Cocktail Napkin Math - Why NBA Teams Should Foul On Defense When Up 2 With 10-24 Seconds To Go (The Cheah Method)
Been a while since I've done one of these. But welcome back to another round of Cocktail Napkin Math. It's been a few days since this take of Cheah's sent the masses over the edge and I've finally found a pocket of time to dive into the play-by-play data to get to the bottom of this. Should NBA teams who are up by two foul their opponent on the floor to send them to the free throw line? I have the answer. Based on history and good ole probability 101.
Pretty much everyone except Cheah thinks the answer is no. Even some randos like Bearcat Randy and Matt "not so" Knicely. Me on the other hand? I was immediately intrigued. In a weird way it reminded me of the "go for two down 14" strategy in football. The specifics are wildly different, but the goal is the same in that you're kind of cheating time to get more information about what you should do. In football - you learn early whether your next touchdown will give you a chance to tie with the bonus of potentially being able to take the lead. In this NBA example - you learn if you need to take another shot to win the game while reducing the chance of losing on a three.
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I don't know. maybe that doesn't make sense. But let's get into this. And for those wondering where this idea came from it was Game 1 Knicks/Pacers when the Knicks found themselves in a similar situation they might do this.
Cocktail napkin time. Sit back and sip on whatever favorite poison you prefer. I like a good Mezcal these days. Had a great one in Sedona a few weeks back. Ginger beer with a little bit of orange juice and whatever else. Unexpectedly surprised how it all came together. Need to go back.
We'll start with the chalk. The road you, New York and probably any team would choose.
Option (A) No Foul
I cut the play-by-play data to show just instances in which a team was down exactly two points with possessions that started between 10 and 24 seconds left in the game. Pretty much didn't want to use situations that required a prayer heave to skew things. Now league average shooting overall is something like 47%. But in these high-pressure moments when time is waning, that drops considerably. At least... it used to.
There were 707 qualifying shots this season under these conditions and overall players made 42.1% of them which is the highest rate in at least the last ten years. But here's the thing. We need more then just the overall percentage. We need the percentage of twos vs threes. Because giving up a three is a nightmare scenario when you could have guaranteed they only have a chance for two.
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Bartender! I'll take some napkins. Neat.
That's right. More threes were made than twos. Welcome to the modern NBA. But to clarify those percentages, it doesn't mean teams were 22 percent from three and 19 percent from two. These are the percentage of total shots taken overall since we are just interested in the likelihood of a two or three occurring. 19-percent of shots in this situation resulted in a made two pointer. And now we can say when you don't foul, you are giving your opponent a 22-percent chance to win the game outright on that possession.
Eyebrows raised yet, haters? They should be.
With this we can figure out what the Knicks odds roughly were when Nesmith inbounded the ball to Haliburton. They only had seven seconds left instead of at least ten, but we'll go ahead and say it's all the same considering they cruised down in the blink of an eye. Let's turn the probabilities of allowing two-point plays and three point plays separately, and find the proper estimated win probability for the team who elected not to foul on defense.
68.5 percent chance of winning by staying pat and playing straight up on defense. That's what option B will have to beat in order for the Cheah method to be properly validated.
Option (B) Foul
The road less traveled will take considerably more work. Yet we will trail blaze it and prepare for whatever this road brings. After all, we have our napkins. So let's use them. What we need to do here is figure out every reasonable game state possible that could occur as a result of fouling, figure the probability of each game state occurring using historical NBA data, then use those probabilities as weights for the chances of winning the game under each condition.
Easy enough, right?
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OK - Let's mind map these possible game states before committing them to the napkin. First - either the free throw shooter is going to make both free throws, or miss at least one. If he makes both, your team will either make a game winning shot or the game will go to overtime. But you have to account for the whammy of them getting an offensive rebound and putting it back in on the second attempt. An uncommon possibility, but not rare enough to ignore. So we'll do that too with the generous assumption the first free throw was also made. Here's what our possible game states look like on the official napkins.
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Opponent makes both free throw game states
Genius stuff. I know.
