Therefore I derived that under both descriptions switching does not help.
By doing that you have assumed that you know one of the situations is actual which is not justified as you don’t know which situation is actual. Thus, the two situations must be considered together and not separately as you don’t know which situation is actual.
I agree that I don’t know. But it does not matter, as for any two amounts, my possible gain and possible loss are the same when I switch. And you agreed with that: (X - Y) = -1 * (Y - X), for every X and Y.
That is true if and only if you have definite information that one of the situations is actual which you don’t have, in the context of the TEP.
In the context of the TEP, there are two and only two distinct possible situations (X, 2X) or (X, 1/2X) for any finite value of X but you don’t know which situation is actual. Thus, you must consider ALL the possible situations together and not separately.
With the two possible situations considered together, the potential gain/loss is not equal (with A as the denoted amount in the selected envelope):
1. If A is the smaller amount, then 2A is in the other envelope. On switching, gain is A
2. If A is the larger amount, then 1/2A is in the other envelope. On switching, loss is 1/2A.
For any finite value of A, the potential gain, A is twice the potential loss, 1/2A.
Another approach to the TEP
In the TEP, that one envelope contains twice the amount in the other (which incidentally, is actually equivalent to double or half), can be interpreted as:
1. Self-referential wrt in any one specific distinct situation. So, with (A, 2A), 2A is double of A and A is half of 2A whereas with (A, 1/2A), A is double of 1/2A and 1/2A is half of A. This is useful in deducing that either (A, 2A) or (A, 1/2A) for any finite value of A, satisfy the TEP, but it has it’s limitations.
However, with two envelopes in play, double or half on switching should now be interpreted as:
2. 2A or 1/2A with A as the common starting reference amount.
2 is not self-referential as A is only the common starting reference amount for comparing gain/loss between two and only two distinct possible situations (A, 2A) and (A, 1/2A) for any finite value of A.
In comparing gain/loss wrt the above two distinct possible situations it is necessary to have a common starting reference amount as in 2 to be coherent whereas 1, being self-referential, is inapplicable because it has no conceptual room for comparing two distinct situations.