I don’t know, I felt the correct sign the first time I read it. I also didn’t get confused by the cognitive reflection test (in the sense that there is no confusion, the correct way of seeing the problem is all there is). It’s really hard to imagine how a person with math training can miss that.
But from what I heard, a sizable portion of math students still manage to get confused by these. Tracing the analogy to cognitive biases, there may be a qualitative difference between a person who just knows about it “in theory” and even done a lot of reading on the topic (“typical” math student), and a person who thought about the techniques at every opportunity for a number of years.
Re. the linked article about the cognitive confusion test:
80 percent of high-scoring men would pick a 15 percent chance of $1 million over a sure $500, compared with only 38 percent of high-scoring women, 40 percent of low-scoring men and 25 percent of low-scoring women.
Wow. That’s the most mind-blowing thing I’ve read in a while. I can’t think of a good explanation for anyone picking the $500, let alone the male-female difference. Maybe they assume that someone might actually give them $500, but the $1M is a scam. And how does this square with the idea that poor people play the lottery more?
I can’t think of a good explanation for anyone picking the $500
Say, you’re starving and if you don’t get a meal today, you’ll die. In such situations, the choice between 15% chance of $1 million and a sure $500 boils down to a choice between 15% chance to survive today and a 100% chance to survive today (assuming that the meal costs less than $500.)
Perhaps the people who chose $500 operate in this ‘starvation mode’ by default.
I doubt this is the case for most of the people who would take the $500 - I’d assume it’s more that most of them couldn’t or just didn’t think to calculate the expectation value of a 15% chance at a million.
Just because someone tells you that something has a 15% chance does not make it so. If someone offers you a 15% chance at $1M for anything less than $150k, then you should be 95% confident that they will try to cheat somehow.
Sure; but it’s posed as a hypothetical. The participants know there’s no real money involved. Are their conscious selves unable to prevent a subconscious defense against being scammed?
If it’s the right answer in reality, then it’s the right answer in a hypothetical. People use their actual cognitive faculties for pondering hypotheticals, not imaginary ones.
This sounds like something that could be tested in a real experiment. We don’t have surplus millions floating around to give away, but surely there has to be a way to arrange so that the subjects either believe it’s just a hypothetical offer as apposed to a real proposition.
They may do, but they are still missing many of the physical reactions one might have to genuinely being offered large sums of money—excitement, adrenalin etc—and these are bound to have some effect on people’s decision making processes.
Perhaps a way around this would be to conduct several thought experiments with a subject in one sitting, and tell them beforehand that ‘one of the offers in these thought experiments is real and you will receive what you choose—although you will not know which one until the end of the experiment’.
This would be a good way to induce their visceral reactions to the situation, and of course, disappointingly perhaps, a more modest-sum-involving thought experiment at the end could provide them with their dividend.
Also worth noting: Deal or No Deal (UK version) demonstrates the variety of reactions and strategies people have to this sort of proposition. The US version is just silly though :)
Maybe they don’t have have the same concept as we do of a “hypothetical”.
Are their conscious selves unable to prevent a subconscious defense against being scammed?
If their conscious selves could shut down the defense, scammers could convince it to. This kind of sphexish paranoia is adaptive, if you’re the sort of person who scores low on the cognitive reflection test. Maybe.
Are their conscious selves unable to prevent a subconscious defense against being scammed?
If anything could let down the defense, scammers could exploit it – so the semiconscious reasoning might go. Epistemic paranoia is adaptive when you’re bad at evaluating arguments.
We’ve had this argument before, and it still looks to me like this couldn’t account for the full effect of risk aversion. The fact that scammers regularly succeed means that people don’t usually base their reasoning on that sort of suspicion.
People who feel secure do not, and people who do not feel secure do. Unfortunately, to someone in the latter camp, genuine opportunity really looks like a scam; it’s “too good to be true”.
This is a really good point. I’d feel much more confident in these sorts of results if the questions were prefaced by disclaimers stating that there is no chance whatsoever of getting ripped off, that the random decision that determines the win or loss is absolutely guaranteed to be secure and accurate, that the $1M is tax-free, will be given in secret, doesn’t need to be reported to the government, etc.
One reason people might pick the $500 is because they’ll come out better off than 85% of people who take the more rational course.
It is little comfort to be able to claim to have made the right decision when everyone who made the less rational decision is waving a stack of money in your face and laughing at the silly rationalist.
People don’t want to be rich—they just want to be richer than their next door neighbour.
I can’t think of a good explanation for anyone picking the $500
I agree with Vladimir Golovin. I definitely think this way—I can think immediately of how useful $500 would be to me, but cannot think of many ways to use a 15% chance of $1M.
Well, I can think of one way—I would take the 15% chance of $1M and then sell it to one of you suckers for $100,000.
Show of hands—who really has $100,000 that they could free up to buy this from thomblake? Personally, I don’t think I actually know what 100k is worth to me because I have never had my hands on that much money.
Actually, if he has the $1M, I’m in. I don’t have $100K liquid in any normal sense but I could certainly raise it for such a deal and divide the risk and return up among a few people.
He does not have (even in this hypothetical situation) a million bucks. Hypothetically, he’s being offered a 15% chance of winning a million bucks.
