of the teams would certainly 874 823 would prefer to tank if the objective it wants be much commented upon in the media and 875 824 to maximize is the probability of winning the among the fans, before and after the picking 876 825 tournament and if, by tanking, it still advances procedure. 877 826 to the knockout stage, but eliminates another • Exciting TV shows: The picking procedure 878 827 team that otherwise would have also advanced would lend itself to exciting, much anticipated 879 828 and (unrealistically) would have beaten every TV shows which would be scheduled right at the 880 829 other advancing team with probability one (and end of the group stage and knockout rounds. 881 830 then would have won the tournament with prob- The main drawbacks are the following: 882 831 ability one), and as a result gets a non-zero 832 probability of winning the tournament. More • Schedule: Adopting the “Choose Your Oppo- 883 833 generally it may prefer to tank if, by tanking, nent” design might lengthen tournaments: 884 834 the global rankings are modified in such a way – in which a minimum number of rest days, 885 835 that it increases its objective, say, the probability say r, is guaranteed to all teams between two 886 836 of reaching a particular round. Mathematically, games; 887 837 this can only happen in the case of extreme, – in which match days and knockout rounds 888 838 unrealistic win probability matrices. (Note that are spread over several days to maximize TV 889 839 by tanking a team likely loses choosing rights, exposure and the value of media rights; 890 840 which typically negatively impacts its objec- – that must be played over a short period of 891 841 tive.) We cannot imagine a realistic situation time. 892 842 where a team would benefit from tanking in the The FIFA World Cup is a typical example of 893 843 “Choose Your Opponent” format. such tournaments. Indeed, consider a tourna- 894 844 • Maximize win incentive during the group ment with the three above characteristics. At 895 845 stage: Teams would have a strong incentive to the end of the group stage or at the end of 896 846 win all group matches, even if they have already a knockout round, a team who played its last 897 847 secured qualification to the knockout stage, and game several days ago may pick a team who just 898 848 even if they have already secured the first place finished playing. This implies that more match- 899 849 in the group. Indeed, even if a team is guaran- free days may be needed between the end of the 900 850 teed to win its group before the last match day group stage and the first round of the knockout 901 851 of the group stage (e.g., Italy during the UEFA stage, and between consecutive rounds of the 902 852 Euro 2016), it still has a strong incentive to win knockout stage, and could lengthen the tour- 903 853 this last match, and with the largest score, to nament. Note, however, that adopting “Choose 904 854 improve its ranking within the n group winners Your Opponent” is not problematic for tourna- 905 855 and be one of the first teams to pick its opponent ments that leave a lot of time between the end of 906 856 during the knockout stage. the group stage and the first knockout round, and 907 857 • Decreased risk of collusion: For the same rea- between consecutive knockout rounds, like the 908 858 sons, teams would be less tempted to collude UEFA Champions League or the UEFA Europa 909 859 during the last match day of the group stage. League. 910 860 The great benefits of being one of the best group Leaving r match-free days between the end of 911 861 winners would deter teams from colluding and the group stage and the first round of the knock- 912 862 being satisfied with a draw or aUncorrected loss. Note, how- out stage, Author and between consecutiveProof rounds of 913 863 ever, that the risk of collusion would not be the knockout stage, is the obvious, simplest way 914 864 canceled. For instance, during the group stage, of ensuring that all teams benefit from r rest 915 865 a team that has nothing to win or lose (this could days between two matches. By proper schedul- 916 866 happen during the last match days) could agree ing, it might be enough to schedule less than 917 867 to lose by many goals/points, which would help r match-free days. The minimum number of 918 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments 11

