A question or two about the two-envelope paradox.
(I’d write about the answer if I had a wee bit more time.)

There are two envelopes, one red and one blue. One
will become the primary envelope and one the secondary envelope. God decides how
many gollars in the primary envelope using the following method. (Gollars are a
rather special form of currency in that their marginal utility is both positive
and constant.)

God tosses a coin repeatedly until it lands heads
the first time. The coin has chance 1/3 of landing heads on any given toss, but
all the tosses are independent. Let n be the number of
that first toss to land heads. (I.e. if it lands heads the first time, n = 1,
the second time n = 2, etc.) He
will put 100 * 2^n gollars in the primary
envelope. He then tosses a fair coin and if it lands
heads He will put half that amount in the secondary envelope, and if it lands
tails He will put twice as much in the secondary envelope. He then chooses
which shall be the primary envelope by tossing a third coin, that has
probability x of landing
heads. If it lands heads, the blue envelope is primary, otherwise the red
envelope is primary. You are then given the red envelope.

Not the real question. God says that you can swap
the red envelope for the blue envelope. When should you accept the offer?
Answer: you should accept when x < ½,
reject when x > ½, and be
indifferent otherwise. (Or so say I. Do you disagree?)

Real question. God says that you can swap the red
envelope for the blue envelope, but it will cost you one gollar. When should
you accept the offer? I.e. for what values of x (which
I presume you know) should the offer be accepted.

Supplementary question. You won’t receive what’s in
your envelope, but the difference between what’s in your envelope and what’s in
the other envelope. Does this make a difference?

Bonus question. Solve the real question when the
probability of the first coin landing heads is not given as 1/3, but is some
variable value x.