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It doesn't however permit the average of long 2s, occurring throughout the shot clock to be worse than that obtained by shots of the last second.

Schtevie: I think we should generally agree to disagree here. But maybe we can find some common ground on this claim. I don't agree with this statement, but I would agree with the notion that a team shouldn't regularly be taking shots throughout the shot clock which are worse than the average value of waiting until late in the clock to shoot. So that would mean comparing the value of long-2s taken at (say) 16 seconds with the average points gained subsequently. But that requires addressing several more elements than your simple comparison, including:
* the likelihood of offensive rebounds at all times;
* the possibility of a turnover after not shooting the long-2 at 16 seconds;
* the quality of the offensive team and the quality of the defensive team.
If teams are routinely taking shots with expected values clearly below what they can expect to have 6-7 seconds later (against that opponent), then that is good evidence of inefficiency.

Good, but not conclusive. Because you also have to account for "keeping the defense honest." The more options they have to defend across all clock times, the harder their job is. If they know a team won't take long-2s at all, or in the first 21 seconds, that allows them to concentrate on defending the alternatives. I fail to see why this inevitable response should not be part of assessing whether teams can improve their performance by a change in shooting strategy.

Jeff,

At this stage, it's hard to draw firm conclusions from what I did, since the model is still so abstract and some of the assumptions are hard to validate. But if you take the model and its conclusions seriously, you come up with a rule that looks like this:



You can argue about how accurate that prediction is quantitatively, but the general shape of the curve is about what I expect. It's not about finding an "optimal time to shoot", it's about finding the optimal tradeoff between shot quality and lost opportunity cost associated with taking a shot early in the clock.

If I were to summarize the finding to a GM, I would say something like this: When there are more than about 7 seconds left in the clock, feel free to take any shot that produces on average more than about 0.9 points per shot. With less than 7 seconds left, for every second that passes you should lower your standards by something like 0.05 points per shot for every second that elapses.

Thanks G+L...

In your first post of this thread you said it was a study "in which I tried to create a theoretical model of the shot selection process and solve for the optimal time to take a shot."

Just above, you said, "It's not about finding an 'optimal time to shoot' "

I was going off the sentence in your first post.

Appreciate your summary, and all the time you've taken to present and explain your study...

Guy wrote:

It doesn't however permit the average of long 2s, occurring throughout the shot clock to be worse than that obtained by shots of the last second.

Schtevie: I think we should generally agree to disagree here. But maybe we can find some common ground on this claim.

I suspect we will end up disagreeing (to some degree) but I'm all for finding as much common ground as possible.

Guy wrote:I don't agree with this statement, but I would agree with the notion that a team shouldn't regularly be taking shots throughout the shot clock which are worse than the average value of waiting until late in the clock to shoot. So that would mean comparing the value of long-2s taken at (say) 16 seconds with the average points gained subsequently.

Common ground!

Guy wrote:But that requires addressing several more elements than your simple comparison, including:
* the likelihood of offensive rebounds at all times;
* the possibility of a turnover after not shooting the long-2 at 16 seconds;
* the quality of the offensive team and the quality of the defensive team.
If teams are routinely taking shots with expected values clearly below what they can expect to have 6-7 seconds later (against that opponent), then that is good evidence of inefficiency.

Let's address these to the extent we can:

* Invoking the existence of offensive rebounds does not have the potential to redeem the long 2. To the contrary, it makes this option less attractive still. The fact of the matter is - and correct me if you believe otherwise - that offensive rebounds are less likely to occur after jump shots. (One citation: http://www.82games.com/rebounds.htm)

* Turnovers are a key feature of the optimal shot selection model too, but once you calibrate it with relevant NBA data (for turnovers, free throws, and offensive rebounds) and work through the numbers, what you clearly see (and this involves no assumption of sub-optimal behavior; just a straight calibration) is that the "observed" shot threshold at the beginning of the half-court shot clock (i.e at 16 seconds) is much higher than almost all NBA players' ability to shoot long 2s. That's just the way it is. And I'd be happy to write this out, long form. (And a teaser: the actual shallowness of the NBA talent pool aside, there is another really good argument, emanating from optimal shot selection theory, as to why this class of shot is a less preferred option to start an offense with)

* As for taking into account the quality of the offensive team and the quality of the defensive team, well, sure. That makes sense if you want to tell stories about particular offensive and defensive matchups. However, the average remains the average. For (approximately) every above average long 2 offense playing against a below average long 2 defense, we also have the opposite. And so it goes. The average does have meaning...on average.

