How to measure the uncertainty of a tournament draw: The case of European football's Champions League

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.