919 rest days can also be decreased by imposing Note that the picking teams could also be 967 920 additional constraints during the “Choose Your allowed to choose the city/stadium where they 968 921 Opponent” procedure, but at the expense of fair- will play their next game. This is illustrated in 969 922 ness. For instance, a team having played its last (Guyon, 2019a) in the case of the UEFA Euro 970 923 group game early may only be allowed to pick 2020, where the rule could have been used to 971 924 an opponent having also played its last group maximize the number of home games during 972 925 game early. For example, in the case of the FIFA the knockout stage. 973 926 World Cup, group winners from the first half • First picked teams possibly feeling offended 974 927 of the groups—those playing early—may only or humiliated: The first teams chosen, as well 975 928 pick their opponents among runners-up from as their fans, could feel offended or even humili- 976 929 the same half. ated to be perceived as one of the weakest teams 977 930 Another scheduling drawback could be the remaining in the tournament. However, if a team 978 931 advantage enjoyed by the teams that play the last is one of the first picks, it probably means that 979 932 group matches. In contrast to the teams from the it really is one of the weakest remaining teams. 980 933 groups that play early, they know which result Moreover, this can actually give those teams a 981 934 will be sufficient to achieve a certain position very strong motivation to prove their pickers 982 935 in the global ranking. Since group labeling is wrong when they face them on the pitch. 983 936 arbitrary, this could be judged unfair—but can • No one wanting to take responsibility for 984 937 be handled to a certain degree by, for example, the pick: It might be that no individual in a 985 938 placing the titleholder automatically in a group team/club wants to take the responsibility of 986 939 that is finished the latest time. Theoretically, all picking the team’s opponent, fearing backlash 987 940 groups can be finished simultaneously, but this if the team is actually eliminated by the chosen 988 941 would decrease the value of media rights. opponent. However, in such a situation, a team 989 942 • Travel plans and logistics: All advancing can easily organize their own draw and decide 990 943 teams would have to wait until the end of a the rules for their draw. For instance, when 991 944 round to know when and where they would a team does not want to make any decision, 992 945 play the next game. For tournaments that must they can organize a uniformly distributed draw 993 946 be played over a short period of time in dis- among all available opponents. A team may also 994 6 947 tant cities, such as the FIFA World Cup, this collectively agree that they want to avoid some 995 948 can be seen as a drawback since teams and fans opponents, which they would exclude from the 996 949 would have less time for planning their trip to draw. If 2/3 of a team prefers Opponent A and 997 950 the next game—or they would have more possi- 1/3 prefers Opponent B, then the team can set 998 951 ble options to consider and plan for—than in the up a draw with a bowl containing three balls, 999 952 traditional system with predetermined bracket two balls “A” and one ball “B”, etc. 1000 953 routes. But it can also be seen as a benefit, as • Distribution of choice options: In the case of 1001 954 all teams would then be put on an equal foot. the full choice implementation, the distribution 1002 955 Moreover, it would not be a problem for tour- of choice options may possibly seem unfair. 1003 956 naments where consecutive rounds are played For example, if Team 1 picks Team 5 in the 1004 957 several weeks apart, such as the UEFA Cham- round of 16 of the UEFA Champions League, 1005 958 pions League or UEFA Europa League, or that and Team 6 has some choice, then Team 5 may 1006 959 are played in one city or neighboring cities, feel aggrieved. Indeed, people usually like to 1007 960 such as Olympic Games tournaments, the final choose even if it has no real effect. 1008 961 stages of basketball and handball national team 962 championships, the “Final Eight” of the UEFA 963 Champions League organized in , Por- Remark 5. (Importance of the seeding rules) Note 1009 964 tugal in August 2020, or the “Major League that since in this system group winners would be com- 1010 965 Soccer is back” tournament organizedUncorrected in July pared across Author groups and the best Proof group winners would 1011 966 2020 in Orlando, Florida. be awarded big benefits, it is crucial that the groups 1012 be well balanced. This can be achieved by a seeding 1013 procedure that fairly reflects the quality of the teams. 1014 6The 2022 FIFA World Cup in Qatar is an exception, as all A poor seeding may be the result of following a 1015 stadiums used for the tournament will be close to each other. poor ranking, see for instance (Guyon, 2014b; Lasek 1016 12 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments

