Stop the Simulations!
A New Approach for Faster and More Accurate Tournament Outcome Predictions.
The following summary critically reviews the research conducted by Ulrik Brandes, Gordana Marmulla and Ivana Smokovic, titled "Stop Simulating! Efficient Computation of Tournament Winning Probabilities". All data, figures, and analysis presented here are drawn from their original work; I do not claim any authorship or ownership of the content. This summary has been written to provide a concise and technically informed synthesis of the paper’s findings, methodologies, and implications, while maintaining fidelity to the authors’ intellectual contributions.
1 Introduction
Predicting the winner of sports tournaments, particularly in football, is both a widespread hobby and a serious analytical task influencing bookmakers and bettors (Winston et al., 2022). Traditionally, these predictions rely on simulations based on single-match outcome models, often repeated tens of thousands of times to produce stable results. While many forecasts exist (ranging from academic institutions to public-facing platforms like FiveThirtyEight) the common bottleneck is computational cost.
This research paper challenges the efficiency of simulation-based methods by introducing an algorithm that computes exact tournament winning probabilities derived from any given single-match prediction model. It circumvents full simulation by enumerating group-stage outcomes and propagating rank probabilities through the elimination rounds. Unlike prior hybrid approaches (e.g., Koning et al., 2003), this method explicitly models intra-group dependencies and exploits conditional independencies to accelerate computation.
By leveraging the limited mixing of team paths in fixed-bracket tournaments such as the FIFA World Cups, the algorithm achieves substantial efficiency gains. It also allows precise empirical assessment of seeding rules and can be extended to other deterministic bracket formats. This framework, though designed for FIFA-style tournaments, is broadly applicable and provides a powerful alternative to simulation-heavy methodologies.
2 Preliminaries
This section outlines the structure and assumptions of the tournament model under study, specifically aligning with the FIFA Men’s World Cup 2022 and FIFA Women’s World Cup 2023 formats, both featuring 32 teams. The competition is split into two phases: a group stage comprising 8 groups with 4 teams each (randomly assigned via seeding pots), and a knockout stage with 16 teams progressing through four elimination rounds. Teams are assumed to be indexed in such a way that group memberships are computationally tractable via integer division.
2.1 Tournament Schedule and Notation
Let N={0,...,n−1} represent the teams. Each group Ng contains four teams indexed sequentially. The top two teams from each group—determined by round-robin matches—advance to the knockout stage, which follows a fixed bracket. Differences in tournament structure, such as trajectory mixing (e.g., possible rematches in finals in the men's but not women's tournament), influence the dependency structure of match outcomes.
2.2 Assumptions
The match outcome model requires three assumptions:
Pre-tournament predictions: Probabilities are computed before the tournament begins, with no updates based on real-time results.
Independence: Match outcomes are statistically independent, relying solely on the competing teams.
Outcome granularity: Only win, draw, or loss outcomes are distinguished; draws are resolved via coin flips in the knockout stage, simplifying the treatment of tie-breakers.
As stated, these assumptions are "instructive and simplify the exposition," but could be relaxed in more elaborate models.
3 Single-Match Outcome Probabilities
The match-level model assumes an upper-triangular matrix M_ij∈{0,1,2} where i<j and values indicate a loss, draw, or win for team i, with M_ji=2−M_ij to preserve symmetry. For knockout matches, draws are treated as fair coin tosses, resulting in the adjusted win probability:
This transformation ensures binary advancement outcomes in elimination rounds without modeling extra time or penalties.
The underlying match outcome model is based on the FIFA/Coca-Cola World Rankings, which follow a variant of the Elo rating system. Each team is assigned a rating ρi, and expected outcomes are derived using a logistic function:
This yields probabilities used in a Bradley-Terry model, extended with the Davidson-Beaver method to include draws in the group stage:
Group stage:
Knockout stage:
The scale parameter σ, which controls sensitivity to rating differences, is recalibrated to improve predictive accuracy. For the Women's World Cup, a data-driven approach set σ=240 by minimizing squared differences with betting odds, and the same 60% reduction (to σ=360) was applied for the men’s tournament.
