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From: Len on 1 Oct 2009 14:13
Can someone offer a resolution?

The content of each of two identical envelopes is determined by a
“St. Petersburg procedure”. A fair coin is tossed repeatedly until
heads first appears. If heads appears on the first toss, $2 will be
placed in one envelope. If heads does not appear until the second
toss, $4 will be placed in that envelope. If three tosses are
required, $8 goes into the envelope, and so on, doubling the amount
for each additional required toss of the coin. So, in general, if n
tosses are required, $2^n goes into the envelope. The envelope is
sealed and the expected value of the content is infinite. That is,
one should be willing to pay any finite amount of money in exchange
for the unknown content of the envelope. (I wish to ignore utility
considerations. Or, simply assume the contents are utils, as opposed
to dollars.) A similar procedure is independently employed to fill
the second envelope. It too is sealed.

You randomly select an envelope and observe its content. You may keep
the content or exchange it for the other, unopened envelope. Should
you be willing to do so?

One could argue “yes”, by the above discussion. But if this were the
case, one should always be willing to make the exchange. Why open the
selected envelope? Just switch! By symmetry this seems ludicrous.

Resolution?

Thanks,

Len
From: Jim Ferry on 1 Oct 2009 15:33
On Oct 1, 2:13 pm, Len <lwapn...(a)gmail.com> wrote:
> Can someone offer a resolution?
>
> The content of each of two identical envelopes is determined by a
> “St. Petersburg procedure”.  A fair coin is tossed repeatedly until
> heads first appears.  If heads appears on the first toss, $2 will be
> placed in one envelope.  If heads does not appear until the second
> toss, $4 will be placed in that envelope.  If three tosses are
> required, $8 goes into the envelope, and so on, doubling the amount
> for each additional required toss of the coin.  So, in general, if n
> tosses are required, $2^n goes into the envelope.  The envelope is
> sealed and the expected value of the content is infinite.  That is,
> one should be willing to pay any finite amount of money in exchange
> for the unknown content of the envelope.  (I wish to ignore utility
> considerations.  Or, simply assume the contents are utils, as opposed
> to dollars.)   A similar procedure is independently employed to fill
> the second envelope.  It too is sealed.
>
> You randomly select an envelope and observe its content.  You may keep
> the content or exchange it for the other, unopened envelope.  Should
> you be willing to do so?
>
> One could argue “yes”, by the above discussion.  But if this were the
> case, one should always be willing to make the exchange.  Why open the
> selected envelope?  Just switch!  By symmetry this seems ludicrous.
>
> Resolution?
>
> Thanks,
>
> Len

You know that whatever envelope you open you will be disappointed,
in that every possible outcome is worse than the "expected" outcome,
so selecting either envelope is equally ludicrous.

On the other hand, the person running the game is probably letting
you play for free, in which case you can only win, so why get upset?
Presumably this person has sold one-in-2^j chances of winning $2^j at
a penny apiece to an infinite number of people (labeled j=1,2,3,...),
and has agreed to pay you what he pays the winner out of that kind of
generosity found among the infinitely wealthy.
From: jillbones on 1 Oct 2009 17:40
On Oct 1, 11:13 am, Len <lwapn...(a)gmail.com> wrote:
> Can someone offer a resolution?
>
> The content of each of two identical envelopes is determined by a
> “St. Petersburg procedure”.  A fair coin is tossed repeatedly until
> heads first appears.  If heads appears on the first toss, $2 will be
> placed in one envelope.  If heads does not appear until the second
> toss, $4 will be placed in that envelope.  If three tosses are
> required, $8 goes into the envelope, and so on, doubling the amount
> for each additional required toss of the coin.  So, in general, if n
> tosses are required, $2^n goes into the envelope.  The envelope is
> sealed and the expected value of the content is infinite.  That is,
> one should be willing to pay any finite amount of money in exchange
> for the unknown content of the envelope.  (I wish to ignore utility
> considerations.  Or, simply assume the contents are utils, as opposed
> to dollars.)   A similar procedure is independently employed to fill
> the second envelope.  It too is sealed.
>
> You randomly select an envelope and observe its content.  You may keep
> the content or exchange it for the other, unopened envelope.  Should
> you be willing to do so?
>
> One could argue “yes”, by the above discussion.  But if this were the
> case, one should always be willing to make the exchange.  Why open the
> selected envelope?  Just switch!  By symmetry this seems ludicrous.
>
> Resolution?
>
> Thanks,
>
> Len

The probability that an envelop contains 2^n utils is 1/(2^n). If
the opened env contained 4 U, then
the probability that the unopened env contained more
the 4 U is only 25%, but the possible gain is unlimited. What if the
UEnv contains 4 U? Lets consider this a gain. Then the probability of
loss is 50% and the probability of gain is 50%. In this case switching
would be correct. What if the OEnv contained 128 U. Now the
probability of loss is
99.21875%. I am ambivalent here I would probably lose by switching,
but I could gain a lot of U's if I did switch.

This doesn't help us mathematically determine if
and when switching is correct.

I will do a simulation to see how the results
compare to the theoretical model

regards, Bill J
From: Bill Taylor on 2 Oct 2009 00:08
There is an automatic upper bound on the number of possible utils.

(And lower bound on the number of negutils.
(Unless you're a Catholic or fundie.))

Different for different folks, but always bounded. Q.E.D.

-- Brutal Bill
From: Tim Little on 2 Oct 2009 00:41
On 2009-10-01, Len <lwapner2(a)gmail.com> wrote:
> Can someone offer a resolution?

There are many resolutions, with different resolutions more or less
acceptable to different people.


> That is, one should be willing to pay any finite amount of money in
> exchange for the unknown content of the envelope. (I wish to ignore
> utility considerations. Or, simply assume the contents are utils,
> as opposed to dollars.)

You do have to assume that utility is unbounded above and that a
linear weighted average is the only rational model for determining the
utility of a probabilistic mix of utilities. Both appear to be very
poor assumptions.

For example, they imply that there exists a positive outcome such that
it is perfectly rational to enter a game in which you cause horrible
painful deaths for everyone you care about, except for a 10^-100
chance of winning the prize.


- Tim
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