A few weeks ago, Zach Lowe of Grantland put out an incredible piece exploring how the Raptors are using their SportVU tracking data. While I was reading through it, I came across a juicy factoid that I wasn’t sure I agreed with:

An example: The analytics team is unanimous, and rather emphatic, that every team should shoot more 3s — including the Raptors and even the Rockets, who are on pace to break the NBA record for most 3-point attempts in a season.

This is a very bold claim – the Rockets already take 28.8 threes per game, a full 35.7% of their shot attempts. And somehow they should be squeezing more three point attempts out of their offense?! In this article, I’m going to test the validity of this claim, and highlight some of the reasons why it is not based on quite as solid a statistical footing as it first appears.

**The Perils of Long Term Thinking**Effective Field Goal Percentage (eFG%) has become a widely used statistic in the modern NBA. More discerning analysts and fans use it in preference to plain old FG% because it comes much closer to calculating the efficiency of a player or team’s shooting habits. Along with its cousins True Shooting Percentage (TS%) and Points per Possession (PPP), eFG% allows people to make long term assessements of what the optimum strategy is. Wielding such figures, you will often hear zealous hoopsheads voice arguments along the lines of: “In the long run, he’ll score more if he turns some of his long twos into threes. He may hit them a few percentage points less often, but the extra point he’s getting means over time he’ll come out ahead”. To go back to Zach Lowe’s article, this is why Toronto’s Director of Analyics Alex Rucker says:

“When you ask coaches what’s better between a 28 percent 3-point shot and a 42 percent midrange shot, they’ll say the 42 percent shot,” Rucker says. “And that’s objectively false. It’s wrong. If LeBron James just jacked a 3 on every single possession, that’d be an exceptionally good offense. That’s a conversation we’ve had with our coaching staff, and let’s just say they don’t support that approach.”

On the surface, this argument seems bullet-proof – in his example, the expected value of a midrange shot is 0.82 points, while the expected value of a three point shot is 0.84 points. Case closed, right? Well let’s just hold on a minute. It’s all very well saying that you should always take the three, but there’s one very obvious counter-example that shows that this line of reasoning is not appropriate for all situations. If your team is down by one and has the final shot, do you take a 28% three point shot or a 42% mid-range two? Clearly, you take the two in this situation: either will win you the game, so you take the shot that’s more likely to go in. This is in direct contradiction to the expected value argument above!

There is a danger to taking the long view, and by association to using expected value, as the basis for your decision-making: the outcome of a single basketball game has nothing to do with the long run. All that counts are the shots that actually go in during that game. Expected value-based numbers are great for telling you what will happen over the course of a season, but they are comparatively lousy for telling you what will happen in any individual contest, or for a specific shot. That is why the using them leads you to the wrong conclusion in situations like the final shot conundrum I described.To work around this issue, we need to think about things slightly differently. Expected value allows us to answer the question “How do I maximise my points total?” But what if we instead ask: “How do I maximise the chance that I score more than my opponent over the course of a game?” The two questions are subtly different, and I would argue that the latter is a better one to ask since it is asking how one wins an individual game rather than winning in the long run. And conveniently, it is possible to work it out! Unfortunately, it requires some slightly more sophisticated mathematics to answer than plain old averages, but provided you don’t mind looking at a few graphs that shouldn’t stop you!**Twos or Threes?**In any given possession, the team with the ball will do one of four things:

- Take a two point shot
- Take a three point shot
- Shoot free throws
- Commit a turnover

For the moment, let’s ignore free-throws and turnovers in order to concentrate on the question – should I take a two pointer or three pointer? The percentages each team shoots from inside and outside the arc are readily available, so let’s use our very own Houston Rockets as a guinea pig. The Rockets shoot 53.6% on two pointers and 36.8% on three pointers (*NB: All statistics from Hoopdata* *unless specified otherwise*) . Furthermore, the Rockets average 98.8 possessions per game. Let’s round that to 99 to make things easier. So let us propose two competing strategies, one where the team shoots only two point shots and one where the team shoots only three point shots. These are the most extreme cases (and obviously are not scenarios that would play out in any actual NBA game), but if you’re trying to determine which is better, then it seems like a logical starting point. We know the percentages and we know the number of shots that will be taken, so we can construct a probability distribution representing the likelihood of achieving any particular score using this strategy. This graph represents the probability that the Rockets are able to make a specific score given their shooting percentages on an individual shot:

As you can see, you can get much higher scores shooting three pointers only, but you are also more likely to get lower scores as well. It appears as though on average you will be scoring more points if you shoot three pointers. However, as I pointed out earlier, the number of points you score is only half the game – you also have to limit the number of points your opponent scores. So what we actually want is the probability that the Rockets score more than their opponent. Fortunately, this is very easy to extrapolate from the above graph – we can add up the probabilities of any individual score occurring to get a cumulative probability distribution like so:

From this graph, we can now read the probability of winning a game depending on which strategy the Rockets employ and the number of points they give up on defense. I’ve marked the team’s actual defensive performance on the graph (99 possessions per game with a defensive efficiency of 103.8 equates to an average of 102.8 points conceded per game). It validates the strategic decisions the Rockets coaching staff and front office have made (and justifies the statements of the Raptors’ analysts) – the all three point strategy is better for them than the all two point strategy, and so maximizing their three point opportunities would seem like a good idea.

