Yup. But I don’t see a good reason to apply it that way here, nor a good principled way of doing so that gives the results you want. I mean, how are you going to calculate the net utility in the “after” situation to arrive at ω+10000000 rather than ω?
no ordinal improvement [...] like a cardinality
It looks to me like it’s the other way around. Surreal integer arithmetic is much more like ordinal arithmetic than like cardinal arithmetic. That’s one reason why I’m skeptical about the prospects for applying it here: it seems like it might require imposing something like an ordering on the people whose utilities we’re aggregating.
the measure would need to be surreal
Despite my comments above, I do think it’s worth giving more consideration to using a richer number system for utilities.
[EDITED to fix an inconsequential and almost invisible typo.]
It does occur to me that while giving the people an order migth be suspicous utilities are a shorthand of preferences which are defined to be orders of preferring a over b. Therefore there is anyways going to be a conversion to ordinals so surreals should remain relevant.
I don’t think I’m convinced. Firstly, because in these cases where we’re looking at aggregating the interests of large collections of people it’s the people, not the possibilities, that seem like they need to be treated as ordered. Secondly, because having an ordering on preferences isn’t at all the same thing as wanting to use anything like ordinals for them. (E.g., cardinals are ordered too—well-ordered, even—at least if we assume the axiom of choice. The real numbers are ordered in the obvious way, but that’s not a well-ordering. Etc.)
I concede that what I’m saying is very hand-wavy. Maybe there really is a good way to make this sort of thing work well using surreal numbers as utilities. And (perhaps like you) I’ve thought for a long time that using something like the surreals for utilities might turn out to have advantages. I just don’t currently see an actual way to do it in this case.
Yup. But I don’t see a good reason to apply it that way here, nor a good principled way of doing so that gives the results you want. I mean, how are you going to calculate the net utility in the “after” situation to arrive at ω+10000000 rather than ω?
It looks to me like it’s the other way around. Surreal integer arithmetic is much more like ordinal arithmetic than like cardinal arithmetic. That’s one reason why I’m skeptical about the prospects for applying it here: it seems like it might require imposing something like an ordering on the people whose utilities we’re aggregating.
Despite my comments above, I do think it’s worth giving more consideration to using a richer number system for utilities.
[EDITED to fix an inconsequential and almost invisible typo.]
It does occur to me that while giving the people an order migth be suspicous utilities are a shorthand of preferences which are defined to be orders of preferring a over b. Therefore there is anyways going to be a conversion to ordinals so surreals should remain relevant.
I don’t think I’m convinced. Firstly, because in these cases where we’re looking at aggregating the interests of large collections of people it’s the people, not the possibilities, that seem like they need to be treated as ordered. Secondly, because having an ordering on preferences isn’t at all the same thing as wanting to use anything like ordinals for them. (E.g., cardinals are ordered too—well-ordered, even—at least if we assume the axiom of choice. The real numbers are ordered in the obvious way, but that’s not a well-ordering. Etc.)
I concede that what I’m saying is very hand-wavy. Maybe there really is a good way to make this sort of thing work well using surreal numbers as utilities. And (perhaps like you) I’ve thought for a long time that using something like the surreals for utilities might turn out to have advantages. I just don’t currently see an actual way to do it in this case.