Monday, January 21, 2013

Should You Foul Intentionally When In The 1&1?

Sometimes twitter discussions spark interesting thought experiments, and I had one of those tonight. Andy Glockner of Sports Illustrated (@AndyGlockner) suggested a rule saying that a team which commits a foul when not yet in the bonus can choose to give the opponent a 1-in-1 instead of the ball out of bounds. This discussion was sparked by the end of the Syracuse/Cincinnati game when Syracuse had the ball up by two but Cincinnati had a couple of fouls to give, meaning that Cincy had foul three times in the final seconds before Syracuse finally went to the line. Andy is right that it's basically punishing teams for not having fouled much all half.

I pointed out that his rule made sense, though I would restrict it to the final minute of the game. I said that otherwise we could have situations where teams hack-an-opponent to send bad free throw shooters to the line for a 1-and-1. Nobody wants to watch that.

While this rule doesn't exist, of course, it does bring up a situation that always exists when a team has been called for 6, 7 or 8 fouls in a half.You basically never see teams in those situations intentionally foul... but should they?

The math on this isn't trivial, and I apologize to those that didn't like math class. I'll have a plot at the end of this if you want to skip the math. But anyway, here is my logic:

Assume that both teams score "P" points per possession, and assume that your opponent hits free throws at some F%. Now, assume that 15% of all missed free throws are rebounded by the opposition. Where did I get that from? I couldn't find NCAA stats for this in my brief search, but this study found a 13.9% offensive rebound rate on missed free throws in the NBA. In general, offensive rebound rates are a couple percentage higher in college than the NBA, so I figured 15% would be a nice round number.

If we assume that, then:

F^2 = probability of hitting both
F(1-F) = probability of hitting the first and missing the second
(1-F) = probability of missing front end

Expected PPP if hitting both FTs = 2
Expected PPP if hitting first, missing second = (1+.15P)
Expected PPP if missing front end = .15P

So throw the expected PPP when sending a guy to the line for 1&1 and set it equal P and you get the break-even point for when the outcome due to fouling is exactly equal to the outcome when playing defense. Here's the plot I made:

Click to embiggen

On the x-axis I have the expected PPP for both you and your opponent (again, assuming it's the same) for a possession. On the y-axis I have the FT% of the guy you fouled. You should foul anytime you are below the line, and you should play out the possession anytime you are above it. I drew the line for an assumed 1 PPP offense (the NCAA average), and it comes to a 56.7% free throw shooter.

Now obviously this curve comes with some caveats. Teams might not have identical offenses (if we assume the x-axis is your opponent's offense, and your offense is better than theirs, then the line shifts to the right). Also, your defensive rebounding (or your opponent's offensive rebounding) might be better or worse than average. Also, you obviously have to make sure that the players committing the intentional fouls are not players that you fear fouling out. And also, using up your 1&1s can harm you later if you unintentionally commit a non-shooting foul and send your opponent to the line for two instead of 1&1.

But you can adjust the equations for you and your opponent for any game. The same general math works - you just change the variable values.

And the conclusion of this analysis is that yes, there are some times where it would make sense to hack-an-opponent prior to the final minutes of the game, either if the opponent has a bad free throw shooter, or if both offenses are swamping the opposing defenses.

Of course, I don't expect this to happen anytime soon. Coaches tend to be risk averse because they don't want to be criticized by the media. It's why all football teams at the NFL and NCAA level punt way too much. But they should be punting less, and NCAA basketball coaches should be aware that there are some situations when fouling the opponent intentionally well before the final minutes of the game are in your best interest.


Anonymous said...

Apply this to specific situations.

Say, you're up 2, but opponent has the ball, shot clock turned off, and opponent will be in the 1 and 1 at next foul.

Anonymous said...

Or same situation, but you're up 1?

Jeff said...

Hm, that's an interesting question. The math might be a bit complicated, but I'll try to do that at some point in the next few days.

Interestingly, if the game is tied and the other team has the ball with the shot clock off then you should foul intentionally. That's an old standard that Ken Pomeroy talks about a lot. Doing it if you're already up 1 or 2, though, is interesting. I'm not sure what the answer is off the top of my head.

Anonymous said...

Rebound rate would be at least somewhat higher than average offensive FT rebounding rate with a poor FT shooter on the line.

Jeff said...

"Rebound rate would be at least somewhat higher than average offensive FT rebounding rate with a poor FT shooter on the line."

Why would you say that?

Anonymous said...

"Rebound rate would be at least somewhat higher than average offensive FT rebounding rate with a poor FT shooter on the line."

I'd think offensive rebound rate would generally be LOWER with a poor free throw shooter at the line, simply because that poor shooter is more likely to be a big man, meaning you're short on the blocks.

Jeff said...

Well, I haven't seen any statistical evidence one way or the other, but your argument for why it would be lower with a terrible FT shooter at the line at least makes logical sense to me.

Adam said...

2nd anonymous' comment makes a good point. So I amend my initial guess an now think O-reb rate would be about even with good and bad ft shooters. My initial thought, however, that a bad FT shooter would lead to more hard caroms off the rim and increase likelihood of O-rebounds.