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  1. #1
    Happy Dancer uumlau's Avatar
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    Default The Exchange Paradox and its interpretations by Ni, Ti, etc.

    Reference "The Exchange Paradox" on wiki. Or just google it.

    In a nutshell, the paradox is as follows:

    There are two envelopes. You know that each contains an amount of money, and that one contains twice the amount of the other.

    • You pick an envelope at random.
    • You open the envelope and see an amount of money, A.
    • If you got the envelope with more money, then the other envelope has A/2 in it.
    • If you got the envelope with less money, then the other envelop has 2A in it.
    • The expectation value of the amount of money in the other envelope must be
      because choosing the other envelope will either halve or double your money.
    • 5/4A is > A, therefore you should choose the other envelope.
    • Except the exact same argument could be made if you chose the other envelope first and there are only two choices, that neither envelope is to be preferred over the other.

    We know that the choice of envelope is random, and "choosing the other envelope" should make no difference at all. So what is the problem with the reasoning above?

    Here's a quick and easy answer


    Here's a more complicated answer


    Here's an even more complicated answer


    One can google for even more complicated answers/explanations. Or read the talk/argument sections of the wiki.

    My question to the forum is this:
    Which explanation (of any that you find from any source) satisfies you the best?

    My purpose in asking is that I look at this problem, and my instant reaction is the first, simple explanation. As far as I'm concerned, I've answered the question fully and succinctly, and any explanation beyond that is either misunderstanding that simple explanation or expressing the same point with much more detail. I believe that is due to my instinctive Ni approach, and that Ti types would want a more methodical approach: Ni cares about "meaning", while Ti cares about "proving". That's my hypothesis at any rate.

    I am interested in anyone's replies, no matter type, though I am especially interested in replies from those who theoretically use both Ni and Ti (INFJ, ISTP, etc.).

    The issue isn't about any sort of "right answer". It's obvious to all parties that something is askew. The question is what degree of explanation of the paradox satisfies oneself.
    An argument is two people sharing their ignorance.

    A discussion is two people sharing their understanding, even when they disagree.

  2. #2
    4x9 cascadeco's Avatar
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    Default

    I liked the first answer too. The other responses seemed in the end kind of irrelevant (in the sense that while I could see response 2 as valid, I found it kind of 'annoying' at the same time. Was drawn to what I saw as 'getting to the point', which was the first. Of course I didn't read the paper! Knew that would go far beyond response 2! hahah)
    "...On and on and on and on he strode, far out over the sands, singing wildly to the sea, crying to greet the advent of the life that had cried to him." - James Joyce

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  3. #3
    Habitual Fi LineStepper JocktheMotie's Avatar
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    I arrived at the first one, first.

    Although the second basically says the same thing and also makes sense. So.



  4. #4
    @.~*virinaĉo*~.@ Totenkindly's Avatar
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    Didn't read the third one yet, but my thought before I read the answers is that the value of the two bags is already set before i know what they are, and the problem proof above tries to define A differently for each scenario. while A already is a fixed amount. I don't really care about what the 'expectation value' is.
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  5. #5
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    My thought was that the answer doesn't match up with intuition and that I ought to approach it in another way. So, I basically 're-derived' that second answer there. I would have rigorously re-checked the logic to that first approach in order to reconcile the approaches, but I'm lazy. (On second glance, it looks like A means different things at different times; it suggests a failure in logic.)
    I didn't really think of that first answer myself; otherwise, I would have gone with it.

    --

    As far as the utility of the explanations? I tend to use the following rule of thumb--the simpler for the application, the better. The first is the 'best' unless we need to dig deeper; we don't need more rigor and detail until we come to situations where the nuances need to be explored. As different perspectives, they're all perfectly sound ways of viewing the situation, and they're all 'right'--but not all are useful for all applications.

    I would not give the Norton paper to my eleven-year-old niece to explain this phenomenon, for instance.

    --

    Nor would I introduce someone to typology by throwing Jung's Psychological Types at them. Nor would I explain my thoughts on 'the spirit of the Christmas season' to a two-year-old; he'll probably be content to think of that 'spirit' as a physical being, Santa. Nor would I want to give a detailed excuse beyond 'because I said so!' to that same two-year-old who questioned why I do not want him to open his presents early.

    The point is that there are more complicated ways of explaining all of these things; things can be explained from many different reference points; different reference points are useful for different reasons and in different situations; let's not delve into details until we need to, but let's also recognize when we do need to do so.

  6. #6
    Lay the coin on my tongue SilkRoad's Avatar
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    You lost me the second any kind of math entered into the...er...equation. Sad, I know.
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  7. #7
    Senior Member INTP's Avatar
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    The sort of math that says that the chances of getting more money if you switch is flawed(atleast for this problem)
    "Where wisdom reigns, there is no conflict between thinking and feeling."
    — C.G. Jung

    Read

  8. #8
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    I simply arrived at the fact that its not possible to know which has more. It doesnt take math or statistics. Simply...the other envelope could have 1/2x or 2x...to me the question is "do you feel lucky?". What is your probability of being lucky...how lucky have you been in the past. How lucky have you been recently "your lucky streak may end...but you may have one or 2 more in your streak". Honestly the analysis you posted was more like "WTF" to me. It takes to much thought...

    Here is my analysis process...
    1. Open one envelope
    2. See how much is possible to loose gain. If there is $100 then the other is either $50 or $200.
    3. Do I even care about losing half of whats in my hand
    3. Am I happy with the amount I have in my hand
    4. If its very little I take the chance
    5. If its alot I walk away happy
    6. Do not even process "what ifs"...make a choice and be happy with it

    edit: oh forgot the math... 0<.5x<2x taken that x > 0. The second I open the first envelope I just guaranteed whether or not X is greater then 0 simply by the fact that the envelope actually contains money. Either way I come out ahead of not opening either one.

    Solve for x: 0<.5x
    answer: x>0

    This is logic(apparently Ti)

    Other people are funny with all there fancy complicated calculations/analysis.
    Im out, its been fun

  9. #9
    can't handcuff the wind Z Buck McFate's Avatar
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    I find the first answer most satisfying, probably because I really don’t see any practical purpose in considering it further. I could easily see myself getting distracted and pulled toward considering the others, though, if I were feeling bored. Or if some possible purpose came to mind (wouldn't have to be practical).

    (edit: I might get distracted by 'is there anything that could possibly be close to this, in which knowing this could change outcome?'- i.e. similar problems, but as long as the outcome couldn't possibly change as a result then it would feel like beating a dead horse.)
    Reality is a collective hunch. -Lily Tomlin

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  10. #10
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    Quote Originally Posted by Z Buck McFate View Post
    I find the first answer most satisfying, probably because I really don’t see any practical purpose in considering it further. I could easily see myself getting distracted and pulled toward considering the others, though, if I were feeling bored. Or if some possible purpose came to mind (wouldn't have to be practical).

    (edit: I might get distracted by 'is there anything that could possibly be close to this, in which knowing this could change outcome?'- i.e. similar problems, but as long as the outcome couldn't possibly change as a result then it would feel like beating a dead horse.)
    beat it...beat it again...again *evil beavis laugh*
    Im out, its been fun

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