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Envelope paradox?

  1. Jul 15, 2008 #1
    envelope paradox??

    Maths can sometimes be nearly as interesting as physics...
    There are two envelopes on a table. Both have cash in, you dont know how much but one has twice as much as the other
    You can pick just one envelope, and you are allowed at most one swap.
    You choose envelope A say, and see that has £100 in it. Are you better off on average to swap and choose envelope B???

    Envelope B could have £50, or £200 in it - 50% chance of each
    So expected value if you swap is 0.5*50 + 0.5*200 = £125
    So on average you are better off swapping - how can that be????
     
  2. jcsd
  3. Jul 15, 2008 #2
    Re: envelope paradox??

    Exactly how you calculated it.

    Look at it this way: 200 is 100 more than what you have, where as 50 is 50 less than what you have.

    So basically, if you lose, you lose 50. If you win, you win 100. So on average you will win more money than you lose, even if you lose and win the same amount of times.

    Say 4 wins and 4 losses, 4 wins = 800, 4 losses = 200, that's 1000, divided by 8 = 125 per win.

    The only reason this works is because you already have a benchmark by looking at the first envelope.
     
  4. Jul 15, 2008 #3

    matt grime

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    Re: envelope paradox??

    No, that is not the explanation. The analysis - that you're best to switch - in the paradox is independent of looking in the envelope. The real problem is that the paradox assumes that there is a uniform distribution on the infinitely many values that may be in the envelopes. This is not possible.
     
  5. Jul 15, 2008 #4
    Re: envelope paradox??

    Right, a different version of that paradox which might clarify the issue is to allow unlimited swaps, but without looking inside the envelopes. If swapping envelopes really yields a 25% expected increase, then swapping again should yield even more increase, and so on without bound.

    I wonder if this paradox has any implications for improper priors that are sometimes used in Bayesian statistics? The same thing is often done there, trying to assign a uniform distribution on infinite support.
     
  6. Jul 15, 2008 #5

    DaveC426913

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    Re: envelope paradox??

    I don't follow the logic here. Are you saying it is better to switch, or it is not?
     
  7. Jul 15, 2008 #6
    Re: envelope paradox??

    The whole point of a paradox is that there is a valid line of reasoning supporting either conclusion.
     
  8. Jul 15, 2008 #7

    matt grime

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    Re: envelope paradox??

    I don't believe that is the case, quadraphonic. The only way that the reasoning is valid to swap is if there is such a thing as the uniform distribution on the natural numbers. There isn't.

    Let's put it this way. Suppose that this is a game show, then the monies in the envelope must be less than the budget on the show. So if you open the envelope and see that the amount is more than half the show's budget, then you stick, otherwise you change. But here we have to observe the money in the envelope (and make a reasonable assumption about the budget of the game show) in order know what to do.


    So, to answer Dave's question: since you know neither the priors nor the posteriors, in the original case switching doesn't do you any good (or harm).
     
  9. Jul 15, 2008 #8

    DaveC426913

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    Re: envelope paradox??

    Ok, we know this intuitively. It seems that the key is to find the flaw in the logic of the OP's analysis. Why does his math seem to show that switching will on average yield a better result?
     
  10. Jul 15, 2008 #9
    Re: envelope paradox??

    Are you assuming you have an infinite amount of envelopes or what? Your rules aren't making any sense to me.

    If you have $100 in your envelope and switch to one and see that it has $200, switching again you would switch back to the first one which had $100, right? You only have 2 envelopes to pick from.
     
  11. Jul 15, 2008 #10

    matt grime

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    Re: envelope paradox??

    For the 4th time: because you have assumed a prior distribution that is uniform on the natural numbers. You have no idea what the correct distributions are, so you can't do any reasoning. If you had some idea you could decide what to do, as I explained above.

    E.g. if you know there is at most 1000 dollars in there, and you open to find 800, don't swap! Dually, if you open to find 1 dollar, and know that the amount is an integer number of dollars, then do swap.
     
  12. Jul 15, 2008 #11

    matt grime

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    Re: envelope paradox??

    The point of the paradox is that the reasoning is independent of the envelope's contents, so you can assume that you didn't open the first envelope. So you swap, but now you're in the same situation, so swap back, do this arbitrarily many times and you have an infinite expected gain.

    This is an utterly bog standard paradox explained in hundreds of places all over the place. Google for an explanation for Devlin, say from the AMA.
     
  13. Jul 15, 2008 #12

    DaveC426913

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    Re: envelope paradox??

    This is too abstract an answer to be meaningful (to me at least). You might as well simply say "The OP's argument is flawed because it contains an error."

