This is something you see all the time. In an important game, a star player picks up a couple of fouls early. The coach then yells at the refs and benches the star player to save him for later in the game.

Intuitively the move makes sense. The coach does not want the player to pick up additional fouls and risk fouling out of the game. But you should ask, is benching a star player really the right move?

The answer is no! In 2010, a compelling argument was presented by Jonathan Weinstein, a professor at Northwestern University, in a post on the economics blog Theory of the Leisure Class. You can read about the statistical theory of foul trouble and some caveats. Below I summarize the reasoning and offer an analogy for why you shouldn’t bench players in foul trouble.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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The argument in a nutshell

Weinstein offers this intuitive explanation for why star players should not be benched.

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.

In other words, the argument is as follows. Let’s say a star player is in foul trouble he is expected to play X minutes before fouling out. What’s the optimal amount of time to bench the star player? The answer is that it’s a trick question: it’s always worse to sit the player than to let him play. The reason is:

–if X is larger or equal to the minutes left in the game, then the player would not have fouled out. So sitting the star player means you lose his playing time and benching is the wrong move.

–if X is less than the minutes M left in the game, it is possible to bench the player (M – X) minutes and still end up with X minutes of playing time. However, there is a risk the coach benches him for too long.

–In practice, X is a random variable and not known. As the two cases above illustrate, benching never results in more minutes for the star player. So any time you bench a star player, the minutes played is a truncation of the distribution for minutes if the player had not been benched.

An example with rolling a die

Let’s use a simple game to illustrate. You are given a die and can roll it up to 40 times. You earn 1 point every time you roll. However, if you roll a 1 you earn a foul, and after 6 fouls you “foul out” and cannot roll any more. The goal is to earn as many points as possible. On any turn you can also “pass” and elect not to roll the die–but you do not get any points as you did not roll the die.

The die game is an analogy for managing a star player: you want to maximize the number of rolls of the die (opportunities for the star player), but each roll risks earning another foul, and you can elect to pass on a roll (benching a star player).

So think about the die game. Is it ever a smart idea to “pass,” if you are in foul trouble early in the game?

Of course not! The only thing passing up on a roll does is limit the number of rolls you have. It is never better than just rolling the die until fouling out.

Let’s do an example. Suppose you pick up 5 fouls in 30 rolls. Would you ever choose to pass on a few rolls, or should you just roll them out? Statistically, here is why you should just roll.

–There’s an 84 percent chance you will not foul out in the last 10 rolls. If you pass for n turns, you are simply losing n points.

–There’s a 16 percent chance you will foul out in the last 10 rolls. In some cases it is possible that passing and rolling would give you the same points. If you would have rolled X times before fouling out, then passing on 10 – X turns would be the same as just rolling the X rolls. However, X is a random variable. There is always a risk of passing for too many turns and losing points. And either way, passing never leads to more rolls than simply playing it out.

In conclusion, the die game illustrates why benching a star player is a bad idea. At best benching results in the same minutes of playing time. But very often it means fewer playing opportunities.

Don’t bench star players!

Obviously there are some caveats to the model. One is a game theory consideration. If a star player is in foul trouble, the other team might target them. However, in that case it is still better to play the star player. Benching means a loss of minutes for sure. A player on the floor, on the other hand, has the ability to avoid fouls. So it stands to reason that benching just to avoid fouls does not make sense.

There are also emotional concerns as players are humans. Some hot-heads get “riled” up and need to be benched. Others might just need the rest–but that is independent of foul trouble. And coaches are obviously playing for their jobs: they would rather use an accepted bad strategy than have a good strategy backfire on them.

But announcers should not repeat the wrong advice of benching star players, and we as fans should expect more of coaches. Announcers always talk about how the referees should let the players decide the game. Now we should expect the coaches to understand that principle too and let star players play even when they are in foul trouble.