Incidentally, in a staggering display of risk aversion, I asked a friend how much she’d pay for a 15% chance of a million dollars and she said maybe twenty bucks because those did not seem like “very good odds” to her. -.-
Just how much is a one in seven chance of a million dollars worth to everyone here, anyway? (Offer me a near certainty of $70,000 and I’d start to have second thoughts about taking the gamble.)
Exactly. If someone has $1.1M, spending $0.1M on a 15% chance of $1M is a good deal. Someone who has $0.05M and has to go into debt to buy the 15% chance is very probably insane.
Precisely the reaction I expected! This model of despair produced by the Singularity Institute for Eliezer Yudkowsky is matching quite well. A rigorous theory of Eliezer Yudkowsky can’t be far off.
Okay, I tested this on a couple of uninvolved bystanders and yes, they would take the $500 over the 15% chance of $1m. Guess it’s true. Staggers the mind.
I tried it on two women at work and they both went for the million, one with no hesitation and the other after maybe 10 seconds. Although they both have some background in finance and are probably 1 to 2 standard deviations above average IQ.
As previous comments have said, it would be possible to sell the 15% chance for anything up to $150k. Once people realise that the 15% chance is a liquid asset, I’m sure many will change their mind and take that instead of the $500.
What does this mean? If the 15% chance is made liquid, that removes nearly all of the risk of taking that chance. This leads me to believe that people pick the $500 because they are, quite simply, (extremely) risk-averse. Other explanations (diminishing marginal utility of money, the $1 million actually having negative utility, etc.) are either wrong, or they are not a large factor in the decision-making process.
Note that the standard explanation for risk-aversion just is diminishing marginal utility (where utility is defined in the decision-theoretic sense, rather than the hedonic sense). However, Matt Rabin pretty convincingly demolishes this in his paper Diminishing marginal utility of wealth cannot explain risk aversion.
I also read someone that humans generally don’t distinguish much between between probabilities lower than 5%. That is, everything below 5% is treated as a low probability event.
Even I, with good mathematical training, I guess if I would prefer $100K at 100% to $1B at 1%.
Although the second alternative has 100X “expected” pay off, I don’t “expect” myself to be lucky enough to get it. :)
And although I’d definitely prefer $1M@15% to $500@100% if you’d multiply it by thousand, I think, I’d take $500K@100% rather than $1B@15% (in this case of course, Bill Gates would laugh at me… :) )
Well, also part of it is that for most people, utility isn’t linear in money.
Imagine someone starving, on the verge of death or such. This offer is more or less their very last chance at this particular moment to be able to survive.
500$ with certainty means high probability of immediate survival. 1 million dollars with 15% chance means ~15% chance of survival.
500$ can potentially get enough meals and so on to buy enough time to get more help.
Again, not everyone is in this situation, obviously. But this is a simple construct to demo that utility isn’t linear in money and that picking the 500$ can, at least in some cases, be rather more rational than the initial naive computation may suggest at first. (Shut up and multiply UTILITIES and probabilities, rather than money and probability. :))
Having said all that, probably for most people in that study, picking 500$ was the wrong choice. :)
Well, also part of it is that for most people, utility isn’t linear in money.
Yeah. This assumption of linearity is annoyingly common; I wish more people were aware of the problems with it when contructing their various thought experiments. Not just with money, either.
I don’t think you can model my preferences with excepted value computation based on a money → utility mapping.
E.g. I’d definitely prefer 100M@100% to any amount of money at less than 100%. Still I’d prefer 101M@100%.
I think that my preference is quite defensible from a rational point of view, still there is no real valued money to utility mapping that could make it fit into an expected utility-maximization framework.
Well, you can use money to do stuff that does have value to you. So while there isn’t a simple utility(money) computation, in principle one might have utility(money|current state of the world)
ie, there’s a sufficiently broad set of things you can do with money such that more money will, for a very wide range of amounts of money, give you more opportunity to bring reality into states higher in your preference ranking.
and wait… are you saying you’d prefer 100 million dollars at probability =1 to, say, 100 billion dollars at probability = .99999?
This kind of super-cautious mindset can’t be modeled with any real valued money X (current state of the world) → utility type of mapping.
If you would trade a .99999 probability of $100M for a .99997 probability of $100B, then you’re correct—you have no consistent utility function, and hence you can be money-pumped by the Allais Paradox.
And as I’ve argued before, that only follows if the a) the subject is given an arbitrarily large number of repeats of that choice, and b) their preference for one over the other is interpreted as them writing an arbitrarily large number of option contracts trading one for the other.
If—as is the case when people actually answer the Allais problem as presented—they merely show a one-shot preference for one over the other, it does not follow that they have an inconsistent utility function, or that they can be money-pumped. When you do the experiment again and again, you get the expected value. When you don’t, you don’t.
If making the “wrong” choice when presented with two high-probability, high-payoff lottery tickets is exploitation, I don’t want to be empowered. (You can quote me on that.)
Yes, but I can’t find it at the moment—it came up later, and apparently people do get money-pumped even on repeated versions. The point about what inferences you can draw from a one-shot stands though.
If making the “wrong” choice when presented with two high-probability, high-payoff lottery tickets is exploitation, I don’t want to be empowered. (You can quote me on that.)
Not very quotable, but I may be tempted to do so anyway.
Aw, come on! Don’t you see? “If X is wrong, I don’t want to be right”, but then using exploitation and empowerment as the opposites instead?