    Again, I know it's wrong but I can't say where the OP's argument contains a flaw.

    I suspect this is an incorrect conclusion: "50% chance of each". Is it because the doubling/halving rule means that the distribution is geometric rather than linear? i.e a 50% chance of each would actually only be the case if the other envelope had L50 or L150 (evenly spaced, not multiplied)?

    The more I ponder the more I think that's what you're saying, but my brain is hurting.
     
    Last edited: Jul 15, 2008
  14. Jul 15, 2008 #13

    matt grime

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    Re: envelope paradox??

    I have explained it to you by example twice, though you may not have seen the second time as it was an edit. Sorry, that you don't get the explanation, but that isn't my fault.

    Let me repeat the example. Suppose you know that there is *at most* 1000 dollars in the envelopes ('cos that is all that the offerer can afford, say). Then clearly opening and discovering 800 Dollars means you don't swap. The posterior (loosely the guess at what the monies might be) has changed 'cos of this information.

    The only way you can say, in the original version, that both possibilities of there being more or less in the other envelope are the same is if you assume that *all* possible dollar amounts might be in the envelopes. Since we can assume that there really are only a finite number of dollars in the world, this is clearly nonsense (and mathematically impossible anyway since there is no such thing as a probability measure on the natural numbers that assigns equal probability to all of them).
     
  15. Jul 15, 2008 #14
  16. Jul 15, 2008 #15
    Re: envelope paradox??

    Right, of course my statement was in the context of the assumptions, which are, as you say, problematic. But I don't think it provides a very satisfying resolution to say that "since you don't know the priors, switching doesn't help you." Because, indeed, its our very ignorance of the priors that leads us to want a uniform prior on the amounts in the first place. On top of that, using an improper prior still leaves us with perfectly sensible posteriors, so we lack the usual easy way of removing them.

    My understanding is that this paradox is still an open question. Perhaps that's not the case...
     
  17. Jul 15, 2008 #16

    Hurkyl

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    Re: envelope paradox??

    No, it doesn't! Lack of evidence for an alternate prior distribution does not constitute evidence for a uniform prior distribution! And even worse, what would ostensibly be "uniform" here isn't even well-defined! What would be uniform in one parametrization of the sample space is non-uniform in other parametrizations.
     
    Last edited: Jul 15, 2008
  18. Jul 15, 2008 #17

    matt grime

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    Re: envelope paradox??

    Perhaps I could have chosen my words better, but under no prior distribution is it always preferable to switch irrespective of the amount in the envelope, and that is provable. With some priors switching is preferable if you know the amount in the first envelope, under others it is not. Under many the expected gain is zero.

    Such a thing can't exist, so this doesn't help.



    Open in what sense? There is a perfectly good explanation for it.
     
  19. Jul 15, 2008 #18
    Re: envelope paradox??

    I find it the most intriguing paradox- many sites give poor and differing explanations.
    For the record, the point about the upper budget is not relevant - I could have said a genie with infinite resources filled the envelope.

    (i) So either its better to swap ....

    or it makes no difference and therefore
    (ii) the probability that envelope B only has £50 in is 2/3 (and not 1/2 as we assume)
    ...
    well (ii) is unlikely to be correct so
    IT IS better to swap if you find ur envelope has £100 in it!!
     
  20. Jul 15, 2008 #19

    matt grime

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    Re: envelope paradox??

    But could not have put a uniform distribution on the possible values which is what the paradox requires. If you believe the 'budget' is the important point then you've not grasped the theory of probability.

    The main point of using a budget is to demonstrate that a non-uniform (and mathematically possible) prior shows that in some cases switching is bad, and in others is good.
     
  21. Jul 15, 2008 #20
    Re: envelope paradox??

    Sure, that's no problem. If I'm given a prior on the amounts in the envelope, the rest is easy to see. But without being given any further information (which is the way that the paradox is invariably stated), we have no basis for assuming any particular prior, and so no way of deciding whether to switch or not.

    And yet, people use improper priors all the time, and get perfectly sensible results. And, anyway, it's easy enough to rephrase the paradox so as to avoid referring to probability at all. The explanation about there being no uniform probability distribution on the natural numbers only tells us that probability theory, as it is formalized, doesn't allow us to know nothing about a distribution on the natural numbers. And yet, it is manifestly the case that we know nothing about the distribution in question. Rather than provide an explanation, this ends up begging the question of whether probability (or, really, Bayesian decision theory) is up to the task in the first place.

    In the sense that, to my knowledge, there is no consensus on a satisfactory explanation. The explanation about uniform distributions on the natural numbers may satisfy you, but it leaves many others wanting.
     
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