The Mathematics of Pythagorean Expectation

Abstract

Pythagorean Expectation treats a team's win probability as a function of its aggregate scoring performance. The formula can be rigorously derived assuming points scored ($PS$) and points allowed ($PA$) follow independent Weibull distributions.

We present the mathematical framework for our two baseline models: 1. Pythagorean Raw: The classical formulation mapping observed point totals to expected win percentage. 2. Pythagorean Adjusted: A regularized linear model (Ridge Regression) that extracts strength-adjusted efficiencies from game-level data.

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1. The Classical (Raw) Model

Derivation

Let $X$ and $Y$ be random variables representing points scored by Team A and Team B (Opponent) in a game. If $X$ and $Y$ are independent and follow Weibull distributions:

$$ f(x; \lambda, k) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k} $$

It can be shown (Miller, 2006) that the probability that $X > Y$ (a win) converges to the Pythagorean form:

$$ P(X > Y) = \frac{\lambda_x^k}{\lambda_x^k + \lambda_y^k} $$

In our application, we estimate $\lambda$ using the observed season totals $PS$ and $PA$.

The Formula

For NCAA Men's Basketball, we use an empirically derived exponent $\gamma = 11.5$:

$$ \text{Win}\% = \frac{PS^{\gamma}}{PS^{\gamma} + PA^{\gamma}} = \frac{1}{1 + \left(\frac{PA}{PS}\right)^{\gamma}} $$

This model serves as our unadjusted baseline. It strictly measures dominance over the observed schedule, without accounting for opponent strength.

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2. The Adjusted Efficiency Model

To account for Strength of Schedule (SOS), we cannot rely on raw averages. A team might score 1.2 points per possession (PPP) because they are elite, or because their opponent has a poor defense.

We model game-level efficiency as a system of linear equations and solve it using Ridge Regression.

Linear Formulation

Let $y_{g}$ be the offensive efficiency (points per 100 possessions) for the home team in game $g$. We posit:

$$ y_{g} = \alpha + \theta_{i}^{Off} - \theta_{j}^{Def} + \epsilon_{g} $$

Where: * $\alpha$: League average efficiency (intercept). * $\theta_{i}^{Off}$: Offensive rating for Team $i$ (above/below average). * $\theta_{j}^{Def}$: Defensive rating for Opponent $j$ (above/below average). * $\epsilon_{g}$: Residual noise (variance unexplained by team strength).

We construct a sparse design matrix $X$ of shape $(2 \cdot N_{games}, 2 \cdot N_{teams})$. For each game, we have two observations: 1. Team $i$ Offense vs Team $j$ Defense. 2. Team $j$ Offense vs Team $i$ Defense.

Optimization: Ridge Regression

This system is often ill-posed or collinear (especially in unconnected graphs or short seasons). To solve it robustly, we apply Tikhonov Regularization (Ridge Regression).

We minimize the cost function $J(\theta)$:

$$ J(\theta) = \sum_{g=1}^{N} (y_g - (\alpha + \theta_{i}^{Off} - \theta_{j}^{Def}))^2 + \lambda \sum_{k} \theta_k^2 $$

Where $\lambda$ is the regularization penalty.

• OLS Term: $\sum (y - \hat{y})^2$ ensures the ratings explain the observed scores.
• Ridge Term: $\lambda \sum \theta^2$ shrinks coefficients towards zero (the league average), preventing overfitting on small samples.

Solution

In matrix notation, let $\beta = [\alpha, \theta^{Off}, \theta^{Def}]^T$. The Ridge estimator is:

$$ \hat{\beta}_{ridge} = (X^T X + \lambda I)^{-1} X^T y $$

Computationally, we solve this using scipy.sparse.linalg.lsqr, which is efficient for our large, sparse operator.

Final Calculation

Once we obtain the optimal coefficients $\hat{\theta}^{Off}$ and $\hat{\theta}^{Def}$, we reconstruct the Adjusted Efficiency for each team:

$$ \text{AdjO} = \alpha + \hat{\theta}^{Off} $$ $$ \text{AdjD} = \alpha - \hat{\theta}^{Def} $$

These values represent how a team would perform against a theoretically "average" opponent. We then feed these adjusted efficiencies back into the Pythagorean formula to generate the final Adjusted Win%:

$$ \text{AdjWin}\% = \frac{(\text{AdjO})^{\gamma}}{(\text{AdjO})^{\gamma} + (\text{AdjD})^{\gamma}} $$

This metric, often called "PythAdj", is widely considered the gold standard for predictive team ranking.