Calculate or simulate?

A couple of weeks ago, Slate published two oddly similar articles within a few days of each other: Extra Points Are For Losers and The Supreme Court Justice Death Calculator.

You might not initially think the articles have that much in common: one’s about football strategy and one’s about the future of the US Supreme Court. But beneath the surface they’re practically the same: both are calculations of joint probabilities for multiple events.

The football article points out that in a certain situation it’s provably the right strategy for an NFL team to go for a two-point conversion instead of a single extra point. Specifically, if you’re down 14 points near the end of the game, and you score a touchdown, you should go for 2.

You can go to the article for details, but here’s the idea. The decision only matters under the assumption that you’re going to get another touchdown and the other team isn’t going to score. So let’s assume that that’s going to happen. If you take the usual extra point each time, you’re going into overtime (assuming your kicker always gets the extra point, which is roughly true in pro football), and you’ve got a 50-50 shot at a win . On the other hand, if you go for the two-point conversion, either you make it and are guaranteed a win, or you miss it, in which case you go for 2 again the next time and have a chance to throw the game into overtime again. You can check by a straightforward calculation that the second strategy gives you better than even odds of winning (for reasonable estimates of the likelihood of a two-point conversion).

The Supreme Court article contains a slightly macabre app that lets you calculate the probabilities of various combinations of Supreme Court justices dying during the second Obama administration. Want to know the odds that Obama will get to appoint a replacement for one of the conservative justices? One of the liberals? One of each? You can play around to your heart’s content.

The math underlying both of these calculations is exactly the same. It’s pretty much just

P(A and B) = P(A) P(B).

That is, to get the probability that two independent events both occur, you multiply together the probabilities of each one.

Oddly, the two articles take different approaches to calculating the probabilities. The football article just calculates them directly by doing the multiplication, but the Supreme Court article estimates them by simulation. If you ask the app for the probability that both Scalia and Kagan will die, it runs 10,000 simulations of the future and counts up the results.

That’s actually not a very good thing to do, especially if you’re interested in low-probability combinations such as this one. If you ask the app that question repeatedly, the numbers bounce around: 0.32%,  0.26%, 0.27%, 0.41%.

In the guts of the program there must be probabilities for each of the individual events, so you could answer this question by simply multiplying them together. The result wouldn’t jump around like that and would be more precise than the simulation-based one (although not necessarily more accurate — as the article points out, the calculation is based on some assumptions that might not be correct).

You could reduce the scatter by raising the number of simulations, of course, but it’s odd to use simulations in this situation to begin with. Estimating probabilities via simulation is a great tool when it’s impossible or difficult to calculate the probabilities exactly, but in a situation like this it’s quicker, simpler, and more precise to just calculate them.

This whole business with the debt ceiling is generally discussed as a political issue, but I’ve been wondering about it as a purely legal issue.

Here’s the question. People often say that, if Congress doesn’t raise the debt ceiling, the US won’t be able to pay its bills and will be forced to default. But as I understand the law (not very well, that is), that doesn’t seem quite right. If Congress does nothing, then, if I’m not mistaken, the law requires two contradictory things:

1. The US government will be required to spend the money allocated by Congress.
2. The US government will be forbidden to execute the mechanism that allows for this spending.

(Let’s follow the President’s lead and leave the platinum-coin option out of the discussion for now.)

When people say that the US will be forced to default in this situation, they’re assuming that the government will obey law 2 and break law 1. Why couldn’t it be the other way around?

Although the question is inspired by the current controversy, I’m curious about the broader legal question: when the law is self-contradictory, is there a legal principle that governs which one takes precedence, or do people get to pick and choose?

Although of course you’d hope that legislatures would avoid making logically incompatible laws, I’d bet that this question has arisen from time to time, in which case it seems to me that there’d be case law laying out a clear legal guideline. Is there?

I know that when it’s a statute vs. the Constitution, the Constitution wins — “supreme law of the land” and all that. But in this case it’s statute vs. statute. (For the sake of argument, let’s assume that both laws are constitutional. I know there’s a 14th-amendment argument about the constitutionality of the debt ceiling, but let’s ignore that for now.)

There’s a famous theorem in symbolic logic of the form

p ^ !p ==> q,

pronounced “p and not-p together imply q.” It says that, once you’ve established both halves of a contradiction, you can logically infer anything you like. (Apparently this is called the paradox of entailment. I was hoping it’d have a cool Latin name like modus ponens, but no such luck.) Maybe the President should use this principle of logic to say that, if the debt ceiling controversy isn’t resolved, he’s allowed to do anything he wants.

Does chocolate cause Nobel prizes?

This has been around for a couple of months, but I just discovered it. An article in the New England Journal of Medicine notes a correlation between per capita chocolate consumption and per capita Nobel prizes in various countries:

The article’s clearly meant to be playful, not serious — something I didn’t now NEJM went in for.

This result came to my attention via a report on the BBC program (or rather programme) More or Less, which includes the following:

Eric Cornell, who won the Nobel Prize in Physics in 2001, told Reuters: “I attribute essentially all my success to the very large amount of chocolate that I consume. Personally I feel that milk chocolate makes you stupid… dark chocolate is the way to go. It’s one thing if you want a medicine or chemistry Nobel Prize but if you want a physics Nobel Prize it pretty much has got to be dark chocolate.”

But when More or Less contacted him to elaborate on this comment, he changed his tune.

“I deeply regret the rash remarks I made to the media. We scientists should strive to maintain objective neutrality and refrain from declaring our affiliation either with milk chocolate or with dark chocolate,” he said.

“Now I ask that the media kindly respect my family’s privacy in this difficult time.”

The program goes on to talk about the actual lesson here, which is our old friend Correlation Is Not Causation. This takes me back to the glorious moment when I first discovered the existence of xkcd:

In the case of the chocolate-Nobel connection, the correlation is highly statistically significant — that is, it’s overwhelmingly unlikely to get such a correlation by chance. That could be explained if chocolate consumption caused Nobel prizes (or vice versa), but it could also be explained if one or more other factors cause increases in both. More or Less explains this pretty well, but oddly doesn’t mention what seems to me to be the most obvious such factor.

Chocolate, despite what you may hear in some quarters, is not a biological necessity but rather a luxury. So you’d expect chocolate consumption to be positively correlated with wealth. And wealthy countries have resources to spend on science, so you’d definitely expect Nobel prize rates to be positively correlated with wealth. So the link between Nobel prizes and chocolate may simply be an artifact of the link between each of these and wealth. I’d bet that that’s all that’s going on here.