Bradley-Terry-Davidson Model

Updated 4 December 2025

The Bradley-Terry-Davidson formulation is a family of paired comparison models extending the classical Bradley-Terry approach by incorporating ties, order effects, and strength-dependent mechanisms.
It employs robust maximum likelihood estimation methods with stringent identifiability and connectivity conditions to ensure unique and reliable parameter inference.
The model has broad applications in sports, games, and psychological assessments, enhancing ranking accuracy and predicting competitive outcomes effectively.

The Bradley-Terry-Davidson (BTD) formulation represents a comprehensive and flexible family of paired comparison models that extend the classical Bradley-Terry model to accommodate ties, order effects (such as home-field advantage), and—recently—strength-dependent mechanisms for both tie and order probabilities. These models serve as foundational statistical tools for inferring latent strength parameters from match or comparison outcomes in domains such as games, sports, and psychological assessment. The BTD framework is distinguished by its structural generalizations, nuanced identifiability constraints, robust maximum likelihood estimation solutions, and detailed existence and uniqueness conditions.

1. Classical Bradley-Terry and Davidson Extensions

The core Bradley-Terry model posits that for any pair $(i, j)$ , the win probabilities are functions of their positive latent strengths $\pi_i, \pi_j$ : $P(i \text{ beats } j) = \frac{\pi_i}{\pi_i + \pi_j}, \qquad P(j \text{ beats } i) = \frac{\pi_j}{\pi_i + \pi_j}$ This logistic-form probability, motivated by paired comparisons, is effective for binary outcomes but does not capture matches with ties or explicit order effects.

Davidson’s extension (Yan, 2014, Whelan et al., 2021, Tsokos et al., 2018) introduces a tie parameter $\gamma \ge 0$ , permitting three-way outcome probabilities: $\begin{aligned} P(i \text{ beats } j) &= \frac{\pi_i}{\pi_i + \pi_j + \gamma \sqrt{\pi_i \pi_j}} \ P(\text{tie}) &= \frac{\gamma \sqrt{\pi_i \pi_j}}{\pi_i + \pi_j + \gamma \sqrt{\pi_i \pi_j}} \ P(j \text{ beats } i) &= \frac{\pi_j}{\pi_i + \pi_j + \gamma \sqrt{\pi_i \pi_j}} \end{aligned}$ This geometric mean tying structure yields tractable likelihood and has been adopted for both frequentist and Bayesian inference (Whelan et al., 2021).

2. Strength-Dependent Extensions and Order Effects

Recent work has identified empirical patterns whereby the probability of a tie and the impact of order effects are themselves functions of competitor strength (Glickman, 30 May 2025). The generalized BTD model expresses parameters on the log-scale ( $\theta_i$ ) and incorporates the following outcome probabilities: $\begin{aligned} P(Y_{ij}=1)\; &= \frac{e^{\theta_i + \frac{x_{ij}}{4}[\alpha_0+\alpha_1\frac{\theta_i+\theta_j}{2}]}}{D_{ij}} \ P(Y_{ij}=0)\; &= \frac{e^{\theta_j - \frac{x_{ij}}{4}[\alpha_0+\alpha_1\frac{\theta_i+\theta_j}{2}]}}{D_{ij}} \ P(Y_{ij}=1/2)\; &= \frac{e^{\beta_0 + (1+\beta_1)\frac{\theta_i+\theta_j}{2}}}{D_{ij}} \end{aligned}$ where $x_{ij}$ encodes the order advantage, and the denominator is

$D_{ij} = e^{\theta_i + \frac{x_{ij}}{4}[\alpha_0+\alpha_1\frac{\theta_i+\theta_j}{2}]} + e^{\theta_j - \frac{x_{ij}}{4}[\alpha_0+\alpha_1\frac{\theta_i+\theta_j}{2}]} + e^{\beta_0 + (1+\beta_1)\frac{\theta_i+\theta_j}{2}}$

Parameters $\alpha_0$ , $\alpha_1$ capture baseline and strength-dependent order effects; $\beta_0$ , $\beta_1$ allow for baseline and strength-dependent tie propensities. This formulation empirically improves fit and captures key phenomena in real competitive settings—for example, that strong-versus-strong matches are more likely to result in ties, and that order effects are more pronounced for high-strength players (Glickman, 30 May 2025).

3. Maximum Likelihood Estimation and Identifiability

For observed outcomes $y_k \in \{1,0,1/2\}$ across $K$ contests, the likelihood functions for BTD models are tractable and permit efficient estimation. The general approach is: $\ell(\{\theta\}, \alpha_0, \alpha_1, \beta_0, \beta_1) = \sum_{k=1}^K \left[ \mathbf{1}\{y_k=1\}\log P_{ij}(1) + \mathbf{1}\{y_k=0\}\log P_{ij}(0) + \mathbf{1}\{y_k=1/2\}\log P_{ij}(1/2) \right]$ Variations such as alternated maximization, Newton–Raphson, or block-relaxation are used for inference, partitioning parameters into strengths and auxiliary components (Glickman, 30 May 2025, Whelan et al., 2021).

