LeBron gets the ball at the top of the arc, quick glance at the clock: 5 seconds left in the fourth quarter, Cavaliers down 2. He shifts his weight and prepares to…
Announcer 1: Jim, LeBron has to stay aggressive here, can’t settle for a jump shot.
Announcer 2: That’s right, Mike, the defense has to make sure not to foul in this situation and risk sending him to the free throw line.
Putanumonit: AARGH! Shut up! That’s not how math works $&#@&%!!!
*smashes my copy of Jaynes into the TV screen*
I love sports, and I love math. Sometimes, the former commits atrocities against the latter, and it makes me very sad.
Let’s go back to LeBron, who this year is shooting 71.6% on free throws (mediocre), 55.3% on two point shots (very good) and 29.9% on threes (really bad compared to the league average of 35%). Announcers praise players like LeBron when they drive to the rim hoping to shoot free throws or score from close range because given these percentages, shooting a pair of free throws nets more points on average than a 2 point shot, and 3 pointers least of all:
Going to the line is worth more than an extra half point. Scoring the most points in expectations leads to winning in most situations. Most, but not every.
When you’re down two points with one shot left to take, you should directly maximize your chances of winning rather than just going for more points on average. If we assume that tying the game will send it to overtime (50% chance of winning), we can calculate the chances of winning for each option.
The picture flipped: a three point shot that was previously the worst option became the best and going to the line became the worst. The defense can increase it’s chance of winning by over 4% by fouling LeBron as soon as he starts a move, and this is true for a pretty bad three point shooter! Fouling Draymond Green when you’re up two increases your chance of winning from 59% all the way up to 78%.
There are other considerations that could swing the decision either way, like the fact that the defense could allow a certain type of shot more easily or that long range shots have a higher chance of an offensive rebound. However, you almost always hear announcers criticize a player for shooting in this situation and I’ve never heard an announcer suggest that the team up two should foul. The coach and the player will usually go for the “safe” play: if a player misses a three pointer (which is likelier than not), everyone will criticize the decision. If the team loses in overtime – oh well, at least he got them there.
The general principle is that when you’re down, you care about maximizing variance more than you care about increasing expectation. You should be willing to trade off a large chance of losing by a lot and a small chance of barely winning, over a certainty of barely losing. A three pointer is a high variance shot, and it doesn’t matter if you lose by one point or two.
Coaches in most sports do often go for high variance strategies when they’re down. Hockey teams pull their goalie, soccer teams substitute attackers for defenders or even send their goalkeepers up the pitch to try and score with an empty net. And yet, when the wining high-variance play requires basic math instead of crazy strategies, coaches often make the wrong decision.
Another rule of thumb I’ve discovered: when something stupid is happening in sports, the NFL is usually at the forefront of stupidity.
Benjamin Morris of 538 lays out a heartbreaking story on NFL coaches’ dumb decisions last weekend.
The first case is straightforward expectation: when Green Bay pulled within one point after a ridiculous drive, then kicked the extra point (94.3%) to go to overtime (win probability 45.5% on the road) rather than go for two (47.2%) and the win. 94.3%*45.5% = 42.6%, which is much less than 47%. What’s weird about this example is that the low chance of winning in overtime doesn’t even play into it: since 94.3% is less than 2* 47.2%, Green Bay should’ve gone for a two point conversion on every single opportunity!
The second case is more interesting, with Kansas City trailing New England by 14 with time running out. As Morris correctly notes, KC has to assume it will score two touchdowns otherwise it’s chances of winning are precisely 0%. Again, by Morris’ numbers KC should have gone for 2 every time anyway, so we’ll assume conservatively that kickers never miss extra points and both teams have an equal 50% chance to win in overtime.
Here’s one way to look at this situation: once KC scores two touchdowns (6 points each) they are “trailing” by 2 points before counting the conversion attempts. What do you do when you’re down 2? Go for the three pointer! In this case every extra point kick is like a free throw: you have to make both just to survive. On the other hand, every two point conversion is like a 3 pointer: you need to make just one to win or tie!
You have to be above a measly 38.4% conversion rate to make going for 2 worth it assuming perfect extra points, or 35.7% with fallible kickers. The league average is 48% – it’s hard to believe any team expects to do worse than 40%.
Here’s a good heuristic to use if multiplying two numbers is too hard to do on the sideline: if expected point are about the same (as they are for extra points and two-point conversions or 3 and 2 point shots in basketball), go for the option that can make you lose by the most points! Going for 2 down 14 in the NFL is worth it because that’s your only chance to lose by a whole two points, so it’s necessarily offset by a double chance of winning by 1.
It is quite remarkable how often NFL coaches make the choice that would be wrong on a middle school math exam: not going for 2 on the first touchdown when down 15, punting inside opponent territory, kicking field goals from the 2 yard line etc. These coaches are paid millions of dollars and supported by staffs of dozens. Here’s the most interesting question: with so much at stake, how come not a single NFL coach can get these decision that require a small chart and a calculator consistently right?