On Sunday night, with about five and half minutes to play and his team down eight points to the Rockets, Denver Nuggets' coach Brian Shaw tried the ol' Hack-a-Howard. It didn't work; Howard went 13 for 19 over a two and a half minute span and doubled the Rockets' lead to 16 points.

ESPN reports that Shaw "lamented using the [Hack-a-Howard] after the game":

"That goes against everything I'm about," Shaw said. "I don't believe in that and I don't think it's in the spirit of the game. So that is exactly what I get for doing that. I'm glad he made his free throws and it shows me to just be true to who you are."

Indeed, losing is exactly what Shaw deserved for resorting to the Hack-a-Howard. But not because the strategy is distasteful or morally questionable. No, Shaw and the Nuggets deserved to lose that game because the Hack-a-Howard *is a terrible strategy*.

The worst thing for Nuggets' fans is that *this is old news*. Patrick wrote a post on on the subject over a year ago, and he isn't the only person to note the ineffectiveness of the strategy; John Hollinger wrote about it in January of 2012, and Dave Berri and Dean Oliver wrote about it well before that. So not only did Shaw use an ineffective strategy, he used one that has been *known *to be ineffective for quite some time.

For those of you looking for a quick recap of the math behind the Hack-a-Howard, let me quote Dave Berri from Patrick's article:

The value of a point is 0.03260287. The value of a free throw attempt is -0.0151305; or, more precisely, 0.45032 * the value of a possession employed (or -0.0336).

Given these values, if a player took 100 free throw attempts, he would have to make 46.4085% to break-even. This is found by multiplying 100 * -0.01513505. Then divide this by the value of a point, 0.03260287.

So any player who makes more than 46.41% of his free throws should to go to the line as much as possible.

Dwight Howard is at 57.6% shooting free throws for his career, which is well north of 46.4%. The decision to go for the Hack-a-Howard looks a bit better when we learn that Howard has shot around 49% on his free throws over the past two seasons (and was at 49% on the season going into tonight's game), but that would still be a bad idea, because 49% still makes it past our 46.4% cutoff.

The list of players that you should try to hack is very small. By my count, since 1979-80, this strategy would probably only work on four players. This because, while 116 players have shot worse than 46.4% on free throws over their NBA careers, only four of them -- Ben Wallace, Chris Dudley, DeAndre Jordan, and Andre Drummond -- ever ended up playing major minutes and would have a chance to be on the court near the end of a meaningful game.

The good news for Brian Shaw is that two of these players are currently playing! So if he actually wants use the Hack-a-Whoever stategy effectively, he'll try it against the Clippers or the Pistons.

Unless Shaw finds that actually winning games isn't being true to himself.

Take an example. Let's say you are losing by 8 with 3 mins to go, and that means that you are only 20% likely to win the game. If you can close it to a 3 point margin in a minute, you are 45% likely to win the game, whereas if you are losing by 13 with 2 minutes to go, you are only 5% likely to win the game. And let's say you employ Hack-a-whoever that is 49% likely to gain you 5 points and 51% likely to lose you 5 points (i.e., it is a losing proposition on an expected value basis). But if I employ the method, I am 49% likely to increase my probability of winning by 25% and 51% likely of decreasing my probability of winning by 15%.

Long story short, if the amount of your potential loss if fixed (at one game), and you are in a losing position, it makes sense to employ an increasingly risky strategy, even if on an expected value basis that strategy is normally less valuable than the normal strategy. What's the worst that happens - you still lose, just by more points? Of course, how this all plays out with Hack-a-whoever, I have no idea. It may even be that Hack-a-whoever is not riskier than normal. But it also may be that, where the free throw probabilities are close to break-even (e.g., with 49% Dwight), the increase in risk makes sense.

Hacking won't improve your expected defensive efficiency (as has been shown here). Hacking won't improve your expected offensive efficiency (it may actually hurt it, as it is difficult to get into transition off free-throws). However, hacking will increase the expected number of remaining possessions.

This is why teams foul at the end of games. Knowing that there is a poor free-throw shooter on the floor who you could foul probably makes the optimal time to start fouling a lot earlier. It would be interesting to bring game time into this and figure out at when to start fouling as a function of time remaining, point differential, and expected free-throw percentage (given league-average efficiency for both teams).

