Thanks Toby, I think this is a useful write up! A random take that I’ve been meaning to write is that all “reasonable” aggregation methods are equivalent to a hyperreal utility function in the following sense:
If you aggregate persons through some mechanism ⊕ which meets the criteria for an abelian linearly ordered group, then there exists some hyperreal-valued utility function u such that ⨁ixi⪰⨁iyi⇔∗∑iu(xi)≥∗∑iu(yi).
Proof sketch:
The Hahn embedding theorem tells us that any abelian linearly-ordered group G=(G,⊕,⪰) can be order embedded into H=(RN,+,≥), where N is the (possibly infinite) number of Archimedean equivalence classes (i.e. “sizes of infinity”) in G, ⊕,⪰ are some unknown aggregation and comparison mechanisms, respectively, + is normal element-wise addition and ≥ is lexicographical ordering (wlog assume it’s right-to-left). We can, in turn, order-embed H into R∗ through ϕ(h0,h1,˙)=h0+h1ω+… provided that N is not larger than the largest ordinal in R∗. It’s clear that this is order-preserving since the way you order the hyperreals is by first comparing the largest “infinity”, then the second largest infinity, and so on.
Say the first embedding is ψ and let u=ϕ∘ψ; u is then our desired utility function, since the group structure gives us that u(⨁ixi)=∗∑iu(xi). ■
Corollary:
In the case when G is Archimedean (i.e. has no “infinite” elements), this just gives us classical (real-valued) utilitarianism: ⨁ixi⪰⨁iyi⇔∑iu(xi)≥∑iu(yi).
Commentary:
I think there are some technical details to be worked out (e.g. ensuring that the order-type of H actually matches the order-type of a subgroup of R∗) but think it probably works?[1] I expect that the controversy comes from the claim that the group axioms are actually reasonable: particularly the assumption that for any population x there exists some population −x such that x⊕−x=0 is something that I think many utilitarians believe but maybe others don’t.
ChatGPT says this proof sketch is valid but that it is more natural to use Hahn series instead of the hyperreals. Its proof does seem more elegant at a quick skim, but I didn’t look closely. Possibly a follow up paper could be about the use of Hahn series in infinite ethics?
Interesting! So this is a kind of representation theorem (a bit like the VNM Theorem) but instead of saying that Archimedean preferences of gambles can be represented as a standard sum, it says that any aggregation method (even a non-Archimedean one) can be represented by a sum of a hyperreal utility function applied to each of its parts.
Thanks Toby, I think this is a useful write up! A random take that I’ve been meaning to write is that all “reasonable” aggregation methods are equivalent to a hyperreal utility function in the following sense:
Proof sketch:
The Hahn embedding theorem tells us that any abelian linearly-ordered group G=(G,⊕,⪰) can be order embedded into H=(RN,+,≥), where N is the (possibly infinite) number of Archimedean equivalence classes (i.e. “sizes of infinity”) in G, ⊕,⪰ are some unknown aggregation and comparison mechanisms, respectively, + is normal element-wise addition and ≥ is lexicographical ordering (wlog assume it’s right-to-left). We can, in turn, order-embed H into R∗ through ϕ(h0,h1,˙)=h0+h1ω+… provided that N is not larger than the largest ordinal in R∗. It’s clear that this is order-preserving since the way you order the hyperreals is by first comparing the largest “infinity”, then the second largest infinity, and so on.
Say the first embedding is ψ and let u=ϕ∘ψ; u is then our desired utility function, since the group structure gives us that u(⨁ixi)=∗∑iu(xi). ■
Corollary:
In the case when G is Archimedean (i.e. has no “infinite” elements), this just gives us classical (real-valued) utilitarianism: ⨁ixi⪰⨁iyi⇔∑iu(xi)≥∑iu(yi).
Commentary:
I think there are some technical details to be worked out (e.g. ensuring that the order-type of H actually matches the order-type of a subgroup of R∗) but think it probably works?[1] I expect that the controversy comes from the claim that the group axioms are actually reasonable: particularly the assumption that for any population x there exists some population −x such that x⊕−x=0 is something that I think many utilitarians believe but maybe others don’t.
ChatGPT says this proof sketch is valid but that it is more natural to use Hahn series instead of the hyperreals. Its proof does seem more elegant at a quick skim, but I didn’t look closely. Possibly a follow up paper could be about the use of Hahn series in infinite ethics?
Interesting! So this is a kind of representation theorem (a bit like the VNM Theorem) but instead of saying that Archimedean preferences of gambles can be represented as a standard sum, it says that any aggregation method (even a non-Archimedean one) can be represented by a sum of a hyperreal utility function applied to each of its parts.
Yes, I think that’s a good summary!