1017 et al. 2013; McHale and Davies, 2007) about pre- 4. Application to the round of 16 of the UEFA 1068 1018 2018 FIFA rankings. A poor seeding may also be Champions League 1069 1019 the result of not following a good ranking; Sec- 1020 tion 4.4 deals with an issue with the seeding of the After describing the flaws of the current draw sys- 1070 1021 group stage of the UEFA Champions League. And a tem of the round of 16 of the UEFA Champions 1071 1022 poor seeding may be also the result of how (legit- League in Section 4.1, we explain how the “Choose 1072 1023 imate) draw constraints are handled. For instance, Your Opponent” policy would work in practice in 1073 1024 Guyon (2014a,2015a) explains that until 2014, due Section 4.2. In Section 4.3 we give a detailed exam- 1074 1025 to how FIFA enforced the geographic constraints that ple using the results of the group stage of the 2017-18 1075 1026 it puts on the draw, the group stage of the FIFA season of the Champions League; we consider both 1076 1027 World Cup was poorly seeded and produced unbal- the full choice and restricted choice implementations. 1077 1028 anced groups. One of his solutions (Suggestion #1 in Finally, Section 4.4 discusses the importance of cor- 1078 1029 (Guyon, 2014)) was adopted by FIFA for the 2018 rectly seeding the group stage. 1079 1030 World Cup. 1031 The fact that the “Choose Your Opponent” design 4.1. Flaws of the current draw system 1080 1032 requires the groups to be very well balanced can be 1033 seen both as a drawback and as a benefit. Indeed, it The “Choose Your Opponent” format could for 1081 1034 can be considered a drawback as tournament orga- instance be used in the UEFA Champions League 1082 1035 nizers sometimes fail to correctly seed the teams, as in order to better reward group winners and make 1083 1036 the above examples show. However, for many sym- the group stage more attractive. In the current for- 1084 1037 metric tournament designs, group balance should be mat, the group stage is made of 8 groups of 4 teams, 1085 1038 a priority of all tournament organizers, and tourna- with the best two teams in each group advancing 1086 1039 ment designs that make group balance very important to the round of 16. Every year in December, the 1087 1040 can be seen as a great way to ensure that the teams draw of the round of 16 is a much anticipated event, 1088 1041 are correctly seeded. Note, however, that asymmetric which attracts a lot of media attention. In order to 1089 1042 designs with voluntarily unbalanced groups do also reward group winners for their group stage perfor- 1090 1043 exist; they have been studied by Csato´ (2020a), who mance, they can only be drawn against runners-up. 1091 1044 has shown potential benefits of those formats. (Two other constraints apply: teams from the same 1092 group or from the same country cannot be drawn 1093 1045 Remark 6. In tournaments such as the NBA, NFL, against each other. This yields interesting draw prob- 1094 1046 MLB, and NHL, where the “ideal” bracket (shown in ability calculations, see (Guyon, 2017a, 2017b, 2018, 1095 1047 Figure 2 in the case where N = 16)—or some varia- 2019b, 2019c, 2020b, 2020c; Kloessner and Becker, 1096 1048 tion of it—has been used for a long time, the first 2013). The draw procedure has also been analyzed in 1097 1049 picking teams may be reluctant to choose a team (Boczon,´ 2018).) 1098 1050 that they were not supposed to meet in the tradi- However, it often happens that some of the 1099 1051 tional bracket, for that might be very embarrassing very best European clubs, considered to be con- 1100 1052 if they lose. In the example of Figure 2, Team 1 may sistently in the top 8 clubs over the past years, 1101 1053 not dare select a team ranked 15 or higher, as that actually finish second in their group. In the recent 1102 1054 bears the risk of people mocking their choice if they years, this was the case of Real Madrid (2016– 1103 1055 lose in the round of 16. However, if Team 1 chooses 17, 2017–18, 2019–20), Bayern Munich (2016–17, 1104 1056 an opponent that is not the lowest-ranking team, it 2017–18), Juventus Turin (2015–16, 2017–18), 1105 1057 is probably because they have determined that they Atletico´ Madrid (2018–19, 2019–20), Manchester 1106 1058 have more chances of beating them than beating the City (2016–17), and Paris Saint-Germain (2015– 1107 1059 lowest-ranking team, and this might indeed be the 16, 2016–17). When group winners are drawn 1108 1060 case. Other factors, such as travel, might also weigh against one of those teams, they can feel bitter 1109 1061 in the choice. Note that this remark does not apply to about the way they are rewarded for winning their 1110 1062 tournaments that do not currently follow the “ideal” group. 1111 1063 bracket, such as the FIFA World CupUncorrected or the UEFA Author Proof The flawed seeding system used by UEFA for the 1112 1064 Champions League. group stage since 2015 made those cases more likely 1113 1065 In the next section, we illustrate the “Choose Your (see Section 4.4). In the 5 years preceding the change, 1114 1066 Opponent” format by showing how it would work for runners-up advanced to the quarterfinals 7 times out 1115 1067 the round of 16 of the UEFA Champions League. of 40 (17.5%), whereas in the 5 years following the 1116 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments 13

