Pickup Basketball: Next!

This is either Steve Nash or Kareem. I can’t tell.

That cocky son of a bitch just drained a three to knock your team off the court. There are three guys on the sideline waiting to play, so you could still get into the next game if you shoot your way in. You and your four other teammates line up to shoot free throws for two survivors, and all five of you make it. Someone cries, “this is going to take forever! Shoot threes.”

In this scenario, your pickup game has obviously determined that every player will get a chance to shoot his way into the next game (otherwise, the first two to make it would have just been in). The question is, is it any faster to shoot threes?

Yes. Well, no. Well, sometimes.

Let’s assume that typical pickup-game players shoot about 35 percent from (an unguarded) three and 65 percent from the line. Simulating a couple scenarios, I came up with the following chart for the expected number of shots required to select the next game’s players.

Waiting Needed Shooting% Shots
5 2 0.35 10.28
5 2 0.65 11.66
7 5 0.35 17.34
7 5 0.65 13.44

You can see that when five losers are shooting for two, it’s a little quicker to shoot threes. But when there are seven fresh players shooting for five, then it’s quicker to shoot free throws. When I adjusted the numbers of players waiting and players needed for other scenarios, I noticed a pattern. The closer the players’ expected shooting percentages are to the fraction of shooters that are needed for the next game, the quicker the process tends to go.

This makes complete sense. If five guys are shooting for two spots, then a quick process would require exactly two guys to make it in the first round. Since you want two out of five to make it, pick a shot where the average player will shoot about 40 percent (2/5), and then you have a much better chance of getting exactly two in the first round.

Of course, if your basketball posse’s method of shooting into the next game is just selecting the first however many make it—with no chance to “tie”—then the above math doesn’t apply. In this case, the harder the shot the longer it takes*:

Waiting Needed Shooting% Shots
5 2 0.35 5.71
5 2 0.65 3.08
7 5 0.35 14.29
7 5 0.65 7.69

By shooting free throws instead of threes, you can cut the total number of shots nearly in half, and your pickup game can get going again.

In the end, if you don’t care about giving everyone an equal chance to get into the next game, go with the first-makes method and shoot free throws (or some other “easy” shot). If equality is your thing, and you allow players the opportunity to “tie” in each round, then try to match the fraction of shooters needed for the next game with the approximate difficulty of the shot.

Or perhaps it’s easier to just say:

If more than half of the shooters are needed for the next game, shoot free throws (easier shots).

If less than half of the shooters are needed for the next game, shoot three pointers (harder shots).

 

*This can be calculated theoretically using the negative binomial distribution.