Sadly, the basketball off-season is here. The NCAA tournament ended a couple of months back, and the NBA season just finished up. It is a good time for thinking about some broader topics related to basketball. Lately, I have been thinking some about how a basketball offense works, and how style of play influences outcomes.

To put more of a Texas spin on this, there is a lot of desire among Texas fans for Rick Barnes to take his offense to the next level. Now, I believe that Rick Barnes already gets pretty good results on the offensive end, but to give Texas the best chance at a championship the offense probably needs to get better.

There are a lot of different ways to improve an offense in basketball. I think it is best to look for easy things that could have a big impact. And for Texas basketball, one of the easiest high impact changes to implement would be to shoot more three point shots. Last season, Texas took roughly 25% of their field goal attempts from beyond the arc. Almost every team in Division I shot the three more frequently than Texas did. It is important to point out that when Texas did shoot the three, they hit a very respectable 37.6%.

I want to illustrate just how valuable the three point shot can be. After the jump, I will try to do this with a simple exercise.

### Two teams

To start out, let's consider two completely made up teams. One team only takes two point shots. They make them 50% of the time. This is a pretty good shooting percentage from two, as the NCAA D-I team median is about 48%. A team that made 50% of their two point shots would rank in the top 25% of Division I. Our second imaginary team only takes 3 point shots, which they do with a shooting percentage of 33%. This means this team is a somewhat below average three point shooting team for Division I basketball (the NCAA median team shot 34.4% from three last year).

I have picked these two shooting percentages for a reason. Both of these teams will average exactly one point per shot. So far, these two teams look even. But there is more to consider. We have to account for the effects of the missed shots.

So let's assign a few more characteristics to these two teams. Let's assume that neither team ever turns the ball over. This assumption will make the argument clearer, and shouldn't have much effect on the conclusions. We will also assume that each team rebounds 35% of their missed shots. This is a pretty typical rate for offensive rebounding of missed field goal attempts in Division I basketball. To keep things simple, after coming up with the offensive rebound, we will assume that each team continues with their strategy at identical shooting percentages. In other words, the three point shooting team continues to take nothing but three point shots, even after an offensive rebound.

Now we can do a simple calculation. We are interested in determining which team will score more points per possession. Since each of the two teams averages one point per shot, all we have to do is determine how many shots per possession they will get. With no turnovers, each team starts off with at least one shot in each possession. The shot either goes in, or the team gets a chance to rebound the miss. So the number of shots per possession becomes

shot/poss = {offensive rebounds/poss} + 1

(1-FG%) is the rate at which shots are missed. (1-FG%) x [shot/poss] is the number of missed shots per possession. If we multiply the number of misses by the offensive rebounding rate, we get

shot/poss = (1-FG%) x (ORB%) x [shot/poss] + 1

which rearranges to give

shot/poss = 1/[1-(1-FG%)(ORB%)]

Now we can see that the team that takes only two point shots will average about 1.2 shots per possession and the team that takes only three point shots will average about 1.3 shots per possession. The three point shooting team averages about an extra 0.1 points per possession than the two point shooting team.

This exercise highlights an important benefit of shooting the three. For an offense, even missed shots bring some value, as some of these misses result in offensive rebounds. These offensive rebounds can turn into extra shots. Shooting more three point shots creates more opportunity for offensive rebounds.

### Some criticisms of this analysis, and how they can be addressed

**1) The scenario is unrealistic. **

There aren't any real teams that only shoot three point shots, or that ignore the three completely. My motivation was to make things as simple as possible, in order to make the effects as clear as possible. In principle, we could do a more complicated analysis using a more realistic scenario, and calculate the benefit of increasing three point shooting that way. Such a calculation actually isn't very hard, just a bit tedious to explain. I have worked out this calculation, and may take a stab at writing about it another time. The more complicated approach shows a similar result; shooting more threes generally is a good thing for most teams to consider, and part of the reason for this is the extra offensive rebounds that can result.

**2) I have assumed that the offensive rebounding rate for both teams would be the same.**

This is a pretty major assumption of what I have done. I have assumed that a team that only takes two point shots and a team that only takes three point shots will likely recover offensive rebounds at the same rate. There have been studies of this. The ones that I can find focused on the NBA. Dean Oliver reported in his book *Basketball on Paper* that at the NBA level 33% of missed two point shots resulted in offensive rebounds and 31% of missed three point shots resulted in offensive rebounds for the shots that he had tracked.

