Go on, take a shot

A mathematical model suggests NBA players should be shooting earlier

Depending on which side of the Atlantic you call home, basketball is either riveting or coma-inducing. Basketball players are both athletes and actors. For sports enthusiasts, much of the excitement comes from waiting for the right scoring opportunity to arise. But a recent study suggests that NBA players may be waiting too long before shooting, and that shooting earlier could add about 4.5 points per game.

For mathematicians, the high scores and frequent shots found on basketball scoresheets give robust data sets. With only 5 players per team and a limited playbook, the interactions between players can be examined using classic models such as game theory. Most sports require quick decisions, and the results of those choices determine the score at the end of the game.

Shot-selection in basketball falls under the broad category of “optimal stopping problems”, the most famous of which is the so-called secretary problem. In the secretary problem, an administrator wishes to hire the best secretary out of n applicants. The applicants are interviewed one-by-one, in random order, and the outcome of the interview is determined immediately. Each applicant can therefore only be ranked relative to those already interviewed. How can the administrator maximize the probability of selecting the best candidate? The secretary problem has a surprisingly simple solution. The best strategy is to interview about 1/3 of the candidates (n/e, to be more precise), reject all of them and then offer the job to the first applicant after that who is better than the unlucky 1/3.

Deciding when to shoot the basketball presents a similar problem, but the solution is more complicated. By shooting early, a team forfeits any shots that would have arisen later in that possession. On the other hand, teams waiting too long pass up opportunities and instead take low-percentage shots in the dying seconds. Brian Skinner of the University of Minnesota constructed a model of the “shoot or pass up the shot” decision. In his model, the optimum time to shoot depends on three factors: the probability that a shot will go in, the distribution of shot quality that the offense will generate in the future, and the time remaining (the NBA allows each team 24 seconds before they have to either take a shot or surrender the ball). The resulting model states, unsurprisingly, that only high-quality shots should be taken early, and that the cut-off for shot quality decreases as the clock ticks down. But NBA players seem to take this too far; with 15 seconds left on the clock, the optimal model predicts about 3 times more shots than are actually taken. NBA players prefer to shoot in the dying seconds.

The model makes a number of assumptions, the most controversial of which is that shot opportunities arise randomly in time. It takes time for a team to set up its offence. Break-away plays may not use the same decision-making process, so shots from the first 7 seconds of ball possession were discounted. After 7 seconds, a team’s offence should be in place. In support of the assumption, there was little correlation in the NBA data between average shot time and the probability that a shot would score.

Under-shooting could be a sign of over-confidence. Players may be unwilling to take moderate-quality shots early in their possession, believing they’ll generate better scoring opportunities in the near future. They may also be underestimating the probability of a turnover and therefore overestimating the time remaining. This model advises ‘ballers to do less acting and more shooting. After all, smug grins are best worn by winning teams.