A probabilistic match classification model for sports tournaments

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.

💡 Research Summary

The paper addresses two major shortcomings of existing match‑classification schemes in tournament design: (i) the binary treatment of “indifference” – a team is deemed indifferent only when its prize is completely independent of any possible match outcome, and (ii) the lack of differentiation among competitive matches with respect to the direction of incentives (whether a team prefers to win or merely to avoid losing). To overcome these issues, the authors propose a probabilistic framework that evaluates each match by simulating all other matches that are played simultaneously or later. For a given match three possible results (home win, draw, away win) are considered; for each result the model runs a large number of Monte‑Carlo simulations (typically 10 000) of the remaining schedule, using a calibrated scoring‑probability model (a variant of the Dixon‑Coles approach) and team strength distributions derived from recent UEFA data. From these simulations the “qualifying probability” of each team – the chance of reaching a relevant prize (e.g., advancing to the knockout stage, qualifying for the Champions League, avoiding relegation) – is estimated under each of the three outcomes.

The key methodological step is to compare, for each team, the incremental change in qualifying probability when moving from a draw to a win versus from a draw to a loss. If the gain from a win exceeds the loss from a defeat, the team has a “win‑seeking” incentive; if the opposite holds, the team has a “loss‑avoidance” incentive; if the two changes are negligible, the team is effectively indifferent. Combining the incentive status of the two participants yields six distinct match categories:

1. Stakeless – both teams indifferent.
2. Defensive asymmetric – one team indifferent, the other loss‑avoidance dominant.
3. Offensive asymmetric – one team indifferent, the other win‑seeking dominant.
4. Defensive (both loss‑avoidance) – both teams prefer to avoid losing.
5. Offensive (both win‑seeking) – both teams prefer to win.
6. Collusive (both share a mutually optimal result) – a specific outcome (often a draw) guarantees the best possible prize for both.

The framework is applied to two tournament formats used by UEFA in the 2024/25 season: the traditional four‑team double‑round‑robin group stage and the newly introduced “incomplete round‑robin” league (36 teams, 6 or 8 rounds). Simulations reveal that the incomplete round‑robin reduces the share of stakeless matches (from roughly 12 % to under 7 %) but markedly increases the proportion of offensive matches (from about 22 % to 35 %). Defensive matches also rise modestly, while asymmetric matches decline when fewer rounds are played. Most importantly, the share of collusive (potentially collusive) matches rises substantially compared with the old format, indicating a heightened risk of match‑fixing or tacit agreements.

These findings have clear policy implications. On the one hand, more offensive matches should enhance spectator appeal and align with UEFA’s claim that the new format “produces more competitive matches”. On the other hand, the increase in collusive opportunities suggests that additional safeguards (e.g., simultaneous final‑round fixtures, more complex tie‑breaking criteria, or financial penalties) may be necessary to preserve integrity. The authors also discuss methodological limitations: reliance on historical strength estimates, the binary classification of incentive strength (thresholds are somewhat arbitrary), and the exclusion of extra‑time or penalty‑shootout outcomes. They propose extensions such as Bayesian updating of team strengths during the tournament, incorporation of multi‑objective incentives (financial rewards, fan engagement metrics), and application to other low‑scoring sports like field hockey or ice hockey.

In conclusion, the paper delivers a robust, simulation‑based classification system that captures nuanced incentive structures in two‑team contests. By quantifying how each possible result reshapes qualifying probabilities, it enables tournament designers to evaluate and fine‑tune formats with respect to competitiveness, offensiveness, and vulnerability to collusion. The approach is general enough to be adapted to any sport where draws are possible and where the choice between offensive and defensive play materially affects both winning and losing probabilities.