Then there is this extremely interesting study by 82games.com, also using NBA data. If we take their numbers, we find that three point shots produce offensive rebounds 31% of the time, two point jump shots from outside the key produce offensive rebounds 23% of the time, and shots in the paint produce offensive rebounds 37% of the time. 82games.com also provides the breakdown of how many shots are taken from each area here. They find that 21% of charted shots are three point shots, 32% are two point jumpers outside the key, and 46% are in the paint. (This is one interesting difference between college and the pros; three point attempts are far more common in the college game.) Using these numbers along with the associated rebounding rates, the offensive rebounding rate for all two point shots is about 32%. The Oliver study and the 82games.com study basically agree. They both indicate that three point shots are rebounded by the offense at a slightly lower rate than two point shots in the NBA.

Given that there is only a small difference between offensive rebound rates on two point and three point shots (at least in the NBA), my assumption that they are the same likely isn't too big of a deal. But as an additional check, we can determine how low the three point shooting team's offensive rebounding rate would have to drop to make their points per possession the same as the two point shooting team. The three point shooting team averages the same number of points per possession as the two point shooting team if their offensive rebounding rate drops from 35% to 26%. This difference is much greater than the drop of 1-2% in offensive rebounding rate suggested by the studies made with NBA shot charting.

**3) Shooting many three point shots is a high variance strategy, and that has to be considered.**

High variance vs. low variance strategies in sports have been discussed in many places. When we talk about variance in this context, we are really discussing the range of possible outcomes for a given strategy. The variance of interest is the variance of total points that can be scored in a particular game. Dean Oliver wrote about it in his book (an interesting quote from the book is here). Gladwell wrote about it in a widely discussed article in *The New Yorker*. It comes up a lot over at Smart Football. It would be possible to read all of this and oversimplify it to say that high variance strategies are best for underdogs. Shooting three point shots is a high variance strategy, so it should be left for underdogs.

I agree with this idea in theory. If two strategies produce the* same expectation value** of total points*, the underdog should choose the higher variance strategy and favorites should stick to the lower variance strategy. But in the three point shot vs. two point shot discussion, the expectation values are quite different. This difference is great enough that it overwhelms the variance effect. To show this, we need to do a more involved calculation.

To settle this issue, we need to know how often the three point shooting team will outscore the two point shooting team. The easiest way to determine this is to simulate a bunch of "games" on a computer. To do this, I assume each game consists of 65 possessions by each team. Changing the number of possessions doesn't really change the results very much, and 65 is a pretty reasonable number for NCAA basketball. In each possession, each team starts off by taking a shot, which they make or miss with a probability based on the field goal percentage. If the shot is missed, then there is an opportunity for an offensive rebound. If they make a shot, or miss and fail to get the offensive rebound, then the possession is over. If they miss and get the rebound, the possession starts over, and they get to take another shot.

If I run this little game 10,000 times, I generally find that the team that shoots only three point shots wins 64% of the time, the team that shoots two point shots wins 33% of the time, and they tie 3% of the time. So the three point strategy seems better, even if it does have a higher variance.

**4) This whole exercise ignores free throws.**

I will be honest. I struggled with how to deal with free throws. In the end, I have decided to analyze this in a very simple manner. If a team only shoots three point shots, they probably won't get to the line very often. For the sake of this exercise, let's assume that the team that only shoots three point shots never gets to take a free throw. Now let's assume that the team that takes two point shots gets to the free throw line in 20% of their possessions.

20% of possessions ending with free throws is a pretty high rate. If we take every team in Division I, estimate how many of their possessions end in a free throw attempt, and estimate their total number of possessions without a turnover, we would find that the median team gets to the free throw line in 16% of their possessions. 20% would be close to the highest rate of any team last year in the NCAA.

The median Division I team shot 69% from the free throw line last year. That means on two shot trips to the line, they averaged 1.38 points. If we ignore one-and-ones and ignore three point plays, then we can estimate that by getting to the line 20% of the time, the two point shooting team raises their points per possession total to 1.25, which still isn't as high as the 1.30 total that the three point shooting team gets without shooting any free throws. To get to 1.30 points per possession, the two point shooting team has to hit about 83% from the free throw line. This would have led all of Division I last season. So free throws can potentially help close the gap between these two approaches, but unless you get to the line a lot and shoot a very high free throw percentage, the three point team still should come out ahead.

### Summary

Shooting three point shots improves offensive efficiency in two distinct ways. Most teams can shoot the three at a high enough percentage to make them more valuable than most two point shots. You also benefit from the potential to get extra offensive rebounds. While shooting a lot of three point shots is inherently a higher variance strategy, the effect of variance is relatively small in comparison to the gains that you get in points per possession. Shooting the three is not just an underdog strategy.