“Choose Your Opponent”: a New Knockout Design for Hybrid Tournaments
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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 tournaments 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, tournament 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 ice hockey 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 ice hockey league 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 Vienna Capitals 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).