Endgames for basketball, the 2-for-1

When should a team try for two shots near the end of a game or quarter?  Here is part of Mr. Winston’s request:

Hypothetically, your team is tied with about a minute left. Your team should shoot the ball with no less than 45 seconds left on the clock, so that if you miss and fail to get the offensive rebound, you are almost guaranteed to get the ball back. Here is his reasoning: “2-for-1 is a basic fundamental premise – I get 2 shots, you get 1. If we are tied before that begins, I am going to win more often than you. Period. Even if the first shot is less-than-great, you have to take a decent look early.

Here are a few points against the 2-for-1:

1. Taking a bad shot too early can lead to long rebounds and leave a team unable to get back to defend.

2. Turnovers and offensive rebounds and fouls are common near the end of games.  No one  knows how many shots are left in the game, so don’t think that backwards induction will work.

3. Your chance of drawing fouls, or inducing sheer defensive lapses, goes down if you take the rapid shot.

4. Taking the bad shot early may disrupt shooting rhythms, dispirit the team if there is a miss, and lead to too much play focused on the shot clock rather than quality execution.

5. You still might get a final shot attempt even if the first two shot attempts, by you and the opposing team, run down the shot clock a fair amount.

Overall I am not one to insist on the 2-for-1.  Basically you are immediately spending a valuable asset — a possession — without that much information about its value.  The deeper economic lesson is that infinite horizon models are more plausible than you think, because no one knows how rapidly events will be taking place.

Here is one paper on endgame strategy in basketball, focusing on the intentional foul.


Related question and one not answered by a scan of the article. Whenever a team decides to hold for the last shot of a quarter or half, why not foul? A possession is worth 1.1 points in an nba game or so. A possession with 2 free throws shot by a 75% shooter is worth 1.5 points. If you foul, though, you get the ball back and a possession yourself. It will be a short possession, so it's possibly not worth a full 1.1 points, but it's not worth nothing. By my assumptions, the possession only has to be worth 0.4 points to make fouling an attractive strategy (the earlier in a last possession you can do it the better, although not so early as to create a possibility of a possession the other way, and of course the worse foul shooter you can foul the better. Also preferable, "going for a steal" aggressive defense that will lead to a foul at high likelihoods but also may force a turnover.)

I've never figured out why teams don't foul in that situation, except perhaps as a tit for tat game theory solution (they don't foul because the other team would just foul back). Any other thoughts?

>except perhaps as a tit for tat game theory solution (they don’t foul because the other team would just foul back)

This doesn't work unless "I foul, you foul" is somehow worse than "I don't foul, you don't foul" for both teams - since basketball is presumably a zero-sum game, this seem unlikely. (Unless, possibly, spectators don't like watching frequent fouls, so the total revenue for the game goes down if they find themselves in this equilibrium).

45 seconds is way too early to take the first of the 2-for-1 shot. I'd put the minimum time at about 32 seconds. If you miss and the opponent rebounds, the shot clock resets around 31 or 30 seconds meaning (with a 24-second shot clock) you get the ball back with about 6 or 7 seconds remaining. I think that's plenty of time (especially if you have timeout and can inbound at half court) to set up a good game-winning shot.

"Turnovers and offensive rebounds and fouls are common near the end of games. "

Are fouls common near the end of a tied game? They are obviously common near the end of a close game where the trailer fouls the leader, but I'm not aware of them being common in a tie game.

I think you are focusing too much on "taking a bad shot". You obviously don't want to take a bad (low-percentage shot). The idea of 2-for-1 isn't to just throw anything toward the rim just to get an extra shot. The idea is to go ahead and take any decent opportunity and don't spend valuable seconds (which can equate to losing a possession) looking for the best shot.

However, I will say that if you work for the best shot possible and make it, you also guarantee yourself another possession because it is common strategy to shoot quickly when you are down one possession with under 30 seconds because the trailer wants the opportunity to foul and get the ball back (or get an offensive rebound) if they miss.

Bad assumption of a 24 second shot clock. The request he was responding to specifically mentioned March Madness, and thus a 35 second shot clock should be assumed.

"Turnovers and offensive rebounds and fouls are common near the end of games. No one knows how many shots are left in the game, so don’t think that backwards induction will work."

Poor logic on fouls. Fouls are common near the end of games because of the strategy of the defense choosing to foul. Incidental fouls are no more common near the end of games than earlier; it's strategic fouls of choice that are common. With less than a minute left, the defense will choose not to foul in general (if there is not an offensive rebound) if it is doing the 2-for-1 strategy, knowing that it will get the ball back.

The 2-for-1 strategy is generally an alternative to choosing to foul, so it's illogical to take the extra fouls of choice into account.

Tyler, your conclusion
"Basically you are immediately spending a valuable asset — a possession — without that much information about its value. The deeper economic lesson is that infinite horizon models are more plausible than you think, because no one knows how rapidly events will be taking place."

reminded me about the conclusion of this article by Ariely and Norton

From thinking too little to thinking too much: a continuum of decision making
Dan Ariely∗ and Michael I. Norton
Due to the sheer number and variety of decisions that people make in their everyday lives—from choosing yogurts to choosing religions to choosing spouses—research in judgment and decision making has taken many forms. We suggest, however, that much of this research has been conducted under two broad rubrics: The study of thinking too little (as with the literature on heuristics and biases), and the study of thinking too much (as with the literature on decision analysis). In this review, we focus on the different types of decision errors that result from both modes of thought. For thinking too little, we discuss research exploring the ways in which habits can lead people to make suboptimal decisions; for thinking too much, we discuss research documenting the ways in which careful consideration of attributes, and careful consideration of options, can do the same. We end by suggesting that decision makers may do well, when making any decision, to consider whether they are facing a ‘thinking too much’ or ‘thinking too little’ problem and adjust accordingly.  2010 John Wiley & Sons, Ltd. WIREs Cogn Sci 2011 2 39–46 DOI: 10.1002/wcs.90

The only problem I see with your advice, as well as with Ariely and Norton's, is that ordinary people may take too long to consider what sort of problem --thinking too much or thinking too little-- they are facing. Indeed, the difference between any Gregg Popovich and any professor is that Popovich has a long experience in the art of living the last minute of a game.