Anyway, do you get the general point about how the money pump only manifests in multiple trials over the same person, which weren’t studied in the experiments, and how Eliezer_Yudkowsky’s argument subtly equates a one-time preference with a many-time preference for writing lots of option contracts?
The above example had no consistent (real valued) utility function regardless off my 100M@.99999 vs. 100B@.99997 preference.
BTW, whatever would that preference be (I am a bit unsure, but I think I’d still take the 100M as not doing so would triple my chances of losing it) I did not really get the conclusion of the essay. At least I could not follow why being money-pumped (according to that definition of “money pumped”) is so undesirable from any rational point of view.
This kind of super-cautious mindset can’t be modeled with any real valued money X (current state of the world) → utility type of mapping.
Yes it can: use the mapping U:money->utils such that U(x) is increasing for x<$100M (probably concave) and U(x) = C = const for x>=$100M. Then expected utility EU($100M@100%) = C*1 = C, and also EU($100B@90%) = C*0.9 < EU($100M@100%). But one of the consequences of expected utility representation is that now you must be indifferent between 20% chance at $100M and 20% chance at $100B.
Call me a chicken, but yes: I would not risk going out empty handed even in 1 out of 100000 if I could have left with $100M.
This kind of super-cautious mindset can’t be modeled with any real valued money X (current state of the world) → utility type of mapping.
Like Vladimir Nesov pointed out, that is false—not the preference being expressed, of course, but the statement that the preference can’t be modeled with the mapping.
Now first let me make it clear that I disapprove of the atmosphere you find in some academic science departments where making a false statement is taken to be a mortifying sin. That kind of attitude is a big barrier to teaching and to learning. Since teaching and learning is a big part of what we want to do here, we should not think poorly of a participant for making a false statement.
But I am a little worried that in 88 hours since the false statement was made, no one downvoted the false statement (or if they did, the vote was canceled out by an upvote). And I am a little worried that in the 81 hours since his reply, no one upvoted Nesov’s reply in which he explains why the statement is false. (I have just cast my votes on these 2 comments.)
It is good to have an atmosphere of respect for people even if they make a mistake, but it is bad IMHO when most readers ignore a false statement like the one we have here when there is no doubt about its falseness (it is not open to interpretation) and it involves knowledge central to the mission of the community (e.g., like the one we have here about the most elementary decision theory). Note that elementary decision theory is central to the rationality mission of Less Wrong and to the improve-the-global-situation-with-AI mission of Less Wrong.
Moreover, if you not only read a comment, but also decide to reply to it, well, then IMHO, you should take particular care to make sure you understand the comment, especially when the comment is as short and unnuanced as the one under discussion. But before Nesov’s reply, two people replied to the comment under discussion without showing any sign that they recognise that the one statement of fact made in the comment is false. One reply (upvoted 3 times) reads, ‘The technical term is “risk-averse”, not “chicken”’. The other introduces the Allais paradox, which is irrelevant to why the statement is false.
I do not mean to single out this comment and these 2 replies or the people who wrote them: the only reason I am drawing attention to them is to illustrate something that happens regularly. And I definitely realize that it probably happens a lot less here on Less Wrong than it does in any other conversation on the internet that ranges over as many subject relevant to the human condition as the conversation on Less Wrong does. And a significant reason for that is the hard work Eliezer and others put into the development of the software behind the site.
But I suspect that one of the best opportunities for creating a conversation that is even better than the conversation we are all in right now is to make the response by the community to false statements (the kind not open to interpretation) more salient and more consistent. Wikipedia’s response to false statements gives me the impression of rising to the level of saliency and consistency I am talking about, but of course the software behind Wikipedia does not support conversation as well as the software behind Less Wrong does. (And more importantly but more subtly, Wikipedia is badly governed: much of the goodwill and reputation enjoyed by Wikipedia will probably be captured by the ideological and personal agendas of Wikipedia’s insiders.)
I disagree that false statements are the sorts of things that should be downvoted. I’m all about this being a place where people can happily be false and get corrected, and that means the ’I want to see fewer comments like this” interpretation suggests that I should not downvote comments merely for containing falsehoods.
“I’m all about this being a place where people can happily be false and get corrected.”
I am, too, until the false statements start drown out the relevant true information so that the most rational readers decide to stop coming here anymore or until the volume of false statements overwhelm the community’s ability to respond to false statements. But, yeah, I am with you.
And you make me realize that downvoting is probably not the right response to a false statement. I just think that there should be a response that is not as demanding of the reader’s time and attention as reading the false statement, then reading the responses to the false statement. (Also, it would be nice to give a prospective responder a way to respond that is less demanding of their time than the only way currently available, namely, to compose a comment in reply to the false statement.)
My original statement was mathematically true. Maybe Vladimir was sloppy reading it (his utilty function satisfied only half of the requirements), but I would not downvote him for that.
Another possible sane motivation for taking the $500 is a familiarity with how commonly lottery winners find their lives ruined by the sudden influx of cash.
It’s possible but doesn’t seem very likely, since given the choice between $1M outright or $500 outright, those same people would almost certainly take the $1M.
I think a more likely explanation is that they conceptualize the problem as having to choose between “probably getting $0” and “certainly getting $500″.
Of course that’s it. $500 is a lot to pay for a lottery ticket, even one with as high a chance of winning as this. Change it to a certain $20 and a 15% chance of $40,000, and I bet (heh) that many more people will take the chance then.
