The Universe of Discourse
Tue, 25 Apr 2006

More on risk
My article on risk was one of the more popular articles so far; it went to the top of Reddit, and was widely commented on. It wasn't the best article I've written. I confused a bunch of important things. But now I find I have more to say on the topic, so I'm back to screw up again.

One big problem with the Reddit posting is that the guy who posted it there titled the post on risk, or why poor people might not be stupid to play the lottery. So a lot of the Reddit comments complained that I had failed to prove that poor people must not be stupid to play the lottery, or that I was wrong on that point. They argued that the dollar cost of a lottery ticket is more valuable to a poor person than to a rich one, and so on. But I didn't say anything about poor people. People read this into the article based on the title someone else had attached to it, and they couldn't get rid of this association even after I pointed out that the article had nothing to say about poor people.

Something I do a lot, in this blog, and in life, is point out fallacious arguments. You get some argument that X is true, because of P and Q and therefore R, and then I'll come along point out that P is false and Q is irrelevant, and anyway they don't imply R, and even if they did, you can't conclude X from R, because if you could, then you could also conclude Y and Z which are obviously false. For example, in a recent article I addressed the argument that:

You can double your workforce participation from 27% to 51% of the population, as Singapore did; you can't double it again.

The argument being that you can't double a participation of 51% because you can't possibly have 102% workforce participation. (Peter Norvig pointed out that he made the same argument in a different context back in 1999.) But the argument here fails, for reasons I won't go into again. This doesn't mean that I believe that Singapore's workforce participation will double again. Just because I point out that an argument for X is fallacious doesn't mean that I believe X is false.

The "risk" article was one of those. I wanted to refute one specific argument, which is that (a) the expected return on a lottery ticket is negative, so therefore (b) it's stupid to buy lottery tickets. My counter-argument was to point out that (a) the expected return on fire insurance is negative, but that you can't conclude that therefore (b) it's stupid to buy fire insurance. It might be stupid to buy lottery tickets, but if it is, it's not because the expected return is negative. Or at least it's not only because the expected return is negative. There must be more to it than that.

I really like that pattern of argument, and I use it a lot: A can't imply B, because if it did, then it would also imply B', and B' is false, or at least B' is a belief held only by dumbasses.

None of this addresses the question of whether or not I think it's stupid to buy lottery tickets. I have not weighed in on that matter. My only argument is that the argument from expected value is insufficient to prove the point.

People have a lot of trouble with second-order arguments like this, though. If I argue "that argument against B is no good," they are likely to hear it as an argument in favor of B. Several of the Reddit people made this mistake. The converse mistake is to interpret "that argument against B is no good, because it can be converted into an argument against B'" as an argument against B'! Some of the Reddit people made this mistake too, and disdainfully explained to me why buying fire insurance is not stupid.

Another problem with the article was that it followed my usual pattern of meandering digression. Although the main point of the article was to refute the argument from expected value, I threw in a bunch of marginally related stuff that I thought was fun and interesting: the stuff about estimating the value one ascribes to one's own life; the stuff about the surprisingly high chance of being killed by a meteor strike. Email correspondents and Reddit commenters mistook both of these for arguments about the lottery, and tried to refute them as such. Well, I have nobody to blame but myself for that. If you present a muddled, miscellaneous article, you can't complain when other people are confused by it.

If I were going to do the article again, one thing I'd try to fix is the discussion of utility. I think my biggest screwup was to confuse two things that are not the same. One is the utility, which decreases for larger amounts of money; your second million dollars has less value than your first million. But another issue, which I didn't separate in my mind, was the administration cost of money. There must be a jargon term for this, but I don't know what it is.

Economists like to pretend that money is perfectly fungible, and this is a reasonable simplifying assumption in most cases. But it's easy to prove that money isn't perfectly fungible. Imagine you've just won a prize. You can have one thousand dollars paid in hundred-dollar bills, or you can have a thousand and one dollars, paid in pennies. Anyone who really believes that money is perfectly fungible will take the pennies, even though they weigh six hundred pounds, because that way they get the one-dollar bonus.

