[pontus]

It depends on how the envelopes are prepared. If the two envelopes have x and 2x in them and one is randomly handed to you, there's a symmetry between them that tells you there is no benefit in switching.

However, suppose x is placed in an envelope and handed to you. Then a fair coin is flipped. If it comes up heads, 2x is placed in another envelope and if it comes up tail, x/2 is placed in that envelope. Then you should switch.

It's a subtle difference in how the envelopes are prepared, but it makes all the difference.

[zmgsabst]

Oh, weird.

I’m not sure I’d have naively realized that the fixed amount in your envelope makes so much difference.

For people confused like me, think about the outcomes:

When the two are prepared together, say $2 and $4, then when you exchange your options are +2 and -2 with 50:50 odds.

When the second envelope is prepared based on $2 in your envelope, then when you exchange your options are +2 and -1 with 50:50 odds.

(I had to diagram this all out; unintuitive to me!)

[pontus]

I did a pretty deep analysis of this problem a while back here in case you're interested: https://mindbowling.wordpress.com/2020/09/14/two-envelope-pa...

[Groxx]

In some ways it's similar to the Monty Hall problem: https://en.wikipedia.org/wiki/Monty_Hall_problem

The introduction of a second "stage" based on your action changes things in a non-obvious way. Your first action is less impactful than your second one, regardless of what you did.

[bottled_poe]

This is new information which changes the definition of the problem though.

[pontus]

Yeah, I agree. This setup though is the one to which the original "naive" argument applies which is why I think it's interesting to ponder.