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Re: Two-envelope problem
Xin lo^~i ca? la`ng vi` ca'i message to^i gu+?i lu'c tru+o+'c la`
HTML format ne^n bi. the server o+? Milwaukee screwed up. Here it
is once more. Vu~
------------------------------------------------------
Aha, ba'c Ha?i dda~ ra tay nha^?y va`o. To^i ddang ba^.n bi.u
go'i ge'm ddo^` dda.c ve^` nu+o+'c, nhu+ng cu~ng muo^'n cho~
mo^`m va`o no'i ba^.y ma^'y ca^u.
>AnHai Doan wrote:
>
>Woa folks, I just took a look at this
>thread. Envelopes! I love envelopes!
>Pls allow me to add to the "mess" :-).
>
>>On Mon, 21 Apr 1997, Tuan V Nguyen wrote:
>>
>> The original problem is:
>>
>> There are two envelopes containing money. One
>>envelope contains twice as much as the other, however since the
>>envelopes look identical you cannot tell one apart from the
>>other. You choose one envelope and you see that it contains
>>$100. Is it better to keep the $100 or choose the other
>>envelope?
>>
>> Firstly, Dung used the expected value concept to work
>> out that the expected value of the 2nd envelope is higher
>> than the first (selected) one, hence it would be wise to
>> take the risk.
>I think his solution is correct.
Meaning that both Dung and Hai are EMV-ers (pronounced i - em - vi o+`.
EMV: expected monetary value).
In general, most peope are not EMV-ers.
In my case, as a matter of fact, i don't know who i am. If somebody
gave me $500,000 and suggested that either i could keep it or put it
on the red/black hole in Casino Continental for double or nothing,
I would fall on my knees thanking her generousity and run away from
Las Vegas as fast as i can before she might change her mind. So i can
be labeled a coward (or more diplomatically - risk averse) can't i.
But once in a while i buy a lottery with the silly hope of getting rich
quickly while avoiding having to marry old, lonely but wealthy women,
paying little attention to the ridicule of my apparently much wiser
fellows who keep calculating to me the expected loss i am bound to face.
I just tell them plainly: "if you don't risk, you can't get rich (or money
doesn't come quick)".
>>Then, Ha Anh Vu disagreed, saying that the
>> idea of expectation can not be applied here since it
>> assumes infinite repeated selection, which is not possible
>> in this case.
>
>I don't think so. Suppose you have exchanged the first envelope for
>the second one. Then if you are to choose between the two actions "keep
>the second one" and "exchange the second for the first one", you still
>have the same expected monetary value of $125 for the first action, and
>$100 for the second one. So you will choose the first action, namely, to
>keep the second envelope.
>
> As long as you HAVE OPENED at least one envelope, you will never face
>the possibility of "infinite switching". This may seems to arise only when
>you don't open any envelope, but even in this case, we have no paradox, and
>the solution of keeping the picked envelope is the correct and intuitive one.
My views on this problem are strongly influenced by a book by R. Neapolitan,
who calls hilmself a non-extreme frequentist. So given the little time i have,
instead of dwelling into further discussions, I quit here but whole-heartedly
recommend this book to anybody who wants to know more about this and several
other famous paradoxes.
>> I have difficulty with the application of expected
>>value concept in this problem as well. In fact, if you
>>apply this concept, then regardless of what the value in
>> the first envelope (say X1), then the second one would be
>> 1.25*X1. Although the mathematics is correct, it does not
^^^^^^^^^^^^^^^^^^^^^^^^^^
>> make sense to me.
Dear anh Tua^'n, just want to note here that subjective expected utility (SEU)
is still in debate. People tend to agree that SEU is an excellent candidate for
a normative decision theory - we should use it, of course with great care -
but seems hopeless as a descriptive one: it is unsuitable to describe human's
decision making behavior.
> In summary:
>
> 1) Some people would say that there is not enough information provided
> in this problem to have a solution. Namely, they object to the fact that
> the probabilities in the problem can't be formulated and calculated in the
> frequentist interpretation (with repeatable trials and so on). I would say
^^^^^^^^^^^
> that this problem is perfectly solvable under the subjective
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> interpretation of probability, and provides a very reasonable solution.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Hey ba'c Ha?i, i think that the problem is still not well-formulated: there
is no obvious way to interpret the premise about the probability of getting
either one of the 2 envelopes. We can interpret it using the principle of
indifference, but again, there are at least 2 ways to interpret this principle.
Once the problem is not well-formulated, the application of SEU is questionable.
> Subjective (also called Bayesian) probability and Bayesian decision
> making have well-grounded (if controversial) theoretical and practical
> foundations, and are rapidly gaining acceptance and use in many
> sciences, inclusing statistics, sociology, artificial intelligence,
> business decision making, just to name a few.
Agree! But SEU (or Bayesian DT) should be used with great care. Failure
by some scientists to pay enough attention to sensitive issues such as
assigning priors has precipated some very sharp criticisms, mostly from
philosophers such as Henry Kyburg, who recently wrote: " Subjective
probability is undogmatic and antiauthoritarian: one man's opinion is as
good as another's .... I conclude that the theory of subjective probability
is psychologically false, decision-theoretically vacuous, and philosophically
bankrupt: its account is overdrawn."
Xin lo^~i ca'c ba'c la` pha?i ta.m hoa~n tham gia tranh lua^.n tre^n vnsa
khoa?ng 2 tha'ng.
Cheers, Vu~