If x and 2x are the amounts in the envelopes, then the expected value at the beginning of the game is (3/2)x.
You choose an envelope.
If you switch envelopes, the expected value of the money in the envelope you switch to is STILL (3/2)x.
The error is that when you say "switching can double your money to 2x or halve it to x/2", you are making a tacit assumption that you have information about your current choice - that you know what x itself is in some sense. This isn't the Monty Hall problem at all - in the MH problem you've gained information before you are offered the option of changing doors. Here, you've gained NO information in between, so there's no advantage to switching.