When Math Fouls Up
I started this morning by reading the latest article by GatorZone (some may call him Chris Harry) and had to apply my state-school graduate brain and Saturday morning fog to try to debunk this take.
Here’s the situation:
Florida basketball is beating Oklahoma by 20 points at the end of the first half.
UF intentionally fouled Oklahoma’s Jeremiah Fears 45 feet from the basket with 10.6 seconds left in the first half.
Fears shoots 84.6% from the free throw line.
The explanation is that by doing so, UF gets the ball back and has a chance to score, so running that simulation of events over and over results in UF being better off.
I have two issues with this:
Bad math
Increasing the variance??
Bad Math
UF gives this math to explain their rationale:
“if we foul an 80-percent free-throw shooter, 80 percent in the one-and-one means there's an 80-percent chance he makes the first and 80-percent chance he gets a second [attempt]. So the value of the first shot is .8. So, .8 multiplied by .8 is .64. That's the value of the second shot. Add them together makes it 1.44”
Note: 1.44 refers to points per possession (PPP) - the number of points a team scores per possession.
This ignores the fact Fears shoots 84.6% from the FT line, so if you update 80% to 84.6%, it’s actually 1.56 PPP. However, that assumes there is 00.00 left on the clock and there is a 0% chance OU gets an offensive rebound (OREB) and scores.
So, the 1.44 or 1.56 PPP(e) assumption is incomplete and missing a huge variable.
It seems like assuming a 15% chance OU gets an OREB after either (possible) miss is reasonable. Fears has a ~15% chance of missing either FT. Let’s assume if OU gets the OREB and immediately shoots the ball with a 50% chance of making the shot. That brings the 1.56 PPP up to 1.60. This ignores two other possibilities that would bring the PPP higher:
UF may have to foul again. If OU gets the OREB, they then the ball again with 9 seconds left now. If it’s still 1 and 1 and you have this philosophy, I’d assume you’d want to foul, especially a big who’s probably a worse FT shooter.
OU shoots a 3 (maybe their PPP(e) after an OREB off a FT is higher than 1.0).
Also, UF may not even get the ball back. What if OU keeps getting offensive rebounds the rest of the half or dribbles the ball out and shoots at the buzzer? There’s a chance UF doesn’t get the ball back, and you have to multiply the 0.85 PPP by that chance.
Almost every situation I can think of either increases the PPP(e) for OU, or decreases the PPP(e) for UF.
Increasing the Variance
Let’s just stick with the 1.60 PPP(e) for OU, and the 0.85 PPP(e) for UF on the final play. The article quotes UF saying:
“we're basically increasing the variance and reducing how much we're supposed to lose that sequence by. Instead of losing it by .85, we lose by .65. We're losing by less, but increasing the variance."
When you update the 1.44 PPP to 1.60 PPP, UF actually goes from losing by 0.59 to 0.75 (compared to the original 0.85 when not fouling). I do agree it increases variance, but when you are the team up 20 points in the first half in a game you were favored to win (i.e. Vegas thinks you’re the better team), you do not want to increase the variance. The underdog wants to increase variance in that situation, not the other way around.
Bonus Issue: OU PPP(e)
In the hypothetical, this is a big assumption:
"The end-of-half possession, when the shot clock is off, is worth a little less than a normal possession because the defense knows about when the shot is coming. So, instead of being worth a normal 1.0 or 1.05 points per possession, it's worth .85."
In the first half of this actual game, OU scored 0.67 PPP (24 points on 36 possessions). I wouldn’t assume the expected value of their final possession is 1.0 or 1.05 if they’re scored 0.67 points per possession in the first half.
If you swap out 0.85 PPP for OU with 0.67, UF actually loses more by fouling than playing the possession out.