Guy wrote:Good, but not conclusive. Because you also have to account for "keeping the defense honest." The more options they have to defend across all clock times, the harder their job is. If they know a team won't take long-2s at all, or in the first 21 seconds, that allows them to concentrate on defending the alternatives. I fail to see why this inevitable response should not be part of assessing whether teams can improve their performance by a change in shooting strategy.

I am a bit mystified why you keep returning to what was an extreme thought experiment (no long 2s at all) as being ultimately relevant to the real question at hand: does NBA shot selection (seen through the lens of actual utilization of the long 2) approximate optimality.

Here's a bit more common ground. I think it is great to keep the defense honest by having folks touching the ball in long 2 range, as part of general offensive flow and also to take the shot under the proper circumstances.

But taking such shots, all else equal, in the first 21 seconds (egads, you don't really mean at 21 seconds, do you? A fast break, ending with a long 2?) all this accomplishes is doing the defense a favor. How does taking a shot that yields fewer points (in expectation) than continuing on with the offense help? It doesn't. It can't.

There are people here with the data to address the long 2 question, right? What percentage of shots taken in the first, say, 8 seconds of the shot clock are long 2s, and what's the accuracy for those shots? While there could be a game theory-related need to take some even early in the shot clock, presumably that number should be low assuming that their accuracy is still below, say, 45%.

About 14% of possessions end in a turnover. From about 11% to 16%, depending on your team.
If you have the opportunity to hoist a long 2 (or any shot), you can deny your opponent that 14% opportunity.
If you don't take the shot, you continue to have an 86% chance of any shot at all.
(Of course, if you miss the shot, you've got some 30% chance to get the OReb.)

So, if you have a 43% FG opportunity, and you don't take it, it's because you're hoping for at least a 50% shot later in the possession?
It may be even a worse tradeoff, if offensive rebounds are less likely toward the end of the shot clock.

Hello. First of all I am optimistic and interested in this article, but I have a question at (22). Why did you choose to take a ratio of the integration of shot quality over the period of time chosen rather than solving for t when F is maximal? It would seem to me that the best basketball strategy would be to cherry pick the best point to take the shot rather than count up all the probabilities of success over the duration and average them out. Even though this doesn't dictate that a player ought to take a shot at a prescribed time in the shot clock, as opportunity will govern the dynamics at every time, but I understand that statistically, you can learn how effectively teams make this decision. Furthermore, I believe that with further discussion, a simple heuristic could be developed from this article that could assist in the decision making of players and the analysis of previous decisions. Thank you for your time.

Since I have written so much now, and lost my goal of brevity, I may as well disclose what I had in mind for this potential heuristic. I was thinking something along the lines of "clock value", something a person can use to judge whether should take the shot opportunity at the time it occurs (this includes the entire play that would likely occur after this moment, not simply if the player jumped up and shot. The former is a better rule of thumb than the latter) or decline the opportunity and convert the tactical advantage into "clock value", which would be something along the lines of how many points that team typically will earn with certain amounts of time. The dynamics of the game are more complicated than this, but this could be used to aid in determining that trading in a shot opportunity would be beneficial. I worded like this because there are other reasons to decline a shot, but something this simple could be a powerful, and predominantly non-compromising tool.

Teams probably feel there is a minimal-quality shot (say, .9 points) that they can nearly always get off before time expires.

A 45% guaranteed shot success seems incredibly optimistic to me. In my observations of basketball games, the vast majority of misses are due to well executed defense, not bad luck. Even a 30% guaranteed shot, I would say, would be along the lines of having Kobe Bryant on your team (although Dirk Nowitzky may in fact be a better candidate for such an example as he almost always takes guaranteed shots).

assuming that all long range 2s are banned

Hmm... what is wrong with long-range 2's? I will agree that they are categorically worse that mid-range shots, but these types of shots are my favorite. On the obvious side, it is the least effective way to score points in any position on the court, but their beauty is that they are very hard to defend, and even if their score volume isn't great, a strong deep 2 threat can really open up other opportunities for other shots. A deep two is something that a tall, athletic player can actually hit consistently, and these types of players are hard to match up against if they possess this talent.

