Monty Hall-esque question

[Masquerade]

Of course as a general strategy it makes no difference whether you switch or not in terms of EV.

To see this imagine an omniscient observer who puts X and 2X into the envelopes and runs the scenario many times.

Of those who never switch half will choose X and half will choose 2X initally so their EV is 3/2 X. Clearly those who always switch have the exactly the same EV.

The fallacy is in thinking that once you've chosen an envelope there is a 50-50 chance of doubling or halving. There is a 100% chance of doubling and 0% of halving - or vice versa. The outcome is completely determined.