Statistical Modeling, Causal Inference, and Social Science: The fallacy of the one-sided bet (for example, risk, God, torture, and lottery tickets)

The fallacy of the one-sided bet (for example, risk, God, torture, and lottery tickets)

As a researcher and teacher in decision analysis, I've noticed a particular argument that seems to have a lot of appeal to people who don't know better. I'll call it the one-sided bet. Some examples:

- How much money would you accept in exchange for a 1-in-a-billion chance of immediate death? Students commonly say they wouldn't take this wager for any amount of money. Then I have to explain that they will do things such as cross the street to save $1 on some purchase, there's some chance they'll get run over when crossing the street, etc. (See Section 6 of this paper; it's also in our Teaching Statistics book.)

- Goals of bringing the levels of various pollutants down to zero. With plutonium, I'm with ya, but other things occur naturally, and at some point there's a cost to getting them lower. And if you want to get radiation exposure down to zero, you can start by not flying and not living in Denver.

- Pascal's wager: that's the argument that you might as well believe in God because if he (she?) exists, it's an infinite benefit, and if there is no god, it's no loss. (This ignores possibilities such as: God exists but despises believers, and will send everyone but atheists to hell. I'm not saying that this highly likely, just that, once you accept the premise, there are costs to both sides of the bet.) See also this from Alex Tabarrok and this from Lars Osterdal.

- Torture and the ticking time bomb: the argument that it's morally defensible (maybe even imperative) to torture a prisoner if this will yield even a small probability of finding where the ticking (H)-bomb is that will obliterate a large city. Again, this ignores the other side of the decision tree: the probability that, by torturing someone, you will motivate someone else to blow up your city.

- Anything having to do with opportunity cost.

- The argument for buying a lottery ticket: $1 won't affect my lifestyle at all, but even a small chance of $1 million--that will make a difference! Two fallacies here. First, most lottery buyers will get more than 1 ticket, so realistically you might be talking hundreds of dollars a year, which indeed could affect your standard of living. Second, there actually is a small chance that the $1 can change your life--for example, that might be the extra dollar you need to buy a nice suit that gets you a good job, or whatever.

There are probably other examples of this sort of argument. The key aspect of the fallacy is not that people are (necessarily) making bad choices, but that they only see half of the problem and thus don't realize there are tradeoffs at all.

P.S. When I was young and stupid, I spent some time trying to convince a student in my intro statistics class that it was a bad idea to play the lottery. In retrospect, I should've told him that it was fine, and just delineated where the probability calculations were relevant (for example, if he were to play the lottery twice a week for a year, or whatever).

Kaiser

Coincidentally, I am looking for any study to prove that most statisticians are averse to playing the lottery. Your blog entry now becomes evidence #1 as even Google has been not much help. More generally, have people studied the relationship between occupation and risk perception? Are traders more risk-taking? Are statisticians more risk-averse? etc.

Jonathan

I used a similar example once in a debate on nuclear power. I pointed out to the audience that, whatever you think the odds of a nuclear accident are, they are much, much lower than chances of death that people take routinely. The other side called me "immoral," which I took as a badge of honor.

Richard

On a bit of a side note, there's no evidence that zero radiation exposure is good. All our data is high dose which is extrapolated to dose zero. Some single cell studies suggest low doses are required for health (we DID evolve in a low-dose environment, not a no-dose environment.) Interestingly, I remember reading that people live slightly longer with better health in Denver. Is it the low dosage of radiation stimulating their immune system or just something about the population of Denver?

Maybe it's something about how people use their imagination? I'm sure if you ask people about the first thing that pops in their head when you mention radiation exposure they'll come up with Chernobyl or something terrible, despite the pretty common knowledge of low-dose sources everywhere. No one remembers their lottery losses, either. But I'm sure they'll remember their wins.

jd

as a professional risk manager I run into this frequently. Our normal tools for understanding risk and return seem to be amazingly good with things that are "familiar." These familiar things are generally high probability (close to the mean) items (both low and high severity). We tend to proverbially fall apart at the seams when dealing with highly improbable yet highly catastrophic risk perception. We also do much worse calculating relative risks between things that we control (driving our car) and things other control (riding in a plane). The things we control we tend to feel more safe in, regardless of the true risks.

This is a fascinating area and I would really enjoy hearing more from you on this topic area.

-JD

Koray

I've been recommending the book How We Know What Isn't So by Gilovich for this sort of stuff. He starts out with the misconception that is the hot hand in basketball (apparently players don't score in streaks, but everybody think so) and moves on to all kinds of wrong beliefs we form.

On the torture question:
What is ignored is that there is a false positive rate of detection of terrorists. For example Senator Stevens wife(Cat Stevens) is not allowed to fly on airplanes because she shares a name with Cat Stevens, the former pop star and converted Muslim, who probably isnt a terrosit, but hey you never know, he could be! The unfortunate Mrs Sen. Stevens is a false positive. Shall we torture her? Even to prevent an H bomb event? What are the societal costs of torturing/banning all of the sad false positives?

Consider the relative sizes of the set of terrorists as compared to the general population of innocents...The false positive rate just kills any argument along these lines.

Andrew

Hey, Senator Stevens is married to a terrorist? That's pretty scary!

Nick

On the plus side, at least Mrs. Stevens can travel home via an express internet tube, on her own luxury dump truck?