Another commentary I once read regarding the “two envelopes paradox”:
In order to make any sense out of this problem, you have to assume some prior probability distribution over the amount of money in the two envelopes. For many of these possible distributions, once you open one of the envelopes and learn how much money is inside, you now know more about whether the other envelope has more or less money than the one you opened. For example, if you assume that neither envelope has more than $1,000 and then open an envelope with $800, the other envelope has to have $400, so, contrary to the line of reasoning in the “paradox”, switching would be bad.
On the other hand, perhaps you only want to think about distributions for which it seems the paradox still holds: ones in which that, regardless of how much money you find in envelope A, envelope B still has an equal chance of being twice as much or half as much. Well, it turns out that you can prove that this criterion also implies that the expected value of the amount of money in envelope A is infinity. This makes the paradox seem much less paradoxical: first, when your expected value is infinity, any specific finite result is disappointing (which is why switching is always correct), and second, any finite number multiplied by infinity is still infinity (which explains how each envelope can have an expected value of 1.25 times the other).
On the other hand, perhaps you only want to think about distributions for which it seems the paradox still holds: ones in which that, regardless of how much money you find in envelope A, envelope B still has an equal chance of being twice as much or half as much
I don’t see your conclusion holding. I am inclined to say: Therefore there are no distributions which that, regardless of how much money you find in envelope A, envelope B still has an equal chance of being twice as much or half as much.
I don’t see your conclusion holding. I am inclined to say: Therefore there are no distributions which that, regardless of how much money you find in envelope A, envelope B still has an equal chance of being twice as much or half as much.
I suppose “numbers selected from all the numbers in the series 2^n” and so forth are ruled out of being distributions based on the “infinities and uncomputable things are just silly” principle? (I am fairly confident that) something on that order of difficulty is going to required to provide the envelopes. A task that is beyond even Omega in the universe as we know it but perhaps not beyond an intelligent agent in the possible universes that represent computational abstractions natively.
Actually there are no uniform distribution in this set (an infinite enumerable set). You may select numbers from this set, but some of them will have higher probability than others.
Moreover, this means that if we want to do this with utility rather than money, you’d need an unbounded utility function, which can’t happen if you’re obeying Savage’s axioms.
Another commentary I once read regarding the “two envelopes paradox”:
In order to make any sense out of this problem, you have to assume some prior probability distribution over the amount of money in the two envelopes. For many of these possible distributions, once you open one of the envelopes and learn how much money is inside, you now know more about whether the other envelope has more or less money than the one you opened. For example, if you assume that neither envelope has more than $1,000 and then open an envelope with $800, the other envelope has to have $400, so, contrary to the line of reasoning in the “paradox”, switching would be bad.
On the other hand, perhaps you only want to think about distributions for which it seems the paradox still holds: ones in which that, regardless of how much money you find in envelope A, envelope B still has an equal chance of being twice as much or half as much. Well, it turns out that you can prove that this criterion also implies that the expected value of the amount of money in envelope A is infinity. This makes the paradox seem much less paradoxical: first, when your expected value is infinity, any specific finite result is disappointing (which is why switching is always correct), and second, any finite number multiplied by infinity is still infinity (which explains how each envelope can have an expected value of 1.25 times the other).
(Apparently, my original source was David Chalmers, of all people.)
I don’t see your conclusion holding. I am inclined to say: Therefore there are no distributions which that, regardless of how much money you find in envelope A, envelope B still has an equal chance of being twice as much or half as much.
I suppose “numbers selected from all the numbers in the series 2^n” and so forth are ruled out of being distributions based on the “infinities and uncomputable things are just silly” principle? (I am fairly confident that) something on that order of difficulty is going to required to provide the envelopes. A task that is beyond even Omega in the universe as we know it but perhaps not beyond an intelligent agent in the possible universes that represent computational abstractions natively.
Actually there are no uniform distribution in this set (an infinite enumerable set). You may select numbers from this set, but some of them will have higher probability than others.
That is what I was getting at with ‘ruled out of being distributions’.
Oh… I misunderstood you then.
Moreover, this means that if we want to do this with utility rather than money, you’d need an unbounded utility function, which can’t happen if you’re obeying Savage’s axioms.