Model Methodology: Bradley-Terry (Static)
Global Bradley-Terry Model
The Bradley-Terry model (often called "Static Elo" or "Global Logistic Regression") estimates a single, time-invariant rating $R_i$ for each team $i$ that maximizes the joint likelihood of all game outcomes observed in the training set. Unlike Elo, this is a batch algorithm that finds a global optimum.
1. Mathematical Formulation
Likelihood Function
We observe a set of games $\mathcal{G}$. For each game $k \in \mathcal{G}$ between team $i$ and team $j$, let $y_k = 1$ if $i$ wins, and $y_k = 0$ if $j$ wins. The model posits:
$$ \text{logit}(P(i > j)) = \beta_{home} \cdot x_{home} + (R_i - R_j) \cdot \ln(10) / 400 $$
This is equivalent to: $$ P(i > j) = \frac{1}{1 + 10^{-(R_i - R_j + \beta_{home})/400}} $$
Objective Function
We solve for the vector of ratings $\mathbf{R}$ and home-advantage parameter $\beta_{home}$ that minimizes the regularized negative log-likelihood:
$$ J(\mathbf{R}, \beta) = - \sum_{k \in \mathcal{G}} \left[ y_k \ln(p_k) + (1-y_k) \ln(1-p_k) \right] + \lambda ||\mathbf{R}||_2^2 $$
This is a convex optimization problem (Logistic Regression with L2 regularization). We solve it using the L-BFGS-B algorithm.
2. Hyperparameters
We tune the regularization strength $C = \frac{1}{\lambda}$ via cross-validation.
- $C = 1.0$: This specifies the prior belief distribution on team strengths.
- $\beta_{home} \approx 201$: The Home Court Advantage is learned from the data as a global parameter, rather than fixed. It is higher than the Elo HCA (150) because the static model must attribute all home wins to this static factor, whereas Elo can attribute some to "temporary form."
3. Properties
- Path Independent: The order of games does not matter. The result is the Maximum A Posteriori (MAP) estimate of team strength given the entire season's data.
- Stationarity Assumption: The model assumes intrinsic team strength $R_i$ is constant throughout the season.
4. Performance
- Validation Log Loss: 0.611
- Accuracy: 62.5%
- Interpretation: Underperforms Elo on prediction because it cannot account for team evolution (injuries, improvement). However, it provides the mathematically "correct" answer to "Who was the best team on average?"