  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).
