RE: Observation selection effects

From: Brent Meeker <meekerdb.domain.name.hidden>
Date: Thu, 14 Oct 2004 15:07:12 -0000

>-----Original Message-----
>From: Stathis Papaioannou [mailto:stathispapaioannou.domain.name.hidden]
>Sent: Thursday, October 14, 2004 7:36 AM
>To: meekerdb.domain.name.hidden; everything-list.domain.name.hidden.com;
>lasermazer.domain.name.hidden
>Subject: RE: Observation selection effects
>
>
>
>Brent Meeker and Jesse Mazer and others wrote:
>
>Well, lots and lots of complex mathematical argument on
>the two envelope
>problem...
>
>But no-one has yet pointed out a flaw in my rather
>simplistic analysis:
>
>(1) One envelope contains x currency units, so the other
>contains 2x
>currency units;
>
>(2) If you stop at the first envelope you choose,
>expected gain is: 0.5*x +
>0.5*2x = 1.5x;
>
>(3) If you open the first envelope then switch to the
>second, your expected
>gain is: 0.5*2x + 0.5*x = 1.5x - as above, just in a
>different order,
>obviously;
>
>(4) If, in a variation, the millionaire flips a coin to
>give you double or
>half the amount in the first envelope if you switch
>envelopes, expected gain
>is: 0.25*2x + 0.25*0.5x + 0.25*x + 0.25*4x = 1.875x.
>
>In the latter situation you are obviously better off
>switching, but it is a
>mistake to assume that (4) applies in the original
>problem, (3) - hence, no
>paradox.
>
>Is the above wrong, or is it just so obvious that it
>isn't worth discussing?
>(I'm willing to accept either answer).
>
>Stathis Papaioannou

It's not wrong - I just don't think it addresses the paradox. To
resolve the paradox you must explain why it is wrong to reason:

I've opened one envelope and I see amount m. If I keep it my gain
is m. If I switch my expected gain is 0.5*m/2 + 0.5*2m = 1.25m,
therefore I should switch.

To say that in another, similiar game (4) this reasoning is
correct, doesn't explain why it is wrong in the given case.

Your (2) and (3) aren't to the point because they don't recognize
that after opening one envelope you have some information that
seems to change the expected value.

In my analysis, it is apparent that the trick of showing the
expected value doesn't change depends on the feature of the
problem statement that the distribution of the amount of money is
scale free - i.e. all amounts are equally likely. If you accept
this, then a Bayesian analysis of your rational belief shows that
the expected value doesn't change when you open the envelope and
see amount m. Intuitively, observing a value from a distribution
that is flat from zero to infinity *doesn't* give you any
information. Solving the paradox is to show explicitly why this
is so.

As Jesse and others have pointed out this scale-free (all amounts
are equally likely) aspect of the problem as stated is unrealistic
and in any real situation your prior estimate of the scale of the
amounts will cause you to modify your expected value after you see
the amount in first envelope. This modification may prompt you to
switch or not - but it's a different problem.

Brent Meeker
Received on Thu Oct 14 2004 - 18:58:32 PDT