But there’s something more interesting lurking in this graph. If the Rockets could hold opponents under 100 points per game, it would actually make more sense to shoot twos rather than threes, despite the averages implying otherwise! This does make some sense when you stop to think about it – if you only need 80 points to win the game, then you don’t need to have a small possibility of making 120 points, especially if it comes with a small chance of only making 75 and losing. But the corollary is eyebrow-raising –**the better you are at defense, the fewer three point shots you should take**!We have to be a little careful with this result. Shooting percentages are still the dominant force on how these curves fall. Each team will have a unique pair of distribution functions and therefore a different sweet spot at which the two curves cross. What is a sufficient level of defense for one team to start avoiding the three point line and driving more will not necessarily be the same for another, so it does not follow that the better defensive teams in the league would necessarily benefit from going inside (in fact, performing these calculations for the league’s best defensive team, the Pacers, shows that they are firmly in the “should shoot more threes” camp). However, what we can say is that for any individual team, tightening up defensively will make it more viable to play closer to the basket. Food for thought.So was this a useful thought experiment? Well, let’s look at some of the assumptions I’ve made:

*Not including free throws.*Trips to the charity stripe are more likely when taking two point shots than three pointers. So including these would probably swing the balance further towards two point shooting.*Not including turnovers*. A turnover is a wasted possession, so this would have the effect of reducing the number of shot attempts in the model. If you need to score the same number of points in fewer possessions, you’ll probably need to take riskier shots (eg. three pointers). This would probably tip things back in the other direction.*Amalgamating all shots inside the arc to a single percentage.*As Kirk Goldsberry says, jump shots tend hover at around 35-45% accuracy across the league. This is distinct from shots at the rim, which tend to be at or above 60%. It’s clearly better to be taking shots at the rim than jump shots (something the Rockets are well aware of), but it does mean that there isn’t quite the homogeneity assumed in my model in real life. You could make this more accurate (and more complicated), by separating out a team’s percentages from various ranges from the basket and thereby generating a more accurate distribution of a team’s score when taking twos. In the end though, I don’t know it would change things by all that much.*Assuming a team can only take all twos or all threes.*Obviously this has little bearing on real-life, where a team is forced to take a mix. But this is an attempt to find an optimum strategy, and that will never be found in the middle. The idea is that teams should be looking at data like this and trying to move their offense towards the end of the scale that they sit.

Working these into the model would certainly change the overall shape of the graphs, but I don’t think any of them would have a significant enough effect to change the headline result.

**The Rest of the League**I was curious, so I decided to see what the results would be like for the other teams in the association. Which teams would benefit from shooting a lot of threes, and which would do better from inside the arc? It turned out that the majority of teams are better off taking more three pointers, but there were a few exceptions:

**Denver Nuggets:**%Win w/2pts*68.9%*, %Win w/3pts*47.1%***LA Clippers:**%W2*71.8%*, %W3*61.3%***Minnesota Timberwolves:**%W2*32.2%*, %W3*17.9%***Oklahoma City Thunder:**%W2*81.4%*, %W3*81.3%***Orlando Magic:**%W2*26.7%*, %W3*25.31%*

The Magic and the Timberwolves are on this list for one simple reason: they are atrocious at three point shooting. These two teams rank 28th and 30th in the league respectively, which would explain why it might be better for the to focus inside the arc. For the Thunder there is very little difference between the two strategies – they are a good enough team to be able to win on most nights no matter which path they choose. However, the two remaining teams require a bit more explanation. Both the Clippers and the Nuggets are excellent 2 point shooters (4th and 5th in the league) and solid defensively (9th and 11th). When you add that to being in the bottom third of three point shooting teams (20th and 24th), you have a recipe for a rare team that don’t necessarily need to rely on the three point shot to succeed.

**Possible Extensions**I’ve found this to be a very interesting avenue of investigation, but I’ve only really scratched the surface of what you can do with this sort of model. There are a number of possible follow-ups that I may look into in the future:

- We could apply similar logic to see how teams should perform at various points throughout a game. How should teams behave going into the 4th quarter ahead by 10? How about down by 5?
- We could look at this on a player-by-player basis – it’s well known that certain players are ‘streaky’, so perhaps we could quantify that?
- Maybe it’s worth filling in some of the assumptions I’ve made to make sure they don’t drastically alter the equations. Does adding in free throws and turnovers make a difference?

Join me next time for more wacky work with probabilities, distributions and hopefully some surprising insights!

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

**@Bored_Trevor**

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