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Brian: Another thought on your model. If I undertand it correctly, you now assume an even distribution of quality shots between f1 and f2 -- say, between 1.1 and .5. So very good shots become available just as often as poor shots. Such an even distribution seems unlikely to me. So you might want to consider other distributions, such as a distribution with relatively few high-quality opportunities, some mid-range opportunities, and a lot of poor opportunities at any given point in time (which seems more intuitive to me).

I suppose in a sense you do this already by modeling a certain frequency of shot opportunities, meaning that shot quality = 0 at all other times. But it's not really binary: in most one-second intervals (after the 1st 6-7 seconds), the player with the ball could get off a non-zero-points shot. Perhaps you might drop this idea of a fixed number of opportunities, and simply posit a distribution of opportunities for any given point in time, which is heavily weighted toward the low end. Just a thought -- I have no idea how that would change your ideal shot frequency curve.

And as mentioned earlier, it seems likely that the distribution changes with time, at least late in the clock.

Guy,

You're right that I also make the sort of arbitrary assumption of a flat distribution of shot quality. This is certainly an arbitrary assumption, but my bet is that it's probably okay as a first attempt. Tweaking the distribution more and more becomes increasingly speculative and introduces all sorts of new free parameters into the theory. A theory with too many parameters runs the risk of having no predictive power at all.

Really, what it comes down to is that I recognize that I have two big assumptions: the shot quality distribution is flat and independent of shot clock time. These are both unjustified, but I'm not willing to make any more complicated assumptions until I/we can figure out some way to get some guidance from the data. If there is some clever way to learn what this "shot opportunity distribution" looks like from data, that would be great, and it would eliminate the need for such arbitrary assumptions. Unfortunately, at the moment I can't think of a way to do so.

EvanZ and mystic,

This weekend I'll sort through the data and separate shooting rates and shot effectiveness by team, shot type (dead-ball vs live), and season. I'll post it on this forum, probably as a new thread.

I don't recall what Bob Chaikin does in his simulation but it might be interesting to hear a bit about it, if he was able & willing or if anyone else remembers / thinks they know. Same with Ben Falk.

Or maybe you could try to reverse engineer the most sound commercial nba game, whichever one is based on nba "dna". Could you run the same game scenario 200 or 2000 times and see what their shot quality distribution is? And repeat for 10, 20, 50, 200 scenarios?

Regrettably, I haven't been able to carve out enough time to fully address the various issues raised in Brian's very interesting (and important) article. But having thought a lot, a lot about this very topic (and approach) starting a very long time ago, I want to get a few points out there before the thread goes to sleep. In particular, regarding data issues.

My first point is that I don't think the article adequately emphasizes (perhaps it isn't made) that the optimal NBA shot selection/threshold curve we would wish to see - the one pertaining to the relevant theory - cannot be generated from unmanipulated NBA data, hence we can make no inferences from such data about how players perform relative to the optimum. To the extent that there are any shot hogs in the league and to the lesser (?) extent that there are any wilting violets unwilling to take really, really good realized shot opportunities early in the shot clock (surely the effect of the former group dominates the latter, right? Not that this distinction is relevant for the point at hand) actual scoring deviates from the potential and cannot be used directly to define the potential.

The second point about data pertains to what was actually presented in the article and in this discussion. Not having included the expected value of offensive rebounds in defining the shot threshold is one problem already noted. Another is the lack of inclusion of foul shots. Am I correct about this assumption? For a player, theoretically forming some rational expectation of future opportunities (based on the mechanism of backward induction), expected points from expected foul shooting opportunities is very important. This omission (again, unless I am misunderstanding what has been presented) is probably more important than that of offensive rebounds, in the sense that the latter's effect is probably uncorrelated with the shot clock whereas the former's isn't.

These issues aside, let me finish with a few comments on the graph and table presented on page two of this discussion, showing the distinction between points after defensive rebounds vs. those after dead balls. First, the points per shot graph.