Well, I don’t buy this in general, but I do know one person who would do this. I was talking with her recently about a lottery winnery who won some huge sum, maybe $100M, and she said, “But it ruined his life.” And I said, “Why? What happened?” And she said, “Oh, I don’t know what happened. But I assume it ruined his life.”
In her mind, I think she had already taken her high prior for B, assumed B, and converted the non-incident into further evidence for B. She did laugh at herself when she realized she’d done this, so there is hope. :)
Being aware of that tendency should make it possible to avoid ruination without forgoing the money entirely (e.g. by investing it wisely and not spending down the principal on any radical lifestyle changes, or even by giving all of it away to some worthy cause).
Unless there’s akrasia involved. I can only imagine how tempting it would be to just outright buy a house if I were suddenly handed a million dollars, no matter how sternly I told myself not to just outright buy a house.
And the best workaround you can come up with is to walk away from the money entirely? I don’t buy it.
If you go through life acting as if your akrasia is so immutable that you have to walk away from huge wins like this, you’re selling yourself short.
Even if you’re right about yourself, you can just keep $1000 [edit: make that $3334, so as to have a higher expected value than a sure $500] and give the rest away before you have time to change your mind. Or put the whole million in an irrevocable trust. These aren’t even the good ideas; they’re just the trivial ones which are better than what you’re suggesting.
Given a million-dollar windfall, buying a house at today’s depressed prices would be one of the best investments you could make. (An additional benefit would be to make the money less liquid, thereby cutting down the temptation to spend it frivolously.)
Perhaps, but owning a house would be a terrible time investment for me the way my life is working. I suppose I could hire a property manager and rent it out, though.
I’m pretty confident that you could sell a true 15% chance to win a million bucks for a lot more than 10k… after all, investment banks make substantially greater gambles regularly.
I’d probably ask for 100k to start and go from there.
It is my understanding that people are in general way too optimistic about how much winning $1M would increase their overall happiness. (Say if they are asked to imagine themselves winning the lottery.)
Re: Maybe they assume that someone might actually give them $500, but the $1M is a scam.
If you were offering this deal, wouldn’t the $1M be based on a deceptive maniuputation of the stated probabilities? Many participants can probably figure that one out.
My model says that there is a big difference between formal education and deep understanding that can only be developed by extracurricular appreciation of the subject.
I was briefly tempted to answer 10 cents to the first problem.
one study was done at University of Toledo″ -- where the mean score on his test was 0.57 out of a possible 3 -- ″and one study was done at Princeton,″ where the mean was 1.63
I’m just thrilled to think of how dumb our elite (top 10% and top 2% respectively?) are.
Almost a third of high scorers preferred a 1 percent chance of $5,000 to a sure $60.
Maybe those are just the high scorers who got 3⁄3 (avoiding the tempting surface error) almost by sheer chance.
I got 3⁄3 and I would take the chance. My rational is that $60 is almost nothing. I can make that very quickly, and it won’t buy much. I won’t notice it in my monthly finances. $5000, on the other hand, is actually worth considering. That could change my month significantly (and impact the rest of my year as well). Would you rather have a 100% chance of getting a nickel, or a 0.01% chance of getting a small diamond?
Exactly! This is gambling, isn’t it? A small expected loss, with a tiny chance of some huge gain.
If your utility for money really is so disproportionate to the actual dollar value, then you probably ought to take a trip to Las Vegas and lay down a few long-odds bets. You’ll almost certainly lose your betting money (but you wouldn’t “notice it in [your] monthly finances”), while there’s some (small) chance that you get lucky and “change [your] month considerably”.
It’s not hypothetical! You can do this in the real world! Go to Vegas right now.
(If the plane flight is bothering you, I’m sure we could locate some similar online betting opportunities.)
That sort of attitude (among my opponents) is very helpful to my poker bankroll. You’re giving up $60 for $50 of expected value. Even given your risk-seeking preference, surely you can find a better gamble. Putting it on a single number in roulette would be a better deal.
By the way, welcome to Less Wrong (I notice you had some comments on Overcoming Bias as well); you should check out the welcome thread if you haven’t already.
I’d be almost guaranteed to lose the diamond before I could liquidate it if I won it. Should I factor that in or not? Diamonds are also notoriously difficult to liquidate unless you are in the relevant cartel...
I would guess that most people who got the first problem correct also had “10 cents” as their initial thought, for about a half or second or so before they had finished reading the question and before they had actually started deliberatively thinking about the problem.
I don’t know, I felt the correct sign the first time I read it. I also didn’t get confused by the cognitive reflection test (in the sense that there is no confusion, the correct way of seeing the problem is all there is). It’s really hard to imagine how a person with math training can miss that.
But from what I heard, a sizable portion of math students still manage to get confused by these. Tracing the analogy to cognitive biases, there may be a qualitative difference between a person who just knows about it “in theory” and even done a lot of reading on the topic (“typical” math student), and a person who thought about the techniques at every opportunity for a number of years.
Re. the linked article about the cognitive confusion test:
Wow. That’s the most mind-blowing thing I’ve read in a while. I can’t think of a good explanation for anyone picking the $500, let alone the male-female difference. Maybe they assume that someone might actually give them $500, but the $1M is a scam. And how does this square with the idea that poor people play the lottery more?
Princeton students scored a mean of 1.63. Heh.
