Can You Outsmart an Economist?

Can You Outsmart an Economist? is an excellent book of puzzles put together by Steven Landsburg. Steve includes a lot of classics such as the Girl Named Florida Problem, the Potato Paradox and Newcomb’s paradox, the former two problems are presented in slightly different and in the first case improved forms so you might not recognize them on first reading. Steve also includes many economic puzzles, Bayes puzzles, common knowledge problems and more. Readers of this blog will certainly know some of the puzzles but will also find lots of novel problems and puzzles. Also included are philosophical paradoxes. For example, the headache problem.

A billion people are experiencing fairly minor headaches, which will continue for another hour unless an innocent person is killed, in which case the headaches will cease immediately. Is it okay to kill that innocent person?

The puzzle here isn’t the answer. The answer is obvious. The puzzle is that smart people can’t agree which answer is obvious.

Overall, this is the best and most diverse collection of puzzles that I know. It’s meant to be dipped into and sampled at leisure. My only complaint is that the puzzles are followed by answers which makes it too easy to fool yourself into thinking you always knew the answer! Answers in the back of the book would have been a minor form of commitment. Recommended.


Can you outsmart an historian?

As soon as you purchase the book, they've won.

that's a good one!

" Can You Outsmart an Economist?"

Yes, just write "Please, turn over" on both sides of a piece of paper.

If you laid all the economist in the world in a long line end to end they would all point in a different direction. They are wrong over 50% of the time. A coin toss can beat an economist.

"A billion people are experiencing fairly minor headaches, which will continue for another hour unless an innocent person is killed, in which case the headaches will cease immediately. Is it okay to kill that innocent person?"

German Economist: What will be the effects on the stock price of Bayer?

Which is the point, I guess.
We already have a calculus along these lines in the form of prescription drug approval. The difference is that the downside of the bargain is delivered by random biological chance, rather than intentional homicide with magical effects.

German Economist: If it's Hitler, ja.

I don't see what is obvious here? I think if an actual human life is on the line there would be a lot of additional questions to be asked before any killing could be justified.

How about if we restructure the question:

If a billion people have a mild headache that can be cured for $1 each, should society spend the resources to cure it?

If 1 person will die if we don't spend $1 billion on saving them should society spend the resources to cure it?

The Utilitarian answer is obvious: The innocent person should die to make the headaches go away.

+1. Revealed preference agrees.

Your situation is different. Choosing a particular innocent person and killing them causes emotional disturbance in surviving people, both the killer and all those who know about it. If a billion people know someone is being killed to cure their headache, the emotional disturbance would likely far outweigh the problem of the headache.

The original question didn't specify who knows what, or how, or who but these things are vital to answering the question. Imagine if a randomly chosen person is tortured to death on live TV. You can argue I'm adding detail not in the original question, but so are you. It's not a well formed question, and so anwering it isn't possible without discussion of the missing information. Not really straightforward at all.

What if it was Hitler's life on the line?

"What if it was Hitler's life on the line?"

And what..... if he were PREGNANT????!!!

Hate to say it, but it becomes a placement of .. Virtue.

Maybe even the optimistic hope that a billion people would happily suffer a one-hour mild headache if they knew the purpose.

I suspect they would if it was a celebrity. Probably not if it was an anonymous donation for a nameless, unknown kid in the third world.

Well, they say that all political philosophy descends from our individual view of human nature.

I think human beings are pretty good, they would do the headache, but they would not cut off a finger.

How much would the average person spend to avoid a one hour mild headache?

Feel free to do the calculations. You'll find that the number multiplied by 1 billion far exceeds the value that human beings will pay to prevent an anonymous persons death.

It would also be interesting if the hypothetical was reversed—“is it okay to give one billion innocent people a headache for an hour if it would save one person from dying?”

My intuition is that both actions are not okay—I would not kill an innocent person to alleviate a headache from billions, but nor would I give a billion innocent people a headache to save one person. I wonder if others agree and if so whether that means there is a role in morality for do-no-harm that sheer utilitarianism can not capture.

utilitarianism is false.

This. There are almost no pure out and out utilitarians. Almost all of of them qualify their utilitarianism in some major way. And that’s exactly as it should be. Pure utilitarianism is bunk.

