Brent Meeker and Jesse Mazer and others wrote:

Well, lots and lots of complex mathematical argument on the two envelope problem...

But no-one has yet pointed out a flaw in my rather simplistic analysis:

(1) One envelope contains x currency units, so the other contains 2x currency units;

(2) If you stop at the first envelope you choose, expected gain is: 0.5*x + 0.5*2x = 1.5x;

(3) If you open the first envelope then switch to the second, your expected gain is: 0.5*2x + 0.5*x = 1.5x - as above, just in a different order, obviously;

(4) If, in a variation, the millionaire flips a coin to give you double or half the amount in the first envelope if you switch envelopes, expected gain is: 0.25*2x + 0.25*0.5x + 0.25*x + 0.25*4x = 1.875x.

In the latter situation you are obviously better off switching, but it is a mistake to assume that (4) applies in the original problem, (3) - hence, no paradox.

Is the above wrong, or is it just so obvious that it isn't worth discussing? (I'm willing to accept either answer).

Stathis Papaioannou

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