This blog is ordinarily read by a small network in the economic theory community, so it was a jolt to see a thousand hits in a day for my “Foul Trouble” post shortly after we were linked by Tyler Cowen’s “Marginal Revolution” blog. The post was also featured by columnists at NBC Sports and ESPN. I even got a kind note from an executive VP of the Houston Rockets. The Rockets were featured in a NYT magazine article by Michael Lewis last year as the “Moneyball” team of the NBA – that is, in recent years they have focused heavily on analytics. The Rockets executive said that he enjoyed my article, but no word on to what extent they agree with me — trade secrets? I’ll continue my thoughts here, but possibly they are way ahead of me.
As economists know, the point of a simple model is not to be “correct” but to serve as a framework for analysis. As many comments noted, I only discussed some of the possible adjustments to the basic theory, and for those I did mention I only offered informal estimates as to whether the adjustment is small. Does the baseline recommendation to ignore foul trouble have “so many caveats, it’s useless” as one comment complained? I don’t think so, and, still not claiming exhaustiveness, will address two major classes of objections in more detail here. One can be rebutted on theoretical grounds; the other has theoretical merit and we have to make some estimates to quantify it.
One class of objections has to do with tactical adjustments that either team makes when a player with foul trouble is in the game. The endangered player may be more careful than usual, and the opponents may go out of their way to create contact with him. While true, these observations don’t hold up as counterarguments. Why? I’ll explain this with game-theory jargon and then without. The baseline argument says that if you bench the player your payoff gets worse for any fixed strategy-pair. In a zero-sum game, this guarantees that the value of the game goes down, which is clear if you look at the definition of minmax. That is, after tactical adjustments you’ll still be worse off than originally.
The more intuitive explanation of this goes as follows: the baseline argument implies that if the player just ignores his foul trouble, the team is better off letting him play. If he *correctly* takes more care to avoid fouling, the team must be *even better* off. Now, we’re dealing with human beings, so we must acknowledge the possibility that he *incorrectly*, or perhaps selfishly, plays too contact-shy. Well, then he needs a kick in the pants: “Look, I want to trust you and leave you in with foul trouble, but if you’re giving up layups I have to put your butt on the bench.” If this doesn’t work, well, sure, you might have to sit him. There is a human element. But I like Jeff’s comment on this: If he knows he’s going to sit with 3, won’t he be timid when he has 2? Also, maybe some players normally foul more than optimal and play *better* D with foul trouble. (Suggested by Ken Nelson .) The human element can cut both ways. In the Michael Lewis article, the Rockets’ GM refers to a foul as the worst outcome of a defensive play, percentagewise. A slight exaggeration (dunks are worse), but trying to avoid fouls can’t be all that bad. A similar chain of reasoning applies to the other team going after your man; I’ll leave that argument to the reader.
Now, a stronger argument, which I dealt with imperfectly in my initial post, has to do with not all minutes being created equal. The best reason behind this has to do with clock management. Some stars are much better than ordinary players at getting off a decent shot quickly, which can be important not only for the trailing team, but for the leading team if they want to burn most of the shot clock and then get something off. (Thanks to the comments, beginning with Jeff, for pointing this out.) Now, while I think all this is true, I want to make some caveats to this caveat:
1. Clock management should apply almost exclusively to the last 2-3 minutes. Here, my sentiments seem to be backed up by many coaches and ex-coach color commentators: when a team goes into a shell, even with a significant lead such as 10 with 5 to play, they are making a huge mistake. Burning an extra 5-10 seconds just doesn’t nearly compensate for getting a bad shot. Similar comments apply to the trailing team: of course you should avoid sheer wasting of time (so, maybe bring it up the floor quicker), but the prime focus should be on getting the best possible shot, until you’re in a real crunch.
2. Given point 1 (even if you want to extend my window by a couple of minutes), the crunch-time argument only supports benching a player with 5 fouls, not with fewer. Third-quarter and second-quarter minutes presumably are still created equal.
3. Ok, when should a player come back in with 5? This depends on how much more value you give to the last few minutes and on his hazard rate for fouling out. Assume that he has a hazard rate of r fouls per minute and that his value per unit time with t minutes remaining is given by a function a(t). Note that I am thinking of a as the contribution to winning percentage, not to points. The average value you get by putting him in with T minutes to go is
and the derivative of this is given by
where the first term represents the cost of shifting his minutes further from the end (we assume a’ is negative) and the a(0) term is the possible benefit from extra total time (once we always put in the shift cost, we can think of the extra time as coming at the end.)
A reasonable value for r is .125 representing 6 fouls in 48 minutes; naturally this would be higher for some players than others. As for the shape of the function a, this is extremely hard to specify, and it makes sense to try various guesstimates. The point of having a formula is to see how various assumptions about endgame importance correspond to policy, not to provide illusory precision.
To get ballpark figures, it’s useful and pretty harmless to let a be piecewise linear. We can normalize it so that a(0)=A, a(t)=1 for t>=M, and a decreases linearly from time 0 to M. That is, “ordinary minutes” are scaled to have value 1 and the very end has value A, with M minutes of “crunch time” whose value increases gradually until the end. Then, for any T>=M, we get
One very interesting feature is that this derivative has the same sign for all T>=M. This means that we should either save the player for crunch time or ignore the 5 fouls, but a compromise where we bring him in with more than M minutes left cannot be right. Now, let’s look at what values of A are needed to support saving the player. If M=3, and he’s unusually foul-prone so r=.15, a quick calculation gives A=4.8 as the value for which V’ becomes negative. This seems unreasonably high to me. Remember, there is a chance the game is not even close in the last minute, and then everyone’s value is much less. I agree that it’s still more on average, but 5 times more is much too rich for me. Anyway, you can plug your favorite values into the formula if you want to explore.
