Friday, April 30, 2010
Tuesday, April 27, 2010
In a professional basketball game, a player is disqualified (“fouls out”) if he is charged with 6 personal fouls. Observers of the NBA know that the direct effect of fouling out actually has less impact than the indirect effect of “foul trouble.” That is, if a player has a dangerous number of fouls, the coach will voluntarily bench him for part of the game, to lessen the chance of fouling out. Coaches seem to roughly use the rule of thumb that a player with n fouls should sit until n/6 of the game has passed. Allowing a player to play with 3 fouls in the first half is a particular taboo. On rare occasions when this taboo is broken, the announcers will invariably say something like, “They’re taking a big risk here; you really don’t want him to get his 4th.”
Is the rule of thumb reasonable? No! First let’s consider a simple baseline model: Suppose I simply want to maximize the number of minutes my star player is in the game. When should I risk putting him back in the game after his nth foul? The phrasing is deceptive, because I shouldn’t bench him at all! Those of you who haven’t been brainwashed by the conventional wisdom on “foul trouble” probably find this obvious. The proof is simple: if he sits, the only thing that has changed when he gets back in is that there is less time left in the game, so his expected minutes have clearly gone down (in fact the new distribution on minutes is first-order stochastically dominated, being just a truncation.)
OK, while I believe the above argument is very relevant, it oversimplified the objective function, which in practice is not simply to maximize minutes. I’ll discuss caveats now, but please note, there is tremendous value in understanding the baseline case. It teaches that we should pay attention to foul trouble only insofar as our objective is not to maximize minutes. I am very comfortable asserting that coaches don’t understand this!
First caveat: players are more effective when rested. In fact, top stars normally play about 40 of 48 minutes. If it becomes likely that a player will be limited to 30-35 minutes by fouling out, we may be better off loading those minutes further towards the end of the game to maximize his efficiency. Notice, though, that this doesn’t lead to anything resembling the n/6 rule of thumb. It says we should put him back in, at the very latest, when he is fully rested, and this isn’t close to what is done in practice. In fact players often sit so long the rest may have a negative impact, putting them “out of the flow of the game.”
Second caveat: maybe not all minutes are created equal. It may be particularly important to have star players available at the end of the game. On a practical level, the final minute certainly has more possessions than a typical minute, but it also has more fouls, so maybe those effects cancel out. I think the primary issue is more psychological: there is a strong perception that you need to lean more on your superstars at the end of the game. I think this issue is drastically overrated, partly because it’s easy to remember losing in the last minute when a key player has fouled out, but a more silent poison when you lose because you were down going into that minute having rested him too long. By the way, my subjective sense is that the last possession is more similar to any other than conventional wisdom suggests: a wide-open John Paxson or Steve Kerr is a better bet than a double-teamed Michael Jordan any time in the game. On a couple of major occasions,
One more psychological caveat: a player who just picked up a foul he thinks is unfair may be distracted and not have his head in the game immediately afterward. This may warrant a brief rest.
Final note: Conventional wisdom seems to regard foul management as a risk vs. safety decision. You will constantly hear something like, “a big decision here, whether to risk putting
There are well-documented cases in the last decade of sports moving towards a more quantitative approach, so maybe there is hope for basketball strategy to change. The foul-trouble orthodoxy is deeply ingrained, and it would be a satisfying blow for rationality to see it overturned.
*Final outcomes are binary, so the classical sense of risk aversion, involving a concave utility function in money, doesn’t apply at all. But there is also a sense of what I call “tactical risk”: a decision may affect the variance of some variable on which your probability of final success depends in a convex (or concave) way. I might write an essay sometime on the different meanings of “risk.” Anyway, here you would presumably should be risk-averse in your star’s minutes if ahead, risk-loving if behind. But this is rendered utterly moot by first-order stochastic dominance!
Monday, April 26, 2010
A friend of mine got “accidentally jobbed” by “accidental UI” in a very simple auction last week. How do you think a director or committee would treat this?
1S-P-3C(nat, inv, alerted)-X-All pass
Now, clearly X here is takeout, right? Well…except the doubler never asked what 3C was. He had a 5-card club stack and assumed 3C, alerted, was
It’s clearly wrong for this to stand, because, however far this was from the players’ intentions, it gives them the ability to make takeout or penalty doubles in the same auction. If you agree, would you trust a director or committee to get this right? (I wouldn’t.)
BTW, it is impossible to avoid conveying UI on this hand, unless you follow the “always ask regardless of your hand” principle, which very few do. If the club stack asks, is told 3C is natural, and passes, the message is clear unless he is an always-asker. That particular UI would have been fairly innocuous on this deal, but the problem remains.
I guess if I were arguing this hand with a committee, one of the strongest points is that doubler ignored the skip bid.
Thursday, April 8, 2010
You know that an ordinary simple squeeze requires that you have all the tricks but one. You probably also know that if you have three threats against one player, or a suit-establishment threat, the squeeze can function with all but two tricks. But did you know that if both conditions hold, n-3 can be enough? Or that you don’t need any entries?
In a rubber bridge game, this 6-card ending happened, with spades as trumps:
North: xx xx Jx ----
West: T ---- xxxx x East: ---- QJ KQ Jx
South: Q x x KT8
As you can see, South has 3 of the last 6. When he plays a trump to his Q, a red-suit pitch by East lets him establish and score a 4th trick, and a club pitch gives two tricks. All this with no entry in any threat suit! Yes, the last trump is a sort of entry-surrogate, but this doesn’t at all resemble a typical ruffing squeeze.
Since this took place at a bar, I hope declarer, Dan Wilderman, doesn’t mind my saying he may have stumbled on this position a bit by accident. Nevertheless, this should definitely be the Wilderman (or, the Wild Man?) squeeze.