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
Wages of Wins Journal 
Harold Almonte
June 12, 2006
I believe in numbers. Maybe that what? is not in the quantity but the distance.
Craig Fratrik
June 12, 2006
The implied coorelation in “the more he shoots, the better he shoots” could be backward. It could be that if he’s shooting well, he shoots more. This would make a lot of sense because there is such an immediate feedback after ever shot.
It would be interesting to try to study how often a good player starts off poorly and keeps shooting ending up significantly better.
crack
June 13, 2006
So, if the correlation is negative why don’t people shoot more? Would Garnett eventually approach 100%? And even if Garnett isn’t improving, why isn’t he taking every shot if he is shooting 52%? Why does anyone on a team take a shot if someone else on their team is shooting better? Is it selfishness? Shouldn’t the only person shooting be the one who has the best pt/shot ratio? According to this data wouldn’t that ratio go up if that person shot more?
Mark C. Foley
June 14, 2006
Hi … I’d like to point out that the definition at the top of the post correctly includes the word “eventually.” Diminishing returns might set in if an NBA game was 100 minutes long. My regular pick-up game provides anecdotal evidence that the longer a game goes the worse the whole group’s shooting % becomes (of there we’re holding constant the total points, games are to 21 always, and allowing time spent running and shooting to vary).
Mark C. Foley
dberri
June 17, 2006
Hi Mark,
Diminishing returns for me always set in much sooner. Usually after the first or second warm-up shot. After that, my shooting percentage continued to decline until people stopped passing me the ball.
Steve Sailer
June 18, 2006
So, if there’s no such thing as diminishing returns on shooting, I guess Pat Riley is an idiot for not having Shaq, who has a very high shooting percentage, just fire away even when he’s double and triple teamed. If Riley understood basketball as well as you do, he’d have Shaq shoot every time, even when he’d be quintuple teamed. He’d put Shaq out on the wing and have him shoot 18 footers over five defenders on every possession. Too bad Pat Riley fell for the myth of diminishing marginal returns on shooting.
dberri
June 18, 2006
Steve,
First question, after leaving so many comments on our work here and around the Internet, have you bought and read the book yet?
Second, are you joking on your comment with respect to Shaq? Surely you realize that if one argued that Player A was a better option than Player B with respect to scoring, the advantage depends upon other factors, such as team defense at a particular point in time.
At this point, given the evidence Marty has presented, you should be saying… “Oh, I guess I was wrong.” Or at least present other evidence to support your point of view.
If you don’t wish to do that, try reading the book we wrote. You seem quite interested, although not very informed. Reading sometimes helps when you are not informed.
At least, that’s what we tell our students.
Steve Sailer
June 19, 2006
“Surely you realize that if one argued that Player A was a better option than Player B with respect to scoring, the advantage depends upon other factors, such as team defense at a particular point in time.”
But that’s precisely what your shots-per-game analysis with which you attempt to disprove diminishing marginal returns fails to take into account.
All else being equal, a player will shoot more in a game where he has a higher chance of making his shots due to weaker defense on him. For example, your coach will insist you shoot more in games where you are being guarded by Steve Nash than in games when you are being guarded by Ron Artest. So, it hardly disproves the Law of Diminishing Marginal Returns to show that shooting percentage doesn’t go down in games where a player shoots more.
Diminishing marginal returns on shooting is true ALL ELSE BEING EQUAL (ceteris paribus).
Diminishing marginal returns on shooting percentage is less a matter of expending energy, as you theorize (although that plays a role), than of utilizing opportunities in order of likelihood of success. If your coach insists that you only shoot in the ten times per game where you have the highest percentage chance of making the shot, you’ll end up with a higher shooting percentage, ceteris paribus, than if he insists you shoot in the 30 likeliest opportunities per game.
Look at Wilt Chamberlain’s statistics for a classic example of diminishing marginal returns. In 1973, he only took 7 shots per game and shot .727. In 1962, he famously took 39 shots per game and only shot .506. Same guy, different coaching strategies about how often per game he should shoot.
Failing to understand that is why you declared Allen Iverson to be the 91st best player in the league in his MVP season when he took 25.5 shots per game. He wasn’t the best, but he was a lot better than 91st. And it’s why you overrate Kevin Garnett, who is a great player, but only took 15.7 shots per game this year on a team with few other offensive options. Garnett would be a tremendous contributor to a strong offensive team, but his current team doesn’t need him to optimize his rating on your system, they need him to take more shots than he took.
dberri
June 21, 2006
Steve,
I take it you have not yet read our book. This is odd, since it seems to have captured so much of your attention.
Well, let me once again try and walk you through where your analysis fails.
Once again you return to the Wilt Chamberlain example. You would think that you could offer more examples of phenomenon that you think so clearly explain much of productivity in the NBA. But this is the evidence you have, so let’s discuss its relevance.
You argue that Marty’s analysis fails to hold all else constant. Your Chamberlain example, though, follows the exact methodology Marty employed. You are simply comparing shot attempts and shooting efficiency. Unlike Marty’s efforts, though, you are taking two data points from one player eleven years apart. Is this what you mean by holding all else constant? Or are you trying to demonstrate your understanding of the importance of sample size? It seems odd that you believe two data points from Wilt Chamberlain from thirty and forty years ago tells us more about Allen Iverson’s productivity than looking at every single game Allen Iverson has ever played. Why do you think Wilt tells us more about Iverson than Iverson’s actual performance tells us about Iverson?
Let’s imagine that looking at every game Iverson played during his career is not persuasive enough. I found another data set that should confirm your story.