Opponent does not make both free throws game states
Not exactly rocket science, but we're building up here. Next we need to find the probability of these individual game states happening. Average NBA free throw shooting is about 78%. Now Cheah argues it should be lower since these are crucial, tense moments. My research suggests that isn't the case. Free throw shooters down 1 or 2 points with 10-24 seconds left in the game since 2016 went 365/465 at the line. 78.5%. Right where you'd expect it. I cut out the final ten seconds to account for the possibility of the purposeful miss if down two. So it turns out clutch free throw shooting isn't really a thing - but there's still only a 61% chance the opponent makes both free throws (.785 * .785).
1a. (Opponent makes both free throws and you make ensuing last shot)
Alright - let's say they do make both. We need the chances your team is able to score right back. Since we already know teams made 22% of all shot attempts from three and another 19% of all attempts from two when down two end of game, I'm just going to use this work here and say that gives us a 41% chance of making any shot in a similar enough situation.
.62 (opponent makes both free throws) * .41 (we make game winning shot) = .25 (25%)
1b (Opponent makes both free throws and we miss ensuing last shot)
Same idea, just using the odds we miss the shot instead. Switch out the .41 for a .59.
.62 (opponent makes both free throws) * .59 (we miss final shot) = .365 (36.5%)
2a (opponent does not make both free throws, your team gets defensive rebound)
Defensive teams grab 78% of rebounds when leading by 1 or 2 points with 10-24 seconds left. Pretty on par with overall averages, but I did the work to make sure this didn't change anything. That gives us…
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.38 (opponent misses at least one free throw) * .78 (defensive team rebound) = .29 (29%)
2b(i) (opponent misses at least one free throw, gets offensive rebound, and makes ensuing shot)
Your worst case scenario that can't be left unaccounted for. Offensive rebounds add an additional sub-element to game state 2a. We don't really care if they get an offensive rebound. We care if they make the shot afterwards. We need to figure out what the odds are of a team scoring on a shot immediately following an offensive rebound. Looking back ten years, teams in the given game situation (down 1-2 with 10-24 seconds left in game) are 43/101 on such shots. 42.5%. A very modest bump from the 41% for teams bringing the ball up the court.
But it's actually at lot worse. This doesn't include the 23 turnovers for those teams who failed to even get up a shot. Makes sense. Not all offensive rebounds are clean and quite frankly, shit happens in the low post. And the napkin accounts for it. Add 23 to the denominator and our offensive rebounding opponents really only have a 34.5% of taking and making their ensuing shot. And the chances of all these sub game states happening are a slim 2 percent.
.38 (opponent misses at least one free throw) * .22 (opponent gets offensive rebound) * .345 (opponent makes put back shot) = .02 (2%)
2b(ii) (same as 2ai except opponent misses ensuing shot)
.38 * .22 * .655 = .055 (5.5%)
All game states add up to 98% which leads me to believe I've covered all possible except the rare oddballs like a stolen inbounds, a full court heave with .6 seconds left after making our shot, or the super rare "Houston Cougars deciding to stop dribbling the ball and let the clock time out at the end of the Final Four".
Red pen time. Let's piece these probabilities of occurrence all together for clarity.
The only thing left to do now is use these percentages as weights to find the answer we've been seeking. The percent chance of winning the game after fouling. Multiply each probability of occurrence by the assumed win expectation for each game state and add those all together.
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77.75%. Crazy. A massive ten percent gain in EV. Haters can think what they want. The Cheah Method is brilliant. Fouling up two pretty much gives you a 50% chance floor by doing everything in your power to make overtime your worst case scenario. The only whammy that fails in doing so is game state 2b(i) which has a mere 2% chance of happening.
Now does this concept suck from an entertainment and suspense perspective? Holy hell yes, it does. I'd banish it forever if I were the commish. But if there's one thing to learn from analytics in sports it's that while it works, it's often a cancer to the game. The fact that this idea sounds lame to watch should really be a sign that it's accurate.
But I don't think that's why everyone was wrong in hating on this take. I think it's a classic case of loss aversion. Our brains are wired to hate losing so much more than loving to win. Every gambler knows this. So the idea of willingly losing a lead breaks the brain from understanding the potential benefits it might provide. Just like old school football hardos who can only think "take the point" when scoring a touchdown while then 14. That's loss aversion in action.
Case closed. This is a brilliant idea and Cheah deserves all the credit for it. From my glass to yours, Steven Cheah. Cheers.