I think DEN/LAL were right to hack Dwight. Houston is GOING to score on DEN/LAL, and not at the "league average" rate, but the Houston Rocket top-5 offense rate. To compound this, DEN/LAL are below average defenses.

It should also be noted that Defensive Rebounding Rates are MUCH higher on missed Free-Throw's then missed Field Goals.

Real quick probably wrong math says, that even if Dwight is shooting 55% or so on FT's, Hack-a-Dwight is still worth it in this situation.

http://www.youtube.com/watch?v=iJBl834dq18

http://www.nba.com/games/20131116/DENHOU/gameinfo.html

Houston is GOING to score? Really. You can check the play by play. Before the Hack-a-Dwight was going on, Houston either turned the ball over or missed on five consecutive plays. Instead of conceding the play (your 55% still gives 1.1 points per possession, not something you want to do down 10) how about play defense?

A much larger sample size indicates that Denver playing defense would lead to a worse outcome then Dwight making 50% of his FT's.

Although if it was my team, and they were apparently playing a good stretch of defense, I wouldn't want them to Hack-Dwight either.

Every argument that starts with "it might" is pointless. And if you want to argue beyond the average / general to a specific, you ned to do at least SOME work, or state what work needs doing.

Otherwise, the average surely has to stand?

This article relates the Hack strategy to the "average" value of a possession. However, we can ALL agree based on very BASIC data, that the Rockets are significantly above average at scoring, and the Lakers/Nuggets are significantly below average at defending.

These "un-average" circumstances significantly raise the break even point of Hacking (46%). How much? I could look into it, but certainly enough to where Hacking a 49% FT-shooter (Dwight) is WELL worth it.

So as soon as the payoffs are non linear in the metric you averaging across - such as points and wins over anything other than a large sample size - the rule of thumb technique stops working.

E[u(X)]u(E[X])

Math'd!

Instead of conceding the play (your 55% still gives 1.1 points per possession, not something you want to do down 10) how about play defense?

Exactly! Nuggets are too busy giving up tons of points to Rockets and Thunder because of hack-a-Dwight too focus on actually playing defense. Re-read the article:

Unless Shaw finds that actually winning games isn't being true to himself.

Math'd!

Based on Houston's Offensive Rating and Denver's Defensive Rating, Houston is expected to average 1.117 Point per Possession on Denver, while Denver tries to "play defense". This equates to a little over 55% FT-shooting.

Coming into the DEN/HOU game, Dwight was a 49% FT-shooter. Hacking him may not have worked, but it was statistically a better option then just "playing defense".

I certainly think this is absolutely a valid strategy against the Drummond's and Deandre Jordan's who are near the break even point. Howard is probably too good of a shooter, but I don't think that its been proven yet.

To those who say you're not disproving math with math. We are saying using an average does not answer the question. The question is not what is an optimum strategy given equally placed teams. The question is, what gets the most outcomes of your distribution curve into a winning position given you are down by 5-7,7-9, etc. Now I'm not going to do that math, because I'm not an NBA math blogger. But I would like the people who have dedicated their time and effort to be focusing on answering the questions that matter. Sorry that came off snarky at the end. Was really more directed at the commenter. You guys are great.

Using this logic, teams should never spike the ball in the NFL. It's also negative play (0 yards vs. 5.2 yards average) versus averages like Hack-a-Howard.

I'm pretty sure, using simulation that takes into account time left, you can make a pretty compelling argument for Hack-a-Howard. Given that the expected points per possession is marginally higher & you can easily more than double the number of possessions left in the game. That way you can greatly increase volatility of results in terms of points scored in the final few minutes, for both teams, at a small cost. I am guessing at least in some scenarios (even with a shooter >46%), that can lead to a higher expected win % for a losing team.

Well, yeah. Even if you want to not believe their arguments that there are instances where the expected point/possession average is greater than the league average, the article is still wrong, if only because the math used to make the argument relies on the general values that set the value of a free-throw to about .45 of a possession to account for and-ones.

And since a "hack-a-whomever" rarely results in an "and-one" situation, you have to actually hit about 52% of your free-throws to score at the league average rate of about 1.05 points/possession (which, it should be noted, is a mark Howard has failed to reach in recent years).

