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Re: Two-envelope problem
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Date: Tue, 22 Apr 1997 14:47:58 +0700
From: Ha Anh Vu <vu@as.nida.ac.th>
Organization: NIDA
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Subject: Re: Two-envelope problem
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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.
<BR>
<BR>
<BR>AnHai Doan wrote:
<BLOCKQUOTE TYPE=CITE>Woa folks, I just took a look at this thread. Envelopes!
I love
<BR>envelopes! Pls allow me to add to the "mess" :-).
<BR>
<BR>On Mon, 21 Apr 1997, Tuan V Nguyen wrote:
<BR>
<BR><I>> The original problem is:</I>
<BR><I>></I>
<BR><I>> > There are two envelopes
containing money. One</I>
<BR><I>> >envelope contains twice as much as the other, however since
the</I>
<BR><I>> >envelopes look identical you cannot tell one apart from the</I>
<BR><I>> >other. You choose one envelope and you see that it contains</I>
<BR><I>> >$100. Is it better to keep the $100 or choose the other</I>
<BR><I>> >envelope?</I>
<BR><I>></I>
<BR><I>> Firstly, Dung used the expected
value concept to work</I>
<BR><I>> out that the expected value of the 2nd envelope is higher</I>
<BR><I>> than the first (selected) one, hence it would be wise to</I>
<BR><I>> take the risk.</I>
<BR>
<BR>I think his solution is correct.
<BR>
</BLOCKQUOTE>
<BR>Meaning that both Dung and Hai are EMV-ers (pronounced i - em - vi o+`.
EMV: expected monetary value).
<BR>
<BR>In general, most peope are not EMV-ers.
<BR>
<BR>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 maney doesn't come quick)".
<BR>
<BLOCKQUOTE TYPE=CITE>
<BR><I>>Then, Ha Anh Vu disagreed, saying that the</I>
<BR><I>> idea of expectation can not be applied here since it</I>
<BR><I>> assumes infinite repeated selection, which is not possible</I>
<BR><I>> in this case.</I>
<BR>
<BR>I don't think so. Suppose you have exchanged the first envelope for the
<BR>second one. Then if you are to choose between the two actions "keep
the
<BR>second one" and "exchange the second for the first one",
you still have
<BR>the same expected monetary value of $125 for the first action, and $100
<BR>for the second one. So you will choose the first action, namely, to keep
<BR>the second envelope.
<BR>
<BR>As long as you HAVE OPENED at least one envelope, you will never face the
<BR>possibility of "infinite switching". This may seems to arise
only when you
<BR>don't open any envelope, but even in this case, we have no paradox, and
<BR>the solution of keeping the picked envelope is the correct and intuitive
<BR>one.
<BR>
</BLOCKQUOTE>
<BR>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.
<BR>
<BLOCKQUOTE TYPE=CITE><I>> I have
difficulty with the application of expected</I>
<BR><I>> value concept in this problem as well. In fact, if you</I>
<BR><I>> apply this concept, then regardless of what the value in</I>
<BR><I>> the first envelope (say X1), then the second one would be</I>
<BR><I>> 1.25*X1. Although the mathematics is correct, it does not</I>
</BLOCKQUOTE>
<I>
^^^^^^^^^^^^^^^^^^^^^</I>
<BLOCKQUOTE TYPE=CITE><I></I>
<BR><I>> make sense to me.</I>
</BLOCKQUOTE>
<BR>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.
<BLOCKQUOTE TYPE=CITE>In summary:
<BR>
<BR>1) Some people would say that there is not enough information provided
<BR>in this problem to have a solution. Namely, they object to the fact that
<BR>the probabilities in the problem can't be formulated and calculated in
the
<BR>frequentist interpretation (with repeatable trials and so on). I would
say
</BLOCKQUOTE>
^^^^^^^^
<BLOCKQUOTE TYPE=CITE>
<BR>that this problem is perfectly solvable under the subjective
</BLOCKQUOTE>
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
<BLOCKQUOTE TYPE=CITE>
<BR>interpretation of probability, and provides a very reasonable solution.
</BLOCKQUOTE>
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
<BR>
<BR>
<BR>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.
<BR>
<BLOCKQUOTE TYPE=CITE>
<BR>Subjective (also called Bayesian) probability and Bayesian decision
<BR>making have well-grounded (if controversial) theoretical and practical
<BR>foundations, and are rapidly gaining acceptance and use in many
<BR>sciences, inclusing statistics, sociology, artificial intelligence,
<BR>business decision making, just to name a few.
<BR>
</BLOCKQUOTE>
<BR>Agre! 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."
<BR>
<BR>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.
<BR>
<BR>Cheers, Vu~
<BR>
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