I’m not familiar with Savage’s theorem, but I was aware of the things you said about the VNM theorem, and in fact, I often bring up the same arguments you’ve been making. The standard response that I hear is that some probability distributions cannot be updated to without an infinite amount of information (e.g. if a priori the probability of the nth outcome is proportional to 1/3^n, then there can’t be any evidence that could occur with nonzero probability that would convince you that the probability of the nth outcome is 1/2^n for each n), and there’s no need for a utility function to converge on gambles that it is impossible even in theory for you to be convinced are available options.
When I ask why they assume that their utility function should be valid on those infinite gambles that are possible for them to consider, if they aren’t assuming that their preference relation is closed in the strong topology (which implies that the utility function is bounded), they’ll say something like that their utility function not being valid where their preference relation is defined seems weirdly discontinuous (in some sense that they can’t quite formalize and definitely isn’t the preference relation being closed in the strong topology), or that the examples I gave them of VNM-rational preference relations for which the utility function isn’t valid for infinite gambles all have some other pathology, like that there’s an infinite gamble which is considered either better than all of or worse than all of the constituent outcomes, and there might be a representation theorem saying something like that has to happen, even though they can’t point me to one.
Anyway, I agree that that’s a more fundamental reason to only consider bounded utility functions, but I decided I could probably be more convincing by abandoning that line of argument, and showing that if you sweep convergence issues under the rug, unbounded utility functions still suggest insane behavior in concrete situations.
Huh, I only just saw this for some reason. Anyway, if you’re not familiar with Savage’s theorem, that’s why I wrote the linked article here about it! :)
I’m not familiar with Savage’s theorem, but I was aware of the things you said about the VNM theorem, and in fact, I often bring up the same arguments you’ve been making. The standard response that I hear is that some probability distributions cannot be updated to without an infinite amount of information (e.g. if a priori the probability of the nth outcome is proportional to 1/3^n, then there can’t be any evidence that could occur with nonzero probability that would convince you that the probability of the nth outcome is 1/2^n for each n), and there’s no need for a utility function to converge on gambles that it is impossible even in theory for you to be convinced are available options.
When I ask why they assume that their utility function should be valid on those infinite gambles that are possible for them to consider, if they aren’t assuming that their preference relation is closed in the strong topology (which implies that the utility function is bounded), they’ll say something like that their utility function not being valid where their preference relation is defined seems weirdly discontinuous (in some sense that they can’t quite formalize and definitely isn’t the preference relation being closed in the strong topology), or that the examples I gave them of VNM-rational preference relations for which the utility function isn’t valid for infinite gambles all have some other pathology, like that there’s an infinite gamble which is considered either better than all of or worse than all of the constituent outcomes, and there might be a representation theorem saying something like that has to happen, even though they can’t point me to one.
Anyway, I agree that that’s a more fundamental reason to only consider bounded utility functions, but I decided I could probably be more convincing by abandoning that line of argument, and showing that if you sweep convergence issues under the rug, unbounded utility functions still suggest insane behavior in concrete situations.
Huh, I only just saw this for some reason. Anyway, if you’re not familiar with Savage’s theorem, that’s why I wrote the linked article here about it! :)