I am having a hard time understanding the explanation for the Two Envelopes Problem, in that some point out that the "wrong proof" falsely uses the variable $A$ and should have used $2x$ and $x$ instead, and others point out the "wrong proof" is false in introduce the variable in the first place because the variable itself is an expected value. Both arguments seems to indicate the misused variable $A$ is the problem.
I can understand what the reasoning is, but consider the following variant scenario:
You are given two indistinguishable envelopes, each containing money. One contains twice as much as the other. You picked and opened one of the envelopes, which contains $10$ dollars. Now you have a chance of switching. Should you switch?
In this variant scenario, no variable is introduced and we have a concrete $10$ dollars to start with. Is the "wrong proof" that states the expected value of switching is now ${1\over2}\cdot5+{1\over2}\cdot20=12.5$ still wrong?
I would think it is still wrong but I am struggling to understand in this no-variable case, where exactly is wrong with the reasoning. Before opening the envelope, I can see how the total amount should be fixed and the envelopes should be assigned $x$ and $2x$. However once I have opened the envelope, the value of the opened one is now fixed and is known information, i.e. $10$ dollars, and the unknown one is either $5$ or $20$ dollars with equal chance.
If the "wrong proof" actually turns correct in this situation, then the two players game version, where
Both players opened their envelopes respectively and found out their own envelopes containing $10$ dollars and $20$ dollars but don't know anything about the other player's envelope. Then both players are presented the chance of switching
The best strategy would then be to switch for both players which sounds odd to me. Anyone can explain what happens in this variant case for me?
EDIT: For anyone still looking at this question, the answer to this question is also given here: Why the equation $E(X \mid X=1000) = 1000$ is false, intuitively?
Quick answer
(1) In the single-player variant version, we don't have enough information to determine whether to switch but it is most likely not $12.5$ because there is no uniform distribution over all numbers and we don't know what the distribution is.
(2) In the 2-player version, both players don't have enough information to determine whether to switch.
The whole envelops "paradox" basically boils down to the fact that humans like to assume uniform distribution when information is not enough. But in the case a uniform distribution is impossible, this natural assumption generates the paradox.