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Sounds like a misuse of relative return (growth rate). If you gain 100% today, then lose 50% tomorrow, you are back where you started. Absolute numbers give you the correct answer: x -> 2x is a gain of x, 2x -> x is a loss of x, so expected value is zero (x is a hard number not a ratio).

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Exactly! The logic of "the average of +100% and -50% is positive" only applies if the denominator is the same. In this case it isn't.

The relative percentage "rule of thumb" doesn't even feel like a primitive in math. It seems wrong to reason from that.

why isn't this listed in wikipedia as an answer.

It is, it's the one where you treat both as having the same expected value

that the expected value is the same that is an outcome of all solutions, so just stating that is reductive.

What is at question is the reason for getting to the expected value is the same.

Because if the larger envelope is changed to contain four times as many dollars, then /2 vs. *4 points back towards swapping indefinitely.

If one envelope contains 4 times the money of the other envelope, then by swapping you either gain 3x or lose 3x, where x is the amount of money in the smaller envelope. Even in this case, the average return is +3x -3x = 0.

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