Tuesday, June 23, 2009

Game Theory and the Wing-Block Dynamic

In 2004, when the Lakers played the Pistons in the NBA Finals, a lot was made of Kobe Bryant continually jacking up outside jumper after outside jumper—none too efficiently, most of the time—while monster center Shaquille O'Neal was taking fewer shots, but making them much more efficiently. On the surface, it sure seemed as though Shaq should have been getting more shots, and of course Shaq, never a wallflower at the quietest of times, was not loathe to point this out.

In 2009, when the Lakers played the Magic in the NBA Finals, a lot was made of Kobe Bryant continually taking jumper after jumper—somewhat more efficiently than before—while his "newly tough" post player Pau Gasol was taking far fewer shots, but making them more efficiently. On the surface, it sure seemed as though Pau should have been getting more shots, and surprisingly Pau, generally a quiet fellow, pointed this out with a certain degree of mordacity.

Obviously, in retrospect, the two series turned out rather differently for the Lakers, which is why the former case was judged by many as the reason the Lakers lost the series, and the latter is just a footnote. Bryant's reputation as a ballhog, already in force before the 2004 Finals, was substantially bolstered by that series, and has only just faded within the last year or two. But is that fair? Is that the only possible interpretation for Kobe's shot-taking? Or could ballhoggery conceivably help a team?

Let me be clear here. There's no question in my mind that Kobe could stand to take fewer shots than he does (unless he's just red hot). The question isn't whether he should take as many shots as he does, but whether he should take shots even when he's shooting them at a lower percentage than the post players. And this really goes for any wing player who dominates the ball (e.g., LeBron, Wade, etc.). I just mention Kobe because I watch all the Lakers games.

I'm going to look at this from a game theory standpoint. Put into elementary game theory terms, Kobe and the Lakers have a set of tactical options, and the defenders have a set of tactical options. If each side optimizes its strategy with respect to the other side, then in the end, the game will reach what's called a Nash equilibrium: Neither side can improve its result by changing its strategy unless its opponent changes it too. (The equilibrium is not named after award-winning point guard Steve Nash of the Phoenix Suns, but John Nash, award-winning mathematician and subject of the award-winning book/movie, A Beautiful Mind.)

Suppose we simplify matters by assuming that the Lakers have just two options: Kobe shoots, or Kobe passes to the post, which then shoots. And the opponents likewise have just two options: double Kobe, or play man-to-man. And naturally, we assume that Kobe shoots a better percentage over man defense than over a double team, and the post shoots better when Kobe draws a double team than when the defense plays man-to-man.

The conditions of the game do not require either side to do the same thing each time. Strategies can be mixed. So Kobe can shoot 60 percent of the time, and pass 40 percent of the time. The defense can double 70 percent of the time, and play man 30 percent of the time. The defense can even have partial strategies like a weak double versus a strong double. Under these simple assumptions, it's fairly straightforward to find the Nash equilibrium, where neither side can unilaterally improve their result. What's interesting about this Nash equilibrium is that both Kobe and the post should shoot exactly the same percentage.

Plainly, that doesn't happen very often. Very often, Kobe shoots a lower percentage than the post (even when factors such as free throws and the three-point line are taken into account); by comparison, it's relatively rare that it happens the other way around. Ostensibly, with Kobe shooting the ball so much, he's not adequately punishing the defense for doubling him. He should instead pass the ball into the post more, gradually causing the defense to double less and play more man defense, up to the point where his shooting percentage rises to match that of the post.

[EDIT: The rest of this post is largely different from what it used to be, because what follows totally swamps in significance what used to be here.]

Having said all that, I'm going to go back and suggest that that strategy actually isn't optimal. How can it be sub-optimal, if it's at the Nash equilibrium? Because the game doesn't stop when the ball hits the rim, so the game theory shouldn't, either.

When players shoot the ball against straight-up defense, the defense has the advantage on rebounding any misses, because they're usually between their man and the basket. However, when a perimeter player shoots against a double team, the rest of the players have a man advantage. In our scenario, this advantage plays out in the post, which means that (a) the chances are much improved for an offensive rebound, and (b) if an offensive rebound is gained, it usually leads to a high-percentage shot.

