[Update 5/17: Thank you for the many interesting comments! Please see my follow-up posted today.]
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? It’s a trick question: 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 of the alternative. This assumes only that his likelihood of getting a foul is time-invariant, which seems reasonable.
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 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!