## Foul Trouble and Game Theory in the NBA

In professional basketball games, the closing minutes of a game are arguably the most important moments in deciding the game, and therefore, any constraint that cripples a team’s hopes of winning in the last few minutes become even more detrimental during the last few minutes than if it occurred in the earlier stages of the game. One such constraint is the number of fouls a player accumulates. As a result, coaches tend to bench a player in the early stages of the game if he accumulates an unreasonable amount of files over a small amount of time, especially for the better players on a team.

While it seems reasonable to do this, this article shows otherwise using game theory. In essence, a coach would like to maximize their payoff, represented as the number of minutes of time their star player is in the game. The traditional argument is that “after his n-th foul, why should we risk putting him back in?” However, if the player sits, he only will have less possible expected minutes to be back on the floor and playing.

Mathematically, let’s suppose that the number of minutes a star player is expected to play before fouling out is X. If X is larger than or equal to the number of minutes left in the game, then he wouldn’t have fouled out, so benching them isn’t optimal here. If X is less than the number of minutes left in the game, then there’s a window of (M-X) minutes (M is the number of minutes left in the game), but benching him for longer is also sub-optimal. However, there’s no real way to determine X, and as we can see from above, benching a player will never result in more expected minutes, so there’s no real reason to bench them.

A simpler way to look at this is a game of dice rolling where you get to roll 40 times, and you earn 1 point every time you roll, but you get a foul if you roll a 1, and you are allowed 6 fouls, and you are allowed to pass a roll. As explained in the article, it’s never a smart idea to pass a roll, as it will just limit the number of rolls you have left.