Friday, April 30, 2010

Random seeding gripe

In the ACBL seeding system, points for winning or placing in a major event decay arithmetically. So, you get 11-n points for winning the Spingold n years ago. Suppose "Bob" won in 2000 and 2001, and "Zia" won in 2003. As of 2004, Bob gets more points than Zia, 15 to 10, which seems fair. But when we get to 2010, suddenly Zia's win is worth more than Bob's two, 4 to 3. To put it differently, it looks crazy that a win 9 years ago is twice as good as a win 10 years ago, while wins initially decay only mildly, 10% a year.

I suggest an exponential decay factor, maybe .9. This would avoid these odd reversals. Given my (lack of) record in major events, my gripe has no self-interest component.

Tuesday, April 27, 2010

Foul trouble

Most game-players are sports fans, so I'm throwing in a little essay I wrote on foul trouble. It was targeted for an econ blog, and uses one or two technical terms, but they aren't central.


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, Jordan agreed. This isn’t to underestimate the star’s importance in scoring and getting other players good shots, just to say that this is not necessarily more important in the final minutes. You do often hear that a team will rise to the occasion when a star is injured or suspended, so even conventional wisdom wavers here. Finally, note that the foul-trouble rule of thumb is applied also to players who aren’t the primary scorer, so that this argument wouldn’t seem to apply. I will give coaches a little credit: they do sometimes seem to realize that they shouldn’t worry about foul trouble for bench players who often don’t play at the end anyway.

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 Duncan back in with 4 fouls.” This is completely the wrong lens for the problem, since the “risky”* strategy is, with the caveats mentioned, all upside! Coaches dramatically underrate the “risk” of falling behind, or losing a lead, by sitting a star for too long. To make it as stark as possible, observe that the coach is voluntarily imposing the penalty that he is trying to avoid, namely his player being taken out of the game! The most egregious cases are when a player sits even though his team is significantly behind. I almost feel as though the coach prefers the certainty of losing to the “risk” of the player fouling out. There may be a “control fallacy” here: it just feels worse for the coach to have a player disqualified than to voluntarily bench him, even if the result is the same. Also, there is a bit of an agency/perception problem: the coach is trying to maximize keeping his job as well as winning, which makes him lean towards orthodoxy.

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

The dog that didn't ask

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 Bergen (and doubled rather quickly, I gather). His partner knew the X was penalty and passed with a club void, because *doubler never asked*. Doubler’s partner never asked either, and probably also thought 3C was Bergen and passed in complete innocence.

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

Entryless Squeeze with Nothing Resembling the Count

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.