Diminishing Returns in the NBA

It probably isn’t worth it at all, but the interwebs have been abuzz recently with discussions of diminishing returns, especially as it relates to rebounding and Wins Produced.  Before anyone gets too excited, I don’t have any data of my own, but I will comment on some other things that people have presented.  This is going to be more of a theoretical/intuitive/procedural discussion.  Consider yourself warned.

First, let’s talk about the concept of diminishing returns.  Broadly speaking, the term means that after a certain amount of something has been produced, addition units provide less benefit than the same number of previous units.  For example, I am a poor graduate student and would love to have $100.  To me, getting $100 might be worth 500 arbitrary happiness points.  But Bill Gates doesn’t need $100; if you gave him another $100, it might be worth 10 happiness points.  That’s diminishing returns.  The rebounding debate isn’t really about this (although it could be); the issue here has to do more with competition.  The claim is that there are only so many rebounds to go around and so if someone gets one, that rebound isn’t there for a teammate (or opponent) to get.  Thus if you put out a line-up with 5 great rebounders, they will cancel each other out to some extent and they will not be as valuable as you might think when they played separately.  The diminishing returns here are in the value of having additional good rebounders (or shot-blockers, or whatever).

Given that we are working under that definition, diminishing returns almost certainly occurs.  In the extreme, it must.  There are x possessions in the game, and y missed shots.  Let’s say you assemble a rebounding team so dominant that they get every rebound.  If there were more missed shots, your team would also get those rebounds, but one player on the team would have to get it at the cost of other players not getting it.  The same reasoning holds for blocks, steals, shots, etc, although there are generally fewer of these per game (excluding shots, obviously).  In fact, we are seeing diminishing returns in action right now with Chris Bosh in Miami; his usage has dropped from an average of 26.2% (excluding his rookie year) to 23.3% so far this season.  The real question is how much diminishing returns occurs for any given statistic and how should it be measured.

So now let’s talk Wins Produced.  The idea is that point differential leads to winning.  Point differential is based on your team’s offensive and defensive efficiencies.  Efficiency is based on possessions, so it is critical to shoot efficiently and get the ball.  Getting the ball happens via rebounding and steals, and losing the ball happens via turnovers.  The question is how valuable the box score stats, which we can measure fairly objectively, are to efficiency, and thus winning.  WP does this with a regression to determine the weights on the different statistics, and it turns out that rebounding is fairly important since it ensures that your team has the ball.

I don’t think anyone has strong objections to this point (at least according to Arturo’s new post and the comments there) (with the exception that it supposedly doesn’t include defense, which isn’t critical to this discussion).  Instead, people are angry with taking these weights and applying them to the player level, which is the real contribution of WP.  Doing so allows us to find the value of individual players, and it has been fairly successful both in retrodicting team performance and predicting future player performance.  But it leads to great rebounders being evaluated as top players (Marcus Camby comes up a lot; you could also look at Ben Wallace) even though they are offensively challenged, and this disagrees with the (oddly absent from statistical textbooks) eye test.  Again, one of the current contentions is that if Camby hadn’t grabbed those rebounds, someone else on his team would have, and so Camby isn’t as valuable as WP thinks he is.

In the comments for Arturo’s post, the interested reader is directed to this article (which refers to this similar article), which purports to find evidence for diminishing returns in defensive rebounds at least, if not offensive boards.  Both of these articles took line-up level data and estimated how they should rebound by adding together the rebound percentages for the players in the line-up.  So player A gets 3% of available offensive rebounds, B gets 5%, etc, and when you add up the five maybe you get 25%.  Then they found the percentage of offensive rebounds actually gathered in by that line-up.  If there are no diminishing returns, a regression line through this scatterplot (predicted versus actual rebound percentage) should have a slope of 1.  If you believe in diminishing returns, you might expect for the line to dip down at high rates (good rebounders get in each other’s way and lower their rates) and rise up at low rates (rebounds will fall more often to players if they don’t have to compete for them with good rebounders).  Both articles find a slope of around .7 for offensive rebounds and around .3 for defensive rebounds, providing evidence for diminishing returns.