Assuming that the notionally omitted effect of foul shooting (or offensive rebounds) isn't responsible, the dip we see at seconds 17 and 18 is a factual anomaly. That is to say that the (general) theory says that these facts are at variance with optimal behavior. Said another way, you should never see an increase in realized scoring opportunities as the shot clock draws down. Period. What the cost is of this possible (I would say probable) inefficiency is another question.

The next observation about the points per shot graph is the obvious difference in performance between dead ball and defensive rebound originating possessions at the end of the shot clock. Given the relatively few shot opportunities left at that point, it isn't a really big deal in terms of potential inefficiencies, but the deviation in the curves sure is a bit strange. To a first, and very good, approximation, there is no reason to assume that the possession origin should matter at all at the end of a shot clock. We are looking in both instances at possessions that settled into half court offenses. Apples to apples. And then there is the more meaningful difference (in terms of potentially foregone points) of the defensive rebound originating possession half court offense consistently underperforming its dead ball counterpart.

It would be interesting to see the plots for different types of dead ball possessions, made final free throw, made basket, timeouts, start of period, etc., for further comparison sake.

If these observed deviations are strange and unexpected (well, they were, a bit, to me) the underlying data in the table below the plots reveals something much stranger and more unexpected. In fact, the data imply something so weird that I am wondering if I am misinterpreting what is written or if the data somehow have been incorrectly related.

Basically, if you look at the disposition of shots over time, you see that the dead ball originating half court offense bears no resemblance, at all, to the defensive rebound originating half court offense. And this isn't a fine point, where you need to manipulate the data on a spreadsheet to find it.

For defensive rebound originating possessions, in the half-court (defined as shots taken from 16 seconds left on the clock) you see a monotonic decrease in shots taken. Every second passed sees fewer shots taken. By contrast, after a dead ball, what should be the "exact same" half court offense (16 seconds to zero) you see something completely different, the shot disposition following a "bellish curve", rising to a peak shooting rate at 8 and 9 seconds remaining, then similarly declining. "Same" half court offense, accordingly approximately the same average efficiency (0.956 vs. 0.965, respectively) but completely different shot selection, in terms of time.

Am I missing something here? If these are the numbers, this is bizarre (and the aforementioned effects of not including the offensive rebounds and foul shooting shouldn't matter on this account).

Anyway, very interesting article, very interesting topic, much more to say...

And let me just add a brief empirical note, touching on the points made above and in a previous reply to Guy regarding the likelihood of $100 bills lying on the sidewalk.

From the same table referred to above, we see that the worst shot in basketball, on average, occurs with one second left on the shot clock and yields 0.843 points on average. Contrast this with data from Hoopsdata which shows that the expected points per shot taken from 16 to 23 feet over the last four years is pretty consistently about 0.8 and that about 20 or so such shots are taken per game. Also, as noted just above, half court offenses have averaged about 0.95 points per shot.

What might we infer from this?

Again, we don't know what offensive potential is in the NBA - only that by definition we know we don't see it, and that it is higher than what is realized. But let's pose a naive, half court counterfactual, assuming that all long range 2s are banned (what currently occupy a minimum of 25% of all shot attempts in the half court offense - about 20 per game) and that the observed offensive efficiency of all other shots holds. If observed half court efficiency is about 0.95 points per shot, the non-long 2s yield 1 point per game, thus there is a wedge of 0.2 points per long 2 shot attempt and the alternative. 0.2 points per shot times 20 shots per game times 2 points per shot is 8 points per game.

But this involves ridiculous assumptions though, right? But this is also a ridiculously large result too (and it is a lower bound in the sense that it utilizes a known below-optimum baseline and because 25% of half court shots being long 2s is a low estimate - I think.) In fact, half of that is a very large result. And half of a half is a large result. And half of a half of a half is still a very meaningful result. One extra point per game is about, what? Three extra wins?