Table 1 Global rankings 1–16 at the end of the group stage of the 2017–18 UEFA Champions League. GD stands for “Goal Difference” and GF for “Goals For”. UEFA uses head-to-head results as the first tie-breakers Group winners Runners-up Rk Team Gp Pts GD GF Rk Team Gp Pts GD GF 1 Tottenham (ENG) H 16 +11 15 9 Bay. Munich (GER) B 15 +7 13 2 Paris S.-G. (FRA) B 15 +21 25 10 Real Madrid (SPA) H 13 +10 17 3 Manchester C. (ENG) F 15 +9 14 11 (SWI) A 12 +6 11 4 Manchester U. (ENG) A 15 +9 12 12 Sh. (UKR) F 12 0 9 5 Barcelona (SPA) D 14 +8 9 13 Chelsea (ENG) C 11 +8 16 6 Bes¸iktas¸ (TUR) G 14 +6 11 14 Juventus (ITA) D 11 +2 7 7 Liverpool (ENG) E 12 +17 23 15 (POR) G 10 +5 15 8 Roma (ITA) C 11 +3 9 16 Sevilla (SPA) E 9 0 12

1117 change the proportion more than doubled, reaching Table 2 1118 15 times out of 40 (37.5%). Round of 16 draw of the 2017–18 UEFA Champions League Manchester United – Sevilla Liverpool – Porto

1119 4.2. The “Choose Your Opponent” format for Bes¸iktas¸ – Bayern Munich Manchester City – Basel Tottenham – Juventus Barcelona – Chelsea 1120 the round of 16 of the UEFA Champions League Paris Saint-Germain – Real Madrid Roma – Shakhtar Donetsk

1121 The “Choose Your Opponent” format would solve 1122 this problem, and could easily be adapted to take the rule and preferring global goal difference within the 1154 1123 country and group constraints into account: UEFA group. Note that goal difference is more attractive 1155 1124 would allow the group winners to choose their oppo- from a theoretical point of view (Berker, 2014). 1156 1125 nent, but group winners could not pick an opponent 1126 from the same country or the same group. The best 4.3. An example 1157 1127 group winner (based on the global ranking 1–16) 1128 would first pick their opponent, then the second best Let us take the 2017–18 season of the UEFA Cham- 1158 1129 group winner would pick theirs, etc. It would make pions League as an example. The results of the group 1159 1130 sense to protect the group winners by enforcing the stage are reported in Table 1, and the outcome of the 1160 1131 restricted choice implementation, where group win- draw of the round of 16 in Table 2. Tottenham and 1161 1132 ners cannot be picked by other group winners. By Paris Saint-Germain were the two best group winners 1162 1133 contrast, the full choice implementation would allow that year, but in the round of 16 they were respec- 1163 1134 group winners to choose other group winners as their tively drawn against Juventus and Real Madrid, two 1164 1135 opponents. clubs which, despite being 1st and 5th in the UEFA 1165 1136 Exactly as is done today for the draw of the round club coefficient ranking at the time, finished second 1166 1137 of 16 of the UEFA Champions League, to make sure in their group. Despite collecting 13 points in Group 1167 1138 that a team’s choice is acceptable, i.e., does not lead H, Real Madrid finished behind Tottenham which 1168 1139 to a dead end, a computer program would provide performed exceptionally well that year (16 points); 1169 1140 the list of admissible opponents ahead of each team Juventus finished second in Group D behind another 1170 1141 pick. A team’s choice leads to a dead end if it makes soccer powerhouse, Barcelona (3rd in the UEFA club 1171 1142 it impossible that the above rules (no match between coefficient ranking at the time). Both Tottenham and 1172 1143 teams from the same country or from the same group) Paris Saint-Germain must have felt bitter and poorly 1173 1144 be met for all the remaining games. The backtracking and unfairly rewarded for their excellent group stage 1174 1145 algorithm is a recursive procedure that does just performance. 1175 1146 that; see (Guyon, 2014a) for an example. 1147 Note that UEFA uses head-to-head results as the 4.3.1. Restricted choice implementation 1176 1148 first tie-breaker. This is impossibleUncorrected to apply across Let us look Author at how the “Choose Proof Your Opponent” 1177 1149 groups. Consequently, it might happen that a runner- design would have possibly worked. We first consider 1178 1150 up has better performance measures in the global the restricted choice version. Here, we assume that 1179 1151 ranking compared to the winner of its group, which teams pick their opponent based on the prevailing Elo 1180 1152 would be quite strange. That is a further argument ratings at the time, see Table 3. Elo ratings, introduced 1181 1153 for not using head-to-head results as a tie-breaking for chess by Arpad Elo in the 1960s, are a reliable 1182 14 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments

