The paradox depends on the possibility of always being able to double the amount. This isn't true.
Swapping once will either double or half the amount; swapping back will just do the opposite.
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Even better: if I put $10 into one envelope, and $20 in another envelope and get you to choose one, what is the probability that the other envelope contains $40? Zero.
Thanks, that is a wonderful way to put it. The swaps are not independent in their effects. That satisfies me.