Discussion

The Two Envelope Paradox
This is a famous
paradox, first framed by the Belgian mathematician, Maurice Kraitchik in 1953
and popularised by Doug Hofstadter in 1982. It has been the
subject of papers in philosophical journals in recent years and its current
status is unresolved, though many claim to have solved it. A Google search nets
12,000 hits. I have looked at the first 40 and found none was satisfactory.
Some people were content with showing that switching doesn’t give any benefit,
others even thought that getting a benefit out of switching is an acceptable
result (at least in the open envelope case). Others thought it enough to show
that the probability of 2x is less than that of x/2. Some answers were so
difficult to follow that I had to give them the benefit of the doubt, though in
these cases I could not see how the solution related to the original problem,
ie what part of the reasoning of Method 2 is in error.
To be honest, my
solution can and should be criticised for being abstruse, given the incredibly
simple nature of the problem. Nor did I reach it unassisted. There are key
ideas I picked up from the Net, without which I probably would never have
solved the paradox. These are: 1) that one must take the two envelopes from a
probability distribution, 2) that any such distribution cannot be uniform, and
3) the example I use in Part 2 (due to Broome), which allows one to see exactly
what happens. The breakthrough was when I realised that the gain for this
distribution is given by an oscillating series that cannot be summed. Is my
solution the first correct one ever? This is a hubristic claim, but it is
possibly the case. Pride comes before the slip, so I’ll probably have to eat
these words. In the meantime I am a legend in my own lunchtime.
Some interesting
treatments of the paradox are:
Two
papers by David Chalmers 2nd paper (2002) and 1st paper
(1994) from the Dept of Philosophy at the University of Arizona
A
paper by Graham Priest and Greg Restall of the Philosophy Dept of
Melbourne University: 
"Envelopes and Indifference" (2003)
The
wikipedia Discussion
of the topic 
A Simple
Explanation by E. Schwitzgebel and J. Dever of the
Universities of California and Texas (2004)
A paper
that nearly gets it right: The Exchange Paradox by John D. Norton of the University of S. California (1998)
A 
paper
in Mind Magazine (Vol 109.435.July 2000) by Michael Clark
(Nottingham University) and Nicholas Shackel (De Montfort University)
Though these are the best solutions I have found, they
all seem to miss the mark. Chalmers states that the gain is positive for each
value of the envelope but that we should not conclude from this that we should
switch. 
Graham Priest and
Greg Restall attack the problem from the point of view of modal logic. Parts of
their answer are excellent, especially the observation that the original
problem is under-determined and their proposal of three mechanisms to generate
the paradox situation. However, I don’t think that they actually dispose of the
kernel of the paradox, but then I am unable to follow their argument (despite
receiving their clarifications). 
The wikipedia
article is essentially correct and shows a distribution similar to mine in Part
2. However, it fails to provide the solution, except for the suggestion that
since the average value in the open envelope and the average gain are both
infinite, we don’t have a paradox. This is wrong (see my Part 2). They frame a
third version of the paradox without using
probability. This is beside the point, as you cannot then talk about expected
gain.
The article by Schwitzgebel
and Dever had me worried for a while. It seemed to give a dead simple solution,
namely that the x in “2x” and the x in “x/2” in the calculation of method 2
have different expectations and hence cannot be used in the same formula.
However, the simple case of (1, 2) or (2, 4) with equal probability of each
invalidates their explanation. There is nothing wrong with applying method 2 to
this case, though the net gain is zero.
Norton’s 25-page paper analyses the problem in essentially
the same way as mine, locating the problem in the indefinite sum of an
oscillating series. Where we differ is in the diagnosis. Norton correctly
concludes that if the amount in the first envelope is unknown then there is
indeed no gain from switching. Where he goes wrong is in his analysis of the
open envelope case. He believes that if we know the contents of the first
envelope then swapping is always advisable. He tries to talk his way out of
this violation of symmetry, but fails to do so. The crucial step he misses is
given in my answer at the end of Part 2:
If we open an envelope and find that it contains 128,
should we switch? Substituting n = 5 in G gives the gain of 27/36
= 128/729. It seems that we can validly extract just one bracketed element from
the oscillating series A. Or can we? Recall that the reason we cannot sum all
of A is that changing the order of the terms changes the sum. By extracting a
single bracketed term out of A we are instantiating the same problem, ie we are
choosing a specific order so as to obtain a specific result. This is arbitrary,
and hence is just as fallacious in the case of a single term as in the infinite
one.
Finally, there is the paper by Clark and Shackel.
They present a case where the gain converges conditionally to 7/12. Essentially
they say that Method 2 is incorrect because it does not respect the symmetry of
the situation, which is true. However, I believe they fail to show at what
point Method 2 goes wrong, which is necessary to resolve the paradox. As for
the open version, they admit that the expected gain is positive for each value,
yet they claim that the overall gain over the whole run will not be. This is
very much like what Chalmers says above, and it makes no sense to me at all.
The problem with discussing this paradox is that
anyone who has spent weeks or months working out their solution is unlikely to
feel motivated to put in the hard work to wade through a densely argued
academic paper of 25 pages in order to rigorously evaluate someone else’s work.
Put crudely, writing about it is more fun than reading another thinker’s solution.
I have made an honest attempt to understand and evaluate the seven papers cited
above. However, my conclusions are circumscribed by my lack of intellectual
prowess. I also, grugingly, admit to a bias towards believing that my solution
is the only correct one.