# The Mathematics of Hack-a-Howard

Ever since the days of Hack-a-Shaq, teams have been on the lookout for big men with poor free-throw shooting percentages so that they can exploit basketball’s free-throw rules to their advantage.  Sporting a career 57.7% free throw shooting percentage, Dwight Howard hovers tantalisingly around the level where such tactics are considered. Unless after 9 years of struggle Howard finally breaks through and lifts his numbers in this area out of the basement, it’s a fair bet that Rockets fans will be seeing more hacking than they have for quite some time in the coming season. The big question, then, is whether or not hacking is an effective strategy to employ – should Rockets supporters be cheering or worried when the free-throw parades inevitably start?

In this post, I’ll be delving into the mathematics of Hacking to try to answer that question. If for some reason you have an aversion to numbers and formulae in general (I can’t imagine why you would), then this may not be the post for you.

Before we begin, let’s step back and consider the strategic reasons why a team would consider indulging in a Hackathon (in the NBA sense of the term):

• It may reduce the rate at which the other team scores.
• It will give them more time with which to mount a comeback.

The first of these is the reason that most people look to in order to justify or dismiss the validity of Hacking. But in order to get to grips with whether it’s truly worth doing, you have to consider the extra possessions as well.

# Expected Value:

Questions of this nature always circle around to Expected Value eventually, and in this instance it’s an appropriate starting point. A naive way of looking at whether Hack-a-Howard (from now on HaH) is worth doing is to compare the expected value of two Dwight free-throws against that of an average Rockets possession. We have Howard’s career free throw percentage of 57.7%, and last year’s putrid 49.2% mark to use.

Rockets 2012-13: 1.067 Points per Possession (PPP)
HaH (career FT%): 1.154 PPP
HaH (last year’s FT%): 0.984 PPP

So on that basis it would seem that while Howard’s career average is enough to take him out of the Hack Zone, last year’s performance isn’t good enough to do so.

Accounting for Offensive Rebounds:

But wait! According to 82games.com (this page is pretty old, but I don’t see a good reason why that number would fluctuate very much), teams rebound 13.9% of their own free-throws.  Offensive rebounds are already factored in to PPP statistics (the formulae involved treat offensive rebounds as a continuation of the original possession rather than a new one), but not into the Expected Value of HaH. While in general teams tend to score pretty heavily on offensive rebounds, let’s assume that the Rockets’ hypothetical opponents are dedicated enough to the hacking strategy that they send him back to the line if the Rockets corral the rebound. This changes the calculation – the expected value of the first free-throw is the same, but on a miss of the second free-throw there is now the possibility of further shots, so it becomes:

`PPP = 2*FT%/100 + (1-FT%)*0.139*(PPP)`

When we fill in the two FT% numbers for Howard and solve for PPP, we get the following results:

HaH (career FT%): 1.226 PPP
HaH (last year’s FT%): 1.059 PPP

With this adjustment we start to edge closer to respectability – there appear to be marginal benefits to Hack-a-Howard if he is as miserable from the line as he was last year, but even a small improvement will tip the scales away from it in the long run. And it’s definitely better to avoid HaH if he’s back to his career average, right? Well, let’s see what happens when we factor in the other half of the equation.

# Accounting for Extra Possessions:

Thinking about the long run is all well and good, but as I’ve pointed out before, basketball games are not decided in the long run. In the context of an individual basketball game, the ideal strategy is not the one that will score you the most points if you were to keep playing until the end of time. Rather, it is the one that will score you the most points before the end of the game, and these are subtly different things.  Having a time-limit alters the strategies involved, sometimes quite dramatically.

How does this change things in the hacking debate? I’m going to analyse with the help of a fairly simple example. The NBA does not allow hacking in the last two minutes of a games. But imagine a situation in which there are about 2:35 on the clock and one team is down 7 having just scored.  In the short term, they would like the scores to be as close as possible when the window for hacking ends at 2:00. Should they hack? If they don’t hack, then each team will probably get one possession before the window for hacking closes. However, if they do hack, then the trailing team will get two possessions to try to close the gap against the other team’s free-throw shooting. So let’s work through the probabilities in each case.