Table 3 Elo ratings of the 16 teams participating in the round of 16 of the 2017–18 UEFA Champions League as of December 11, 2017, day of the draw Group winners Runners-up Elo Rk Team Gp Elo rating Elo Rk Team Gp Elo rating 1 Barcelona (SPA) D 2025 2 Real Madrid (SPA) H 2024 3 Manchester C. (ENG) F 1968 4 Bay. Munich (GER) B 1947 7 Manchester U. (ENG) A 1910 5 Juventus (ITA) D 1945 8 Tottenham (ENG) H 1909 6 Chelsea (ENG) C 1910 9 Paris S.-G. (FRA) B 1895 12 Sevilla (SPA) E 1789 10 Liverpool (ENG) E 1875 13 Porto (POR) G 1769 11 Roma (ITA) C 1865 14 Sh. Donetsk (UKR) F 1754 15 Bes¸iktas¸ (TUR) G 1733 16 Basel (SWI) A 1710

1183 measure of team skills. For European soccer clubs, the same group as Barcelona. For instance, 1220 1184 they are maintained by the website clubelo.com. Barcelona chooses Chelsea. 1221 • Then Bes¸iktas¸ (Team 6) can choose among: 1222 Bayern Munich, Real Madrid, and Juventus. For 1223 1185 • First, Tottenham (Team 1) would choose their instance, they pick Juventus. 1224 1186 round of 16 opponent among runners-up. The • Next, Team 7 (Liverpool) must choose between 1225 1187 available teams for Tottenham are: Bayern Bayern Munich and Real Madrid. For instance, 1226 1188 Munich, Basel, Shakhtar Donetsk, Juventus, they pick Bayern Munich. 1227 1189 Porto, and Sevilla. Tottenham cannot choose • Finally, the last round of 16 matchup is Roma– 1228 1190 Real Madrid, as both teams were in the same Real Madrid. 1229 1191 group. They cannot play against Chelsea either, 1192 due to the country constraint. For instance, Tot- The resulting round of 16 table is shown in Table 1230 1193 tenham picks Basel (the runner-up with the 4. In this example, the backtracking algorithm does 1231 1194 lowest Elo rating at the time, see Table 3). not detect any future dead end. However, if Bes¸iktas¸ 1232 1195 • Then it is Team 2’s turn (Paris Saint-Germain) (Team 6) chose Real Madrid instead of Juventus, then 1233 1196 to choose their opponent among the remaining Liverpool (Team 7) would have had no choice: they 1234 1197 runners-up. The available teams for Paris Saint- would have had to play against Juventus. Indeed, the 1235 1198 Germain are: Real Madrid, Shakhtar Donetsk, backtracking algorithm would not allow Liverpool to 1236 1199 Chelsea, Juventus, Porto, and Sevilla. Paris choose Bayern Munich, as in this case the last match- 1237 1200 Saint-Germain cannot choose Bayern Munich, up would be Roma–Juventus, which is prohibited by 1238 1201 as both teams were in the same group. For the country constraint. 1239 1202 instance, Paris Saint-Germain picks Shakhtar 1203 Donetsk. 1204 • The following team to choose their opponent is 4.3.2. Full choice implementation 1240 1205 Manchester City. The available opponents are: As for the full choice implementation, it may have 1241 1206 Bayern Munich, Real Madrid, Juventus, Porto, worked this way: 1242 1207 and Sevilla. Manchester City cannot choose • 1208 Chelsea, another English team. For instance, First, Tottenham (Team 1) would choose their 1243 1209 they pick Porto. round of 16 opponent among all other quali- 1244 1210 • Then it is Team 4’s turn (Manchester United). fied teams (group winners or runners-up). The 1245 1211 The available teams are: Bayern Munich, Real available teams for Tottenham are: Paris Saint- 1246 1212 Madrid, Juventus, and Sevilla. Manchester Germain, Barcelona, Bes¸iktas¸, Roma, Bayern 1247 1213 United cannot choose Chelsea either, because Munich, Basel, Shakhtar Donetsk, Juventus, 1248 Porto, and Sevilla. Tottenham cannot choose 1249 1214 of the country constraint. For instance,Uncorrected they pick Author Proof 1215 Sevilla. Real Madrid, as both teams were in the same 1250 1216 • Then it is Barcelona’s turn (Team 5). They can group. They cannot play against the 4 other 1251 1217 only pick Bayern Munich or Chelsea. They qualified English teams either (Manchester 1252 1218 cannot play against Real Madrid, another Span- City, Manchester United, Liverpool, Chelsea). 1253 1219 ish team, nor against Juventus, which was in For instance, Tottenham picks Basel. 1254 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments 15

