Tuesday, April 27, 2010

Foul trouble

Most game-players are sports fans, so I'm throwing in a little essay I wrote on foul trouble. It was targeted for an econ blog, and uses one or two technical terms, but they aren't central.


In a professional basketball game, a player is disqualified (“fouls out”) if he is charged with 6 personal fouls. Observers of the NBA know that the direct effect of fouling out actually has less impact than the indirect effect of “foul trouble.” That is, if a player has a dangerous number of fouls, the coach will voluntarily bench him for part of the game, to lessen the chance of fouling out. Coaches seem to roughly use the rule of thumb that a player with n fouls should sit until n/6 of the game has passed. Allowing a player to play with 3 fouls in the first half is a particular taboo. On rare occasions when this taboo is broken, the announcers will invariably say something like, “They’re taking a big risk here; you really don’t want him to get his 4th.”

Is the rule of thumb reasonable? No! First let’s consider a simple baseline model: Suppose I simply want to maximize the number of minutes my star player is in the game. When should I risk putting him back in the game after his nth foul? The phrasing is deceptive, because I shouldn’t bench him at all! Those of you who haven’t been brainwashed by the conventional wisdom on “foul trouble” probably find this obvious. The proof is simple: if he sits, the only thing that has changed when he gets back in is that there is less time left in the game, so his expected minutes have clearly gone down (in fact the new distribution on minutes is first-order stochastically dominated, being just a truncation.)

OK, while I believe the above argument is very relevant, it oversimplified the objective function, which in practice is not simply to maximize minutes. I’ll discuss caveats now, but please note, there is tremendous value in understanding the baseline case. It teaches that we should pay attention to foul trouble only insofar as our objective is not to maximize minutes. I am very comfortable asserting that coaches don’t understand this!

First caveat: players are more effective when rested. In fact, top stars normally play about 40 of 48 minutes. If it becomes likely that a player will be limited to 30-35 minutes by fouling out, we may be better off loading those minutes further towards the end of the game to maximize his efficiency. Notice, though, that this doesn’t lead to anything resembling the n/6 rule of thumb. It says we should put him back in, at the very latest, when he is fully rested, and this isn’t close to what is done in practice. In fact players often sit so long the rest may have a negative impact, putting them “out of the flow of the game.”

Second caveat: maybe not all minutes are created equal. It may be particularly important to have star players available at the end of the game. On a practical level, the final minute certainly has more possessions than a typical minute, but it also has more fouls, so maybe those effects cancel out. I think the primary issue is more psychological: there is a strong perception that you need to lean more on your superstars at the end of the game. I think this issue is drastically overrated, partly because it’s easy to remember losing in the last minute when a key player has fouled out, but a more silent poison when you lose because you were down going into that minute having rested him too long. By the way, my subjective sense is that the last possession is more similar to any other than conventional wisdom suggests: a wide-open John Paxson or Steve Kerr is a better bet than a double-teamed Michael Jordan any time in the game. On a couple of major occasions, Jordan agreed. This isn’t to underestimate the star’s importance in scoring and getting other players good shots, just to say that this is not necessarily more important in the final minutes. You do often hear that a team will rise to the occasion when a star is injured or suspended, so even conventional wisdom wavers here. Finally, note that the foul-trouble rule of thumb is applied also to players who aren’t the primary scorer, so that this argument wouldn’t seem to apply. I will give coaches a little credit: they do sometimes seem to realize that they shouldn’t worry about foul trouble for bench players who often don’t play at the end anyway.

One more psychological caveat: a player who just picked up a foul he thinks is unfair may be distracted and not have his head in the game immediately afterward. This may warrant a brief rest.

Final note: Conventional wisdom seems to regard foul management as a risk vs. safety decision. You will constantly hear something like, “a big decision here, whether to risk putting Duncan back in with 4 fouls.” This is completely the wrong lens for the problem, since the “risky”* strategy is, with the caveats mentioned, all upside! Coaches dramatically underrate the “risk” of falling behind, or losing a lead, by sitting a star for too long. To make it as stark as possible, observe that the coach is voluntarily imposing the penalty that he is trying to avoid, namely his player being taken out of the game! The most egregious cases are when a player sits even though his team is significantly behind. I almost feel as though the coach prefers the certainty of losing to the “risk” of the player fouling out. There may be a “control fallacy” here: it just feels worse for the coach to have a player disqualified than to voluntarily bench him, even if the result is the same. Also, there is a bit of an agency/perception problem: the coach is trying to maximize keeping his job as well as winning, which makes him lean towards orthodoxy.

