## 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).

1. Kaiser says:

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.

2. Jonathan says:

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.

3. Richard says:

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.

4. jd says:

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

5. Koray says:

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.

6. Anonymous says:

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.

7. Andrew says:

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

8. Nick says:

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

9. Sascha says:

Sorry for being late, but… @Jonathan: rather than calling you immoral, the other side should have dissected your argument as obviously fallacious and methodically unsound. In other words: perfect nonsense. While it might(!) well be true, that the odds of a nuclear accident (please define!) are lower than those of individual death you ignore, amongst other things, that (1) the individual risk of dying is directly affected by the risk of nuclear accidents, (2) a (large-scale, I assume) nuclear accident bears much higher cost for a large number of citizens than any individual death, (3) its probability consequently should be compared not to occurence of individual but of mass death (or at least severely damaged health) – you are comparing “things” within extremely different scales –, and (4) a low probability still means, that you just do not know *when* an accident will occur. Under _normal_ circumstances, that is. Further: what about each of those risks, when *not* employing nuclear energy?

In short: you appear to be actually falling quite well into the trap of the one-sided bet yourself (and not only that). A “similar example”, smarty-pants, huh? ;-p

10. Sean says:

The lottery ticket example is a bit silly. Clearly what matters is not the expectation (which is negative obviously) but the expected marginal utility (the sum of the marginal utilities of the payoffs * their respective probabilities – the marginal utility of the ticket price).

That calculation is going to vary from person to person but it's still positive for most people assuming the ticket price is low and the payoff is large with teensy probability. The idea that you shouldn't buy a ticket for a single lottery draw because "most people buy more than one ticket" is itself a fallacy and actually doesn't affect the reasoning above at all.

The payoff of the multiple tickets multiple draws scenario is just a sum as above with more terms and you need to judge the expected marginal utility of the combined payoff distribution vs the ticket prices for yourself based on your personal economic circumstances.

11. Doug Ransom says:

Lotto players easily get \$1 of utility from a ticket — the anticipation of the draw, the excitement of checking their numbers…

12. Rocky says:

This could easily be applied to the security theater that goes on at airports.

The costs of screening: all the machinery, the staff, plus lost productivity of all the people standing in line vastly outweighs the benefits.

13. Nathan says:

One could argue that the anticipation and excitement that lotto players experience are based on an irrational overestimation of their odds of winning. Those with a more realistic view would not feel as much excitement, so it wouldn't be worth it for them.

14. David says:

People live slightly longer with better health in Denver because people with chronic lung disease do not move to Denver, and people who develop chronic lung disease in Denver move away.

Hidden biases, gotta love 'em.

15. Chris says:

Evaluating the expected utility of a lottery ticket based on the expected financial payoff is misleading, anyway – utility is nonlinear, so you have to put the money-to-utility function *inside* the sum/integral over all possible outcomes, otherwise you're averaging the wrong variable.

I think that makes the lottery look worse, not better, though, which means the real explanation of why people play it lies in their subjective expectations which overrate the probability of very-low-probability events (like winning).

P.S. Pascal was wrong – believing in god is zero cost only if you never spend time talking about your god or convincing others to believe in your god, never donate money to your church, never volunteer for activities that promote the church's interests, and never restrict your actions based on the church's teachings; this describes approximately no believers.

16. meno says:

"utility is nonlinear"

Indeed. Psych studies measuring of enjoyment vs cost tend to show a log-linear correlation between cost of item and enjoyment. Ie, marginal enjoyment decays exponentially with cost.

Which suggests that it's more rational to live a live frittering your money away on icecreams, eating out and buying amusing bits and pieces instead of saving up for a better house and car…

17. dilbert dogbert says:

Pacal may have had a point if you accept the idea that there is only a choice between one or no god. What happens if there are multiple gods and they are all like that guy in the OT and have their specific chosen people and hells?

18. Doug K says:

'the lottery is a tax on people who don't understand statistics'
is what I used to believe.

However I am now of Doug Ransoms' mind. It's properly understood as entertainment. We suspend disbelief willingly in the theatre for the purpose of being entertained: why not for the lotto ?

19. Wouldn't you have to consider more possible actions than just buying or not buying lottery tickets?

There are other things you might do with the money, e.g., add to your investment in an index mutual fund.

Someone who bought \$1000 of lottery tickets a year, after 40 years would end up with \$1000, assuming that the expected value of a ticket is 50% of its purchase price (this number is about right, taking into account multiple winners of jackpots, the fact that the winner of a jackpot has to reduce the amount received because of annuitization, and the taxes on large pots; it's also consistent with the fact that lotteries generally send about half their "take" to the state that sponsors them). So, \$40,000 in, \$1000 out.

Someone who put the same amount into an index fund, assuming (conservatively) an average 7% return, would end up with in excess of \$200,000. The same \$40,000 in ends up being worth well over \$200,0000

Is the utility of the fun of buying \$40 of lottery tickets every week for 40 years worth a \$200,000 shortfall at the end? Maybe to some people, but my students generally decide that the lottery isn't a very good option. A maximum contribution to an IRA invested in stocks would likely leave those of my students who made that choice with a million dollars or more by the time they retire. (Inflation will kill some of that, but still…)

There are other options, but decision theory says that we ought to consider a reasonable range of the things we might do with the money (actions). Most decisions are not simple binary choices.

Even if a "utility function" doesn't really exist (because it can't really be estimated coherently, if for no other reason), the basic idea can be applied usefully to illustrate such concepts.

20. I still don't see the fallacy in buying the lottery tickets. It is a chance, a chance to win big, and that chance is worth the dollar. No fallacy in my point of view.

21. Basil says:

Pascal's Wager shouldn't really be compared to winning the lottery. Just our existence on earth decreases the probability of their not being a God significantly. Even if you wanted to assume a very low probability of this existence, you have an option of having or not having a ticket when it's all said and done. Comparing Pascal's Wager to the lottery and it's affect on humans should not be considered either. Your argument of believing in God is degrading and a cause to lower one's intellectual capabilities or social economic status is uncomparable. Many of our "educated folks" seem to find it degrading to acknowledge the fact of our purely impropable life. Almost like their intellect would be questioned. The documentary by Ben Stein in 2008 "Intelligent Design" discusses this fact. I believe we need to find a middle ground and agree NOT do disagree on the issue!

22. Andrew Gelman says:

Ben Stein, huh?

23. Paul says:

I think you're conflating Pascal's wager with Faith. The post wasn't that you shouldn't believe in God, just that Pascal's postulation of why you should believe is bad statistics.

I mean, if you truly only believed in God because you wanted in with him if he did exist, do you think Saint Peter would be impressed? "Hey, that was a clever hedging of your bets, come on in". Or would he be angry that you only had faith out of a greedy sense of self preservation? Would God prefer a moral nonbeliever to a subscriber to Pascal's Wager? Pascal said "absolutely, 100% not. Totally, statistically impossible. There's no possible God who wouldn't prefer a Christian to a Muslim, or to a nonbeliever." That's bad statistics.

None of that means its wrong to believe in God, you can build your own priors on that one. But if you're only believing in him because of what Pascal said, you might want to rethink what "Faith" means.

24. ang says:

The lottery is nicely summed up in an old Dilbert cartoon.
http://dilbert.com/strips/comic/1994-04-08/

25. Basil says:

Here's a link to the documentary online. http://topdocumentaryfilms.com/expelled-no-intell

26. Basil says:

That's a cute comic!