I posted the following as an answer on thisThis old thread began along the lines I had been thinking about recently:
Now, I realise this is old hat to many, but please bear with / be gentle if possible... Essentially my experience of playing with thisthe 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 whata sub puzzle or perhaps a related puzzle which I asked:think is of interest in itself and might shed some light on aspects of the two envelope puzzle.
Step 1 as stated by the OP inIn the link above. Step 2 as originally stated, but it is made explicit that a coin toss was used to fillstandard presentation of the secondtwo envelope usingpuzzle, the double or half rule. Step 3 both are sealed andway in which the amounts are unknown to the player, but it is also stated that a coin toss was usedselected and the player is toldorder in which envelope contains the original amount, soenvelopes are filled is not specified. In the amount that startedsub/related puzzle I present below (Scenario 3), the process.
It seems clear they should choose the envelope created by tossingway in which the coin. The process of puttingamounts are selected, the moneyorder in which the envelopes in a given wayare filled and the information about whichfact that one envelope contains the starting amount makes it identical to Step 1, in contentis opened, it seemsare specified.
BUT: when the player holds the envelope created by tossing the coin and focuses on the amount it contains What is more, or when theyin addition to getting to open it and see an actual amountenvelope, the other envelope appears 'better' - itplayer is double or half with equal probability (coin toss used, let's not forget). So, in two steps (Steps 2told how they were filled and 3) we have gone from a clear expectation of gain, which I think we all agree is correct, to the two envelope paradoxone was filled first. Where is the false step?
Perhaps, to begin to answer my own question,But consider first Scenarios 1 and 2. In Scenario 1 the false stepplayer is in my claimgiven 10 dollars. He is then told that the playerhe can reasonably worrykeep that the envelopeor he didn't choose is doublecan have 20 or half with equal probability. Sure,5 depending on the toss of a coin toss was used, but perhaps the prior question. We presume he's willing to consider is how wasplay the original amount determinedodds and what doesaccordingly he takes up the valueoffer as there's an expectation of gain. In Scenario 2 it is as in the envelopeScenario 1 except that is openedwe tell us about this? Ihim that we have read overalready performed the mathematical treatments of this. I guess they always seem slightly unconvincingcoin flip and put the resulting amount in a sealed envelope. What if I just boldly say: 'I thought of an original amount'? I can do thisNo tricks, so it is 20 or 5 again. IfThe player goes for it helps, let's say I say the amounts are all whole number powers of two (including negative whole number powers)again, and rightly so envelope pairs differ by one number in.
Now, Scenario 3, the powersmain event. In this case Ihe 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 boldly assert that I thought'thought of a whole number. The player is then flippedtold that another envelope we are prepared to offer him instead contains an IOU for an amount created by tossing a coin to adddouble or subtract one from this.
EDIT (ADD)halve the value in the first envelope. This may help to clarify my point / puzzle: what I called StepWhich one aboveshould he take home with him? Following the logic of Scenario 2, he should swap. There is a definite amount in other words what the OP on the other thread called Problemfirst one one, is just a coin flip to double or halve anand because of the way the amount. The player should take this as there is in the second one was created it has an expected gainvalue of 1.25 times the original amountfirst one. Problem 3 or Step 3, as I formulated it above, is to take
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, put it in an envelopethe one he started with, then flip a coin to doublemust contain either half or halve that amount and putdouble the resulting sum in another envelope. The player is offered both envelopes sealed and told whichamount he is whichlooking at. It seems obviousalso occurs to him that he should takeif the one resulting fromodds of this are 50-50 he seems to have chosen the coin flip as, followingwrong envelope. But he can't have because the logic of the first situation, its expected value isScenarios 1.25 times, 2 and 3 lead him to the amount infirm conclusion that the othersecond envelope was the better one.
Here'sSo, he reasons, the rub / question:odds can't be 50-50. In fact, he reasons, the player openschance 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 seesreckons that for there to be an amountexpectation that the second envelope would contain 1. At this point should he start worrying about his choice25 times the amount in the first, and if he shouldn'tthe likelihood that the first one would contain twice as much as the second one must be a mere 20%, why not? Supposewith 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, is he supposed to thinksupposes that there wasis 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?
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.