Or do they?  Issue number one: using percentage of rebounds.  I would generally support the idea of using percentage of rebounds available, because this helps correct for both pace and shooting quality.  Players on the Warriors have many more chances to get rebounds because they usually play fast and miss a lot of shots, whereas the Spurs play slowly and make a lot of shots.  Comparing David Lee to Tim Duncan on rebounds per game may not be a fair comparison.  However, percentages have a disadvantage in that they are bounded; you can only get 100%.  If I ran out the imaginary line-up of this year’s top rebounders so far and added them up, Reggie Evans, Joey Dorsey, Kevin Love, Marcus Camby, and Dominic McGuire would be expected to get 120% of available rebounds.  Some might find this unlikely.

The issue still arises even if you don’t have the top five rebounders in the league; it is difficult to move closer to 100% as you start closer to 100% (something I’ve mentioned before).  Thus it makes total sense for the disparity to be larger for defensive rebounds.  The projected values are as high as 85% in one article and 96% in the other, so it’s virtually guaranteed that the actual values will not be that high.  On the other hand, offensive rebounds only range up to about 40% and the boundary issue isn’t a problem.  What should you use instead of rebound probability?  It’s a tough call.  In this case, since you’re essentially comparing players to themselves (what you think they should get to what they actually get), I think using the straight box score rebound counts would be ok.  Like I said, I think you still might find diminishing returns, but I don’t think it will be as dramatic.

Issue number two: properly attributing rebounds.  Let’s remember, “no one” is arguing that rebounds aren’t worth .03 wins.  So if your team gets 50 rebounds in a game, they have gained 1.5 wins (balanced out by some number of possessions lost and points scored/surrendered).  The articles linked above, which find regression weights of .7 and .3, suggest that those values should be used for individual players.  That is, when Marcus Camby gets 3 offensive boards and 8 defensive boards, he should be credited with 2.1 offensive boards and 2.4 defensive boards.  But that would also apply to all of his teammates.  So if Camby played in that 50 rebound game, and 8 of those were offensive, the players would end up credited with 5.6 offensive + 12.6 defensive boards = 18.2 rebounds.  We just lost 31.8 rebounds, or almost a full win.

For this enterprise to mean anything at all, individual player rebounds must add up, both literally and in terms of credit, to team rebounds.  So those 32 rebounds have to go somewhere.  WP takes the apparently erroneous approach of crediting them to the people who got them.  What would diminishing returns people suggest?  There are 9 other players on the court, so maybe we should give (1-.7)/9 offensive rebounds to everyone else when Camby gets a rebound.  Or maybe we should give the .3 rebounds to Camby’s back-up, since he would have gotten that rebound if he had been in the game.  Or we could watch tape of every game and see who was near Camby.  We would then take each player’s distance from Camby and combine it with their height and gumption level (adjusted for clutch rebounding ability in the last two minutes of close games) to calculate a probability of having grabbed the rebound if Camby hadn’t been so rude as to diminish their returns, and multiply that percentage by .3 for teammates and .7 for opponents.  This would ensure that player credited rebounds (PCR) would add to team rebounds and thus wins would be calculated and attributed correctly and fairly.  If someone with game tapes does this, I’d like to be credited with the PCR idea.

Hopefully that came across as sufficiently sarcastic.  I can’t really see any reasonable, objective manner for attributing potential rebounds to other players.  We could wave our hands and give up and use a team adjustment (players on good rebounding teams all get equal credit for that rebounding) as WP does for defense, but we would then have to do the same thing for any statistic that suffers from diminishing returns and soon we wouldn’t be measuring individuals at all.  But I’m willing to listen to suggestions, hopefully from proponents of diminishing returns/detractors of WP.  Hopefully this PCR measure would then be consistent as players move from team to team, thus reflecting their true ability to rebound/block out for teammates/whatever their contribution is.  I look forward to seeing the new model!

