The Bradley–Terry Model in Sports Prediction & Betting

Introduction

Predicting outcomes in sports competitions has become increasingly relevant — not only for coaches and analysts, but especially for the betting and sports-lottery market, where each fraction of a percentage point of accuracy can be a competitive edge.

The Bradley–Terry model emerges here as an elegant and powerful tool. Originally conceived for paired comparisons in psychometrics, it was quickly adopted by sports statisticians for its ability to turn head-to-head results into quantitative strength ratings. Its conceptual simplicity — the probability of one team beating another is proportional to the ratio of their relative strengths — hides a wealth of practical applications, from ranking teams to generating predictive odds comparable to those of bookmakers.

This article presents, rigorously but accessibly, the mathematical foundations of the model and its applications in sports lotteries — from the basic formulation to advanced extensions, estimation methods, predictive validation and empirical evidence.

1. Mathematical foundations

1.1 Basic formulation

Each competitor k has a latent strength parameter α_k > 0. When teams i and j meet, the probability that i beats j is the ratio of their strengths:

P(i \text{ beats } j) = \frac{\alpha_i}{\alpha_i + \alpha_j}

and, equivalently, the probability that j beats i:

P(j \text{ beats } i) = \frac{\alpha_j}{\alpha_i + \alpha_j}

The two sum to one. A common reparameterisation sets β_k = log(α_k), turning the odds ratio into a difference of parameters:

\log\!\left(\frac{P(i \text{ beats } j)}{P(j \text{ beats } i)}\right) = \log(\alpha_i) - \log(\alpha_j) = \beta_i - \beta_j

This logistic form links the model directly to logistic regression and to rating systems such as Elo.

1.2 Identifiability and normalisation

The α_k are not uniquely identifiable: multiplying all of them by a positive constant leaves the probabilities unchanged. To fix the scale, a normalisation constraint is imposed:

\sum_{k=1}^{N} \alpha_k = 1

where N is the number of teams. This anchors the scale and allows direct comparison between competitors’ ratings.

1.3 The Bradley–Terry–Davidson model for ties

In sports like football, where a draw is a frequent outcome, the basic model must be extended. Davidson’s (1970) extension adds a parameter θ > 0 to accommodate ties. The probability of a draw is:

P(\text{tie}) = \frac{\theta \sqrt{\alpha_i \alpha_j}}{\alpha_i + \alpha_j + \theta \sqrt{\alpha_i \alpha_j}}

and the win probabilities become:

P(i \text{ beats } j) = \frac{\alpha_i}{\alpha_i + \alpha_j + \theta \sqrt{\alpha_i \alpha_j}}, \qquad P(j \text{ beats } i) = \frac{\alpha_j}{\alpha_i + \alpha_j + \theta \sqrt{\alpha_i \alpha_j}}

The larger θ, the higher the draw probability for a given pair of strengths.

2. Parameter estimation

2.1 Maximum likelihood

Parameters are typically estimated by maximum likelihood. Given observed results where w_ij is the number of wins of i over j, the likelihood is:

L(\alpha) = \prod_{i \neq j} \left(\frac{\alpha_i}{\alpha_i + \alpha_j}\right)^{w_{ij}}

and the corresponding log-likelihood:

\ell(\alpha) = \sum_{i \neq j} w_{ij}\left[\log(\alpha_i) - \log(\alpha_i + \alpha_j)\right]

subject to the normalisation constraint on the α_k.

2.2 Zermelo’s iterative algorithm

The log-likelihood has no closed-form maximiser, but it can be solved iteratively. Zermelo’s algorithm (1920s) updates each parameter as:

\alpha_i^{(t+1)} = \frac{W_i}{\displaystyle\sum_{j \neq i} \frac{n_{ij}}{\alpha_i^{(t)} + \alpha_j^{(t)}}}

where Wᵢ is i’s total wins and n_ij the number of meetings between i and j. It converges to the maximum-likelihood estimates and is efficient even for large datasets.

2.3 Bayesian approach

Alternatively, the model can be estimated in a Bayesian framework, incorporating prior information and quantifying uncertainty. A common prior is:

\beta_k \sim \mathcal{N}(0, \sigma^2)

or, in the original parameterisation:

\alpha_k \sim \text{Dirichlet}(1, \dots, 1)

Inference then proceeds via Hamiltonian Monte Carlo, producing full posterior distributions for team ratings (for example, the footBayes package on Stan).

3. Incorporating contextual effects

3.1 Home advantage

One of the best-documented effects in sport is home advantage. The model can incorporate it with a parameter γ > 0 multiplying the home team’s strength:

P(i \text{ beats } j \text{ at home}) = \frac{\gamma \alpha_i}{\gamma \alpha_i + \alpha_j}

where i is the home team; γ > 1 means playing at home raises the win probability.

3.2 Surface and context-specific effects

Empirical studies show the playing surface (grass, clay, hard court) can significantly affect performance. The model can include surface-specific parameters:

P(i \text{ beats } j \text{ on surface } s) = \frac{\alpha_{i,s}}{\alpha_{i,s} + \alpha_{j,s}}

where α_{k,s} is team k’s strength specifically on surface s. In tennis, surface-specific ratings have produced rankings superior to the official ATP ones for certain players.

