The relative wages of offense and defense in the NBA: a setting for win-maximization arbitrage?

Justin Ehrlich; Shane Sanders; Christopher J. Boudreaux

doi:10.1515/jqas-2018-0095

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The relative wages of offense and defense in the NBA: a setting for win-maximization arbitrage?

Justin Ehrlich , Shane Sanders und Christopher J. Boudreaux

Veröffentlicht/Copyright: 29. Mai 2019

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Aus der Zeitschrift Journal of Quantitative Analysis in Sports Band 15 Heft 3

Abstract

In basketball, a point scored on offense carries a nearly identical on-court (win) value as a point denied on defense (e.g. within the Pythagorean expected wins model). Both outcomes bear the same score margin implication. As such, a win-maximizing team is expected to value the two outcomes equally. We ask whether the salaries of NBA players reveal such an equality among NBA teams. If not, a win-maximizing team would enjoy a disequilibrium arbitrage opportunity, whereby the team could improve, in expectation, even while reducing roster payroll. We considered the 322 National Basketball Association (NBA) players during the 2016–2017 season who were on a full-season contract for which the salary was not stipulated under the NBA Collective Bargaining Agreement. We estimated the implied marginal wage of an additional point created on offense (denied on defense) per 100 possessions. Namely, we constructed a set of fixed effects, ordinary least squares regression models that specify a player’s pre-assigned 2016–2017 player salary as a function of primary team fixed effects, offensive adjusted plus minus, defensive adjusted plus minus, position-of-play, and control variables such as age. We conclude that a win-maximizing NBA team currently faces a substantial arbitrage opportunity. Namely, one unit of offense carries the same estimated implicit salary as approximately two and a half to four units of defense. We also find moderate between-team variation in adjusted plus minus return on payroll allocations.

Keywords: adjusted plus minus; basketball; payroll; win maximization

Acknowledgement

Thank you to Sean Forman (Sports Reference), Layne Vashro (Kroenke Sports & Entertainment), and participants of the 2017 Midwest Sport Analytics Conference for helpful comments and suggestions. We would especially like to thank two anonymous reviewers, an anonymous Associate Editor, and Editor-in-Chief Steve Rigdon for their advice and suggestions.

Appendix

Marginal Effects of Offense and Defense within the Pythagorean Expected Wins Model

The (generalized) Pythagorean Expected Wins model is presented as follows:

(8)E(Wi,j)=Ptsi,jαPtsi,jα+Opp_ptsi,j

where E(W_i,j) represents expected wins for team i in season j, Pts_i,j represents average points scored per game for team i in season j, Opp_pts_i,j represents average opponent points scored per game for team i in season j, and α is a returns to scale parameter that determines the expected win effect of a marginal point scored or allowed. Morey et al. (1993) estimates α as equal to 13.91 for NBA basketball. Within this Appendix, we will not specify α other than to make the restriction that α > 0 (such that points scored are productive toward winning in expectation).

The marginal expected win value of a point scored on offense is equal to:

(9)∂E(Wi,j)∂Ptsi,j=αPtsi,jα−1⋅Opp_ptsi,jα(Ptsi,jα+Opp_ptsi,jα)2

The marginal expected win value of a point denied on defense is equal to:

(10)∂E(Wi,j)∂(−Opp_ptsi,j)=−∂E(Wi,j)∂(Opp_ptsi,j)

where the equality in (10) is attributable to the chain rule. Further, we have that:

(11)−∂E(Wi,j)∂(Opp_ptsi,j)=αPtsi,jα⋅Opp_ptsi,jα−1(Ptsi,jα+Opp_ptsi,jα)2

Then, we have that:

(12a)∂E(Wi,j)∂Ptsi,j=∂E(Wi,j)∂(−Opp_ptsi,j)forPtsi,j=Opp_ptsi,j

and

(12b)∂E(Wi,j)∂Ptsi,j≈∂E(Wi,j)∂(−Opp_ptsi,j)forPtsi,j≈Opp_ptsi,j

Hence, the marginal implication of a point scored on offense and a point denied on defense are roughly equivalent across the observed set of NBA team seasons. Further, we have from our previous calculations that:

(13)∂E(Wi,j)∂Ptsi,j=Opp_ptsi,jPtsi,j∂E(Wi,j)∂(−Opp_ptsi,j)

For a below average team (one for which expected Opp_pts_i,j > Pts_i,j), then, the Pythagorean model estimates marginal offense to be more win valuable than marginal defense, where the scalar difference in relative values is equal to Opp_ptsi,jPtsi,j. For a team with an expected points ratio, Opp_ptsi,jPtsi,j, of 1.1, the marginal win value of offense would be 1.1 times that of defense.^[10] At the league level, these value distortions would tend to equal out, as there is a clear equality of points scored and points allowed at the league level.

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Published Online: 2019-05-29

Published in Print: 2019-08-27

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adjusted plus minus; basketball; payroll; win maximization