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16 Responses to Diminishing Returns in the NBA

  1. EvanZ says:

    “I can’t really see any reasonable, objective manner for attributing potential rebounds to other players.”

    Actually, there is a very simple way. Give credit to the defense for “creating” a missed field goal. Let’s say each player gets 0.14 points of credit (0.7/5) for defense, and the rebounder gets an extra 0.3 for the rebound. The rebounder then ends up with 0.44 in this simple distribution method. The points still add up, just as they do the other way by giving all the credit to the rebounder. That’s the way I’m heading these days, anyway…

    • Alex says:

      I don’t think that’s a reasonable way of divvying up rebounds though. Do all players on the floor equally contribute to someone missing a shot, or gathering the rebound? If Battier forces Kobe into a tough jumper and Scola gets the rebound I might see some reason for crediting Battier (although he already gets credit in the defensive adjustment), but what did the other three Rockets do? Battier, in fact, might have missed the box out on Kobe who almost got the ball before Scola did. Or maybe Kobe dumps it in to Pau, who turns and misses and the ball goes straight to Yao. None of the other Rockets did anything. Why should anyone other than Yao get credit? Broadly speaking, it seems like giving guards and bigs equal credit is wrong by default.

      There are infinitely many ways you could divide up the credit for a rebound amongst the five players on a team, so long as they add up to one. Dividing it equally is one, giving it all to the rebounder is another, and the method you suggest is a third. The question is which one is most correct? Giving the rebounder all the credit has all the supporting evidence that goes with WP associated with it; is there another set of weights that does better?

  2. nerdnumbers says:

    Alex actually a great point with the divvying idea is the team defense. Right now the Wins Produced metric does account for defense (for all naysayers, seriously look it up). However it does this at the team level. The effect is that the team wins are more accurate and each player gets a little of the credit (based on time on the floor). Now if I drop the team aspects of the Wins Produced formula, guess what? Even on the worst teams in the league the WP48 of each player looks very similar. So with say rebound divvying the only way this will get a huge influence on the players (remember DJ tried a 0.7 weight on it at one point btw.) is to give most of the credit for a rebound to the non-rebounding players for the rebound. If this “tweak” makes logical sense to anyone, please step forward.

    A big question here is does the model tell us an accurate representation? I argue yes, some people argue no, but hey everyone’s entitled to their opinion. A model’s strength should be if it represents the data right, not if everyone likes it (As I’ve said before, the Laugh Test is a terrible test of any metric.)

    By the way, awesome article. I tried to write something up and couldn’t get something I liked. I was pleasantly surprised to see this awesome piece of writing up when I woke up this morning.

    • EvanZ says:

      Now if I drop the team aspects of the Wins Produced formula, guess what? Even on the worst teams in the league the WP48 of each player looks very similar.

      Dre, your calculation raised a red flag in my mind, since Dave has said (from the web page on calculating wins produced):

      The average value, in absolute terms, of DEFTM48 is 0.011, so again this is a very small adjustment. And as we saw with MATE48, DEFTM48 has very little impact on our assessment of individual players. The correlation coefficient between P48 and Adj. P48 in 1977-78 was 0.9977.

      Why the disconnect? Or am I misunderstanding what you did?

      • nerdnumbers says:

        I’m saying the same thing Dr. Berri is. Namely if you take a team adjustment and cut it up among all of the players then the impact is small. So doing some kind of 30% rebound reattributer would likely make very little impact on each player’s WP48, be confusing at best and again as Alex points out, why do it?