My intuition is that giving people a mild headache for an hour is a good trade-off to save a life, though after reading your comment I'm not sure why I feel that way. Maybe due to the finality of death vs the transience of a headache? For instance I don't think I would give people a permanent headache to save a life.

"The puzzle is that smart people can’t agree which answer is obvious."

The puzzle is that smart people can't see that the headache example and the coal-mining example are not equivalent (killing a specific person is simply not the same as asking a large number of people to run a small risk -- a request that they can decline, that they are compensated for if they accept, and that they may be able to moderate by being conscientious).

You are correct, but this is just equivalent to Tyler's statement. The puzzle is that some smart people get the wrong answer mainly because they don't see that these examples are different. But it is also not a puzzle to me (anymore).

Alex, not Tyler.

Sometimes the good of the one outweighs the good of the many, and the coal mining vs. headache analogy is poor on it's face.

Suck it up you 1/7 of the planet and take some Aleve.

Le Guin never cared about outsmarting economists, and her short story 'The Ones Who Walk Away from Omelas' does a much better job exploring the idea that those who reject the very premise of the question are the ones that do not care about answering such questions, or living among people who think such questions matter.

Admittedly, Le Guin was inspired by Dostoyevsky's Brothers Karamazov to an extent, though she credited William James.

Who also has a perspective on such questions - 'Or if the hypothesis were offered us of a world in which Messrs. Fourier's and Bellamy's and Morris's utopias should all be outdone, and millions kept permanently happy on the one simple condition that a certain lost soul on the far-off edge of things should lead a life of lonely torture, what except a sceptical and independent sort of emotion can it be which would make us immediately feel, even though an impulse arose within us to clutch at the happiness so offered, how hideous a thing would be its enjoyment when deliberately accepted as the fruit of such a bargain?'

We know the answer to James's question, sadly enough.

@clockwork_prior - so you are or are advocating being a vegitarian?

A billion people are willing to pay $1 to alleviate the headache.

Could you find a million people willing to take a one-in-a-million chance of dying in return for $1,000.?

I'd guess that you would, as practically everyone will translate "one-in-a-million" as "essentially zero" yet value $1,000. as "substantially more than zero."

And finding people willing to voluntarily assume the risk changes the moral calculus. Doesn't it?

What if those who had the headache were offered a remedy that combined a 100% cure rate with a one-in-a-billion fatality rate?

Or does distributing risk (aka insurance) violate the hoax?

Based on my commute, my salary, and average vehicle deaths per billion passenger miles in the U.S., my revealed price to take on a one-in-a-million chance of dying is no higher than $1,500, and I'm sure the actual price is lower still. You could find a million people just in the Bay Area who would take that trade-off.

I think the "fairly minor headaches" phrasing uselessly obfuscates the issue.

Change the phrasing to "1 billion people have to work 1 hour extra to afford the cure to prevent 1 innocent person from dying".

Smart people can't agree on the headache problem because it's asking (at least) 3 different questions:

(1) If a billion people value headache reduction more than 1 innocent person values his life, is it ok for the billion people to pay the one innocent person to die to alleviate their headaches?

(2 and 3) Suppose a team of (governmental or non-governmental) experts makes an imperfect determination that the benefits of headache reduction for 1 billion people exceed the cost of dying for 1 innocent person. (2) Is it ok to kill the innocent person without compensation? (3) Is it ok to tax the 1 billion people for headache reduction and use the proceeds to compensate the 1 innocent person before killing him, without giving him the choice about whether to accept the deal and without giving the 1 billion people a choice about whether to pay the tax?

Cost-benefit analysis is fine as long as one remembers that no central panel of experts has sufficient information to determine costs nor benefits.

If you can't suspend disbelief for the framing of this play question, you might as well dive into the perfect tax rate.

In other words, it's entire strength is in being imaginary.

If it were obvious that the question's "entire strength is in being imaginary", then "smart people" would not be struggling with it. The "framing of this play question" obscures that it is assuming away all the interesting aspects of the question. It's like asking, "If we knew for sure that the sky was blue, then do we need to determine its color?" Here, the question can be transparently rephrased as, "If we knew for sure that 1 innocent person would accept payment to give up his life to alleviate 1 billion people's headaches, then is it ok to effectuate the same outcome that the 1B + 1 would have chosen voluntarily for themselves anyways?"