This is all to reemphasize the point that the numbers should show significant and competitively meaningful inefficiencies with regard to shot selection over time, they just need to be massaged and tortured to do it.

Edit/P.S. To be clear, I am not gratuitously riding my "long 2s suck" hobby horse here. I only bring up them up because it is a known category of on average sucky shots whose precise suckiness is known...and...these happen to be dispersed across the half court shot clock...and...they are on average necessarily sub-optimal.

This is not to say that highly contested mid-range shots taken early in the shot clock (as they occasionally are) don't also suck - they do - or that 3s taken by folks who can't hit them (what happens) don't suck - they do - etc. And it is also not to say that all long 2s suck. Some are clearly of quality and rational, but that only implies that the others in the group are even suckier and more sub-optimal.

And let me just add a brief empirical note.... Again, we don't know what offensive potential is in the NBA - only that by definition we know we don't see it, and that it is higher than what is realized.

I think the word you are looking for is not "empirical," but rather "tautological."

But let's pose a naive, half court counterfactual, assuming that all long range 2s are banned (what currently occupy a minimum of 25% of all shot attempts in the half court offense - about 20 per game) and that the observed offensive efficiency of all other shots holds.

I would say the naive counterfactual is that efficiency on all other shots falls (and turnovers rise) such that average points per possession is unchanged. In fact, even that is a very generous assumption (for your view) -- in reality, a team that was instructed never to shoot a long 2 would surely see it's average points per possession fall. The interesting question is whether there is any point between current long-2 frequency and zero at which offensive production would increase. I assume your view is that the answer is "yes," and that may be true, but I'd need to see some evidence to be persuaded.

Guy wrote:

And let me just add a brief empirical note.... Again, we don't know what offensive potential is in the NBA - only that by definition we know we don't see it, and that it is higher than what is realized.

I think the word you are looking for is not "empirical," but rather "tautological."

Well, um, no, that is the word you imposed by manipulating the quote. (Naughty, naughty.) The last sentence was written (later) to emphasize that the estimate provided of the opportunity cost is a lower bound on this particular dimension.

Guy wrote:

But let's pose a naive, half court counterfactual, assuming that all long range 2s are banned (what currently occupy a minimum of 25% of all shot attempts in the half court offense - about 20 per game) and that the observed offensive efficiency of all other shots holds.

I would say the naive counterfactual is that efficiency on all other shots falls (and turnovers rise) such that average points per possession is unchanged. In fact, even that is a very generous assumption (for your view) -- in reality, a team that was instructed never to shoot a long 2 would surely see it's average points per possession fall.

I am a bit surprised that you would infer (and suggest) that the extreme of my little thought exercise was my invoking "reality". Despite specific words to the contrary. All you would have had to do is clip the quote like you did above and the interpretive error would have been avoided. Whatev.

That said, it is an interesting (extreme) thought experiment as to what would happen if all 16-23 footers were verboten. Some additional provisos would have to be added (specifically, would the defense know the policy) but whatever the case, it isn't obvious to me that average points per possession would fall. There are a lot of really bad shots that would be eliminated.

Guy wrote:The interesting question is whether there is any point between current long-2 frequency and zero at which offensive production would increase. I assume your view is that the answer is "yes," and that may be true, but I'd need to see some evidence to be persuaded.

Dang. No need to assume. Just read what was written (and, by definition, of course, it may be true). Semantics aside, it is an interesting issue as to what evidence should be required to be persuasive in this instance and on whom should the burden be placed to provide it?

The theory introduced in Brian's article is true. As in, not false. The general result is that all else equal (same players, conditions, rational expectations, etc.) there is a "curve" defining optimal offensive performance, and a related one, showing the threshold for optimal offensive decision making each step of the way. Given this (what is true - and frankly highly intuitive) to the extent that in practice any player at any time deviates from such decision-making, offensive production is lower than it would otherwise be.

