Jeff Haynes/Reuters

Shoot! A new article provides mathematical guidelines about when a basketball player should attempt to score and when the team is better off waiting for another chance.

The Mathematics of Basketball

To shoot, or not to shoot, that is the question. Whether 'tis nobler to try to score right away or wait for a better chance.

Professional basketball players face that quandary multiple times in every game. And in an article posted at arXiv.org on 29 July, Brian Skinner, a graduate student in theoretical physics at the University of Minnesota, Twin Cities, provides some mathematical guidance for the best time to take aim.

Skinner, an avid basketball fan, was inspired to analyze the game when he heard a talk at an American Physical Society meeting in 2007 on the flow of traffic. Every driver tries to minimize his or her commuting time rather than reduce the average travel time of all drivers, resulting in a paradoxical situation: Closing a road may actually reduce congestion by forcing drivers to take a route many had avoided, speeding up the average commute.

That paradox reminded Skinner of the Patrick Ewing theory in basketball, named after the high-scoring player for the New York Knicks. Analysts had noticed that in games from which Ewing or other big scorers on a team were absent, that team was more likely to win. In addition, the diagrams and flow of players in basketball also resembled the traffic models Skinner had seen.

In an article published last year in the Journal of Quantitative Analysis in Sports, Skinner proposed that virtually every equation and variable from traffic theory could be reassigned and interpreted as a description of a basketball game, with each route in the traffic model transformed into a different basketball play.

In the new paper, Skinner wanted to see if his model could predict the best time to take a shot. His equations follow the ball from the inbound pass until it reaches the hoop. He factors in the probability that a given shot will go in, the quality of future shots the team is likely to generate, and the number of seconds left before the players must either shoot or forfeit the ball to the opposing team. (NBA teams must shoot within 24 seconds of possessing the ball, whereas men's college basketball teams have 35 seconds.)

Skinner finds that, as expected, the more seconds left on the clock, the choosier a player ought to be, passing up all but the highest quality shots. But his model also points to some less obvious conclusions.

Suppose, Skinner says, that team A and team B each have the same chance of scoring on a given shot but that A passes the ball twice as fast as B. Skinner further assumes that both teams have the same ball turnover rate and have plenty of seconds left on the shot clock. Conventional wisdom would indicate that team A's best prescription for success would be to shoot twice as often as B. However, Skinner's equations show that for a scenario in which if team B shoots, on average, every 20 seconds, team A should shoot every 13 seconds rather than every 10. The extra 3 seconds allows team A to be more selective about which shots to take, which turns out to be the winning strategy.

"Brian's paper poses an intuitive theoretical model of the way basketball players should select shots in a variety of situations," says Matthew Goldman, a graduate student in economics at the University of California, San Diego, who also models basketball games. "The most interesting case is the conventional situation of NBA basketball with a ticking shot clock," which analysts often ignore in evaluating the role and contributions of a player, he adds.

Theory is one thing, but putting the models into practice is whole other ballgame. "There's more receptiveness" to these ideas among basketball coaches and managers than ever before, says sports statistician Dean Oliver, who became director of production analytics at ESPN in March after working with the Denver Nuggets since 2006. Expecting a player to judge whether to shoot or wait for a better setup requires split-second decision-making that not everyone can do, he adds.

Because of the money at stake, professional teams are also reluctant to share data that might be compared with the models, Skinner says. "They don't want to comment on what they've found in their research."

In his own pickup basketball games, Skinner says he's always thinking whether the best player is shooting too much or whether the worst player should shoot more. "When you're a nerd, you can't turn it off."

Posted in Math