GravAndLev, this graph could be a result of the diminishing returns garnered by teams trying improve an already below average quality possession, meaning the data used in the later period of the shot clock is biased towards these already poor possessions.

It seems to me that you have the concept that shot quality is a function of time left on the clock, and decreases from left to right. This may be apparent and true statistically, but is not very useful analytically because this function is in fact a complex result. I can think of two major contributors to this function.. possession quality, and the team's ability to improve on this quality over time.

Consider this statistical analysis:
You are a psychic who has a threshold ability to generally predict a coin flip.
Someone is flipping a coin 2 times, and you are trying to either: call heads on the first flip and predict correctly; decline calling heads on the first flip, and succeed when the second flip is a head.

On the first test and first flip, you are predicting that there is a 70% chance the coin will land on a head, so you call heads.
70% of the time you will have won the game.

On the second test and first flip, you are predicting that there is a 30% chance the coin will land on a head, so you decline on the first flip.
50% of the time you will have won the game.

Try doing this analysis with more data, or take this conclusion with this simple case and you will come to the same result that you will win more often *on* the earlier flips, however it is simple to conclude that you can potentially lose an advantage by "aiming" earlier in those tests that you predicted the outcome to be unfavorable up to that point.

This simple probability analysis is the reason your data appears as it does. The shots taken earlier in the clock naturally have a higher chance of success, but you should not conclude that you gain by actively trying to hasten your shot. In fact, the opposite is true. The more you prolong your shot, the more good opportunities you will likely have.

And this is where the theoretical statistics and math is above me. Perhaps you could take this coin flip analysis, try some different constants and trends of prediction, and see if you can come up with a theory on how to succeed at it. After this is understood, find similar mathematics in basketball and see if you can apply it.

just a straight calibration) is that the "observed" shot threshold at the beginning of the half-court shot clock (i.e at 16 seconds) is much higher than almost all NBA players' ability to shoot long 2s.

This data is biased. Consider a simple random sample of possession qualities. Some of these possessions are going to be good, leading to open 3's or layups and dunks, while others are going to be bad and have varying results. Almost none of these good possessions are going to, in procedural manner, end up into being one of these long 2 data points. On the other hand, almost all your data points from these long 2's will be the result from poor initial possession qualities.

This has nothing to do with my inclination to believe that medium and long range 2's are valuable.. this is the nature of the sampling system you are using, which is flawed.

EvanZ wrote:So, the question I would ask you guys is which play types do you think are influenced by or depend on the shot clock? Could it be that ISO and BALL are plays that are created more towards the end of shot clock and thus contribute to the effect being discussed? Or are all these plays evenly distributed throughout the shot clock. This is similar to schtevie's line of thought, except drilling a bit deeper to get down to the level of individual plays.

It would be interesting to know if there is any correlation between play types and elapsed clock time. I suppose my prior is that there needn't be any such correlation, in the sense that, to a first approximation, each (half court) play type probably takes about the same amount of time, as a realized NBA average. This is surely not to say, however, that the amount of time taken by each type of play is a constant, nor that realized play time isn't related to its productivity (http://www.82games.com/dribbles.htm).

But let me take a step back and add a few general words of context about plays (and play types) and their general relationship to optimal shot selection theory. Simple stuff. Intuitive stuff. But my guess is not an approach that directly guides much NBA thinking.

Back in the days of yore, when I was pondering and scribbling about these issues, I found plays or "looks", rather than shot clock time remaining, to be the preferred mode for addressing offensive optimization. Though shot clock time, of course, is the ultimate disciplinarian, the relevant framework is how many looks an offense can generate within that allotted period.

