[AaronBrown]

I'll stay out of the meaning of probability discussion, but I have to defend the envelope paradox. It is considerably deeper than suggested here. Like all good paradoxes, the point is not to explain it, but to peel away the layers.

You do not need to assume that the amount in the envelope is equally likely to be any real number. It is true that from a Bayesian perspective if seeing the amount of money in one envelope does not change your estimate of the probability of it being the larger amount, then your prior probability distribution must have an infinite expectation. If you expect infinity, any finite number is a disappointment, so you should always switch.

That's not a logical inconsistency, although it's hard to think of an example that doesn't involve God or the Devil.

But in the first place, that argument only makes sense to Bayesians. In the second place, you can make the paradox work with finite expectations as well.

One example is an American bets a European that the US dollar will buy less that 0.80 Euros in one year (or set any rate such that the probability is 50%). If the American wins, he gets 800 Euros. If he loses, he pays US $1,000.

The American will either lose $1,000 or win Euros he can exchange for more than $1,000. The European will either win 800 Euros or dollars he can exchange for more than 800 dollars. So both parties have positive expectations. In fact, the American can convert his bet to a 50/50 proposition for $1,000 and get paid $50, by selling an option in the financial markets. The European can get a 50/50 800 Euro bet and be paid 40 Euros.