# The Law of Diminishing Returns in the NBA

Very few ideas in economics rate the title “Law.” The idea of diminishing marginal returns is one of these. For those who did not take an introductory economics class in high school or college, or simply forgot the formal definition, here is what this law states:

As one increases a variable input (such as labor), holding all other inputs in the production process constant, each additional input added will eventually yield less and less additional output.

Basically, this means that as you hire more and more workers, holding all other factors (machinery, land, raw materials, …) constant, the additional output you receive from workers, while still positive, should decline. The important part of the definition is that all other factors are being held constant. Because the new workers have to share the fixed quantities of machinery, land, raw materials, etc… with existing workers, this necessarily, at some point, decreases everyone’s productivity.

In basketball almost all inputs, i.e., the length of the game, the number of basketballs (yes, there is just one), are held constant. This is why when very good players are paired together on the same team their individual performance declines. And this point is clearly detailed, both theoretically and empirically, in The Wages of Wins.

A type of diminishing marginal returns has been suggested in some of the comments on our blog and elsewhere – that one reason a player shoots inefficiently is because he shoots too much. The fixed input here, I imagine, would be energy. If a player has only a fixed amount of energy, each additional shot decreases the remaining amount of energy, fatigue sets in and he is less successful.

Now if this were truly diminishing returns, as it is defined, then the next shot you take should be less likely to go in than the shot you just took. Of course, a player’s energy (or his shot selection) does not continually decline as the game progresses. So in other words, there is no reason to think that the first shot taken isn’t the worst shot the player will take in a given game. So the theoretical basis of this story is suspect.

Still, the proof is in the data. If this supposed application of diminishing marginal returns has legs, one might suspect a negative relationship between shot attempts and shooting percentage, i.e., the more shots a player takes, the lower his field goal percentage.

Using box score data for every NBA game played between 1991-2005 seasons (Thanks to Justin  Kubatko of Basketball-Reference.com – Basketball Statistics, Analysis, and History. http://www.basketball-reference.com/ for providing the box score data), Jeff Gerlach and I looked into whether such a relationship exists between shoot attempts and shooting efficiency.

First some specifics, while there are more than 400,000 player observations over the period, some players failed to record a field goal attempt during some games. These players were removed from the analysis. We also removed, somewhat arbitrarily, anyone who recorded 5 or less field goal attempts. We are then left with a total of 255,854 observations.

In the end, we did find a relationship. Only it was positive!

Specifically, if a player took 1 extra shot during the game, his field goal percentage would increase by a whopping 0.0017. 10 more shots during the game increased field goal percentage by 0.017. Not much of an effect. As we say in the book, not much oomph!

A positive relationship might make sense here because of the more than 5 shots cutoff. If good shooters shoot the ball more, one might suspect that their shooting percentage would be higher or that coaches prefer better shooter shooting more. With this in mind, we changed the cutoff to more than 10 shots. This yields 131,932 total player observations. And while the relationship is still positive, its impact has decreased fivefold to 0.0003.

What if the cutoff is more than 20 field goal attempts in a game? Now we have limited our sample to 15,150. One might suspect that this is the group most likely to experience the diminishing marginal returns phenomenon people expect. When we look at the relationship between total number of shots and shooting percentage for this group, there is no relationship. Nothing, Zilch, nada,…

Now I’m of the opinion that making the cutoff twenty shots in a game should control for player quality, but let’s get specific and look at the two players people have focused upon, Kobe Bryant and Allen Iverson.

Kobe took at least five shots in 739 games. For these games we find a significant and POSITIVE relationship between shot attempts and field goal percentage. The more he shoots, the better he shoots. When we look at the 258 games where he took at least twenty shots we find no significant relationship at all.

What about The Answer? Iverson took at least five shots in 663 games. Just like Kobe, we find a significant and POSITIVE relationship between shot attempts and field goal percentage. Again, the more he shoots, the better he shoots. And when we look at games where he took at least twenty shots we again find no significant relationship at all.

What does all this tell us? The story that shot attempts and shooting efficiency have a negative relationship is not in the data. In sum, the diminishing marginal returns to shooting tale told by some is suspect theoretically. And empirically, it doesn’t seem to have legs.

- MBS