How to measure the uncertainty of a tournament draw: The case of European football's Champions League
László Csató, András Gyimesi, Dries Goossens, Karel Devriesere, Roel Lambers, Frits Spieksma
TL;DR
This study tackles draw uncertainty in tournament draws by introducing a simulation-based measure of the variance in teams' qualifying probabilities across draws, using $1000$ draws and $1000$ outcome scenarios per draw to compare the old UEFA Champions League group-stage format with the new incomplete round-robin league phase. It combines integer-program-based draw generation, a Poisson goals model calibrated by Elo-based win expectancy, and a three-component decomposition of the reform's effects into inaccurate seeding, the knockout play-offs, and first-stage structure changes, formalized via $\Delta V_i$ with components $\Delta V1_i$, $\Delta V2_i$, and $\Delta V3_i$. The main finding is that the incomplete round-robin design substantially reduces draw uncertainty, especially when seeding is misaligned with actual strength, with the seeding effect driving most of the improvement and the play-offs contributing a robust additional reduction; when seeding is perfectly aligned, the advantage diminishes. The work provides actionable insights for tournament design and seeding policies, highlighting the importance of robustness to seeding accuracy in achieving fairer draws across sports.
Abstract
According to recent empirical studies, the group draw of major sports tournaments can imply a high level of uncertainty, and some lucky teams enjoy an unfair advantage over the other teams. We propose a novel technique to quantify this draw uncertainty, which, arguably, has an optimal level of zero. Our simulation-based approach requires generating a high number of random draws to compute the variance of qualifying probabilities for each team. The method is applied to compare draw uncertainty in the former group stage and the current incomplete round-robin league phase of the UEFA Champions League. We also break down the impact of the 2024/25 reform into various components. The new format is found to decrease draw uncertainty; the reduction can mainly be attributed to the inaccurate seeding system used by UEFA. Our results reveal that the primary benefit of an incomplete round-robin tournament compared to the standard group stage lies in the robustness of its draw uncertainty to the seeding of the teams, which is a crucial aspect of fairness.