As such, the evidence required is the identification of sub-optimal shots. Just one, to establish the truth of the proposition. Many, many to make it competitively relevant. So, along this line, I provided the "tautology" of long 2s, a very common shot that, on average, is worse than the worst shot available on average within the shot clock, when one either shoots the ball or turns the ball over. And then I noted that there is nothing special about long 2s. There are lots and lots of other terrible shots taken on average in a basketball game. Just watch!

But lets stick with the prima facie evidence provided by the long 2s. For this evidence not to be persuasive, a very curious and, frankly, highly unpersuasive argument would have to be made and accepted in its stead. (And please correct me if I'm not specifying it precisely.) The argument would have to be that all long 2s taken (in the sense of the average efficiency) were the best possible shots available during their particular runs of the shot clock. Said another way, that subsequent opportunities, on average, would yield less than 0.8 points per shot, less than what is currently observed when forced to shoot before the shot clock runs out.

So, every time some knucklehead jacks up a long jumper with a defender in his face with 15 seconds left on the shot clock and a teammate open on the block, no problemo! That must be optimal. How so? We know, a priori, it would have been impossible to generate a superior shot with the remaining time. This is the strict alternative, and it isn't so persuasive.

Sometimes abstract theory is useful for structuring and guiding one's intuition, and I think that this is a particular case in point.

Is it possible that G&L's quest to "solve for the optimal time to take a shot" will end up being something like "your first look at an opportunity that represents 1.1 points per attempt or better" rather than a point in time on the clock?

Shooter + Spot on Floor + Guarded or Unguarded

It doesn't really matter if it's 18 seconds, 15 seconds, 10 seconds, or 5 seconds...successful offenses take the first opportunity that represents 1.1 points per attempt or better. The charts show that it gets more difficult the longer you wait...but there wouldn't be any evidence that a poor shooter forcing up a guarded shot from a bad location is making a good choice at 16 seconds or something.

Shooting at the most successful time on the clock doesn't turn a bad opportunity into a good one. What's "optimal" involves the convergence of shooter/spot/space, and the clock's just there in the background.

Well, um, no, that is the word you imposed by manipulating the quote. (Naughty, naughty.)

It certainly wasn't my intention to "manipulate" your words, which of course are there for everyone to see. You began by promising some "empirical" information bearing on the issue. I don't see any -- did I miss the empirical part of your post? What I see are strong assumptions that inefficiencies must exist ("by definition," no less!) .

Some additional provisos would have to be added (specifically, would the defense know the policy) but whatever the case, it isn't obvious to me that average points per possession would fall. There are a lot of really bad shots that would be eliminated.

To think that offenses might improve if deprived of one of their 3 weapons is, to me, wildly implausible. For one thing, every team has invested resources in players whose main talent is shooting long 2s, a skill which now has no value. So at least until a team overhauled its roster to match the new strategy, you are wasting a lot of talent. Second, the idea that teams could always create a better shot than a long two seems far-fetched -- where's the evidence for this idea? And then there's the defensive response (how would you propose to stop them from noticing the lack of long-2s?): a team with only 2 shooting options will be at a huge disadvantage, because defenses only have to worry about two possibilities rather than 3. To doubt this is to completely ignore the certainty that defenses will adjust.

Given this (what is true - and frankly highly intuitive) to the extent that in practice any player at any time deviates from such decision-making, offensive production is lower than it would otherwise be. As such, the evidence required is the identification of sub-optimal shots. Just one, to establish the truth of the proposition.

OK, here is the crux of the disagreement. Hundreds or even thousands of "sub-optimal" shots, by your definition, is not evidence of inefficiency. By that logic, players are being inefficient so long as there is any difference in the average value of different shot types. And that is simply wrong, or at least, is a definition of inefficiency that has no practical meaning or use.