Agreed, at the very least society doesn't give equal weighting to a voluntary death versus a deliberate death.

As for these paradoxes, it seems both the the Girl Named Florida and the Newcomb paradoxes are actually paradoxes due to ambiguity in the language of the problem. Once you clear up the ambiguity, there's no paradox just a simple math problem (like the Potato 'paradox' is).

It reminds me of economists who play games with what is the "short run", the "intermediate run" and the "long run" in their models. If the short run in monetarism is a few minutes, days or perhaps a few weeks, as I claim it is, then money is largely neutral (which the data seems to support). If not, then the monetarists are right about the Fed having great power over the economy.

It seems like nonsense to me, but maybe I'm just not bright enough.

My method would be that if you're told a couple has two kids, and one is a girl, you take that as the prior state of the world, and the odds of two girls becomes the odds for the remaining birth, which is 50-50.

All this stuff with names and color curtains seems like piffle.

> and one is a girl

That's the part you are missing. The above statement doesn't specify which child is being discussed, and that changes everything.

If the statement said "the older one is a girl" or "the taller one is a girl" ... then NOW it specifies which one, and all that is unknown is the gender of the other one, which is 50-50.

If the statement does not specify which child is being discussed, more things are unknown, and the odds change. Piffle, it is not.

Maybe this will help.

If you have twins, you could have 2 boys, 2 girls, or 1 of each. Those are the only three possibilities. And since life is random, you have a 1/3 chance of each possibility, right?

Wrong -- having one of each is the most likely scenario. The kids could be BB, BG, GB, or GG. A 50% chance for having one of each, compared to 25% for two boys and 25% for two girls.

One cannot say that BG and GB are the same thing, and mistakenly group them together as "having one of each." They are distinctly separate cases. If you don't believe me, have and son and daughter and tell them to swap genders, because it's all the same to you, because you still have one of each.

Now -- since you are more likely to have one of each, you could also say that in most families with a twin girl, she will have a twin brother instead of a twin sister. That is undeniably true. Just look at the letter pairings: in 2 of the 3 families with girls, she has a twin brother.

And so -- If I have twins and I bring one to show you, and it's a girl, you can safely assume the other one is probably a boy.

It sounds strange -- which is what makes it fun -- but it is true.

You're not helping. First, let's clarify that we are talking about fraternal twins.

If I SHOW YOU a girl, the probability of two girls is 1/2.

If I TELL YOU truthfully that one is a girl, the probability of two girls is 1/3.

"If I SHOW YOU a girl, the probability of two girls is 1/2."

No, I don't think that is true. If I take 4 families with a BB, BG, GB & GG set of fraternal twins and ask the mom to bring a daughter to meet me. Then I'm going to have 3 moms and 3 girls. If I then ask the Shown girl about their sibling, 2 of the 3 are going to have brothers.

No. If we assume independence, once I SHOW YOU a girl, the equation simplifies to: what is the sex of the other child? The answer is 50/50. You have attained maximum specificity, just like if you say the OLDER child is a girl.

"just like if you say the OLDER child is a girl."

Yes, I was going to use that as an example of what would result in a 1/2 result. But what is 'wrong' about my example where I SHOW a girl.

We throw out the BB twins, right?

Of the remaining babies, there are 4 girls, and two of them have a female sibling (each of the GG pair.)

Agreed, I think to phrase it in a way that avoids confusion. It's 'SHOW a Sibling of two who then turns out to be a Girl' versus 'SHOW a Sibling of two who Is a Girl'.

So, on bring your Girl to work day, 2/3rds of the Girls with 1 sibling have a brother. But on bring your Child to work day, only 1/2th of the Girls with 1 sibling have a brother.

As I SAID, it depends on how you word the problem. Then it becomes trivial.

Bonus trivia (not trivial): in fact, the probability of having all boys or all girls is not "50%" as assumed but biologically it depends on the parents. I notice mixed race Asian mother and Causasian father almost always have girls (and if they have boys the boys are very feminine looking). And I know of several Greeks who literally had almost a dozen boys and hardly any or no girls. If it was "50-50" for B/G, the odds of that happening would be astronomically low.

