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.

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