Readers of this blog know that I view that as a very good thing.

Justin Rao from UCSD analyzes shot-making decisions by the Los Angeles Lakers over the course of 60 games in the 2007-2008 NBA season.  He collected data on the timing of the shot and identity of the shooter and then recorded additional data such as defensive pressure and shot location by watching the games on video.  The data were used to check some basic hypotheses of the decision theory and game theory of shot selection.

The team cooperatively solves an optimal stopping problem in deciding when to take a shot over the course of a 24 second possession.  At each moment a shot opportunity is realized and the decision is whether to take that shot or to wait for a possibly better opportunity to arise.  Over time the option value of waiting declines because the 24 second clock winds down and the horizon over which further opportunities can appear dwindles.  This means that the team becomes less selective over time.  As a consequence, we should see in the data that the success rate of shots declines on average later in the possession.  Justin verifies this in the data.

Of course, the shot opportunities do not arise  exogenously but are the outcome of strategy by the offense and defense.   The defense will apply more pressure to better shooters and the offense will have their better shooters take more shots.  Both of these reduce the shooting percentage of the better shooters and raise the shooting percentage of the worse shooters.  (For example when the better shooter takes more shots he does so by trying to convert less and less promising opportunities.)

With optimal play by both sides, this trend continuues until all shooters are equally productive.  That is, conditional on Kobe Bryant taking a shot at a certain moment, the expected number of points scored should be the same as the alternative in which he passes to Vladimir Radmanovic who then shoots.  To achieve this, Kobe Bryant shoots more frequently but has a lower average productivity.  Also the defense covers Radmanovic more loosely in order to make it relatively more attractive to pass it to him.  This is all verified in the data.

Finally, these features imply that a rising tide lifts all boats.  That is, when Kobe Bryant is on the court, in order for productivities to be equalized across all players it must be that all other players’ productivities are increased relative to when Kobe is on the bench.  He makes his teammates better.  This is also in the data.

The equal productivity rule applies only to players who actually shoot.  In rare cases it may be impossible to raise the productivity of the supporting cast to match the star’s.  In that case the optimal is a corner solution: the star should take all the shots and the defense should guard only him. On March 2, 1962 Wilt Chamberlin was so unstoppable that despite being defended by 3 and sometimes 4 defenders at once, he scored 100 points, the NBA record.