There’s a family of approaches here, but it’s not clear that they recreate the same behaviour as the parliament (at least without more arguments about the parliament). Whether they are more or less desirable is a separate question.
Incidentally, the version that you suggest isn’t quite well-defined, since it can be changed by adding a constant to the theory of a function. But that can easily be patched over.
I like the originality of the geometric approach. I don’t think it’s super useful, but then again you made good use of it in Theorem 19, so that shows what I know.
I found the section on voting to need revision for clarity. Is the idea that each voter submits a function, the outcomes are normalized and summed, and the outcome with the highest value wins (like in range voting—except fixed-variance voting)? Either I missed the explanation or you need to explain this. Later in Theorem 14 you assumed that each agent voted with its utility function (proved later in Thm 19, good work by the way, but please don’t assume it without comment earlier), and we need to remember that all the way back in 4.0 you explained why to normalize v and u the same.
Overall I’d like to see you move away from the shaky notion of “a priori voting power” in the conclusion, by translating from the case of voting back into the original case of moral philosophy. I’m pretty sold that variance normalization is better than range normalization though.
There’s a family of approaches here, but it’s not clear that they recreate the same behaviour as the parliament (at least without more arguments about the parliament). Whether they are more or less desirable is a separate question.
Incidentally, the version that you suggest isn’t quite well-defined, since it can be changed by adding a constant to the theory of a function. But that can easily be patched over.
I’ve argued that normalising the variance of the functions is the most natural of these approaches (link to a paper giving the arguments in a social choice context; forthcoming paper with Ord and MacAskill in the moral uncertainty context).
I like the originality of the geometric approach. I don’t think it’s super useful, but then again you made good use of it in Theorem 19, so that shows what I know.
I found the section on voting to need revision for clarity. Is the idea that each voter submits a function, the outcomes are normalized and summed, and the outcome with the highest value wins (like in range voting—except fixed-variance voting)? Either I missed the explanation or you need to explain this. Later in Theorem 14 you assumed that each agent voted with its utility function (proved later in Thm 19, good work by the way, but please don’t assume it without comment earlier), and we need to remember that all the way back in 4.0 you explained why to normalize v and u the same.
Overall I’d like to see you move away from the shaky notion of “a priori voting power” in the conclusion, by translating from the case of voting back into the original case of moral philosophy. I’m pretty sold that variance normalization is better than range normalization though.
Thanks for the feedback!