A probabilistic match classification model for sports tournaments
László Csató, András Gyimesi
TL;DR
The paper addresses how to classify two-player matches when incentives to attack or defend are not strictly binary. It introduces a probabilistic, simulation-based framework that computes $P_{m-0}$, $P_{0-0}$, and $P_{0-m}$ for each match and assigns six incentive-based categories. Applied to both the pre-2024 UEFA group-stage and the post-2024 incomplete round-robin formats, it finds a reduction in stakeless matches but a rise in defensive and offensive matches, with elevated collusion risk in some scenarios. The framework offers practical insights for scheduling, incentives, and stakeholder decision-making in football and can extend to other two-player contests with draws and offensive/defensive dynamics.
Abstract
Existing match classification models in the tournament design literature have two major limitations: a contestant is considered indifferent only if uncertain future results do never affect its prize, and competitive matches are not distinguished with respect to the incentives of the contestants. We propose a probabilistic framework to address both issues. For each match, our approach relies on simulating all other matches played simultaneously or later to compute the qualifying probabilities under the three main outcomes (win, draw, loss), which allows the classification of each match into six different categories. The suggested model is applied to the previous group stage and the new incomplete round-robin league, introduced in the 2024/25 season of UEFA club competitions. An incomplete round-robin tournament is found to contain fewer stakeless matches where both contestants are indifferent, and substantially more matches where both contestants should play offensively. However, the robustly higher proportion of potentially collusive matches can threaten with serious scandals.