Table 4 Possible round of 16 of the 2017–18 UEFA Champions League under the “Choose Your Opponent” design with restricted choice Tottenham – Basel Barcelona – Chelsea Paris Saint-Germain – Shakhtar Donetsk Bes¸iktas¸ – Juventus Manchester City – Porto Liverpool – Bayern Munich Manchester United – Sevilla Roma – Real Madrid

1255 • Then it is Team 2’s turn (Paris Saint-Germain) against Chelsea. For instance, they pick Sevilla. 1290 1256 to choose their opponent among the remaining • Roma (Team 8) has already been chosen, there- 1291 1257 teams. The available teams for Paris Saint- fore it is Team 9’s turn (Bayern Munich). They 1292 1258 Germain are: Manchester City, Manchester can play against Real Madrid, Chelsea, and 1293 1259 United, Barcelona, Bes¸iktas¸, Liverpool, Roma, Juventus. For instance, they pick Chelsea. 1294 1260 Real Madrid, Shakhtar Donetsk, Chelsea, • Finally, the last round of 16 matchup is Real 1295 1261 Juventus, Porto, and Sevilla. Paris Saint- Madrid–Juventus. 1296 1262 Germain cannot choose Bayern Munich, as both The corresponding round of 16 table is shown in Table 1297 1263 teams were in the same group. For instance, 7 5. Note that in this case runners-up could face each 1298 1264 Paris Saint-Germain picks Shakhtar Donetsk. other in the round of 16. 1299 1265 • The following team to choose their opponent is 1266 Manchester City. The available opponents are: 1267 Barcelona, Bes¸iktas¸, Roma, Bayern Munich, 4.3.3. Comparison 1300 1268 Real Madrid, Juventus, Porto, and Sevilla. In the full choice version, powerful clubs that failed 1301 1269 Manchester City cannot choose the other to win their group would probably not be picked by 1302 1270 English teams. For instance, they pick Porto. group winners, and as a result they would have to face 1303 1271 • Then it is Team 4’s turn (Manchester United). each other in the round of 16 (in our example, Bayern 1304 1272 The available teams are: Barcelona, Bes¸iktas¸, Munich–Chelsea and Real Madrid–Juventus). There- 1305 1273 Roma, Bayern Munich, Real Madrid, Juven- fore in the full choice version it would be extremely 1306 1274 tus, and Sevilla. Manchester United cannot important for soccer powerhouses to win their group; 1307 1275 choose the other English teams. For instance, otherwise they would likely eliminate each other in 1308 1276 they pick Bes¸iktas¸, a group winner, whereas in the round of 16, and also possibly in the following 1309 1277 the restricted choice version they had picked rounds. This is because the full choice version gives 1310 1278 Sevilla. more options to the group winners, who are given 1311 1279 • Then it is Barcelona’s turn (Team 5). They the possibility to avoid a strong runner-up and prefer 1312 1280 can pick among: Roma, Bayern Munich and picking a group winner that they believe they have 1313 1281 Chelsea. They cannot play against the other more chances to eliminate. This is for instance the 1314 1282 Spanish teams (Real Madrid and Sevilla), nor choice made by Manchester United and Barcelona 1315 1283 against Juventus, who was in the same group in our example, who avoid Sevilla and Chelsea and 1316 1284 as Barcelona. For instance, Barcelona chooses prefer Bes¸iktas¸ and Roma, respectively. 1317 1285 Roma. Therefore, by giving more choice options to the 1318 1286 • Since Bes¸iktas¸ (Team 6) has already been group winners, the full choice implementation max- 1319 1287 picked, it is directly Team 7’s turn (Liverpool). imizes win incentive in the group stage, and might 1320 1288 They can only choose between Bayern Munich, seem preferable in this regard. 1321 1289 Real Madrid, and Sevilla, since they cannot play

