fouls as resources in basketball

This one is in honor of March Madness, even though I am now officially in last place in both brackets I filled out. Oy, vey!

A common scene in the first half of a college basketball game: a player is called for his third foul and, in the words of the announcer, “he’ll rush to the bench.” Ditto with the fourth foul in the second half, because the fifth foul is when the player “fouls out” and must leave the game. Fouling out is relatively uncommon, largely because most coaches follow this rigid rule of removing players after their fourth fouls (or third if during the first half).

My question is: why?! It seems to me that fouls in basketball are a bit like pollution in a cap-and-trade system: licenses to commit a certain amount of a kind of semi-forbidden behavior that can often lead to gains in a field. The sanction for using too many of these licenses is–not having access to the gains available if one used more. (Of course I’m not talking about intentional, technical, or flagrant fouls here.) So removing a player before he can commit his fifth foul is voluntarily withholding team talent from a field whose whole existence is legitimated by the successful deployment of that talent.

Time would be a legitimate reason to withhold such talent but for one thing: greater deployment of talent earlier in the game tends to diminish the need for that talent later in the game. So if a star player is kept in until his fifth foul, it is less likely that the game will enter “crunch” time later on and require that player’s star power to rescue the team. In other words: a basket with 30 seconds left is no more valuable than a basket with 10 minutes left.

Author: andrewperrin

University of North Carolina, Chapel Hill

9 thoughts on “fouls as resources in basketball”

  1. I think that there are a lot of factors whether it makes sense to pull a player. One is whether they are an offensive or defensive specialist. If a center has four fouls, but his/her strength is defense, then I think that you are right. If, on the other hand, s/he (I really want OW’s gender-neutral pronouns right now) is an offensive specialist, then substitutions become easier at the end of the game because more fouls and time-outs and a slower flow of the game make it easier to substitute the player in only on offense than earlier in the game when the flow is more continual.

    Also, my guess is that most teams have game plans for both offenses and defenses without certain players who tend to get a lot of fouls. And, in all likelihood, if a single player is picking up fouls, then the current game plan probably isn’t working so it might help to switch the game plan up (i.e., switching from man-to-man where the center is picking up fouls to a zone defense).


  2. Players have to adjust their play according to the ratio of their remaining fouls to the time remaining. A (male) player with three fouls at halftime has to play tentatively. If you bench him till the last five minutes he can come out and play aggressively later, maximizing his contribution per minute on the court. If you keep him out till the last two minutes, and he never gets another foul, you kept him out too long.


  3. To say that a basket with 30 seconds left is not more valuable than one with 10 minutes left forgets that there is agency in the world. Teams can — and do — adjust strategies in response to the score.

    If we want to keep it in a quasi-stats language, the basic point would be that there are some approaches to the game that reduce mean improvements in relative score but increase variance. In real terms: pressing and fouling at the end of the game will probably increase the amount by which you lose, but are the only way in which you get the chance to lob up a lot of 3s. Particular players (e.g. point guards) are particularly important when you want to be able to vary your approach.

    The one thing I would say that you have right here — drawing on the opinion of someone I know quite well who is in the hoops world — it is that coaches do operate too often by a heuristic on fouls without enough attention to who is in trouble and the likelihood that said player will commit more fouls. An interior defender is more at risk than a point guard, generally speaking. Put another way, if a guy who average 2 fouls per game and has minimal game-to-game variance gets 2 in five minutes, you should worry less.

    That said, there are coaches who think about this.


  4. Yep, you can thank the late Jimmy Valvano for that. The realization that fouls can be good, or at least useful if used strategically.


  5. Note also that games are not really of equal length for purposes of this sort of analysis. What really matters is number of possessions. Ceteris paribus, when you are behind, you want to lengthen the game; when ahead the opposite. Again, who you can have on the floor impacts your ability to affect that variable.

    Another issue, of course, is that teams are not only additive. They are also interactive. Combinations matter. E.g. they throw pressure at you, you need ballhandlers.

    Put that all together and what do you get…, it’s really complicated. Which is a recipe for heuristics.


  6. The problem with looking at fouls as cap and trade system is that it isn’t that a player is only able to foul four times. A player can foul as many times as he wants, he can only get called on it a certain number of times.

    After a police officer tickets you for speeding, you don’t gun the engine and speed off. Later the same day you may choose to drive over the speed limit at the same intersection.

    But when an officer pulls you over and gives you a reprimand, you in essence bench yourself. You may get away with some reckless driving twenty minutes later, but you assume that even the slightest hint of recklessness (the kind of “foul” that would normally be ignored) immediately after a reprimand will get you pulled over again.


  7. A couple of thoughts:

    1. Fouls lead to player frustration. Player frustration leads to increased likelihood of fouling. Thus, one reason to take a player out is so that s/he may calm down and therefore be less likely to foul upon returning to the game.

    2. Fouls lead to increased scrutiny by refs. If we assume there are fouls that don’t get called or might be borderline and the refs let them pass, which seems pretty safe to assume, then simply by virtue of having been called for a foul you are more likely to get called for a subsequent foul because the refs are watching you more closely.

    I suppose underlying both of the above points is the fact that fouls aren’t just resources to be deployed for some kind of gain. They aren’t always rational. I’m not sure about the ratio but a lot of fouls are “stupid” and therefore bypass the resources logic.


  8. These are smart ideas, and thanks all for suggesting them as solutions, albeit partial ones, for the problematic I raised. A couple of points, though:

    1.) These are, in general, reasons why it may make sense to pull players as they accumulate fouls, not reasons to implement a rigid rule as is not only implemented but assumed in the game.

    2.) I think it was a mistake to understand the foul itself as the resource being limited. As in cap-and-trade, the foul is the negative by-product of the good being limited, which is the playing time of the player. Thus to remove a player on foul 4 is to forego access not to the fifth foul but rather to the playing time between foul 4 and the (usually hypothetical) foul 5.


  9. Andrew:

    RE: (1)– yes. And in fact, you will occasionally see teams play players in ways that run contrary to the standard heuristic. E.g a secondary big man may be allowed to play with lots of fouls; or a team with little hope of beating a much better team may ask an experienced star player (i.e. Stephon Curry) to take his chances. But again, faced with the complex calculation, coaches do mostly just do what most people do in such situations, which is that they turn to rules of thumb. Some of it is surely risk aversion (if you fail, fail in the standard way and nobody says anything).

    Note also that since conditioning means that most players see declining returns to minutes around minute 30 (sooner if played continuously), foul trouble can also be taken as a reason to move rest intended for later to earlier in the game.


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