Nevermind, that's exactly what you are saying below.

"If you have twins,"

You need to specify non-identical twins.

"and one of them is a girl"

If the one being mentioned is the older child, then the chance that the other one is a girl is 50-50. If the one being mentioned is the younger child, then the chance that the other one is a girl is still 50-50. Therefore the change is 50-50 overall.

Nope. The answer is 1/3. Four equally likely possibilities for two kids: BB BG GB GG. Since you know one is a girl, you throw out BB. From the remaining three equally likely outcomes, GG is just 1/3.

That's the easy part: understanding how knowing something specific about the one girl drives the possibility asymptotically toward 1/2 depending on the specificity is the tricky part. I haven't fully absorbed the intuition, but the math is plain enough.

Assume 20% of girls have freckles. Knowing there is at least one girl with freckles makes the likelihood of two girls 47.4%.

Knowing there is at least one girl without freckles puts the likelihood of two girls at 37.5%.


I get the 4-cell truth table, but I have this strong feeling that when anyone "sees a girl," a column, rather than a cell should be eliminated, making it a one child problem.

The intuition is that rare events that happen to girls are asymptotically twice as likely to happen to the GG family (because they have two girls) as to a GB or BG family.

Suppose we hear that a girl in our town has been struck by lightning, and all the girls were equally likely to be struck. There's a 25% chance it was in the BG family, a 25% chance it was the older girl in the GG family, a 25% chance it was the younger girl in the GG family, and a 25% chance it was in the GB family. So if a girl is struck by lightning, there's a 50% chance she had a sister. Similarly if she's named Florida. The rare event means we're no longer conditioning across families, but across girls.

But I should admit I was slow to grasp the Monty Hall Problem for perhaps the same reasons.

Me too. It feels like a very similar error.

You think that because you and Tyler disagree on which answer is obvious, that is evidence that "smart people" disagree? LOL

Is there any evidence that IQ and morality are correlated?

Other words, less than smart people probably disagree as well.

That 2006 post is embarrassing. I can't believe it hasn't been deleted in shame by now.

Alex thinks you should murder an innocent person to alleviate brief and minor headaches... because it's the same cost-benefit as having a coal miner die in a job accident when the job provides indescribably staggering benefits.

Remember, folks, this is the kind of person who "teaches your child how to think" and charges you $45K per year.

"Answers in the back of the book would have been a minor form of commitment."

Ie, the answers are behind the table of contents that States the problems?

In most cases, the answer should be longer than the problem statement.

In my favorite textbook, Knuth grades his problems, with some problems given a rating that he says the answer will warrant a Nobel, or other prize.

But I look at problems where the teacher/professor is not as interested in the final answer, but the work leading to the answer: "show your work."

"Is it okay to kill that innocent person?" OK to whom? And whose Mother-in-law gets killed?

Alex's argument is terrible, and he should feel terrible. Yes, it's a decision that's already being made, but in the other direction! Millions of people decide every weekend that a slightly more fun few hours is worth a severe headache for 4 to 6 hours the next day. Given the category difference between a severe headache and a mild one, and the category difference between having a somewhat more enjoyable evening and the opportunity to exist at all, this should be a no brainier (category differences offset the magnitude increases in the number of headache sufferers). Especially given Alex's insistence that fore knowledge of who receives the harm/benefit is inconsequential, this seems like an inescapable conclusion from his own stated logic.

Intent matters. It's the difference between vehicular homicide and accepting the real risk that allowing people to drive will kill some number of them.

The big flaw in Tabarrok's argument is that we don't choose between horrible deaths in coal mining for a few and mild convenience for the many, we choose between horrible deaths for the many who depend on energy to produce food/heat/etc, and horrible deaths for a few in coal mines (or mining rare earth elements for solar panels, if you prefer). That's why mining has gotten so much safer as living standards rise -- the tradeoffs have to remain rational.

The question is silly, really. You might just as well ask if it's moral to trade a light slap on one person who will barely notice it for an absolute certainty of eternity of torment for an infinite number of people. These choices never really exist.

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