Match me if you can - Mathematische Gedanken zur Champions-League-Achtelfinalauslosung

The UEFA Champions League seeding is not strategy-proof since the 2015/16 season

“…For the sake of credibility and transparency, the draw is implemented by a specific mechanism. Although the unfairness of the draw procedure has been uncovered ten years ago (Kiesl, 2013;Klößner and Becker, 2013), it is near-optimal according to a recent paper published in a leading journal of management science (Boczoń and Wilson, 2022). This result has been refined by showing that reversing the default draw order can significantly reduce the level of unfairness.…”

“…The (standard) UEFA mechanism is distinct from a uniform draw over all admissible matchings (Boczoń and Wilson, 2022;Guyon, 2014b;Kiesl, 2013;Klößner and Becker, 2013). Even though the distortions in pairwise match probabilities seem to be small, it is relatively frequent that a certain club i plays against club j with a higher probability than against club k according to an evenly distributed draw, but a match between i and k has a higher chance to occur than a match between i and j according to the UEFA mechanism (Klößner and Becker, 2013).…”

“…This procedure always leads to a valid matching if there is an assignment satisfying all constraints. The existence can be proved by Hall's marriage theorem analogously to Kiesl (2013), but now the maximal number of teams in the Round of 16 is five for at most two countries. The result appears in Haigh (2019, Chapter 3.6) as an application of mathematics in everyday life.…”

“…Naturally, there is no "free lunch" and the randomisation mechanism chosen by UEFA is unfair, i.e., the feasible outcomes are not equally likely (Kiesl, 2013;Klößner and Becker, 2013). Nonetheless, utilising the main result of Budish et al (2013), Boczoń and Wilson (2022) have recently demonstrated that the design of the UEFA Champions League Round of 16 draw is near-optimal and close to a constrained-best solution as it resembles the fairest possible lottery over the constrained assignments.…”

The effects of draw restrictions on knockout tournaments

“…On the other hand, draw restrictions in knockout tournaments have received far less attention. According to Kiesl (2013) and Klößner and Becker (2013), the mechanism used in the UEFA Champions League Round of 16 draw implies that every result of the draw could not have the same probability. However, the UEFA draw procedure remains close to a constrained-best in terms of fairness (Boczoń and Wilson, 2022).…”