Relieving and Readjusting Pythagoras

TL;DR

This paper introduces a new, more accurate Pythagorean expectation formula for baseball winning percentage, based on a linear combination of Weibull distributions, improving prediction accuracy by about 25% with theoretical support.

Contribution

It proposes a novel linear combination of Weibull distributions to better estimate team run production and winning percentage, with theoretical justification for parameters.

Findings

Increases prediction accuracy by approximately 25%.

Performs as well or better than existing formulas.

Easily computed from publicly available data.

Abstract

Bill James invented the Pythagorean expectation in the late 70's to predict a baseball team's winning percentage knowing just their runs scored and allowed. His original formula estimates a winning percentage of ${\rm RS}^2/({\rm RS}^2+{\rm RA}^2)$, where ${\rm RS}$ stands for runs scored and ${\rm RA}$ for runs allowed; later versions found better agreement with data by replacing the exponent 2 with numbers near 1.83. Miller and his colleagues provided a theoretical justification by modeling runs scored and allowed by independent Weibull distributions. They showed that a single Weibull distribution did a very good job of describing runs scored and allowed, and led to a predicted won-loss percentage of $({\rm RS_{\rm obs}}-1/2)^\gamma / (({\rm RS_{\rm obs}}-1/2)^\gamma + ({\rm RA_{\rm obs}}-1/2)^\gamma)$, where ${\rm RS_{\rm obs}}$ and ${\rm RA_{\rm obs}}$ are the observed runs scored and allowed and $\gamma$ is the shape parameter of the Weibull (typically close to 1.8). We show a linear combination of Weibulls more accurately determines a team's run production and increases the prediction accuracy of a team's winning percentage by an average of about 25% (thus while the currently used variants of the original predictor are accurate to about four games a season, the new combination is accurate to about three). The new formula is more involved computationally; however, it can be easily computed on a laptop in a matter of minutes from publicly available season data. It performs as well (or slightly better) than the related Pythagorean formulas in use, and has the additional advantage of having a theoretical justification for its parameter values (and not just an optimization of parameters to minimize prediction error).