I think Vasco already made this point elsewhere, but I don’t see why you need certainty about any specific line to have finite expectation. If for the counterfactual payoff x, you think (perhaps after a certain point) xP(x) approaches 0 as x tends to infinity, it seems like you get finite expectation without ever having absolute confidence in any boundary (this applies to life expectancy, too).
Ya, I agree you don’t need certainty about the bound, but now you need certainty about the distribution not being heavy-tailed at all. Suppose your best guess is that it looks like some distribution X, with finite expected value. Now, I suggest that it might actually be Y, which is heavy-tailed (has infinite expected value). If you assign any nonzero probability to that being right, e.g. switch to pY+(1−p)X for some p>0, then your new distribution is heavy-tailed, too. In general, if you think there’s some chance you’d come to believe it’s heavy-tailed, then you should believe now that it’s heavy-tailed, because a probabilistic mixture with a heavy-tailed distribution is heavy-tailed. Or, if you think there’s some chance you’d come to believe there’s some chance it’s heavy-tailed, then you should believe now that it’s heavy-tailed.
(Vasco’s claim was stronger: the difference is exactly 0 past some point.)
I would be less interested to see a reconstruction of a proof of the theorems and more interested to see them stated formally and a proof of the claim that it follows from them.
Hmm, I might be misunderstanding.
I already have formal statements of the theorems in the post:
Stochastic Dominance, Anteriority and Impartiality are jointly inconsistent.
Stochastic Dominance, Separability and Impartiality are jointly inconsistent.
All of those terms are defined in the section Anti-utilitarian theorems. I guess I defined Impartiality a bit informally and might have hidden some background assumptions (preorder, so reflexivity + transitivity, and the set of prospects is every probability distribution over outcomes in the set of outcomes), but the rest were formally defined.
Then, from 1, assuming Stochastic Dominance and Impartiality, Anteriority must be false. From 2, assuming Stochastic Dominance and Impartiality, Separability must be false. Therefore assuming Stochastic Dominance and Impartiality, Anteriority and Separability must both be false.
Ya, I agree you don’t need certainty about the bound, but now you need certainty about the distribution not being heavy-tailed at all. Suppose your best guess is that it looks like some distribution X, with finite expected value. Now, I suggest that it might actually be Y, which is heavy-tailed (has infinite expected value). If you assign any nonzero probability to that being right, e.g. switch to pY+(1−p)X for some p>0, then your new distribution is heavy-tailed, too. In general, if you think there’s some chance you’d come to believe it’s heavy-tailed, then you should believe now that it’s heavy-tailed, because a probabilistic mixture with a heavy-tailed distribution is heavy-tailed. Or, if you think there’s some chance you’d come to believe there’s some chance it’s heavy-tailed, then you should believe now that it’s heavy-tailed.
(Vasco’s claim was stronger: the difference is exactly 0 past some point.)
Hmm, I might be misunderstanding.
I already have formal statements of the theorems in the post:
Stochastic Dominance, Anteriority and Impartiality are jointly inconsistent.
Stochastic Dominance, Separability and Impartiality are jointly inconsistent.
All of those terms are defined in the section Anti-utilitarian theorems. I guess I defined Impartiality a bit informally and might have hidden some background assumptions (preorder, so reflexivity + transitivity, and the set of prospects is every probability distribution over outcomes in the set of outcomes), but the rest were formally defined.
Then, from 1, assuming Stochastic Dominance and Impartiality, Anteriority must be false. From 2, assuming Stochastic Dominance and Impartiality, Separability must be false. Therefore assuming Stochastic Dominance and Impartiality, Anteriority and Separability must both be false.