      • Alex says:

        If I’m following the discussion, including Andres’ comment from today, I think there is one important difference. If you remove the teammate or defense team adjustments you aren’t necessarily disadvantaging anyone in particular. For example, Chris Paul is good because he shoots well, gets lots of assists, doesn’t turn the ball over, etc. None of these is impacted by the teammate or defense adjustments, and they are presumably mostly independent of these adjustments. But, if you start taking rebounds away from good rebounders and giving them to his teammates, this will drastically affect their wins produced. You will systematically lower the contribution of players who get lots of rebounds while raising the level of all other players. I imagine that if you made this change (let’s call it MWP for modified wins produced) and correlated it with WP, you would get something far less than .9977.

  3. dberri says:

    This really is an excellent article. Diminishing returns does exist (as was noted in The Wages of Wins, Stumbling on Wins, and various academic articles). But to account for this in a metric like WP –as this article notes — is problematic. I think it is better to “credit” a player for the stats he accumulates. When you think about why he got those stats, though, you need to think about the factors that cause performance to change. And diminishing returns is on the list (although the impact on per-minute performance does appear to be small).

    • StL Reflections says:

      I do think this is the core of the question-is it ‘better’ (that is, a more accurate representation of a player’s contribution to a winning basketball team, a useful measure to decide which players to pursue as a GM, etc.) to give a rebounder full credit for a rebound? In WoW, the rebounder gets credit for the change of possession, and they are only partially responsible for that (along with the other defenders). The data suggests that prolific rebounders leaving a team do not have the negative effects on rebound margin you would expect based on their numbers, JUST LIKE the loss of prolific shot takers doesn’t really hurt a team’s usage rates.

      • bduran says:

        So what if a passer has multiple choices of open players and he chooses and that guy makes a shot. Shouldn’t the other open guy get some love? Remember, according to Dberri’s research every stat has diminishing returns.

        • Alex says:

          Maybe, but then you’d have to give the other open guy partial credit for a missed shot too, since we don’t know if he’d make it or not. Which means we’d have to take away credit from the guy who made the assist, because if he had passed to the other guy and he missed it, there wouldn’t have been an assist. If he had missed, there would be a rebound to account for. I think the hypotheticals get out of hand quickly.

      • bduran says:


        I didn’t intend that comment as an endorsement of the suggestion.

        • Alex says:

          That’s fine; I don’t mean to be dismissive. I honestly would be happy to see someone generate a reasonable, powerful alternate set of weights that attribute credit to all the players who deserve it, for any given stat. Before I heard about plus/minus and all it’s variants, I thought that would be a great way to account for everything and wondered why people didn’t do it, especially with some kind of regression. If player X is on the court and his team gets a lot of rebounds then, even if he didn’t get them, it seems like he should get credit for those rebounds. Then I read about adjusted plus/minus and all the issues it has, and felt kind of let down (especially since I would love to do the same thing with football, but the problems there are even worse). So I feel that WP is the next best thing practically speaking, and I think it makes sense to attribute all the credit to the player who gets the stat rather than intuiting or making up weights. I think we all know that WP is an incomplete description of what happens on the court, but until someone can demonstrate a better model I think it’s what we should go with.

  4. Pingback: How About Those Jazz? (and answers to other questions) | The Wages of Wins Journal

  5. Devin says:

    Yeah, this is awesome. It’s going on the FAQ list!

  6. Austin says:

    Maybe I’m just missing something, but it seems to me that to account for diminishing returns, the easy solution (in terms of calculating a WP type metric) is to not use a linear weight.

    That is, if each subsequent defensive rebound is worth less, then use (dreb)^.5 or (dreb)^.7 or whatever fits the data.

    Would this be doable?

    • Alex says:

      Austin, it isn’t so much that the rebounds become worth less as that the rebounders (the players themselves) become worth less if they are good at rebounding (since they get rebounds that teammates might have otherwise gotten). So the debate is more about how to value rebounds at the player level while still having them add up to team-level rebounds, which everyone is happy to say is linear and accounted for correctly.

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