I don't think its a matter of someone being right or wrong, its just a different way of looking at it.

This article was looking at the Hacking strategy in an average situation, and in an average situation, it was spot on. Its almost never worth it.

But I think its also worth looking at some individual situations, since many of these Hackathons take place in "un-average" situations, where hacking can sometimes be well worth it.

"It's a little more complicated than that" - Ze Frank https://www.youtube.com/watch?v=xuiQrffcC28

The only cases where it makes sense are:

1. Where you have an advantage.

2. ??

That's the equation so far.

I'm unconvinced by a marginal argument that can't be quantified. "Sure it doesn't work most of the time, but there may be a case where it works, so there's that".

> & you can easily more than double the number of possessions left in the game

Can you explain that? It only works before the 2 minute mark of the fourth quarter. So you'd have to start it pretty early - at worst with 3 minutes left - to double the possessions.

"This analysis is clearly seriously flawed and the authors should re-evaluate."

Presumably you have the expertise to explain these serious flaws? Because there's no way someone would just throw about the words "seriously flawed" without thoroughly understanding the math behind those flaws, right?

The author of the piece is not impressed with the criticisms that have been made to date. I haven't seen any math, only descriptions of imaginary math. If anyone actually wants to do the work and form a coherent and evidence-based argument, I will be happy to read it and re-evaluate my thinking on this issue if necessary.

One issue that many critics have forgotten (and one that I and the other writers did not account for) is the fact that, as Howard gets multiple attempts in rapid succession, his ability to make free throws undoubtedly improves. Shooting many free throws consecutively, Dwight Howard has been known to hit upwards of 80%.

http://sports.yahoo.com/blogs/nba-ball-dont-lie/dwight-howard-actually-makes-80-percent-free-throws-002057899--nba.html

Just for example, if you're making a math based argument and you're assuming the wrong distribution, the fact that someone points out, with plain english, that your model's assumed distribution is unlike the thing you're trying to study is valid!

The important counter-arguments here are of a similar form: It's not reasonable to use whole-game values for specific end-game situations, and net-negative strategies can become best-bets if they increase the volatility of the outcome AND your team is likely to lose.

Let me be as crystal clear as possible. On "average", Hack-a-Howard is statistically un-sound, for the reasons discussed in the article. However, MANY times the Hack-a-Howard situation isn't an "average" situation (HOU vs. DEN), and in such a situation, it IS statistically sound (see my above comments).

Devin,

Math can be done to specify the EXACT break-even point on FT%, but the overall premise is above dispute (based on what has been presented so far). A great offense playing a poor defense will score at a significantly higher clip then the "average" situation described above. This raises the "average" break-even FT% of hacking. Pretty simple.

"as Howard gets multiple attempts in rapid succession, his ability to make free throws undoubtedly improves"

I'd like to see the "math" behind this. The article you linked below simply shows that Howard is a good FT-shooter in practice, not that successive FT-attempts will "undoubtedly" increase accuracy.

Another way of thinking about it that might explain why you haven't proven much here that is a little more Gaussian in nature is: Say the average point spread goes down 1 point without hacking but the 3 sigma of the point spread goes down 2 points. You have officially reduced your odds of winning.

Concerning doing the math. I am not the one putting myself out there as the man - you are. I am not going to spend 8 hours making the big Monte-Carlo model you really need here to disprove your theory. I and several others are willing to explain some logic as to why we think your theory is flawed and if you aren't willing to stop and listen that is your problem. It is probably a hint that this is the first post I can remember from Wages of Wins, NBA Geek or here where you have 4 or 5 loyal readers giving meaningful well thought out arguments to the contrary. Might wanna step back and say "whoa".

This example, as much as anything, shows why the simple logic set forth in the post is insufficient. If the clock is under 24 seconds and you are losing, then you need to foul ANYBODY. Even if they are a 90% free throw shooter. When you are losing and there is less than 24 seconds left on the game clock, the value of a possession isn't 0.0336, it's infinite.

"imaginary math" is more popularly known, by mathematicians, as "algebra", and it's pretty important for considering whether the equations you are calculating bear the correct relationship to the thing you hope to measure. Yours don't, which kind of ends the whole "math" debate before it starts.