This old thread began along the lines I had been thinking about recently:

https://stats.stackexchange.com/questions/95694/two-envelope-problem-revisited/122757#122757

Now, I realise this is old hat to many, but please bear with / be gentle if possible... Essentially my experience of playing with the two envelope puzzle is that it generates sub-puzzles which sometimes (or even often) loop back to the main one. So, with that in mind, here's a sub puzzle or perhaps a related puzzle which I think is of interest in itself and might shed some light on aspects of the two envelope puzzle.

In the standard presentation of the two envelope puzzle, the way in which the amounts are selected and the order in which the envelopes are filled is not specified. In the sub/related puzzle I present below (Scenario 3), the the way in which the amounts are selected, the order in which the envelopes are filled and the fact that one envelope is opened, are specified. What is more, in addition to getting to open an envelope, the player is told how they were filled and which one was filled first.

But consider first Scenarios 1 and 2. In Scenario 1 the player is given 10 dollars. He is then told that he can keep that or he can have 20 or 5 depending on the toss of a coin. We presume he's willing to play the odds and accordingly he takes up the offer as there's an expectation of gain. In Scenario 2 it is as in Scenario 1 except that we tell him that we have already performed the coin flip and put the resulting amount in a sealed envelope. No tricks, so it is 20 or 5 again. The player goes for it again, and rightly so.

Now, Scenario 3, the main event. In this case he is given an envelope and told it contains an unspecified amount of money in the form of an IOU. That amount is in there already. It was chosen by the master of ceremonies who literally just 'thought of a number. The player is then told that another envelope we are prepared to offer him instead contains an IOU for an amount created by tossing a coin to double or halve the value in the first envelope. Which one should he take home with him? Following the logic of Scenario 2, he should swap. There is a definite amount in the first one one, and because of the way the amount in the second one was created it has an expected value of 1.25 times the first one.

The player then opens the second envelope. He sees an amount of money on the IOU slip. It occurs to him that the other one, the one he started with, must contain either half or double the amount he is looking at. It also occurs to him that if the odds of this are 50-50 he seems to have chosen the wrong envelope. But he can't have because the logic of Scenarios 1, 2 and 3 lead him to the firm conclusion that the second envelope was the better one.

So, he reasons, the odds can't be 50-50. In fact, he reasons, the chance that the first envelope contains twice as much as the second envelope must be less than 50%, by quite a way. He does a quick calculation and reckons that for there to be an expectation that the second envelope would contain 1.25 times the amount in the first, the likelihood that the first one would contain twice as much as the second one must be a mere 20%, with an 80% chance it contain half as much (suppose he sees 250 on opening. The logic of the way the envelopes were created seems to mean he should think that the expected value of the other envelope is 200. It can't be 200 of course, it can only be 125 or 500. So he supposes that there is a 4/5 chance that the original sum was 125 and a 1/5 chance that it was 500).

Is the player wrong in thinking that his interpretation of Scenario 3 follows from Scenarios 1 and 2? And / or is he wrong in his calculation of the likelihood of the original values? Or is something else going on?

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Comment

The attraction of this sub problem for me is that the way the envelopes are filled constrains some of the possibilities while at the same time amplifying some of the (apparent?) paradoxes of the two envelope puzzle: if mathematical calculations (distributions) are to be used to resolve the matter they have an even bigger job to do than in the general two envelope puzzle in the sense that in this sub puzzle there is a clear (and valid?) expectation of gain one way (swapping), which only seems to work if larger values for the original amount are a lot less likely than smaller ones.