These sorts of brain-teasers are of limited help in developing your critical thinking skills for dealing with real world problems. Here the problem is presented to you and you just have to figure out what went wrong with a train of thought. In the real world, the BIG problem is noticing that there is a difficulty in the first place.
From advertising via a friend: Apparently a specific technique used in advertising a product with a known weakness is to promote it as a strength. Eg when first feedback from consumers shows that the taste of a particular toothpaste is disliked, the response may be to put a prominent "Great New Taste!!!" label on the pack.
This was made famous by Heinz Ketchup in the 1970s. They surveyed consumers and found they were losing market share because their ketchup was so hard to get out of the bottle because it was so thick. So they made a series of "It's Slow Good!" commercials implying that pouring slower was, for some reason, a good thing. And it worked.
While I'd seen the missing dollar problem before, I think I have a new appreciation for it now. I seem to recall puzzle books presenting this problem, but even when they present the solution, they present it in terms of "here's where the missing dollar is". But as you, Wikipedia, and Paulos point out, the whole problem is that the dollar is only "missing" relative to an invalid comparison.
So, to solve the problem by finding a missing dollar is to fail to learn from it.
...This riddle made me remember reading about how beginning magici
One technique is to look carefully for fallacies and/or gaps in the reasoning by summarizing the key theses of the argument and then considering what assumptions (and definitions) have to be made for the theses to be accepted and what has to be true for each to follow deductively from what has already been given. A book on "critical thinking" (e.g., this one) will have lots of exercises to develop this kind of skill. They typically have lots of examples drawn from politics just because political discussion is so frequently chock full of fallacies...
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 rea...
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.
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.
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. -.-
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
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".
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.
I remembered shortly after writing this that there was an example of outright lying in an attempt to get one to conform to a certain pattern of thought in Initiation Ceremony (hooray fictional evidence).
I had heard it before a while back but I decided to think through it myself and see what was going on. The first thought that occurred to me was that the bellhop's two dollars should be subtracted, just as each of the 1 dollar bills given back to the guests was (to get 27 from 30). Then, I imagined that the guests had not paid with $10 dollars bills, but in 30 ones. The hotel has 30, then each of the three guests is given 1and the bellhop takes two. Here is where the money is:
It is basicall...
A while back on here somebody mentioned that they were at a College, and then later on somebody else mentioned MIT, so I drew the conclusion that they were at MIT. This is a power which must be used wisely...
Can anyone think of good ways to notice when outright deception is being used? How could a rationalist practice her skills at a magic show?
Most "rationalists" are quite smart people, so tricks that are designed by a trickster to fool the masses rarely work on us. For example, I doubt that many on this site would invest heavily in a pyramid scheme or get fooled by a used car salesman. This is because these tricks are targeted at the average idiot.
However, I have recently noticed that there is, for each of us, a stalker who stalks us and at eac...
Can anyone think of good ways to notice when outright deception is being used?
I suspect one of the best ways may be to try to re-create their reasoning from the beginning, so you engage the logical inference part of your brain in trying to (re)reason, rather than the 'listening to and believing' part, as we now know that we first believe, and have to consciously reject, new ideas, rather than the other way around.
Where money is concerned, I suppose you could check whether income matches expenditures.
other examples of flagrant misdirection
I can think...
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.
Related: Trust in Math
I was reading John Allen Paulos' A Mathematician Plays the Stock Market, in which Paulos relates a version of the well-known "missing dollar" riddle. I had heard it once before, but only vaguely remembered it. If you don't remember it, here it is:
Three people stay in a hotel overnight. The innkeeper tells them that the price for three rooms is $30, so each pays $10.
After the guests go to their rooms, the innkeeper realizes that there is a special discount for groups, and that the guests' total should have only been $25.
The innkeeper gives a bellhop $5 with the instructions to return it to the guests.
The bellhop, not wanting to get change, gives each guest $1 and keeps $2.
Later, the bellhop thinks "Wait - something isn't right. Each guest paid $10. I gave them each back $1, so they each paid $9. $9 times 3 is $27. I kept $2. $27 + $2 is $29. Where did the missing dollar go?"
I remembered that the solution involves trickery, but it still took me a minute or two to figure out where it is. At first, I started mentally keeping track of the dollars in the riddle, trying to see where one got dropped so their sum would be 30.
Then I figured it out. The story should end:
Later, the bellhop thinks "Wait - something isn't right. Each guest paid $10. I gave them each back $1, so they each paid $9. $9 times 3 is $27. The cost for their rooms was $25. $27 - $25 = $2, so they collectively overpaid by $2, which is the amount I kept. Why am I such a jerk?"
I told my fiance the riddle, and asked her where the missing dollar went. She went through the same process as I did, looking for a place in the story where $1 could go missing.
It's remarkable to me how blatantly deceptive the riddle is. The riddler states or implies at the end of the story that the dollars paid by the guests and the dollars kept by the bellhop should be summed, and that that sum should be $30. In fact, there's no reason to sum the dollars paid by the guests and the dollars kept by the bellhop, and no reason for any accounting we do to end up with $30.
The contrasts somewhat with the various proofs that 1 = 2, in which the misstep is hidden somewhere within a chain of reasoning, not boldly announced at the end of the narrative.
Both Paulos and Wikipedia give examples with different numbers that make the deception in the missing dollar riddle more obvious (and less effective). In the case of the missing dollar riddle, the fact that $25, $27, and $30 are close to each other makes following the incorrect path very seductive.
This riddle made me remember reading about how beginning magicians are very nervous in their first public performances, since some of their tricks involve misdirecting the audience by openly lying (e.g., casually pick up a stack of cards shuffled by a volunteer, say "Hmm, good shuffle" while adding a known card to the top of the stack, hand the deck back to the volunteer, and then boldly announce "notice that I have not touched or manipulated the cards!"1). However, they learn to be more comfortable once they find out how easily the audience will pretty much accept whatever false statements they make.
Thinking about these things makes me wonder about how to think rationally given the tendency for human minds to accept some deceptive statements at face value. Can anyone think of good ways to notice when outright deception is being used? How could a rationalist practice her skills at a magic show?
How about other examples of flagrant misdirection? I suspect that political debates might be able to make use of such techniques (I think that there might be some examples in the recent debates over health care reform accounting and the costs of obesity to the health care system, but I haven't been able to find any yet.)
Footnote 1: I remember reading this example very recently, maybe at this site. Please let me know whom to credit for it.