Say, you’re starving and if you don’t get a meal today, you’ll die. In such situations, the choice between 15% chance of $1 million and a sure $500 boils down to a choice between 15% chance to survive today and a 100% chance to survive today (assuming that the meal costs less than $500.)
Perhaps the people who chose $500 operate in this ‘starvation mode’ by default.
The general term for “people who operate in starvation mode” is “the poor”.
I doubt this is the case for most of the people who would take the $500 - I’d assume it’s more that most of them couldn’t or just didn’t think to calculate the expectation value of a 15% chance at a million.
I think people are flipping the offer in their minds and comparing a sure $500 to an 85% chance of zilch.
If that’s actually the way you think, I’ve got some food to sell you.
Just because someone tells you that something has a 15% chance does not make it so. If someone offers you a 15% chance at $1M for anything less than $150k, then you should be 95% confident that they will try to cheat somehow.
Sure; but it’s posed as a hypothetical. The participants know there’s no real money involved. Are their conscious selves unable to prevent a subconscious defense against being scammed?
If it’s the right answer in reality, then it’s the right answer in a hypothetical. People use their actual cognitive faculties for pondering hypotheticals, not imaginary ones.
This sounds like something that could be tested in a real experiment. We don’t have surplus millions floating around to give away, but surely there has to be a way to arrange so that the subjects either believe it’s just a hypothetical offer as apposed to a real proposition.
They may do, but they are still missing many of the physical reactions one might have to genuinely being offered large sums of money—excitement, adrenalin etc—and these are bound to have some effect on people’s decision making processes.
Perhaps a way around this would be to conduct several thought experiments with a subject in one sitting, and tell them beforehand that ‘one of the offers in these thought experiments is real and you will receive what you choose—although you will not know which one until the end of the experiment’.
This would be a good way to induce their visceral reactions to the situation, and of course, disappointingly perhaps, a more modest-sum-involving thought experiment at the end could provide them with their dividend.
Also worth noting: Deal or No Deal (UK version) demonstrates the variety of reactions and strategies people have to this sort of proposition. The US version is just silly though :)
Maybe they don’t have have the same concept as we do of a “hypothetical”.
If their conscious selves could shut down the defense, scammers could convince it to. This kind of sphexish paranoia is adaptive, if you’re the sort of person who scores low on the cognitive reflection test. Maybe.
Most people don’t really get hypotheticals. Even most high IQ people seem not to.
If anything could let down the defense, scammers could exploit it – so the semiconscious reasoning might go. Epistemic paranoia is adaptive when you’re bad at evaluating arguments.
We’ve had this argument before, and it still looks to me like this couldn’t account for the full effect of risk aversion. The fact that scammers regularly succeed means that people don’t usually base their reasoning on that sort of suspicion.
People who feel secure do not, and people who do not feel secure do. Unfortunately, to someone in the latter camp, genuine opportunity really looks like a scam; it’s “too good to be true”.
This is a really good point. I’d feel much more confident in these sorts of results if the questions were prefaced by disclaimers stating that there is no chance whatsoever of getting ripped off, that the random decision that determines the win or loss is absolutely guaranteed to be secure and accurate, that the $1M is tax-free, will be given in secret, doesn’t need to be reported to the government, etc.
One reason people might pick the $500 is because they’ll come out better off than 85% of people who take the more rational course. It is little comfort to be able to claim to have made the right decision when everyone who made the less rational decision is waving a stack of money in your face and laughing at the silly rationalist. People don’t want to be rich—they just want to be richer than their next door neighbour.
I agree with Vladimir Golovin. I definitely think this way—I can think immediately of how useful $500 would be to me, but cannot think of many ways to use a 15% chance of $1M.
Well, I can think of one way—I would take the 15% chance of $1M and then sell it to one of you suckers for $100,000.
Show of hands—who really has $100,000 that they could free up to buy this from thomblake? Personally, I don’t think I actually know what 100k is worth to me because I have never had my hands on that much money.
Actually, if he has the $1M, I’m in. I don’t have $100K liquid in any normal sense but I could certainly raise it for such a deal and divide the risk and return up among a few people.
He does not have (even in this hypothetical situation) a million bucks. Hypothetically, he’s being offered a 15% chance of winning a million bucks.
Incidentally, in a staggering display of risk aversion, I asked a friend how much she’d pay for a 15% chance of a million dollars and she said maybe twenty bucks because those did not seem like “very good odds” to her. -.-
I don’t have $100,000. I only have about $30,000.
Just how much is a one in seven chance of a million dollars worth to everyone here, anyway? (Offer me a near certainty of $70,000 and I’d start to have second thoughts about taking the gamble.)
Exactly. If someone has $1.1M, spending $0.1M on a 15% chance of $1M is a good deal. Someone who has $0.05M and has to go into debt to buy the 15% chance is very probably insane.
Agreed. That made my eyes water quite a bit. Alicorn’s large-amounts-of-money-can-have-negative-utility explanation snapped me out of it.
Just wait until Eliezer sees this...
AAIIIIIEEEEEAAARRRRRGGGHHH.
Just when you think your species can’t possibly get any more embarrassing.
Precisely the reaction I expected! This model of despair produced by the Singularity Institute for Eliezer Yudkowsky is matching quite well. A rigorous theory of Eliezer Yudkowsky can’t be far off.
