alt Hacker NewsYou're correct. I realized that I completely misunderstoond the original problem after reading McDonnell's paper more carefully.
I thought you meant the strategy can make the winning chance always >50% even after the player opens the first envelope, which isn't possible.
However you actually meant the strategy can make the expected winning chance >50% before the player opens the first envelope, for any well-defined distribution of real number, even the distribution is not known to the player, which now I realize is true.
(I haven't thought through some edge case like Cantor distribution, but now I incline to it's true not just for "many distributions". Of course for a discrete distributions, we need to specifiy the two envelopes can't have the same number. Besides that, it seems to hold true for any distribution?)
Exactly right. After you've picked the envelope, you may be nearly guaranteed to be wrong. For example both numbers are large positives and you picked the smaller. Now you're almost guaranteed to guess larger, and be wrong.
But before you pick, your odds were still bigger than 50%. Just not by much.