Coaching Below Expectation

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:

LBJ expect
Shot options for LeBron James

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.

LBJ expect 2
Shot options when down 2

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.

Goalie Peter Schmeichel scores a tying goal in the final minutes in 1995.

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!

LB exp KC
After touchdown options for Kansas City

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.

andy reid 1.jpg

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?

6 thoughts on “Coaching Below Expectation

  1. You answered your last question earlier in your article. Loosing in overtime is less likely to put a coach’s job security in jeopardy then gambling and loosing will.

    I remember when the Patriots under Belichick went for it on fourth and short in the fourth quarter and failed the conversion. It ended up losing them the game. Belichick defended his call citing statistics and probability but by that point (he’s won three Supper Bowls, had only one losing season with the Patriots, and an undefeated regular season) his job was not at all in jeapordy. At this point, Belichick would probably have to have back to back winless seasons to get fired. In any case even he faced criticism for that call.

    Doing what Belichick did would probably get most other NFL coaches in hot water. By itself, it’s not likely to get a coach on the hot seat or fired but it can contribute. It’s not about weather or not the general manager or the owner understands the math but whether fans collectively do. A team has to sell itself as a team fans can invest in. Punting the ball away to lose the game or losing in overtime damages confidence less than turning it over on downs and losing the game or missing the 2 point conversion and losing in regulation. The increase in confidence from a successful gamble isn’t as much as the decrease from a failure.

    Any NFL coach that takes his team to the playoffs less than 75% of the time (that is the say all but Belichik) doesn’t have as much freedom to gamble. But there are coaches that do. Notably #1 Texas went for two when down 6-14 (after the (TD) to beat #2 Arkansas in 1969. I have seen teams go for two at the end of the game to win or lose in regulation more than once. It happens but the right calculation to make wouldn’t be “what’s more likely to win the game?” but rather “what’s more likely to get me fired?” and the two aren’t always the same.


    1. That’s a good way to look at it, which brings me to another point: having a tenured coach in any team sport is a gigantic advantage. If a coach is secure he can “afford” to maximize wins (Belichik), the players know that they have to fall in line or no one will take their side (contra Kelly with the Eagles) and the team doesn’t have to deal with constant “hot seat coach” chatter (half the league). I don’t think that Mike Tomlin is a very good coach, but I’d rather have a safe-in-his-place Tomlin than whomever Cleveland hires who knows his half-life is 10 months. As a GM, it seems that just saying “This is our coach for the decade, deal with it” is more important that trying to luck into a genius.

      One last thought: all the fans who must be “appeased” by sub-optimal decisions, do they make the same decisions when playing Madden in the comfort of their couch?


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