Consistency and Pythagoras
Pythagorean win share has been one of the fundamental contributions to Sabermetrics. Several hundred articles, both academic and non-academic, have explored variations on Bill James' original formula and its fit to empirical data. This paper considers a variation that is previously unexplored on any systematic level, consistency. After discussing several important contributions to the line of literature, we demonstrate the strong correlation between Pythagorean residuals and several notions of run distribution consistency. Finally, we select the "correct" form of consistency and use it to construct a simple regression estimator, which improves RMSE by 11%.
Decomposing Pythagoras
Abstract The Pythagorean win expectancy model developed by Bill James remains one of the most celebrated results in sports analytics. Many have extended the application of this model from its original use in baseball to other sports. Others have shown technical scoring conditions that imply the equivalence of win probability and the Pythagorean model. However, no explanation has been offered for why different sports yield different results beyond “that’s what the data say.” This article presents a theoretical analysis of the Pythagorean model by first deducing an exact within-team equation relating win percentage to seasonal scoring records, and then reconciling mathematically this result with the Pythagorean model which is cross-sectional across teams in a league. We derive a complete decomposition of the Pythagorean coefficient γ in terms of the exact model, and show that γ captures two key quantities – average points per game, and the average margins of victory and defeat – that together explain why different sports yield different results. We demonstrate this decomposition using the past decade of seasonal results from MLB baseball, NBA basketball, NFL football, and NHL hockey, and show that the data do reflect the properties deduced in our analysis.
The Collective Bargaining Effects of NBA Player Productivity Dynamics
We apply quintile regression methodology to player pay and performance data from the 1985-86 to 2005-06 seasons of the National Basketball Association (NBA). In addition to confirming a finding from Hakes and Turner (2007) of systematic bias in pooled OLS regressions of career paths for salary and productivity, the quintile analysis presents two important results regarding NBA salary structure. Unlike Major League Baseball (MLB), the highest ability veteran NBA players suffer salary suppression relative to the lesser-talented players in their debut-year cohort, indicating rents have been transferred from the most able players to players of lesser abilities. Also, while young NBA players in general suffer from salary suppression relative to free agents, as is well-reported in baseball, our regression results show that the highest-ability young players suffer the most salary suppression, and that the effects of the rookie salary cap in the 1995 NBA Collective Bargaining Agreement depressed salaries for young players of all ability levels.