Despite the normalization, women’s rankings still exhibit greater point disparities and skewness, resulting in sharper outcome probabilities.
4 Tournament Probabilities
In fixed-bracket tournaments like the FIFA World Cup, team trajectories become deterministic after group rankings are known. The group stage is modeled as eight independent round-robin groups of four teams. Each group produces 6 matches with 3^6=729 possible outcome combinations. Under the assumption of match independence, the probability of each outcome sequence is computed as the product of the six match probabilities. To resolve ties (e.g., identical points), rankings are randomly assigned among tied teams, and the probability of each first-second team pair advancing is averaged across all relevant sequences.
4.1 Group Stage
For each group, all 729 outcome sequences are evaluated to calculate the joint advancement probabilities G_ij for team pairs i,j. These probabilities are stored efficiently by indexing ternary-encoded sequences and leveraging pre-computed lists of possible rank combinations. Due to randomness in tie-breakers, multiple team-pair outcomes may result from the same match sequence. The overall complexity of group stage enumeration is modest: only 15,912 operations are needed across all groups.
4.2 Knockout Stage
The knockout stage progresses through four rounds: Round of 16, Quarterfinals, Semifinals, and Final. Due to limited trajectory mixing, dependencies are localized (typically between only two groups per match).
Advancement probabilities are computed pairwise at each round:
Round of 16: Uses group pairings (e.g., A1 vs B2) and computes all cross-pair matchups, accounting for the dependency of one team's rank on another’s
Quarterfinals: Similar structure but with broader pairing (8x7 combinations).
Semifinals and Final: Differ between men’s and women’s tournaments—men’s allow full mixing, women’s restrict finals to teams from different bracket subtrees.
The total computational workload remains tractable: 80,452 operations for the men’s tournament and only 23,256 for the women’s.
The efficiency arises from fixed structure and independence assumptions, enabling exact computation with a fraction of the time needed for simulation-based methods.
5 Results
Using the proposed algorithm atop the FIFA rating-based single-match model, exact winning probabilities for the 2022 and 2023 FIFA World Cups were derived. These reflect both team strength and bracket structure.
In the men’s tournament, Brazil had the highest winning probability (18.01%), followed by Belgium and Argentina. The results aligned with expectations, though notable deviations—like Morocco’s semifinal run or Germany’s early exit—demonstrate the stochastic nature of the format
In the women’s tournament, stronger rating disparities led to more concentrated probabilities: the USA (21.12%) and Germany (12.58%) were top contenders.
The bracket structure significantly affected these probabilities; for instance, France and Brazil’s shared group placement led to divergent paths depending on final group rank, influencing their final-stage chances.
Figure 8 quantitatively compares approximation errors of Monte Carlo simulations to the exact method. To achieve comparable accuracy, over 10,000 simulations are needed; even with 100,000, some probabilities retain >0.1% error. The exact method outperforms simulation in accuracy and speed, completing in the time of just a few hundred simulations.
6 Conclusion
This study demonstrates how limited mixing in tournament structures, such as those of recent FIFA World Cups, can be exploited to compute exact team winning probabilities. The approach is model-agnostic, requiring only pre-tournament, pairwise match outcome probabilities under mild independence assumptions. Compared to simulation-based methods, it is significantly faster and more accurate (requiring computation time equivalent to just a few hundred simulations, yet yielding exact results).
The method generalizes to other tournament formats defined by parameters governing group sizes, advancement rules, and bracket constraints. However, formats involving greater trajectory mixing or larger group sizes, such as the upcoming 48-team FIFA World Cup with 6-team groups, pose computational challenges due to combinatorial explosion in outcome sequences. Nonetheless, for tournaments with fixed seeding paths and bounded inter-group interaction, the algorithm remains scalable and practical for pre-tournament evaluation and live updates.
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References
Brandes, U., Marmulla, G., & Smokovic, I. (2023). Stop Simulating! Efficient Computation of Tournament Winning Probabilities. arXiv preprint arXiv:2307.10411. https://arxiv.org/abs/2307.10411
To keep this article concise, please refer to the original paper for the full list of references.