4.4. Importance of the seeding rules 1322 7Bes¸iktas¸ had a lower Elo rating than Shakhtar Donetsk at the time, but Paris Saint-Germain would maybe have considered The above example highlights the importance of 1323 that it is more difficult to get a good result when they play the first leg in Turkey, due to the famous “hot” ambianceUncorrected in Turkish correctly seeding Author the group stage Proof and illustrates one 1324 stadiums, rather than in . Paris Saint-Germain may also of the important flaws of the current seeding system 1325 have considered that if they chose Bes¸iktas¸, many Bes¸iktas¸ fans of the UEFA Champions League. UEFA modified the 1326 would have attended the second leg in Paris, as there are many seeding system of the group stage in 2015–16. Before 1327 football fans of Turkish descent living in the Paris area. For similar reasons, Manchester City (Team 3) may have also preferred to the 2015–16 season, the 32 teams participating in the 1328 choose Porto. group stage would be placed in 4 pots of 8 teams, with 1329 16 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments

Table 5 Possible round of 16 of the 2017–18 UEFA Champions League under the “Choose Your Opponent” design with full choice Tottenham – Basel Barcelona – Roma Paris Saint-Germain – Shakhtar Donetsk Liverpool – Sevilla Manchester City – Porto Bayern Munich – Chelsea Manchester United – Bes¸iktas¸ Real Madrid – Juventus

1330 Pot 1 being made of the 8 best teams (based on the despite being the best group winner with 16 points, 1360 1331 UEFA club coefficient, which measures performance had to face a strong Juventus in the round of 16—and 1361 1332 in European club competitions in the last 5 years), were eliminated. 1362 1333 Pot 2 the following 8, etc. Since each group of 4 is As noted in Remark 3.2, it would also be very 1363 1334 made of one team drawn from each pot, this procedure important that groups be well balanced in the “Choose 1364 1335 guaranteed that groups were well balanced, and two Your Opponent” design. For instance, under the cur- 1365 1336 top 8 teams could not face each other in the group rent seeding rules of the UEFA Champions League, 1366 1337 stage. Since the 2015–16 season, in order to reward there is usually at least one Pot 1 team (the champions 1367 1338 domestic champions, Pot 1 is made of the domestic of the domestic leagues ranked 6th or beyond) with 1368 1339 champions of the 8 highest ranked domestic leagues a UEFA club coefficient or Elo rating much lower 1369 1340 (even if they have a low UEFA coefficient, i.e., a poor than those of the other Pot 1 teams (both numbers 1370 1341 performance in European club competitions in the measure team skills based on performances in the 1371 1342 last 5 years), while Pots 2 to 4 are still built based on past few years). For instance, in 2017–18, Spartak 1372 8 1343 decreasing UEFA club coefficient. This means that () was seeded in Pot 1, as the champi- 1373 1344 two top 8 teams (based on the UEFA coefficient) can ons of the 6th highest ranked domestic league, despite 1374 1345 now face each other in the group stage, and in this the fact that their UEFA club coefficient was only 1375 1346 case it is likely that one top 8 club will finish second 18.6, much lower than those of the other Pot 1 teams 1376 1347 in their group. The group winner facing that team in (the average of the 7 other club coefficients in Pot 1377 1348 the round of 16 (under the current draw system) will 1 was 120), and as a matter of fact the third lowest 1378 1349 feel unfairly rewarded for winning its group. UEFA club coefficient of the 32 clubs that qualified 1379 1350 For example, in 2017–18, despite being ranked for the group stage. No Russian team had a club coef- 1380 1351 3rd and 5th in the UEFA club coefficient ranking, ficient larger than 87.1 at the time. As a result, under 1381 1352 Barcelona (Pot 2) and Juventus (Pot 1) were placed in “Choose Your Opponent”, the winner of the group of 1382 1353 the same group. As a consequence it was very likely Spartak Moscow would have been awarded an undue 1383 1354 that during the draw of the round of 16 one of the advantage: since it faced a weak Pot 1 opponent due 1384 1355 runners-up would be from the top 5 of the UEFA club to the poor seeding, it could have more easily been 1385 1356 coefficient ranking, either Barcelona or Juventus, and ranked as one of the best group winners and thus 1386 1357 one group winner would have to face a very strong been one of the first teams to choose their round 1387 1358 opponent in the round of 16. Juventus finished sec- of 16 opponent. This again underlines the impor- 1388 1359 ond in the group behind Barcelona, and Tottenham, tance of adopting seeding rules that fairly reflect team 1389 abilities. 1390 8To be precise, in both seeding systems, the Champions League Guyon (2015b) suggested a better way of seed- 1391 title holder is seeded in Pot 1, as an exception. Since the 2018– ing the UEFA Champions League group stage that 1392 19 season, the Europa League title holder is also seeded in Pot would reward the whose teams 1393 1. For instance, when none of the two title holders have won performed best in Europe in the past years, and for 1394 one of the 6 highest ranked domestic leagues, only the domes- tic champions of the 6 highest ranked domestic leagues are seeded each country would also reward the teams that per- 1395 in Pot 1, along with the two title holders. A good summary of the formed best in their domestic league the previous 1396 UEFA Champions League seeding policy is (Csato,´ 2020d), which year. Corona et al. (2019) have simulated the UEFA 1397 also identifies an issue of incentive incompatibility in the current Champions League under the pre-2015 and post-2015 1398 rule. The Champions League titleholder wasUncorrected automatically allo- Author Proof cated into Pot 1, independently of its UEFA club coeffcient even seeding regimes. Dagaev and Rudyak (2019) have 1399 before the 2015–16 season (UEFA website, Regulations of the also obtained numerical estimates for competitive- 1400 UEFA Champions League 2012–15 Cycle, 2014/15 Season). This ness changes in the UEFA tournaments caused by 1401 advantage enjoyed by the title holder dates back to at least the the 2015–16 seeding reform. Note that Engist et al. 1402 1997–98 season (Wikipedia, 1997–98 UEFA Champions League group stage). (2021) find no evidence that seeding itself contributes 1403 J. Guyon / “Choose your opponent”: A new knockout design for hybrid tournaments 17

