Abstract
Every shot in basketball has an opportunity cost; one player’s shot eliminates all potential opportunities from their teammates for that play. For this reason, player-shot efficiency should ultimately be considered relative to the lineup. This aspect of efficiency—the optimal way to allocate shots within a lineup—is the focus of our paper. Allocative efficiency should be considered in a spatial context since the distribution of shot attempts within a lineup is highly dependent on court location. We propose a new metric for spatial allocative efficiency by comparing a player’s field goal percentage (FG%) to their field goal attempt (FGA) rate in context of both their four teammates on the court and the spatial distribution of their shots. Leveraging publicly available data provided by the National Basketball Association (NBA), we estimate player FG% at every location in the offensive half court using a Bayesian hierarchical model. Then, by ordering a lineup’s estimated FG%s and pairing these rankings with the lineup’s empirical FGA rate rankings, we detect areas where the lineup exhibits inefficient shot allocation. Lastly, we analyze the impact that sub-optimal shot allocation has on a team’s overall offensive potential, demonstrating that inefficient shot allocation correlates with reduced scoring.
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The first and second authors contributed equally to this work.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. This paper represents partial fulfillment of the first author’s PhD thesis requirements.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
A.1 Empirical implementation
To illustrate some important considerations associated with our approach, we present an example of LPL and PLC using empirical FG% and FGA rates. This example demonstrates that our metrics are agnostic to the underlying FG% model.
We examine the Cleveland Cavaliers’ starting lineup as discussed in the main text. In order to obtain FG% and FGA rate estimates, we divide the court into 12 discrete regions and calculate the empirical FG%s and FGA rates for each player within these regions. We defined these regions based on our understanding of the court, but it is worth noting that defining these regions requires many of the same considerations as with any histogram style estimator; namely, that increasing the number of regions will decrease bias at the expense of increasing variance. In some cases, a player may have only one or two shots within an area, resulting in either unrealistically high or low FG% estimates. As an ad hoc solution to this, we give all players one made field goal and five field goal attempts within each region, which means that players with just a handful of shots in a region will have their associated FG% anchored near 20%. For the field goal attempt estimates, we simply count up the number of attempts for each player within each section, and normalize them to get the attempts per 36 min. With these FG% and FGA rate estimates, we can replicate the analysis detailed in Section 3.
Figure 14 shows the empirical ranks for this lineup, as well as the rank correspondence. Generally, it shows the same patterns as the model-based analysis in Figures 5 and 6. However, there are some key differences, including Tristan Thompson having a higher field goal percentage rank from the right midrange and a corresponding reduction in rank for Kevin Love in the same area. This pattern is also manifest in Figure 15, which shows the empirical LPL. We observe that most lineup points appear to be lost in the right midrange and in above the break three point shots. Finally, considering the empirical PLC in Figure 15, we notice that in addition to the Love-Thompson tradeoff in the midrange, JR Smith appears to be overshooting from the perimeter, while Kyrie Irving and LeBron James both exhibit undershooting.
Histogram of
Left: 20% quantile LPL surfaces for the Cleveland Cavaliers starting lineup. Middle: median LPL surfaces. Bottom: 80% quantile LPL surfaces. The top rows show LPL per 36 min while the bottom rows show LPLShot.
Posterior distribution of the effect for TGLPL in model (17)–(18) described in Section 4.2.
The persistence of the Love-Thompson connection in the midrange in this empirical analysis, and its divergence from what we saw in the model based analysis, merits a brief discussion. Kevin Love and Tristan Thompson both had a low number of shots from the far-right midrange region, with Love shooting 8 for 26 and Thompson shooting 4 for 6. Because they both shot such a low amount of shots, even with the penalty of one make and four misses added to each region, Thompson appears far better. This highlights the fact that although LPL and PLC are model agnostic, the underlying estimates for FG% do matter and raw empirical estimates alone may be too noisy to be useful in calculating LPL. One simple solution may be to use a threshold and only consider players in a region if the number of their field goal attempts passes that threshold.
A.2 Additional figures
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