In 2004 Iverson was the leading scorer on the U.S. Olympic Team. On this team Iverson didn’t have to take 25 shots a game since he was obviously not the only player who could score (in fact he only took about eleven field goal attempts per contest). Furthermore he was not consistently facing NBA talent. So we are observing Iverson with better teammates and worse competition. His shooting efficiency should have skyrocketed.
That is not what we observed. From three point range, Iverson converted on 36.6% of his shots, about average for an NBA player, although not quite so good when one considers that the international three point line is closer than the NBA’s arc. From inside the arc Iverson only shot 39%, well below the performance of an average NBA player, and even below what Iverson normally offers in an average NBA contest. So in this example, much like we see when we look at Iverson’s entire career, we do not see efficiency rise when Iverson takes fewer shots.
I would add, and had you read the book you would know this, that NBA productivity is not strictly about shooting efficiency. You are obsessing on one aspect of performance and ignoring all else. You are demonstrating a point we make in the book, though. Analysis of the NBA often begins and ends with scoring, a point you will see when you get around to reading chapter ten.
Although there is more to productivity than scoring, scoring and shooting efficiency do matter. And increases in efficiency will raise a player’s overall productivity. At the end of the day you are correct to argue that if Iverson hit more of his shots, he would be a more productive player. And if Garnett hit fewer shots, he would be less productive.
Where your argument lacks evidence is your explanation for why Iverson fails to hit his shots. You keep arguing that if he took fewer shots he would be more efficient. The data disputes your conclusion. At least, the data on Iverson disputes your conclusion.
Let me make one final observation, which I have made before in this forum. The Wins Produced model is designed to tell us how productive a player has been. It does not tell us why that player is productive. Even if the evidence supported your Iverson story (which it doesn’t) it would not invalidate our measure of productivity. All we are telling you is what it means for Iverson to miss so many shots. Now I think he shoots poorly because he is not as good a shooter as other NBA players. You think it is because he has to take so many shots. Either way, Iverson still misses those shots and his productivity level is still lower because of this lack of efficiency.
Carlos
June 22, 2006
First, I haven’t read the book, but the methodology on this particular point strikes me as a bit odd. The point is that looking at the games of the same regular season to see if the player in question shoots better or worse when he shoots more is probably not the right way to look at it. After all, you would guess that guys will shoot more if they are shooting better (or maybe defenses will collapse on them if they shoot a lot). A better idea could be to look at what happens to guys who suddendly have to increase or decrease dramatically their possesion usage. ¿What happens when a guy who has been taking 10 shots a game has to take 20 shots a game? Also, you could make a graph with shots taken and TS% as the y and x axis and see what happens. I remember doing it a while ago for Kobe and Iverson and it was interesting. Also, I would point that Kobe and Iverson are probably the worst guys to make this kind of study, because they are so good at creating their own shot.
Carlos
June 22, 2006
And of course, the assumption that EVENTUALLY diminishing marginal returns set in it’s still true. After all, there is a reason why Shaq doesn’t take every single Heat shot. What must be taken into account is that a basketball game is not a shooting contest but that there are two sides who are changing tactics and strategies all the time and it makes interpreting this data difficult. Also, it’s entirely possible (in fact, quite likely) that some players experience much less acutely the law of diminishing returns than others. Iverson in particular has never been a great shooter but it’s excellent at creating his own shot which means that he can take an ungodly amount of shots and still make a decent amount, while if he takes less shots, he doesn’t improve much. Each player has a unique profile and achieves maximum efficiency and productivity at different points, probably.
Jake
June 23, 2006
I feel that simply considering shots taken and FG% does not yield accurate results. For instance, consider these two scenarios:
if iverson makes only 3 of his first 10 shots in a game, he may only take another 5 shots. lets say he hist 2 of them, for a total of 5 out of 15, 33%
if iverson makes 6 of his first 10 shots of a game, he is likely to shoot another 10, because he is doing so well. but if he only makes 2 of those next 10, his total of 8 for 20 gives him 40% efficiency.
more shots, more efficiency?
instead of looking at total shots in a game, compare dto shooting percentage, i would propose to look at total shooting percentage of each shot taken. what i mean to say is, how many times does iverson make his first shot of the game? second? third? etc.
moreover, one could look at the entire NBA, what would the leaguewide fg% on first-shot-of-the-game look like? would it change over the course of a game? these statistics, to my knowledge, have not been tracked, ans it may be a tough task. but if we could look at the fg% of a player, or the league as a whole, for each succesive shot attemp, over an entire season or career, we could then answer this question definitively.
Huey
March 5, 2007
This was a pretty interesting thread. I’m disappointed Steve didn’t respond.
Sam Cohen
December 8, 2007
A bit late (i.e. a few months late) to be asking a question on this post, but did this study look at FG% as a function of shots per minute for each player as opposed to total number of shots?
Lontum Nchadze
January 26, 2010
This example is a gross representation of the law of diminishing returns. A crash course in physiology will tell you that energy can not be a fixed factor. In fact, it is suppose to decrease as the number of shuts increase. However, additional factors such as the player’s enthusiasm, proximity of opponents at the time of aim, just to name a few are not constant, hence decrease energy may not lead to poor aims. Now, this is a beter way to observe the phenomena.
let the size of the field, the number of basket balls and number of baskets be constant, and let the number of players be varied. then use any parameter to measure performance of players as their number increases. NB, first observe increasing, constant and then diminishing returns. This experiment will do justice to economic theory.
Terry
May 28, 2011
I never saw 1 person play a basketball game unless you only look at freethrows. There are 10 people on the floor and if you don’t consider the efficiency of the other 4 on your team you can’t evaluate the response of the defense. If you make most of your shots and the other 4 can’t hit a shot the defense is going to adjust and double team you and your efficiency WILL go down.