You establish inefficiency when, and only when, you demonstrate that a new decision rule ("shoot when X is true," or "never shoot unless X is true") does the following: A) is something NBA players are capable of implementing, B) will increase offesnive productions, and C) cannot be offset by a logical change in defensive strategy. Your theoretical musings are a long way from meeting this test.

I think it might be helpful to bring in the efficiency and rate at which various different play types are actually run in the NBA. I wrote a post last year on Synergy offense, which included the following two box plots:

Rate (% time) at which plays are run


Efficiency (PPP) of different play types



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.

Hi everyone,

I appreciate the points that are being raised here. I only wish that I had as much apparent confidence as schtevie does that my paper is "important"!

schtevie, I also appreciate you emphasizing the sort of dramatically pronounced difference between possessions beginning with a defensive rebound and possessions beginning with a deadball. Like you, I expected that the state of the offense during the last few seconds of the shotclock would be completely independent of how the possession started. The best hypothesis I have to explain this difference is that there is some inherent time scale (about 16 seconds, apparently) associated with how long it takes for a team to run through its well-practiced offensive set. If the team goes through that without taking a shot, then there is a rapid decrease in the quality of shots taken, because the team starts "grasping at straws", and the shots being taken are improvised rather than the product of a well-rehearsed offense. Since possessions beginning with a defensive rebound start more quickly, they tend to reach the "improvised" stage during the last few seconds of the shot clock, while possessions beginning with an inbounds pass start more slowly and perhaps never really get to the improvised stage.

Of course, all of the paragraph above is just speculation. I will be very glad if someone explores this issue more deeply in a follow-up study. (And, if you find something interesting, I imagine that there's a good chance of it being well-received at PLoS.)

Also, I just wanted to say that I haven't forgotten my promise (on Feb. 1, above) to break down the shooting data more finely by possession type, team, and season. I'll get around to that before long, and when I do I'll post it on this forum.

If a possession consists of an offense trying to find a good shot...as a ballhandler pursues a set of options within a called play or during improv...based on how the defense is guarding that ball and that set of options...and the play clock isn't an issue until the last few seconds when it becomes a sixth defender (because it can force a turnover)...what does this study tell us beyond what we already knew?

Isn't the "optimal time to shoot" when an opportunity at or above a certain threshold presents itself? And, isn't that disconnected from the clock?

It can be a dunk on a break in the first few seconds...it can be an open FT line jumper off a screen at the 15-second mark...it can be an open three off a set of passes at the 10-second mark. It's based on what a defensive failure gives you, not on what the clock is saying.

Does this study suggest that's an erroneous line of thought?

Guy wrote:

Well, um, no, that is the word you imposed by manipulating the quote. (Naughty, naughty.)

It certainly wasn't my intention to "manipulate" your words, which of course are there for everyone to see. You began by promising some "empirical" information bearing on the issue. I don't see any -- did I miss the empirical part of your post? What I see are strong assumptions that inefficiencies must exist ("by definition," no less!) .

If you don't like the word empirical, use whatever adjective you feel comfortable with to describe the average efficiency of long 2s (of which many, many, many are taken) being lower than the average efficiency of a last second shot (of which not so many are taken). Your point, and I hope I am being fair, is that absolutely no inference can be drawn, given the theory of optimal shot selection, that any of those long 2s taken earlier in the shot clock were they not taken would have resulted in scoring opportunities with a higher return.

Guy wrote:

Some additional provisos would have to be added (specifically, would the defense know the policy) but whatever the case, it isn't obvious to me that average points per possession would fall. There are a lot of really bad shots that would be eliminated.

To think that offenses might improve if deprived of one of their 3 weapons is, to me, wildly implausible.

Again, we can fixate on the extreme case, but that isn't really the point. But what the heck? Let the parlor game commence! I'll play devil's advocate. (But this really, really, really isn't on point.)

Guy wrote:For one thing, every team has invested resources in players whose main talent is shooting long 2s, a skill which now has no value. So at least until a team overhauled its roster to match the new strategy, you are wasting a lot of talent.

