Iterative Elo Model

The Iterative Elo model estimates the time-varying strength of team $i$ at time $t$, denoted as $R_i^{(t)}$. It is an online learning algorithm equivalent to Stochastic Gradient Descent (SGD) on the log-likelihood of paired comparison outcomes, assuming a non-stationary data generating process.

1. Mathematical Formulation

Probability Model

The probability that Team A beats Team B is modeled by the logistic function (sigmoid) of the rating difference, adjusted for Home Court Advantage ($HCA$):

$$ P(A > B) = \sigma\left(\frac{R_A - R_B + HCA_{A}}{400} \ln(10)\right) $$

where: - $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid function. - $HCA_A$ is the Home Court Advantage term ($+150$ if A is home, $-150$ if B is home, $0$ if neutral). - The factor $400$ and base-10 scale are historical conventions from chess, implying a 400-point difference corresponds to a 10x change in odds ($P \approx 0.91$).

Simplified standard form: $$ E_A = \frac{1}{1 + 10^{-(R_A - R_B + HCA_{A})/400}} $$

Update Rule (Learning)

After observing the outcome $S_A \in {0, 1}$ (where 1=Win, 0=Loss), the ratings are updated:

$$ R_A^{(t+1)} \leftarrow R_A^{(t)} + K \cdot (S_A - E_A) $$ $$ R_B^{(t+1)} \leftarrow R_B^{(t)} + K \cdot (S_B - E_B) $$

Equivalence to SGD

The log-loss for a single match prediction is: $$ \mathcal{L} = - [S_A \ln(E_A) + (1-S_A) \ln(1-E_A)] $$

The gradient with respect to the rating $R_A$ is proportional to the residual $(S_A - E_A)$. Thus, the Elo update is exactly a gradient descent step with learning rate $\eta \propto K$.

2. Hyperparameters

We determine optimal hyperparameters via time-series cross-validation (minimizing out-of-sample log-loss on the 2025-26 season).

• $K = 64.0$: A relatively high learning rate, reflecting the high non-stationarity of college basketball (teams improve/regress significantly during the season due to coaching, injuries, and chemistry).
• $HCA = 150.0$: The home team is treated as having a rating advantage of 150 points.

3. Properties

• Path Dependent: The order of games matters. $G_1$ then $G_2$ yields a different final rating than $G_2$ then $G_1$. This is desirable for tracking current strength.
• Memory: The model effectively maintains an exponentially weighted moving average of performance. The "half-life" of a game's influence is determined by $K$.

4. Performance

• Validation Log Loss: 0.5757
• Accuracy: 69.6%