Maximizing offensive efficiency

During Kobe’s “hot streak,” I’ve been writing that he’s actually inefficient compared to Andrau Gasnum, the Lakers’ superb tw0-man post presence. I’ve said that he should give up some shots until his efficiency equalizes with Gasnum’s. Adrian the Canadian was quick to send me a Sloan Sports Analytics Conference paper arguing that teams  might equalize offensive efficiency too much already. The author (Brian Skinner) uses some network theory for unknown reasons (it’s not related to his point), but the paper boils down to a comparison of the marginal cost and benefit of changing shot selection from one player to another.

Assume that players’ efficiency decreases as they take more shots (due to more defensive attention, fatigue, etc.). Imagine you are Kobe. Giving Andrau Gasnum one of your shots has the following marginal benefits:

  • higher efficiency on all your other shots
  • higher efficiency of this specific shot (since Gasnum is more efficient than you).

The marginal cost:

  • lower efficiency for Gasnum on all his other shots.

If the benefits exceed the costs, shifting shots is the smart move. Skinner argues in the paper above that these costs might be less than the benefits at extreme values, but that the costs will overtake the benefits long before efficiency equalizes (which is the point that teams might aim for).

I think he’s technically correct — I was wrong in my earlier posts to say that the Lakers should shift shots to Gasnum until the efficiencies are exactly equal. For example, if Kobe gives all his shots to Gasnum and, in doing so, equalizes their efficiencies, there will be no gain on that individual shot (efficiencies are now equal) and no gain in Kobe’s efficiency on other shots (he has none left). Meanwhile, the shot will lower Gasnum’s efficiency on ALL other shots. The right point to stop shifting shots will surely happen even sooner than this.

However, Skinner’s examples depend heavily on the functional form that he assumes for the shot attempt/efficiency relationship and on teams’ starting points (he assumes teams start from the point where all efficiencies are equalized). He has no data — just a model. In his example with Ray Allen, he assumes that the less efficient player (Kobe in our case) does not improve his shooting percentage when he gives up a shot to the more efficient player (Gasnum). Kobe might take bad shots no matter what since he’s more macho than smart, but this completely eliminates one large marginal benefit of shifting shots away from him!

So, while I agree with Skinner that efficiency should not be equalized completely, I still think there are opportunities to improve efficiency by shifting shots and bringing efficiencies closer together. In the Lakers’ case, Kobe has taken at least 28 shots in all four of his 40 point games! Gasnum didn’t shoot nearly as many. An increase in Kobe’s efficiency would affect a ton of shots, while a decrease in Gasnum’s would affect far fewer, suggesting that the benefits outweigh the costs for the Lakers. (Again, all of this depends on the true relationships between efficiency and shot volume.) The lesson is a two parter: (1) teams shouldn’t equalize efficiencies exactly, but (2) there’s probably some benefit from bringing efficiencies closer together when they are way out of whack.


8 responses to “Maximizing offensive efficiency

  1. Nice post. One thing to think about is during what situations shots are taken. I’m a critic of Kobe’s slot selection as well. However, you can imagine that at certain points – little time left on the shot clock, maybe against certain defenses – Kobe will take low probability shots because they’re low probability shots for everyone on the court and Kobe has the highest low probability.

  2. Yes, this is definitely true. There are some possessions on which there is no high percentage shot, and guards often end up with the ball on these possessions (i.e., the ball is far from the hoop late in the shot clock). However, you and I know from observation that Kobe forces lots of shots early in the clock as well.

  3. Nice post Tyler. Isn’t this an issue of marginal vs. average efficiency? Teams should allocate shots to equalize marginal efficiency across players, but this might lead to differences in average efficiency. In concrete terms, the Lakers should run plays that get layups and dunks for every player when they can, then move to open jump shots, on down to contested jump shots, etc. If Gasnum’s efficiency falls off quickly on plays that aren’t layups and dunks, then they’ll end up with higher per-play average efficiency than Kobe even if the Lakers are playing optimally. That said, I totally agree with you that at the margin, Kobe is likely to be less efficient than Pau.

  4. I think Skinner has the right maximization problem in his paper (sum of x_i * f_i(x_i), where x_i is the number of shots taken by player i, and f_i(x_i) is his efficiency on all those shots, as long as he gets the modeling right. One key assumption to his (and, implicitly, my) approach is that the efficiency of all your shots, even those already taken, is affected by taking one more shot (this is like a long run equilibrium efficiency from adding that extra shot I guess). So, in that case, maximizing average efficiency, which is what Skinner and I do, ends up being the right thing to do I think. This might not be exactly fair, but I think it’s important to consider the effect of each shot on future shots as part of the marginal effect of that shot at least.

    Concerning your example, I might argue that the Lakers should keep running plays for Gasol to get layups and dunks until the average efficiency of that play came down to the level of other plays. Would you agree?

  5. I agree, that’s the right objective function, and I also agree that maximizing average team efficiency is the right way to go. However, I don’t agree with you that equalizing average efficiencies across players or plays is right.

    To see my argument, consider a stylized example. Suppose that Kobe shoots 50% on every shot. Suppose you can run a play for Gasol that results in dunks the first 3 times, but every time you try beyond that results in a turnover (maybe the defense figures it out). Then if you equalized average efficiencies, you’d run the Gasol play 6 times, resulting in 3 dunks and 3 turnovers, and each player would have 50% efficiency. But this is clearly a bad idea — you want to run the Gasol play only 3 times, resulting in a 100% shooting percentage, and give the rest of the shots to kobe who has some chance to make them, leaving him with a lower 50% shooting percentage.

    I think Skinner’s paper captures this. His Ray Allen example is all about how marginal efficiency, not average efficiency, should be equalized, and how this might result in Ray optimally shooting a higher percentage on average than his teammates because his inframarginal shots are better.

  6. Yes, sorry, my last comment suggests equalizing average efficiency again, which is not right (that was my original mistake in my previous post). Using Skinner’s linear functions, average efficiency (which is the only stat we have) is somewhat a proxy for marginal efficiency, thus my use of it in the last few posts, but as you say, marginal efficiency is what needs to be equalized.

    It’s my subjective opinion that the Lakers’ current shot allocation is inefficient. The question is which side of the hump they are on in the graph Skinner gives for his Ray Allen example. As I described above, his functional form artificially supports his argument, so I think the maximum should be located at a point that gives Ray Ray more shots. In your example, average efficiency wouldn’t help you figure out where they are on the graph, since marginal efficiency changes abruptly, but I think with a smoother function, comparing average efficiencies and shot counts has some value.

    • I think we’re in agreement here. The linearity in Skinner’s paper helps with the calculations but I think the intuition that optimal shot selection is about marginal efficiency isn’t dependent on functional form.

      That said, I share your belief that Kobe shoots too much and the Lakers are inefficient. One could play devil’s advocate and argue that Gasol and Bynum cannot create their own shots, so their high shooting percentages are driven by putbacks and dunks they get out of double-teams on Kobe. But I think this would be selling them short; both are very effective in the post and usually get double-teamed, which creates good shots for the Lakers’ guards as well. More of this and less contested jump shots by Kobe would probably be good (though he’s shooting 51% from 16-23 feet this season according to!).

  7. Pingback: Steve Kerr agrees | Causal Sports Fan

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