Goodness, where does the "no value" come from? Good shooting would remain an exceptionally important skill. The unrealistic thought experiment we are engaged in is eliminating long 2s, not 3 pointers or jump shots closer in, the point being to eliminate the shots that good shooters take that actually are sub-optimal. There is no issue of roster overhaul or even talent wasting. The issue is redirection of existing resources in the short term, then, of course, relevant investments on the margin over time - investments which don't directly imply changes of roster (just as we've seen with players improving their long range shooting to realize the opportunities from the 3 point line - but, oh wait, that can never have happened, as that would have implied exploitable inefficiencies). This shouldn't be a controversial point. Please see: 3 point shot, history of.

Guy wrote:Second, the idea that teams could always create a better shot than a long two seems far-fetched -- where's the evidence for this idea?

Where does this notion of "always" come from? Not from the theory in question. To the contrary. Not from anything I said. We are thinking through the wacky counterfactual of pretend all long 2s don't exist. In such a world, the points from some long 2s (those currently "optimal") would not be able to be regained, but other long 2s would "surely" be improved upon (in particular those within the mix that yield below 0.8 points per shot that occur well before the end of the shot clock, where the revealed potential of the half court offense, in expectation, is to provide better opportunities). The question at hand is the net effect, given the extreme assumptions we are operating under.

Guy wrote:And then there's the defensive response (how would you propose to stop them from noticing the lack of long-2s?): a team with only 2 shooting options will be at a huge disadvantage, because defenses only have to worry about two possibilities rather than 3. To doubt this is to completely ignore the certainty that defenses will adjust.

Look, now I concede for the silliness. If a defense really, truly is promised and knows that from 16 to 23 feet from the basket the opposing team will never shoot, well, that would impose great difficulties on the offense. However, relax the extreme assumption just a wee bit to allow the Ray Allens of the NBA world the opportunity to take wide open catch and shoot long 2s if left wide open, such that defenses would be obliged to defend this zone, and I would be willing to continue. But again, all this is beside the real point here.

Guy wrote:

Given this (what is true - and frankly highly intuitive) to the extent that in practice any player at any time deviates from such decision-making, offensive production is lower than it would otherwise be. As such, the evidence required is the identification of sub-optimal shots. Just one, to establish the truth of the proposition.

OK, here is the crux of the disagreement. Hundreds or even thousands of "sub-optimal" shots, by your definition, is not evidence of inefficiency.

Yes, I am saying that thousands of "sub-optimal" shots, by my definition, is evidence of inefficiency.

Guy wrote:By that logic, players are being inefficient so long as there is any difference in the average value of different shot types. And that is simply wrong, or at least, is a definition of inefficiency that has no practical meaning or use.

This is simply and completely incorrect. Full stop. There must be some misunderstanding here. The logic of optimal shot selection simply says that shots taken yield in expectation more points than would occur, in expectation, by continuing the offensive play. In such an "optimal" world, it perfectly well allows for dunks yielding more points per shot than long 2s. 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.

Guy wrote:You establish inefficiency when, and only when, you demonstrate that a new decision rule ("shoot when X is true," or "never shoot unless X is true") does the following: A) is something NBA players are capable of implementing, B) will increase offesnive productions, and C) cannot be offset by a logical change in defensive strategy. Your theoretical musings are a long way from meeting this test.

I am trying to think of the best way to respond to this categorization. On the one hand, but for point C, I basically agree, but on the other, I find this construction completely misguided.

Previously, I raised the issue of on whom the burden of proof should lie. If the (ahistorical) null is that we live in a world of offensive basketball perfection, then your construction is plausible (and I'll take A through C in a moment). But the fact is that we have demonstrably never yet lived in such a world. (I refer again to the history of ginormous changes in pace and productivity through the decades, then overlapping with the slooow adoption of the 3 point shot, as the most conspicuous examples of "deviations" from optimality.) Then there is the basic theory described in Brian's article - what is completely true - which defines the terms of optimality. Against which the "non-empirical" evidence cited is highly relevant, never mind all the evidence our lyin' eyes provide recurrently when watching the occasional, highly conspicuous boneheaded NBA shot selection.

