Kkwan, remember we are dealing with a case similar to the TEP. What I said is we need to move to this case and once we have the answers to apply it to the TEP.
I think I can show you the answer that way but you do need to play along to do it. It shouldn’t take too long.
So back to my example, what should you do and why?
Your example is a particular instance of the TEP and should be treated as such.
So, without knowing what is in the selected envelope, we can denote the amount in the selected envelope as A and as A is either the smaller or the larger amount, either (A, 2A) or (A, 1/2A) are in the two envelopes.
We then proceed to consider whether to switch or not to switch as follows:
1. If we do not switch, we get A.
2. If we do switch, gain/loss is A/1/2A.
So, we should switch because the potential gain is twice the potential loss with the probability of gain/loss as 1/2. Also, with switching, the long run average is 5/4A which is more than A.
The rationale was stated in my post 1745 which I hope you have carefully read and digested.
Furthermore, with total ignorance, we cannot assume we can select either $10 or $40 because that implies we know what amounts are actually in the two envelopes. That is begging the question.
However, selecting $20 is not begging the question as the implication is either ($20, $40) or ($20, $10) are the two envelopes and we do not assume anything because do not know which event is actual.
This is the rationale of considering both the two possibilities together and not separately as by doing that, we are begging the question which is philosophically unsound and/or unjustified.
For the sake of argument, if you are omniscient and/or the above philosophical consequences are ignored (which is unwise and which contravenes the fundamental nature of the TEP), then:
In your example, if we can select $10, then $20 must be in the other envelope and if we can select $40, $20 must be in the other envelope.
So, if either $20, $10 or $40 can be selected, there will be 3 possible situations:
1. $20 is selected, gain/loss on switching is $20/$10.
2. $10 is selected, gain on switching is $10.
3. $40 is selected, loss on switching is $20.
In 1 and 2, one should switch. In 3, one should not switch.
The problem is, we don’t know which situation we are in.
But, as the probability of 1 is 1/2 and the probability of either 2 or 3 is 1/4, it is still beneficial to switch because the total probability of 1 and 2 is 3/4.
So, is that your “solution” of the TEP?