The thing you called “pseudograding” is normally called “filtration”.
In practice, because of the complexity of the world, and especially because of the presence of probabilistic uncertainty, an agent following a non-Archimedean utility function will always consider only the component corresponding to the absolute maximum of I, since there will never be a choice between A and B such that these components just happen to be exactly equal. So it will be equivalent to an Archimedean agent whose utility is this worst component. (You can have an I without an absolute maximum but I don’t think it’s possible to define any reasonable utility function like that, where by “reasonable” I roughly mean that, it’s possible to build some good theory of reinforcement learning out of it.)
The thing you called “pseudograding” is normally called “filtration”.
Ah, thanks! I knew there had to be something for that, just couldn’t remember what it was. I was embarrassed posting with a made-up word, but I really did look (and ask around) and couldn’t find what I needed.
...Although, reading the definition, I’m not sure it’s exactly the same...the severity classes aren’t nested, and I think this is probably an important distinction to the conceptual framing, even if the math is equivalent. If I start with a filtration proper, I need to extract the severity classes in a way that seems slightly more convoluted than what I did.
In practice, because of the complexity of the world, and especially because of the presence of probabilistic uncertainty, an agent following a non-Archimedean utility function will always consider only the component corresponding to the absolute maximum of I, since there will never be a choice between A and B such that these components just happen to be exactly equal. So it will be equivalent to an Archimedean agent whose utility is this worst component.
If I understand what you do correctly, the severity classes are just the set differences Vα∖⋃β<αVβ, where {Vα}α∈I is the filtration. I think that you also assume that the quotient Vα/⋃β<αVβ is one-dimensional and equipped with a choice of “positive” direction.
Yes! This is all true. I thought set differences of infinite unions and quotients would only make the post less accessible for non-mathematicians though. I also don’t see a natural way to define the filtration without already having defined the severity classes.
Two comments:
The thing you called “pseudograding” is normally called “filtration”.
In practice, because of the complexity of the world, and especially because of the presence of probabilistic uncertainty, an agent following a non-Archimedean utility function will always consider only the component corresponding to the absolute maximum of I, since there will never be a choice between A and B such that these components just happen to be exactly equal. So it will be equivalent to an Archimedean agent whose utility is this worst component. (You can have an I without an absolute maximum but I don’t think it’s possible to define any reasonable utility function like that, where by “reasonable” I roughly mean that, it’s possible to build some good theory of reinforcement learning out of it.)
Ah, thanks! I knew there had to be something for that, just couldn’t remember what it was. I was embarrassed posting with a made-up word, but I really did look (and ask around) and couldn’t find what I needed.
...Although, reading the definition, I’m not sure it’s exactly the same...the severity classes aren’t nested, and I think this is probably an important distinction to the conceptual framing, even if the math is equivalent. If I start with a filtration proper, I need to extract the severity classes in a way that seems slightly more convoluted than what I did.
See my response to Dacyn.
If I understand what you do correctly, the severity classes are just the set differences Vα∖⋃β<αVβ, where {Vα}α∈I is the filtration. I think that you also assume that the quotient Vα/⋃β<αVβ is one-dimensional and equipped with a choice of “positive” direction.
Yes! This is all true. I thought set differences of infinite unions and quotients would only make the post less accessible for non-mathematicians though. I also don’t see a natural way to define the filtration without already having defined the severity classes.