Pythagoras at the Bat
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The Pythagorean formula is one of the most popular ways to measure the true ability of a team. It is very easy to use, estimating a team’s winning percentage from the runs they score and allow. This data is readily available on standings pages; no computationally intensive simulations are needed. Normally accurate to within a few games per season, it allows teams to determine how much a run is worth in different situations. This determination helps solve some of the most important economic decisions a team faces: How much is a player worth, which players should be pursued, and how much should they be offered. We discuss the formula and these applications in detail, and provide a theoretical justification, both for the formula as well as simpler linear estimators of a team’s winning percentage. The calculations and modeling are discussed in detail, and when possible multiple proofs are given. We analyze the 2012 season in detail, and see that the data for that and other recent years support our modeling conjectures. We conclude with a discussion of work in progress to generalize the formula and increase its predictive power without needing expensive simulations, though at the cost of requiring play-by-play data.
KeywordsPythagorean Formula Play Worth Simple Linear Estimator Structural Zeros Sabermetrics
The first author was partially supported by NSF Grants DMS0970067 and DMS1265673. He thanks Chris Chiang for suggesting the title of this paper, numerous students of his at Brown University and Williams College, as well as Cameron and Kayla Miller, for many lively conversations on mathematics and sports, Michael Stone for comments on an earlier draft, and Phil Birnbaum, Kevin Dayaratna, Warren Johnson and Chris Long for many sabermetrics discussions. This paper is dedicated to his great uncle Newt Bromberg, who assured him he would live long enough to see the Red Sox win it all, and the 2004, 2007 and 2013 Red Sox who made it happen (after watching the 2013 victory his six year old son Cameron turned to him and commented that he got to see it at a much younger age!).
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