What effect does that have? Suppose that the man advantage on rebounding leads to an increase of 15 percent in the offensive rebound rate; for example, if the offensive used to get 20 percent of the rebounds, they now get 35 percent. And suppose also that this leads to a successful shot 60 percent of the time. If the wing player misses, let's say, 60 percent of his shots against a double team, and he faces a double team 50 percent of the time, the offensive rebounds effectively amount to an increase in shooting percentage of 0.5 × 0.6 × 0.6 × 0.15, or 2.7 percent. That doesn't sound like much, perhaps, but it's about a standard deviation's worth, the difference between a top-10 guard and a middle-of-the-road guard. And it's how much worse the wing should shoot than the post at the true optimal strategy.

Again, I'm not suggesting that this is how Kobe thinks (although I'm pretty sure he does think that his misses can lead to easy baskets for his team), or that Kobe shoots exactly as much as he ought to. But it might explain why, even if he's shooting a lower (true) percentage than his post players are, he shouldn't necessarily shoot it less.


  1. I have two probability-esque questions for you, both unrelated to this last post of your (though the first was inspired by a line in it) -

    1) I often hear people say a shooter is "red hot" or "cold", but given any random process of course we know that there will be long strings of "hits" or "misses", even if it's 50/50. In fact, I've heard that humans tend to underestimate how often these strings will occur and how long they will be - for instance, in a high school assignment that I barely recall we were instructed to go home and flip a coin 100 times and record the results. Of course the temptation is to just make up 100 results and save yourself the trouble, but the point is that you tend not to make up realistic results, which should involve some strings of four or five H or T in a row (and thus your teacher catches you). Anyway, the point is that I wonder if shooters really do get hot or cold or whether it's just our perception based on this tendency to mis-estimate a poisson process. This would not be difficult to analyze given a simple chronological list of the outcome of every shot taken by a player. You could consider both within-game and across-game. The analysis would be to plot a distribution of the lengths of the strings of hits or misses and determine whether it's exponential (i.e. poisson, i.e. all shots independent events) or not. What do you think? I'm not sure if such a data source exists, or if there would be too many lurking factors, like the defense becoming tougher or easier for long stretches, for instance.

    2) Minnesota's just been lambasted for their choice of drafting two PGs (Rubio and Flynn) with their top two picks. But looking back at past drafts, the success rate of any player is so low, even at picks 5 and 6, that it seems like this would actually be a *great* strategy, if your goal was really to assure yourself at least one solid PG. E.g. if Rubio's got a 50% chance to be an NBA starter and Flynn the same, the wolves are still only at a 75% chance to have filled their need at PG. Maybe they should have kept Lawson with that later pick, just to up the chances a little more (despite undoubtedly getting railed even more by the media). What is your take on this? I see it as a tendency to over-value the draft picks, or to selectively remember successful picks and forget bad ones, but maybe I'm missing something. This observation really didn't have much to do with probability after all, sorry :)

    Hope this comment isn't annoying; it just seems like you're someone who might appreciate these observations.

  2. No, it's not at all annoying. They're just the things I like to think about, too.

    As far as the hot hand goes, Henry Abbott mentioned a large-ish study recently in TrueHoop. It was within the last couple of months, I'm pretty sure. I forget the details, but it looked for but failed to find any evidence of a hot hand. In fact, jump shooters who had just made a shot proceeded to shoot a lower percentage on their next shot.

    However, one should take care in drawing any hard-and-fast conclusion from that last observation. In particular, the study also found that jump shooters who had just made a shot also became significantly more aggressive in taking their next shot. So it's quite possible that they were in fact a bit hot, but that was more than counteracted by their poor shot selection. The study was careful to say that the hot hand had not been disproved--only that no evidence had been found for it.

    As far as Minnesota's World of Point Guards is concerned, it might be a reasonable strategy if (a) you were only concerned with getting a good PG, and (b) the draft were the only way to get one. I'm not sufficiently informed about Minnesota's situation to know if (a) is true, but (b) certainly isn't; you can either further develop one of the players you already have, or you can trade for one, potentially using one of the players you just drafted. Such situations are a bit too contextual to attach much confidence to a statistical analysis, but I think you can see that there's more to the situation than just flipping two coins and hoping at least one comes up heads.

    That doesn't mean that Minnesota necessarily screwed up. They might have felt that those were the best players available at those times, regardless of need. But I seem to recall that they traded for the fifth pick, so I'm not sure what their overall strategy is. Doubtless that's partly by design. But the fact that most people (i.e., not just me) are having trouble wrapping their heads around the parade of PG picks suggests that there's probably something screwy going on, too.