But let's take your preferred mode for establishing the proposition and see where it takes us.

A: Are NBA players and coaches capable of implementing changes to offenses that tend more towards the optimal? Well. Sure. Of course. What's not clear or attainable about: "Hey you! Don't shoot from location X with more than Y seconds on the shot clock. You do that and you're riding the pines." Of course, more sophisticated presentations are possible; we're talking about professionals here.

But that doesn't mean players and coaches will, or will do so conspicuously or quickly. Such is the history of the NBA. Coming to recognize what is better shot selection and reining in poor shot selection is absolutely nothing new (per the history referred to). It's been going on since mostly forever, but at a very slow pace relative to the competitive opportunities.

B: Will allegedly better shot selection improve offensive performance? Optimal shot selection charts on a team-wise basis can be constructed with a bit of work. And experiments could ensue.

C: Are potential defensive responses relevant? I think the correct answer here is no. The entire point of the optimal shot selection analysis is to say that offenses are using time inefficiently. That is they are taking shots that should be passed up for better opportunities later in the shot clock (or not taking shot opportunities that cannot be expected to be improved upon later in the shot clock) given the current state of "defensive technology". It is illegitimate, for the purposes of this debate, I think, to stipulate that if an offensive player passes up a (bad) shot and moves the offense along that somehow defenses would newly become better than they had previously demonstrated, on the pass and after. The whole point of the exercise is to determine optimal choices in terms of what exists - ceteris paribus.

If you want to expand the conversation and talk about dynamic defensive consequences, that is another discussion. But to anticipate, I think if the stipulated outcome of an improvement in offensive shot selection were to be no net gain in offensive efficiency, proper accounting is to acknowledge both offensive and defensive improvements, resulting in no net gain for either side, not no improvement whatsoever. Such a decomposition is essential for telling the story straight.

Jeff,

Isn't the "optimal time to shoot" when an opportunity at or above a certain threshold presents itself? And, isn't that disconnected from the clock?

The point of the study is that the threshold is not disconnected from the clock. When you have plenty of time, you can afford to be extremely selective and hold out for only a very good shot. When time is scarce, you can't afford to be selective and you have to take essentially whatever you can get. The point of my paper was to try and derive a relation for how selective you need to be as a function of shot clock time.

Appreciate the response G+L...

I'd suggest that time isn't scarce until the last few seconds. And, from what I've seen on the floor (and I've tried to watch this closely since this thread started)...teams generally take the first good look they get from a shooter in a position to shoot...or from a driver in position to attack the basket (as free throw potential increases their point-per-shot potential). If a study like this shows the "optimal" time to shoot is at x seconds...that would still seem to be trumped by the fact that it's extremely suboptimal to put up an attempt at x seconds if the guy with the ball at that point is a non-shooter being guarded at a spot on the floor he's not comfortable from.

If you start a stopwatch at the beginning of each pitcher/batter showdown in baseball, it could turn out that the most productive moment for hitters is at the 22 second mark or something...but the batter should still be looking for a pitch he can hit...or at least a favorable count where he can make an aggressive guess at what's coming. The 22 second mark is coincidental in terms of real-world decision-making. Pro basketball is different because the shot clock does become an extra defender very late in the possession, and nothing like that happens in baseball. But, we already knew that forced shots late in the shot clock were sub optimal.

Do you believe your work will lead to an "optimal" time on the clock when shots should be taken in a way that would trump the convergence of shooter/spot/space? If you were to sum up your work to an NBA general manager in one or two sentences...what would those sentences be? (I'm asking that in a friendly way, tone can be tough to present online)