Money has a physical manifestation, even when it's just numerals written in a ledger somewhere, and managing the physical manifestation of money has an associated cost. The cost of managing a penny is a significant fraction of the value of the penny, to the point that many people throw away pennies or dump them in jars just to avoid the cost of dealing with them. In some circumstances, like the lottery ticket purchase, the non-fungibility of money is important. Blowing one dollar on a lottery that pays a thousand dollars is not the same as blowing a thousand dollars on a lottery that pays a million dollars, and it's not the same as blowing your whole paycheck on a big stack of lottery tickets. Partly it's the risk issue, and partly it's this other issue, that I don't know the name of, that a single dollar is worth less than one one-thousandth of a thousand dollars, because the cost to administer and manage it is proportionately higher. I didn't make this clear in the original article because it wasn't clear in my mind. Oh well, I'm not yet a perfect sage.

One last point that has come up is that a couple of people have written to me to say that they would not take the Russian roulette bet for any amount of money at any odds. (Here's a blog post to that effect, for example.) One person even suggested that I only assumed he would take the bet at some odds because I'm an American, and I can't conceive of anyone refusing a big pot of money.

Well, maybe that's true, but I don't think that's why I assumed that everyone would take the bet for some amount of money. I assumed it because that is what I have observed people to do. I now know there are people who say that they would not play Russian roulette at any odds for any payoff. And I think those people are fooling themselves.

If you think you're one of those people, I have this question for you: Do you own a bicycle helmet? And if you do, did you buy the very top-of-the-line helmet? Or did you buy a mid-price model that might offer less protection? What, just to save money? I offered you a million dollars at million-to-one odds. Do you think that fifty dollars you saved on your bicycle helmet is paying you off for less risk than my million-to-one Russian roulette bet?

Well, maybe you don't own a bicycle, so you think you have no need of a helmet. But if the people who wrote to me were as risk-averse as some of them said they were, the lack of a bicycle wouldn't stop them from wearing helmets all the time anyway—another reason I think they are fooling themselves. I've met some of these people, and they don't go around in helmets and padded armor all the time.

Or maybe you do own the very safest helmet money can buy, since you have only one head, after all. But I bet you can find some other example? Have you ever flown in a plane? Did you refuse to fly anywhere not served by Qantas, like Raymond in Rain Man, because every other airline has had a crash? If you had a choice to pay double to fly with Qantas, would you take it? Or would you take the cheap flight and ignore the risk?

One comment that replies to the blog I cited above really hits the nail on the head, I think. It says: "you don't get paid a million dollars to get in your car and drive somewhere, but what are the chances you'll be killed in an auto accident?" My Russian roulette game is a much better deal than driving your car.

I'm going to end this article, as I did the last one, with an amusing anecdote about risk. My great-uncle Robert E. Machol was for a time the chief scientist of the Federal Aviation Administration. The regulations for infant travel were (and still are) that an infant may make an air trip on its parent's lap; parents do not need to buy a separate ticket and a seat for the infant.

In one air disaster, an infant that was being held on its parent's lap was thrown loose, hurtled to the end of the corridor, and died. The FAA was considering changing the rules for infants to require that they purchase a separate ticket, entitling them to their own seat, into which would be installed an FAA-approved safety car seat. Infants in their own restraint seats would be much safer than those held on their parents' laps.

Dr. Machol argued against this rule change, on the following grounds: If parents are required to buy separate tickets for their infants, air travel will be more expensive for them. As a result, some families will opt to take car trips instead of plane trips. Car trips are much more dangerous than plane trips; the fatalities per passenger per mile are something like twenty times higher. More babies can be expected to be killed in the resulting auto crashes than can be expected to be saved by the restraint seat requirement.

As before, this is not intended as an argument for or against anything in particular, except perhaps that the idea of risk is complex and hard to understand. Probably people will try to interpret it as an argument about the fungibility of money, or whatever the next Reddit person decides to put in the article title. You'd think I would have learned my lesson by now, but, as I said, I'm not yet a perfect sage.

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