The latter. It doesn’t even make sense to speak of maximizing the expectation of an unbounded utility function, because unbounded functions don’t even have expectations with respect to all probability distributions.
There is a way out of this that you could take, which is to only insist that the utility function has to have an expectation with respect to probability distributions in some restricted class, if you know your options are all going to be from that restricted class. I don’t find this very satisfying, but it works. And it offers its own solution to Pascal’s mugging, by insisting that any outcome whose utility is on the scale of 3^^^3 has prior probability on the scale of 1/(3^^^3) or lower.
The latter. It doesn’t even make sense to speak of maximizing the expectation of an unbounded utility function, because unbounded functions don’t even have expectations with respect to all probability distributions.
There is a way out of this that you could take, which is to only insist that the utility function has to have an expectation with respect to probability distributions in some restricted class, if you know your options are all going to be from that restricted class. I don’t find this very satisfying, but it works. And it offers its own solution to Pascal’s mugging, by insisting that any outcome whose utility is on the scale of 3^^^3 has prior probability on the scale of 1/(3^^^3) or lower.