Or, should the shot clock go to 10 seconds in the final minute?

It's simpler to focus on the question of whether to go 2-for-1 at the end of the first half (or the first or third quarter in the NBA). Once you're done with that question, then focus on the more complicated endgame scenario, where fouling and the exact score become relevant.

Without worrying about basketball specific values, the logic would run as follows. Let t, p, and f represent the values of the two-for, penultimate, and final shots. The two-for-one strategy has value t - p + f and the wait for best has value p - f. These are equal when t + 2f = 2p.

Is it possible to get more than 2/3 the value of a teams normal offense in two-for-one and final shot situations (when timing the shot for a brief window)? Going two-for-one is a good idea if the answer is yes.

I agree with distinguishing between endgame and end half or quarters. That said, one has to take better account of the fact teams can and should prepare for these situations.

"4. Taking the bad shot early may disrupt shooting rhythms, dispirit the team if there is a miss, and lead to too much play focused on the shot clock rather than quality execution."

If a team were to practice for this situation, this wouldn´t be a problem. Everyone would know why a shot was taken, even if it wasn´t would normally be considered a quality shot, and quality execution would less important because you would be looking for a shot in transition or reasonably open. There are plenty of early shots to be had, because defenders know many players aren´t supposed to shoot on the first pass. You have the advantage that the defending team has to be a group of incredibly smart basketball players to change their strategy for the case that the offensive team is going 2-for-1.

If I were coaching, with a 24-second shotclock, It would be automatic to get a shot with 30-34,35 seconds left, having started the possesion with, let´s say, 54 seconds or less on the clock.

The NBA status quo seems to be, if you start the possession with 35 seconds or less (at the end of quarter, but not endgame) you shoot as late as possible. The reasoning here is, I guess, that the other team´s quality of possession will be severely diminished if they get the ball back with under 8 seconds or so. This gives you:

[normal possession] - [bad possession] > [quick possession] + [bad possession] - [normal possession]

2[np] > qp +2bp; I would say the worst case scenario is that your quick possession is worth the same as what I called the bad possession (in the NBA I would have a hard time believing that it is not worth more), but lets take qp = bp

So, if you want to stick with the NBA stratgey, and not go 2-for-1 then you have to believe that

np > 1.5 bp

Taking the number above np = 1.1 bp has to be less than about .75. And that´s the most optimitic view for this strategy.

NBA teams obviously have plenty of resources to think about, and research this, so its difficult question what they do without hard data about their scoring. But I know the Phoenix Suns do this all the time, and I find it hard to believe its the best strategy, especially considering they are very good offensively which increases their scoring on normal possession compared to the league average, but also on quick, or bad, possessions which, given there are two of them, is more important.

btw, I didn´t address the fouling situation, because the situation I described is the one I think of, when hearing about 2-for-1 in bball, and it frustrates the hell out of me!

Also, my basic feeling is that there is a resistence to thinking about the game in these terms from both players and coaches, that it somehow disrupts "players rhythm" or "feel" or whatever... An example of this attitude is a comment from Grant Hill I heard to a question about is reaction to help from Shane Battier on guarding Kobe Bryant in last years playoffs. Battier is well known for studying detailed breakdowns of shot selection, field goal percentages (from which spot, in which situtations) of prominent players he has to guard -- there is a great NYT article about this from a year or two back. Hill said something to the effect that is just "Shane being Shane", he himself is not a numbers guy and cant play basketball that way.

As a (former) player, I can somewhat sympathise, but only somewhat. There´s nothing wrong with a little rationality coaching for these guys at the college level...someone will probably make sure they get their econ 101 credits whether they learn the stuff or not

I see that no one has remarked on the consequences of the officials' practice (more pronounced in the NBA than in college) to "swallow their whistles" during the last couple of minutes of a close game. As a consequence, tight defense can mean considerably more physical contact in those closing seconds; a bump or shove that would have been a foul in the first few minutes of the game is simply not called during the final 60 seconds.

The implication is, in my mind, that the chances of getting a good shot on the second possession of the 2-for-1 are considerably less than they have been all game, decreasing the value of that possession.

It's trivial to say you shouldn't take a shot that you will not make. The decision should be to accept a lower probability shot to the extent the odds are offset by the expected value of the final possession, which you could determine based on % chance of scoring, getting a defensive rebound or turnover and not commiting a turnover yourself. Since these types of mathemtical questions are purely hypothetical and can't be solved in real time, it makes sense to coach players to hurry and take a decent shot vs. wait and take a better shot.

I know what Bo Ryan would say.....

Not quite the same situation, but Gus Johnson got roasted for suggesting that BYU foul Florida when Florida had the ball in a tie game with :15 left. (IIRC Florida had Vernon Macklin, a career 45% FT shooter, on the floor.)

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