Identifiability hinges on the translation invariance of $\theta_i$ —the likelihood is unchanged under shifts $\theta_i \rightarrow \theta_i + c$ for all $i$ . Commonly, the constraints $\sum_i \theta_i = 0$ or $\theta_1=0$ are imposed (Glickman, 30 May 2025, Gyarmati et al., 2023).

Bayesian extensions utilize improper “Haldane” priors uniform in the log-parameters, or plausibly Laplace posteriors based on Hessian approximations around MAP points (Whelan et al., 2021).

4. Existence and Uniqueness of the MLE

The existence and uniqueness of the MLE are tightly tied to data structure, particularly connectivity in the induced win/loss graphs. For two-outcome BT models, Ford’s criterion requires strong connectivity—every partition of the objects must observe wins across both directions (Gyarmati et al., 2023, Yan, 2014). This result extends to all strictly log-concave comparison distributions.

For three-outcome (tie-inclusive) models, necessary and sufficient conditions are more complex:

Davidson’s DC-conditions require at least one tie and cross-edges in both win directions across every partition.
The MC-conditions (Orbán–Mihályko) refine this by requiring bidirectional outcomes within a single pair and connectivity via tie or dual win-loss results.
The SC-conditions (Gyarmati–Orbán–Mihályko) further generalize: the object graph must display cycles among winners, and every partition must be “linked” by either win-loss or tie edges.

Empirical simulations demonstrate that SC-conditions provide the highest detection rate for MLE existence across varied scenarios—substantially outperforming DC and MC (Gyarmati et al., 2023). This reflects deep structure in the parameter space for three-outcome paired comparison models.

A related challenge arises when connectivity is insufficient for classical MLE existence. The “ $\varepsilon$ -singular perturbation” addresses this by adding a small pseudo-count to every observed comparison, guaranteeing existence and robustness of the penalized MLE under a weak connectivity condition (Yan, 2014).

5. Extensions to Multiple Outcomes and Computational Solutions

Paired comparison models can extend to multi-category outcomes beyond ternary win-loss-tie (e.g., regulation win/loss, OT wins/losses). Generalization is achieved by parameterizing each result with fractional win values and outcome type indicators. The likelihood generalizes as: $P_I(i, j) = \frac{\theta_i^{p_I} \theta_j^{1-p_I} \nu^{o_I}}{\sum_{J} \theta_i^{p_J} \theta_j^{1-p_J} \nu^{o_J}}$ with maximum likelihood and Bayesian inference (Laplace/MAP or Hamiltonian Monte Carlo) tractable via gradient-based optimization or exact posterior sampling frameworks such as Stan (Whelan et al., 2021).

Computational methodologies include block-relaxation maximization, BFGS, EM (Caron–Doucet latent Gamma variables), and MM algorithms (Hunter). Penalized and ordinal cumulative-link forms are implemented for feature-rich extensions and league-calibrated fits. The rankings are robust to perturbations, and convergence is fast for practical examples (Yan, 2014, Whelan et al., 2021, Tsokos et al., 2018).

6. Applications and Empirical Significance

Bradley-Terry-Davidson models are widely adopted for ranking, strength assessment, and probabilistic forecasting in sports tournaments, chess, and psychological testing (Glickman, 30 May 2025, Whelan et al., 2021, Tsokos et al., 2018). Strength-dependent models notably improve predictive fit when competitors span a wide strength range, as shown in US Chess Open game analyses (Glickman, 30 May 2025). In college hockey and soccer, BTD variants—especially those leveraging semi-parametric feature-based architectures—match or outperform complex hierarchical alternatives in ranking accuracy and probabilistic scoring (Tsokos et al., 2018).

Large-scale simulations confirm that advanced existence criteria (SC-conditions) and $\varepsilon$ -perturbed estimators yield robust and reliable rankings even in nonbalanced or sparse data designs (Gyarmati et al., 2023, Yan, 2014).

A plausible implication is that further structural generalizations—incorporating covariate effects, time dynamics, and higher-order match features—will continue to extend the utility of the BTD family in competitive analysis and decision sciences.

Model Variant	Tie Handling	Existence/Uniqueness Criterion
Bradley-Terry	No ties	Strong connectivity (Ford’s criterion)
Davidson extension	Fixed tie parameter	DC/MC/SC-conditions (see text)
BTD (strength-dep.)	Strength-dependent ties/orders	SC-conditions, ε-perturbation if necessary

The BTD formulation sits at the intersection of latent variable modeling, statistical ranking, and applied comparison science, offering a rigorous, theoretically underpinned, and empirically validated toolset for pairwise outcome analysis.