By the way, my result that you either save the 5-foul player for crunch time or ignore foul-trouble surprised me, especially as it is quite robust to different specifications for a. When this happens I always stop and describe the math to myself verbally to see if it makes sense. I have a profound distrust of formulas I can’t describe verbally; maybe this is what makes me an economist rather than another kind of mathematician. Anyway, what the math is saying is that, by looking at the playing time until foul-out as out of your control, you can decompose the effect of putting him in an instant earlier as 1) shifting his minutes earlier (bad) and 2) possibly getting an extra instant of time at the end (good). For instants prior to M, both of these marginal effects decay *at the same rate*. That is, shifting his minutes earlier matters less because he may not even reach crunch time, and reaching the last minute is also less likely, but the overall sign of the derivative cannot change. This actually made sense once I thought about it. I think theorists live for the cases where the math teaches us something that we can understand by verbal logic, but would have likely missed without the formulas.
This entry has gone on long enough that I will not (as least right now) try to address many more of the many intelligent comments. I will take a moment to agree with one point: after a foul, especially if it is a *stupid* foul or your guy is upset by the ref’s call, you might bench him briefly to try to get him back in the proper flow. This is a far cry from sitting him for the whole second quarter with 3 fouls, though.
5 comments
May 24, 2010 at 5:08 pm
johnnybravo79
By admitting that saving someone for the last few minutes can have some merit, you’ve basically given up the farm, or at least all but perhaps a nice, small organic garden. The reason is that a player’s readiness to contribute to a game at any given point must be taken into account. A player who sits on the bench for a significant period of time after playing all out earlier in the game could very well start to get stiff. A stiff player is not likely going to have maximum level of performance when entering a game with a couple minutes left.
Consider this outworking of your strategy: Star player X gets three quick fouls in the first quarter. Using your strategy, you’d aim to maximize that player’s minutes by continuing to play him in the second quarter with his regular rotation of minutes. But for whatever reason, he picks up 4 and 5 before the half. Now after second half warm ups, he has to sit for an hour before coming in at crunch time. At that point, do you really think he’d be able to play at full speed and ability?
I’d argue that the best way to keep that player ready to play at the maximum level, the player should see some sort of action at least every few minutes of game time (or ~fifteen minutes of actual time). One solution: for a player in foul trouble, sit him on the bench for awhile, bring him in after a few minutes to keep him warmed up, and if he gets no more fouls, great, leave him in; if he does pick up another foul, sit him and wait a few more minutes to bring him back in to get him warmed up again. Which is basically a version of what coaches do. I’m not arguing that coaches always make these kinds of choices using the best evidence, or that they aren’t a bit too risk-averse when choosing who plays and who doesn’t. I’m not even saying that strategy to bring a player back in at the n/6 point of the game is a good one (and coaches probably should put players in foul trouble back into the game a bit earlier, a move justified by the fact that players don’t foul out that often, meaning they likely had at least some more time left available in that game). But to the extent that you’re saving up a player for the most important part of a game, to use their skills when they might best be put to good use, taking a more gradual schedule for re-entering a foul-beleaguered player into the game is called for. Which means that the conventional wisdom probably needs to be only tweaked, not radically changed.
True, the mileage may vary based on player. A jump shooter who has to run around a double screen to take a three probably could do that job cold, whereas someone who has to play post-up defense and rebound against an all-star power forward is almost certainly going to get beat the first couple times down the court.
May 26, 2010 at 9:08 pm
Brendan
I still don’t buy your assertion that the likelihood of getting a foul is time-invariant.
July 7, 2010 at 12:39 pm
Charrua
Fascinating discussion, but I haven’t seen anybody mention the obvious; ¿why did the guy got into foul trouble in the first place?
If it was a case of “too much energy” (ie: a player who is overactive and fouls trying to get blocked shots or steals) it might be wise to sit him and let him calm down.
It might be a bad matchup too, a guy playing against somebody who is especially good at getting calls against him. In which case, it may be wise to sit him, let somebody else try to handle that guy and put him again when the other guy leaves.
In any case, a guy who gets into foul trouble is a sign of a defensive problem and it would be natural for coaches to react to that.
November 25, 2011 at 10:52 pm
David
Another factor exists that I don’t think was mentioned in the comments that were made here or in your earlier blog. The stamina of not only the star player in foul trouble but also the second stringer who might have to replace the star must be considered. Let’s say a team has two centers: a starting center and a backup center. Maybe the starting center has the stamina to play almost all of 48 minutes (if not in foul trouble) at peak performance. But the backup guy fades much more quickly. If you let the starting center foul out at any time other than the last few minutes, you have to use the backup guy for the (substantial) rest of the game with no breaks. The backup center may not be up to playing effectively for that long at an NBA level without a break. I can see a coach wanting to keep his options open and that requires that he avoid anyone actually fouling out–or, at worst, delaying fouling out until the final minutes.
November 27, 2011 at 12:13 am
Jonathan Weinstein
I think that’s a good point, David. It seems to come into play only if the star might actually foul out really early, so I don;t think it’s too much in play on the 3rd or 4th foul.