When to Shoot a Basketball Now Has an Answer, Mathematically Speaking

August 24, 2011

Brian Skinner (University of Minnesota) has taken a shot at providing mathematical guidance for the optimal time to take shots in basketball games.

His article, "The Problem of Shot Selection in Basketball: The Shooter's Sequence," was posted at arXiv.org (July 29, 2011) and featured in a recent article in Science.

Assuming that the quality of shot opportunities generated by the offense would fall randomly within a uniform distribution, Skinner shows that the "lower cutoff" for shot quality f depends on the number n of shot opportunities remaining (say, before the shot clock expires), whereby the larger n demands that only higher-quality shots should be attempted.

The function f(n), which Skinner dubs the "shooter's sequence," turns out to constitute a recursive quadratic map with no known analytical solution. Skinner derives f(n) in the presence of a finite turnover rate, and uses it to predict the hazard rate for an optimal-shooting team as a function of time.

“[Skinner's paper] poses an intuitive theoretical model of the way basketball players should select shots in a variety of situations," said Matthew Goldman (University of California, San Diego), who also models basketball games.

Last year, in an article in the Journal of Quantitative Analysis in Sports, Skinner hypothesized that any route in a traffic model could be transformed into basketball play. That is, equations and variables in traffic theory could be reinterpreted as descriptions of basketball games.

Source: arXiv.org, Science