Decomposing Pythagoras

Kaplan Edward H. () and Rich Candler ()
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Kaplan Edward H.: William N. and Marie A. Beach Professor of Operations Research, Yale School of Management, New Haven, CT, USA
Rich Candler: Applied Mathematics Program, Yale University, New Haven, CT, USA

Journal of Quantitative Analysis in Sports, 2017, vol. 13, issue 4, 141-149

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.

Keywords: applied probability; Pythagorean method; regression decomposition; sports analytics (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1515/jqas-2017-0055

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