According to this general theoretical construct (aside from an independent improvement in the scoring chances at any particular look) there are three ways in which an offense can improve itself. It can:

(1) Increase the number of plays/"looks" within the shot clock. Each additional look affording the offense one more opportunity for an above average scoring opportunity.
(2) For a "set" offense (i.e. a given number of looks) improve shot selection at each look (what has been the primary topic of discussion here, and I think, it is fair to say, the primary point of emphasis of Brian's paper).
(3) Optimize within the offense (defined by a set number of looks) in terms of reshuffling the order of looks.

It is this last point that relates to the concept of "play types". A simple example provides an illustration. Suppose an offense consists of only two looks before the shot clock runs out (and we are abstracting here from fouls, offensive rebounds, etc.). These are a possible alleyoop and a jump shot, each having the exact same expected efficiency, ex ante, of 0.5.

But these play types are not otherwise equal. The jump shot can be attempted regardless of circumstance. The alleyoop however cannot. Either the pass can be made (in which case the passer makes it) which happens 50% of the time (at which point the ball is dunked with a success rate of 100%) or the scorer doesn't get free and the pass isn't made.

So, the question is what is the optimal order of looks for this offense, and the answer is obvious. You run the alleyoop first followed by the jumper and get an offense that overall shoots 75%. The reverse, as an alternative, yields but 50%.

This example, of course, is contrived, but it illustrates an important point. What it says is that you want to organize an offense so as to give early looks to those players that have greater probability mass in the upper tail of the shot distribution (as long as they aren't sinks, of course, who won't move the ball along if the good shot opportunity doesn't obtain, or are more likely to turn the ball over, counteracting the potential scoring gains). All else equal, picking off these "upper tails", in rank order, is the key to optimizing on (3).

Hopefully, in the glorious future, we will begin to see data organized so as to address some of these issues. Until then, I suspect no one will be dislodged from their prior beliefs about how optimal NBA offensive organization/decision-making is.

schtevie wrote:
This example, of course, is contrived, but it illustrates an important point. What it says is that you want to organize an offense so as to give early looks to those players that have greater probability mass in the upper tail of the shot distribution (as long as they aren't sinks, of course, who won't move the ball along if the good shot opportunity doesn't obtain, or are more likely to turn the ball over, counteracting the potential scoring gains). All else equal, picking off these "upper tails", in rank order, is the key to optimizing on (3).

Right, and this is what I assume explains the PPS vs. time distribution. Worse shots (those nasty long 2-pters) are saved for "later", after other better options have been eliminated (due to defensive pressure, offensive ineptness, whatever). The shot clock time is not causative. It's an effect. Your mid-range jumpshot won't have significantly greater efficiency simply because you take it earlier.

In fact, that last assumption/hypothesis of mine would presumably be simple enough to test and relatively straightforward to interpret. Really, we just need to categorize each shot in a way that I do the PSAMS metric (inside shots, mid-range, 3-pt, and foul shots) and plot the time-distribution of each type. I think that would go a long way to informing this discussion.

Honestly I think this thread is heading away from a valid theory, rather than narrowing in on one. I thought I described this clearly, but it hasn't yet been picked up on. The assumption that this thread is leading to conclude is that an offense generally begins with a better shot than later, and players miscalculate, don't capitalize on this, and end up taking a junk shot at the end of the clock. My observations imply that there is a beginning quality of a possession and it progresses from there. There may be a very good opportunity at the beginning.. the best that will occur.. and thus it should be taken. Other times, you can force a defense to react to this weakness, and the offense can exploit that shift into a better opportunity after some proper execution. These junk shots you are referring to, that everybody hates, are not due to undervaluing early shots, because in these scenarios, these early shots had, in fact, been largely of little value, and thus even a low value shot that is greater than this (the junk shot), is better.

No, that's not what I said, nor is it what schtevie said, as far as I understand it. What I assume is that teams look for a good shot first, and if there are no good shots available, they settle for a lower quality, but available shot. I did not say anything about players "miscalculating".

Thanks Evan, I appreciate the reply. But that still doesn't relate to my understanding. What I'm getting at, is players will sometimes settle for the early shot instead of working for a later shot. This is not the same as passing up an early shot and settling for a later shot. Why I say that this thread is going the wrong direction is because the second principle is seemingly being accepted as the principle, where I see no grounds for it. I have only ever seen grounds for the first.