--Delta, your friendly neighborhood Friendly AI
Okay, I tested this on a couple of uninvolved bystanders and yes, they would take the $500 over the 15% chance of $1m. Guess it’s true. Staggers the mind.
I tried it on two women at work and they both went for the million, one with no hesitation and the other after maybe 10 seconds. Although they both have some background in finance and are probably 1 to 2 standard deviations above average IQ.
That’s not very surprising. You could see if they passed all three questions on the reflection test.
My fiance (who has a more advanced degree than I) thought I was trying to trick her and made me restate the problem several times.
As previous comments have said, it would be possible to sell the 15% chance for anything up to $150k. Once people realise that the 15% chance is a liquid asset, I’m sure many will change their mind and take that instead of the $500.
What does this mean? If the 15% chance is made liquid, that removes nearly all of the risk of taking that chance. This leads me to believe that people pick the $500 because they are, quite simply, (extremely) risk-averse. Other explanations (diminishing marginal utility of money, the $1 million actually having negative utility, etc.) are either wrong, or they are not a large factor in the decision-making process.
Note that the standard explanation for risk-aversion just is diminishing marginal utility (where utility is defined in the decision-theoretic sense, rather than the hedonic sense). However, Matt Rabin pretty convincingly demolishes this in his paper Diminishing marginal utility of wealth cannot explain risk aversion.
OK, you have to think like reality too. How many times am I going to post this same sentence on one thread?
Off topic, but I just wanted to draw your attention to a comment made about you here:
http://www.kurzweilai.net/mindx/frame.html
Over halfway down page is a topic with your name. In this topic one commenter says unkind things about you. Your response ?
For a person who doesn’t expect to get many more similar betting chances, the expectation value of the big win is unphysical.
Yes. Probably that must be one of the reasons.
I also read someone that humans generally don’t distinguish much between between probabilities lower than 5%. That is, everything below 5% is treated as a low probability event.
Even I, with good mathematical training, I guess if I would prefer $100K at 100% to $1B at 1%.
Although the second alternative has 100X “expected” pay off, I don’t “expect” myself to be lucky enough to get it. :)
And although I’d definitely prefer $1M@15% to $500@100% if you’d multiply it by thousand, I think, I’d take $500K@100% rather than $1B@15% (in this case of course, Bill Gates would laugh at me… :) )
Well, also part of it is that for most people, utility isn’t linear in money.
Imagine someone starving, on the verge of death or such. This offer is more or less their very last chance at this particular moment to be able to survive.
500$ with certainty means high probability of immediate survival. 1 million dollars with 15% chance means ~15% chance of survival.
500$ can potentially get enough meals and so on to buy enough time to get more help.
Again, not everyone is in this situation, obviously. But this is a simple construct to demo that utility isn’t linear in money and that picking the 500$ can, at least in some cases, be rather more rational than the initial naive computation may suggest at first. (Shut up and multiply UTILITIES and probabilities, rather than money and probability. :))
Having said all that, probably for most people in that study, picking 500$ was the wrong choice. :)
Yeah. This assumption of linearity is annoyingly common; I wish more people were aware of the problems with it when contructing their various thought experiments. Not just with money, either.
I don’t think you can model my preferences with excepted value computation based on a money → utility mapping.
E.g. I’d definitely prefer 100M@100% to any amount of money at less than 100%. Still I’d prefer 101M@100%.
I think that my preference is quite defensible from a rational point of view, still there is no real valued money to utility mapping that could make it fit into an expected utility-maximization framework.
Well, you can use money to do stuff that does have value to you. So while there isn’t a simple utility(money) computation, in principle one might have utility(money|current state of the world)
ie, there’s a sufficiently broad set of things you can do with money such that more money will, for a very wide range of amounts of money, give you more opportunity to bring reality into states higher in your preference ranking.
and wait… are you saying you’d prefer 100 million dollars at probability =1 to, say, 100 billion dollars at probability = .99999?
Call me a chicken, but yes: I would not risk going out empty handed even in 1 out of 100000 if I could have left with $100M.
This kind of super-cautious mindset can’t be modeled with any real valued money X (current state of the world) → utility type of mapping.
The technical term is “risk-averse”, not “chicken”.
If you would trade a .99999 probability of $100M for a .99997 probability of $100B, then you’re correct—you have no consistent utility function, and hence you can be money-pumped by the Allais Paradox.
And as I’ve argued before, that only follows if the a) the subject is given an arbitrarily large number of repeats of that choice, and b) their preference for one over the other is interpreted as them writing an arbitrarily large number of option contracts trading one for the other.
If—as is the case when people actually answer the Allais problem as presented—they merely show a one-shot preference for one over the other, it does not follow that they have an inconsistent utility function, or that they can be money-pumped. When you do the experiment again and again, you get the expected value. When you don’t, you don’t.
If making the “wrong” choice when presented with two high-probability, high-payoff lottery tickets is exploitation, I don’t want to be empowered. (You can quote me on that.)
This is what I’m thinking, too. Curious, since you say you’ve argued this before, did Eliezer ever address this argument anywhere?
Yes, but I can’t find it at the moment—it came up later, and apparently people do get money-pumped even on repeated versions. The point about what inferences you can draw from a one-shot stands though.
Not very quotable, but I may be tempted to do so anyway.
Aw, come on! Don’t you see? “If X is wrong, I don’t want to be right”, but then using exploitation and empowerment as the opposites instead?
