Let's say you open the envelope. And there's $20. That means the set of two envelopes is either $20 and $10 or $20 and $40. In one case, the amount of money in play is $30, in the other, $60.
Now, you have no idea which it is. And the fact that you're holding a $20 doesn't tell you anything.
The $20 is fake information. It hasn't revealed any real info, as you knew you'd open something. The key is not to combine a world where the total envelope value is $30 and where the total envelope value is $60 which is what happens if you add up probabilities.
Instead, there's a 50% chance you're holding 1/3 of the total value, 50% chance you are holding 2/3. So if you switch, half the time, you add 1/3, half the time, you lose 1/3. net expected gain from switching: 0.
Now, you have no idea which it is. And the fact that you're holding a $20 doesn't tell you anything.
The $20 is fake information. It hasn't revealed any real info, as you knew you'd open something. The key is not to combine a world where the total envelope value is $30 and where the total envelope value is $60 which is what happens if you add up probabilities.
Instead, there's a 50% chance you're holding 1/3 of the total value, 50% chance you are holding 2/3. So if you switch, half the time, you add 1/3, half the time, you lose 1/3. net expected gain from switching: 0.