Don't think that's Heisenberg you can smell, it's Schrodinger's cat's decomposing body.

Let me try again. Adding probabilites (the .5x + 2x part) is for mutually exclusive events. If we were flipping a coin to double or halve our money it works. In this case though, doubling or halving our money are not strictly mutually exclusive. The amount we are playing for when we choose to switch is contingent on our initial choice. The two amounts are interdependent, not mutually exclusive.

The case where it is positive EV is when after you have picked, the other guy says "Ok now i'm going to give you a third envelope which is either double or half whatever you have now. Do you want to switch?" In that case, you should (and you should do it forever since you can never go bust.)

You don't have enough information for a formula. It's a pure guess. It's a fifty-fifty prop. If there is a hundred in one and two hundred in the other, you could play one hundred and fifty for a chance to draw. That would be an even money gamble. Nobody wants an even money gamble except an all day sucker.