Anyway, do you get the general point about how the money pump only manifests in multiple trials over the same person, which weren’t studied in the experiments, and how Eliezer_Yudkowsky’s argument subtly equates a one-time preference with a many-time preference for writing lots of option contracts?
Yep.
Rockin.
The above example had no consistent (real valued) utility function regardless off my 100M@.99999 vs. 100B@.99997 preference.
BTW, whatever would that preference be (I am a bit unsure, but I think I’d still take the 100M as not doing so would triple my chances of losing it) I did not really get the conclusion of the essay. At least I could not follow why being money-pumped (according to that definition of “money pumped”) is so undesirable from any rational point of view.
Yes it can: use the mapping U:money->utils such that U(x) is increasing for x<$100M (probably concave) and U(x) = C = const for x>=$100M. Then expected utility EU($100M@100%) = C*1 = C, and also EU($100B@90%) = C*0.9 < EU($100M@100%). But one of the consequences of expected utility representation is that now you must be indifferent between 20% chance at $100M and 20% chance at $100B.
I also made the requirement that 101M@100% should be preferred to 100M@100%.
Your utility function of U(x)=C for x>100M can’t satisfy that.
Like Vladimir Nesov pointed out, that is false—not the preference being expressed, of course, but the statement that the preference can’t be modeled with the mapping.
Now first let me make it clear that I disapprove of the atmosphere you find in some academic science departments where making a false statement is taken to be a mortifying sin. That kind of attitude is a big barrier to teaching and to learning. Since teaching and learning is a big part of what we want to do here, we should not think poorly of a participant for making a false statement.
But I am a little worried that in 88 hours since the false statement was made, no one downvoted the false statement (or if they did, the vote was canceled out by an upvote). And I am a little worried that in the 81 hours since his reply, no one upvoted Nesov’s reply in which he explains why the statement is false. (I have just cast my votes on these 2 comments.)
It is good to have an atmosphere of respect for people even if they make a mistake, but it is bad IMHO when most readers ignore a false statement like the one we have here when there is no doubt about its falseness (it is not open to interpretation) and it involves knowledge central to the mission of the community (e.g., like the one we have here about the most elementary decision theory). Note that elementary decision theory is central to the rationality mission of Less Wrong and to the improve-the-global-situation-with-AI mission of Less Wrong.
Moreover, if you not only read a comment, but also decide to reply to it, well, then IMHO, you should take particular care to make sure you understand the comment, especially when the comment is as short and unnuanced as the one under discussion. But before Nesov’s reply, two people replied to the comment under discussion without showing any sign that they recognise that the one statement of fact made in the comment is false. One reply (upvoted 3 times) reads, ‘The technical term is “risk-averse”, not “chicken”’. The other introduces the Allais paradox, which is irrelevant to why the statement is false.
I do not mean to single out this comment and these 2 replies or the people who wrote them: the only reason I am drawing attention to them is to illustrate something that happens regularly. And I definitely realize that it probably happens a lot less here on Less Wrong than it does in any other conversation on the internet that ranges over as many subject relevant to the human condition as the conversation on Less Wrong does. And a significant reason for that is the hard work Eliezer and others put into the development of the software behind the site.
But I suspect that one of the best opportunities for creating a conversation that is even better than the conversation we are all in right now is to make the response by the community to false statements (the kind not open to interpretation) more salient and more consistent. Wikipedia’s response to false statements gives me the impression of rising to the level of saliency and consistency I am talking about, but of course the software behind Wikipedia does not support conversation as well as the software behind Less Wrong does. (And more importantly but more subtly, Wikipedia is badly governed: much of the goodwill and reputation enjoyed by Wikipedia will probably be captured by the ideological and personal agendas of Wikipedia’s insiders.)
I disagree that false statements are the sorts of things that should be downvoted. I’m all about this being a place where people can happily be false and get corrected, and that means the ’I want to see fewer comments like this” interpretation suggests that I should not downvote comments merely for containing falsehoods.
“I’m all about this being a place where people can happily be false and get corrected.”
I am, too, until the false statements start drown out the relevant true information so that the most rational readers decide to stop coming here anymore or until the volume of false statements overwhelm the community’s ability to respond to false statements. But, yeah, I am with you.
And you make me realize that downvoting is probably not the right response to a false statement. I just think that there should be a response that is not as demanding of the reader’s time and attention as reading the false statement, then reading the responses to the false statement. (Also, it would be nice to give a prospective responder a way to respond that is less demanding of their time than the only way currently available, namely, to compose a comment in reply to the false statement.)
My original statement was mathematically true. Maybe Vladimir was sloppy reading it (his utilty function satisfied only half of the requirements), but I would not downvote him for that.
Another possible sane motivation for taking the $500 is a familiarity with how commonly lottery winners find their lives ruined by the sudden influx of cash.
It’s possible but doesn’t seem very likely, since given the choice between $1M outright or $500 outright, those same people would almost certainly take the $1M.
I think a more likely explanation is that they conceptualize the problem as having to choose between “probably getting $0” and “certainly getting $500″.
Of course that’s it. $500 is a lot to pay for a lottery ticket, even one with as high a chance of winning as this. Change it to a certain $20 and a 15% chance of $40,000, and I bet (heh) that many more people will take the chance then.
