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:
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.