I’m going to use the stats of the Oklahoma City Thunder as the hacking team in these calculations, because they were the last team to try it out against the Rockets. On any given possession, a team will score between 0-4 points. We can calculate the probability of each score happening and generate a distribution of the change in score over that possession. The number crunching involved is outside the scope of this article, but I went and calculated these numbers for both the Rockets and Thunder using numbers from NBAwowy and Hoopdata. (NB. Unfortunately I couldn’t find any good data on the number of 4 point play opportunities the teams get each year, so I’ve had to ignore them and just give find the probabilities of scoring between 0-3 instead).

Rockets:

```0 point possession: 54.48%
1 point possession: 3.77%
2 point possession: 30.24%
3 point possession: 11.51%```

Thunder:

```0 point possession: 51.43%
1 point possession: 3.14%
2 point possession: 36.60%
3 point possession: 8.83%```

Option 1: No Hacking

Using the above probabilities, we can generate the distribution of how we would expect the score to change if the Thunder decided not to do any hacking:

As you might expect given the Thunder’s excellent offense, in a 1 possession competition they are very slightly more likely to do well:

• 31% of the time they will make some inroads into the deficit,
• 40% of the time there will be no change in score differential,
• 29% of the time the Rockets would pull further away.
• The mean of the distribution is 0.04, so ever so slightly in the Thunder’s favour in the long run.

Option 2: Hacking

In this scenario, the Thunder are going to get two possessions to try to score and the Rockets are going to get 4 free throws from Howard. First let’s look at how things would shake out using Howard’s numbers from last season:

The first thing to notice here is that the probability of the score staying the same is much lower. This is a virtue of there being more possessions to work with – the more shots that are put up, the less likely that the score will stay the same. Overall, this distribution is skewed in the Thunder’s favour:

• 41% of the time the Thunder will make gains,
• 20% of the time the score will remain unchanged,
• 39% of the time the Rockets will extend their lead.

So in this scenario the Thunder have significantly improved their chances of improving their position (34%->41%). They are ahead on average as well – the mean of the distribution is 0.09. So if Howard’s free throw shooting remains at last season’s numbers it is well worth hacking. But we already knew that from the Expected Value calculations we did earlier.

The more interesting scenario is to run the same calculations, but with Howard’s career free throw percentage instead:

Eyeballing the graph would make it seem as though the Rockets are coming out ahead on balance. Here are some numbers from this graph:

• 34% of the time the Thunder will make gains,
• 20% of the time the score will remain unchanged,
• 46% of the time the Rockets will extend their lead.
• The mean of the distribution is -0.25, firmly in the Rockets’ favour.

Note though, that just because on average this strategy favours the Rockets does not mean that it’s not worth the Thunder doing. The numbers reveal a very interesting point: OKC are still more likely to make a gains using this strategy than they would be by not hacking (34% to 31%)! This gain is at the expense of giving the Rockets more of a shot at pulling away, but if you are losing by 7 then staying that far behind may be just as bad as falling further adrift. So there is a legitimate argument that it might still be better to employ this strategy than to let the game continue in the same vein.

The reason why this analysis works out differently is that the extra possessions make a big difference. This is not something you can see by only looking at expected value – you have to delve into the related distributions to see it properly. Hacking is a situational tool, and it is impossible to give a single threshold beyond which a team should hack. That calculation must take into account the number of additional possessions that will be gained by employing the tactic and the state of the game.

Ideally, you would combine this sort of analysis with data about the percentage chance of winning given the time left in the game and the score differential. Such data is available – for example, John Schumann uses it in his series of posts about the most important plays of the NBA finals. With tools like that you could make a fairly persuasive case about whether or not it made sense to hack in a given situation. When the season starts and the hacking begins, it may be possible to judge whether opposing teams are deploying it properly.

In the mean time, Rockets fans are going to have to live with the fact that there will almost certainly be situations where it is worth the other team’s while to send Howard to the free-throw line, even if he can improve his free throw shooting over the summer.

Rob Dover writes for Red94. You can find him on Twitter @Bored_Trevor.