Alicorn can still rescue her justification by saying that winning a lot of money by luck can ruin your life :-)
Well, I don’t buy this in general, but I do know one person who would do this. I was talking with her recently about a lottery winnery who won some huge sum, maybe $100M, and she said, “But it ruined his life.” And I said, “Why? What happened?” And she said, “Oh, I don’t know what happened. But I assume it ruined his life.”
In her mind, I think she had already taken her high prior for B, assumed B, and converted the non-incident into further evidence for B. She did laugh at herself when she realized she’d done this, so there is hope. :)
Being aware of that tendency should make it possible to avoid ruination without forgoing the money entirely (e.g. by investing it wisely and not spending down the principal on any radical lifestyle changes, or even by giving all of it away to some worthy cause).
Unless there’s akrasia involved. I can only imagine how tempting it would be to just outright buy a house if I were suddenly handed a million dollars, no matter how sternly I told myself not to just outright buy a house.
And the best workaround you can come up with is to walk away from the money entirely? I don’t buy it.
If you go through life acting as if your akrasia is so immutable that you have to walk away from huge wins like this, you’re selling yourself short.
Even if you’re right about yourself, you can just keep $1000 [edit: make that $3334, so as to have a higher expected value than a sure $500] and give the rest away before you have time to change your mind. Or put the whole million in an irrevocable trust. These aren’t even the good ideas; they’re just the trivial ones which are better than what you’re suggesting.
Ha! Buying a house and even more so moving is hard work, even with hired help. No way I’d do that right away.
Given a million-dollar windfall, buying a house at today’s depressed prices would be one of the best investments you could make. (An additional benefit would be to make the money less liquid, thereby cutting down the temptation to spend it frivolously.)
Perhaps, but owning a house would be a terrible time investment for me the way my life is working. I suppose I could hire a property manager and rent it out, though.
How sure are you that you know more than the market on this one? What information do you have that (still) rich property speculators don’t have?
Housing markets aren’t even theoretically efficient. Too big, diffuse, illiquid, etc.
ok, but are you arguing that Matt’s skepticism is unwarranted? are you heavily invested in realestate?
Don’t forget bad zoning regulations: http://www.heartland.org/publications/environment%20climate/article/10635/Landuse_Regulation_Makes_Housing_less_Affordable_Harvard_Study_Finds.html
So they should keep $10,000 if they win. Or just sell a stake in the winnings to a less risk-averse third party for $10,000, risk free.
What $10,000?
EDIT: Never mind; I thought he said “the $10,000”.
You could probably sell somebody a 15% chance to win a million bucks for 10k. It’s worth fifteen times that to a risk-neutral agent.
I’m pretty confident that you could sell a true 15% chance to win a million bucks for a lot more than 10k… after all, investment banks make substantially greater gambles regularly.
I’d probably ask for 100k to start and go from there.
It is my understanding that people are in general way too optimistic about how much winning $1M would increase their overall happiness. (Say if they are asked to imagine themselves winning the lottery.)
Re: Maybe they assume that someone might actually give them $500, but the $1M is a scam.
If you were offering this deal, wouldn’t the $1M be based on a deceptive maniuputation of the stated probabilities? Many participants can probably figure that one out.
kudos for linking to Virginia Postrel.
Think like reality. If it’s hard to imagine how something could happen update your model.
My model says that there is a big difference between formal education and deep understanding that can only be developed by extracurricular appreciation of the subject.
I was briefly tempted to answer 10 cents to the first problem.
I’m just thrilled to think of how dumb our elite (top 10% and top 2% respectively?) are.
Maybe those are just the high scorers who got 3⁄3 (avoiding the tempting surface error) almost by sheer chance.
I got 3⁄3 and I would take the chance. My rational is that $60 is almost nothing. I can make that very quickly, and it won’t buy much. I won’t notice it in my monthly finances. $5000, on the other hand, is actually worth considering. That could change my month significantly (and impact the rest of my year as well). Would you rather have a 100% chance of getting a nickel, or a 0.01% chance of getting a small diamond?
What if we transform the problem, so that you have the opportunity to pay $60 for a 1% chance to gain $5000?
Exactly! This is gambling, isn’t it? A small expected loss, with a tiny chance of some huge gain.
If your utility for money really is so disproportionate to the actual dollar value, then you probably ought to take a trip to Las Vegas and lay down a few long-odds bets. You’ll almost certainly lose your betting money (but you wouldn’t “notice it in [your] monthly finances”), while there’s some (small) chance that you get lucky and “change [your] month considerably”.
It’s not hypothetical! You can do this in the real world! Go to Vegas right now.
(If the plane flight is bothering you, I’m sure we could locate some similar online betting opportunities.)
That sort of attitude (among my opponents) is very helpful to my poker bankroll. You’re giving up $60 for $50 of expected value. Even given your risk-seeking preference, surely you can find a better gamble. Putting it on a single number in roulette would be a better deal.
By the way, welcome to Less Wrong (I notice you had some comments on Overcoming Bias as well); you should check out the welcome thread if you haven’t already.
I’d be almost guaranteed to lose the diamond before I could liquidate it if I won it. Should I factor that in or not? Diamonds are also notoriously difficult to liquidate unless you are in the relevant cartel...
I would guess that most people who got the first problem correct also had “10 cents” as their initial thought, for about a half or second or so before they had finished reading the question and before they had actually started deliberatively thinking about the problem.