1404 positively to a team’s success in the UEFA Champi- the world; the “Choose YourOpponent” easily adapts 1452 1405 ons League. to the constraints that UEFA currently puts on the 1453 draw. In (Guyon, 2019a), we have also studied how 1454 1406 Remark 7. The implementation of the “Choose Your the “Choose YourOpponent” format could have natu- 1455 1407 Opponent" policy in the UEFA Europa League Round rally been adapted in order to maximize the number of 1456 1408 of 32 is nontrivial because 24 teams come from home games during the knockout round of the UEFA 1457 1409 the Europa League and 8 come from the Cham- Euro 2020, which for the first time will be hosted by 1458 1410 pions League. Currently, the twelve group winners 12 countries. 1459 1411 and the four third-placed teams from the Champi- This new knockout format could create logistics 1460 1412 ons League group stage with the better group records issues for tournaments that are played in distant cities 1461 1413 are seeded. To decide the Round of 32, a draw over a short period of time, such as the FIFA World 1462 1414 is organized that matches up seeded teams with Cup, but it looks particularly well suited to competi- 1463 1415 unseeded teams, with the restriction that teams from tions where knockout draws/picks can be organized 1464 1416 the same group or the same association cannot be well ahead of the knockout matches, like the UEFA 1465 1417 drawn against each other. In one simple implemen- Champions League or the UEFA Europa League; or 1466 1418 tation, the 12 Europa League group winners would to competitions that are organized in one city or in 1467 1419 be ranked 1–12 based on their group stage perfor- close cities. We encourage sports governing bodies 1468 1420 mance, and the 4 best third-placed teams from the and leagues around the world to consider it as a way 1469 1421 Champions League would be ranked 13–16; and sim- to build fairer, more exciting tournaments. 1470 1422 ilarly for ranks 17–32. Alternatively, teams coming 1423 from the Champions League could get ranks 25–32, 1424 or ranks 1–8, or ranks 1, 5, 9, 13, 17, 21, 25, 29, or Acknowledgments 1471 1425 ranks 4, 8, 12, 16, 20, 24, 28, 32, etc.

We are very grateful to Laszl´ o´ Csato,´ Dries 1472 Goossens, and Frits Spieksma for their interesting 1473 1426 5. Conclusion comments on the preprint (Guyon, 2019a), and to two 1474 anonymous referees for their very helpful comments 1475 1427 In this article, we have introduced and investigated on a first version of this article. We would also like 1476 1428 a new, exciting, and simple design for hybrid tour- to thank Nicholas Hall for pointing out to us (very 1477 1429 naments, in which the teams that have performed unrealistic but mathematically possible) situations in 1478 1430 best during a preliminary group stage can choose which teams may still be tempted to tank under the 1479 1431 their opponent in the subsequent knockout rounds. “Choose Your Opponent” knockout design. 1480 1432 The main benefit of this format is that it significantly 1433 increases win incentive during the group stage, by 1434 rewarding the best group winners with first picks, References 1481 1435 thus making the group stage more exciting. It also

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Uncorrected Author Proof