There are well-documented cases in the last decade of sports moving towards a more quantitative approach, so maybe there is hope for basketball strategy to change. The foul-trouble orthodoxy is deeply ingrained, and it would be a satisfying blow for rationality to see it overturned.

*Final outcomes are binary, so the classical sense of risk aversion, involving a concave utility function in money, doesn’t apply at all. But there is also a sense of what I call “tactical risk”: a decision may affect the variance of some variable on which your probability of final success depends in a convex (or concave) way. I might write an essay sometime on the different meanings of “risk.” Anyway, here you would presumably should be risk-averse in your star’s minutes if ahead, risk-loving if behind. But this is rendered utterly moot by first-order stochastic dominance!


Memphis MOJO said...

This is a great post. Is there any way to test it?

Ken said...

"Also, there is a bit of an agency/perception problem: the coach is trying to maximize keeping his job as well as winning, which makes him lean towards orthodoxy." More than a bit. The lag from statistical insight to change in practice can be decades for this precise reason. C.f. the early-inning sacrifice bunt (first computer-aided debunking in the 1970's), which has some other similarities as well (a double play is the "riskier" and less-"controlled" outcome).

The Pretender said...

My (long) response posted on my blog. I think at a basic conceptual level, I disagree with you for the same reason that I disagreed with your tennis post a while back.

Jonathan Weinstein said...

Kenny: Definitely there are major obstacles to adopting unorthodox strategy. Aside from media/ownership backlash, the players have to feel good about implementing it. If they are unused to playing with foul trouble, they may be tentative and ineffective (as Henry said). But change can happen, as with 4th-down strategy in the NFL, if high-prestige coaches like Parcells and Belichick implement it knowing they can weather the criticism and others eventually copy.

Henry: Sorry I haven't had time to give this the response it deserves. Our thinking may not be all that far apart. We agree the rest thing is not a major issue (we both mentioned the rust possibility which counteracts it.) One real disagreement is how different the last minute is from previous minutes. My Paxson/Kerr examples certainly don't prove anything, they are if anything an attempt to dispute the CW that on your last possession your star should force it over a double-team rather than pass, which is tangential to the foul trouble debate. No doubt you're better off with the star on the floor to create, on the last *or* any other possession. Anyway, I've watched a lot of games too, and I'm just not all that convinced that the last minute is all that special, other than the obvious fouling by the trailing team. There are even some stars who become less valuable in the final minute when the other team is in a must-foul situation, if they are big men who can't shoot FTs. It's the big men we're usually talking about with foul trouble anyway.

The other thing you point out is that players may lose effectiveness if playing with foul trouble. This could be a serious issue; the coach would need to make a judgment here, but if I am right that you are better off with the star in as long as he plays "normal," perhaps a coach with high enough prestige and trust can teach him to stay intelligently aggressive when he keeps him in. If he is able to *intelligently* play differently from normal, that only enhances my argument. And if the other team distorts their strategy to try to foul him out, that can backfire. I remember teams looking foolish force-feeding their mediocre center to try to foul out Shaq.

The Pretender said...

I did not say that the last minute is different from previous minutes. I just said that not all minutes are alike, and the point at which later minutes are more important than earlier minutes does not necessarily start in the final minute.

In a vaccuum, I don't disagree with the your main gist. For a must win game, the foul trouble thing really doesn't become a problem. Eg. Dallas @ San Antonio, game 6, Dirk got 3 fouls early, Carlisle left him in, he still played aggressively, got a 4th stupid foul, took him out, played him 2nd half, didn't get 5th foul till very end. However, the grind of an 82 game season affects a lot of decisions and so the majority of any sample will not conform to any simple theoretical model. Whereas the fourth down situation eventually resolved itself in football because each game is so important in a 16 game season.

Another thing is that most of what you consider conventional wisdom (this is apparent in your essay) comes from the sports announcer. The sports announcer is not an expert. That's why they hire a former coach or player to fill in the actual basketball knowledge.

Most, if not all, coaches don't really have that "conventional wisdom" flaw to their thinking. As far as "prestigious" coach is concerned, another thing is that the NBA is the most star-oriented league of all the sports.