3.3 Dynamic models with temporal variation

Team strength is not constant over time. Dynamic versions let the strength parameters evolve smoothly, typically as a random walk:

\beta_{k,t} = \beta_{k,t-1} + \varepsilon_{k,t}, \qquad \varepsilon_{k,t} \sim \mathcal{N}(0, \sigma^2)

This lets relative strength update each round, capturing form, injuries and other changes across a season.

4. Application in lotteries and sports betting

4.1 Generating predictive probabilities

The most direct betting application is generating predictive probabilities for future fixtures. Once the α_k are estimated, the probability that i beats j is:

\hat{P}_{ij} = \frac{\hat{\alpha}_i}{\hat{\alpha}_i + \hat{\alpha}_j}

For sports with draws, the Davidson extension gives win and tie probabilities:

\hat{P}_{ij}^{\text{win}} = \frac{\hat{\alpha}_i}{\hat{\alpha}_i + \hat{\alpha}_j + \hat{\theta}\sqrt{\hat{\alpha}_i \hat{\alpha}_j}}, \qquad \hat{P}_{ij}^{\text{tie}} = \frac{\hat{\theta}\sqrt{\hat{\alpha}_i \hat{\alpha}_j}}{\hat{\alpha}_i + \hat{\alpha}_j + \hat{\theta}\sqrt{\hat{\alpha}_i \hat{\alpha}_j}}

4.2 Comparison with bookmaker odds

A key step is comparing the model’s probabilities with those implied by the odds. Odds O imply a probability P_book = 1/O (adjusted for the bookmaker’s margin). When the model’s probability significantly exceeds P_book, a potential value bet appears. The expected value of a bet is:

EV = \hat{P} \times O - 1

If EV > 0, the bet has positive expected value in the long run — never a guarantee on any single bet.

4.3 Empirical evidence

Empirical studies support the model. In an analysis of 3,439 men’s tennis matches (Jan 2019–Sep 2020), it reached 62.46% classification accuracy against the best bookmaker odds, with a reported 26.44% ROI across surfaces (25.74% on clay). In football, Bradley–Terry-based models have performed comparably to hierarchical Poisson log-linear models, and dynamic versions applied to the Australian Football League have correctly predicted up to 71.5% of results.

These figures illustrate potential, not a promise: results observed in a study do not guarantee future returns, and all analysis carries uncertainty.

5. Connection to rating systems and other models

5.1 Equivalence with the Elo system

There is a direct link between Bradley–Terry and the Elo rating system. In Elo, the expected win probability of a player rated Rᵢ against one rated Rⱼ is:

P(i \text{ beats } j) = \frac{1}{1 + 10^{(R_j - R_i)/400}}

which is exactly Bradley–Terry with α_i = 10^{R_i/400}. Elo can thus be seen as a sequential implementation of Bradley–Terry.

5.2 Relationship to logistic regression

The logistic reparameterisation shows the model is essentially a logistic regression with dummy indicators for each competitor — which makes estimation easy with standard statistical packages and allows extra covariates.

5.3 Comparison with the Thurstone–Mosteller model

The Thurstone–Mosteller model differs in the assumed distribution of latent strength differences: Bradley–Terry assumes a logistic distribution (logit link), Thurstone–Mosteller a normal one (probit link). In practice both give similar results, with Bradley–Terry preferred for its computational tractability.

6. Limitations and challenges

Despite its elegance, the model has important limitations:

Independence of comparisons: it assumes matches are independent, which may fail when the same teams meet repeatedly.
Stationary parameters: the basic version assumes constant strength over time, an unrealistic simplification.
Ignores margin of victory: it uses only the result, not the scoreline.
Tie handling: Davidson’s extension assumes a symmetric draw probability, which may not fit every sport.
Sparse data: with many teams and few meetings per pair, estimates can be unstable.

Recent work addresses these via hierarchical Bayesian models, dynamic approaches and additional contextual information.

Conclusion

The Bradley–Terry model is a powerful, versatile tool for analysing sports competitions and, in particular, for generating the predictive probabilities used in lotteries and betting. Its solid theory and flexibility to include contextual and dynamic effects make it a natural choice for analysts.

For the analyst it offers a systematic, quantifiable way to assess probabilities and identify value. But no model removes uncertainty: it estimates probabilities, not certainties. Use it to inform judgement, always within responsible, affordable limits.

References

Bradley, R. A., & Terry, M. E. (1952). Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika, 39(3/4), 324–345.
Davidson, R. R. (1970). On extending the Bradley–Terry model to accommodate ties in paired comparison experiments. JASA, 65(329), 317–328.
Zermelo, E. (1929). Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 29(1), 436–460.
Tsokos, A., et al. (2018). Modeling outcomes of soccer matches. Machine Learning, 108(1), 1–35.
Macrì Demartino, R., Egidi, L., & Torelli, N. (2024). Alternative ranking measures to predict international football results. arXiv:2405.10247.
Whelan, J. T., & Klein, J. E. (2021). Bradley–Terry modeling with multiple game outcomes. arXiv:2112.01267.
Egidi, L., Macrì Demartino, R., & Palaskas, V. (2025). Fitting football models with the footBayes package. CRAN.
Baio, G., & Blangiardo, M. (2010). Bayesian hierarchical model for the prediction of football results. Journal of Applied Statistics, 37(2), 253–264.
Maher, M. J. (1982). Modelling association football scores. Statistica Neerlandica, 36(3), 109–118.

References gathered as editorial background only; the text above is original to G10Tips.

The Bradley–Terry Model in Sports Lottery Analysis