FWIW—here (finally) is the related post I mentioned, which motivated this observation:
Natural Abstraction: Convergent Preferences Over Information Structures
The context is a power-seeking-style analysis of the naturality of abstractions, where I was determined to have transitive preferences.
It had quite a bit of scope creep already, so I ended up not including a general treatment of the (transitive) ‘sum over orbits’ version of retargetability (and some parts I considered only optimality—sorry! still think it makes sense to start there first and then generalize in this case). The full translation also isn’t necessarily as easy as I thought—it turns out that ≥nmost is transitive specifically for binary functions, so the other cases may not translate as easily as IsOptimal. After noticing that I decided to leave the general case for later.
I did use the sum-over-orbits form, though; which turns out to describe the preferences shared by every “G-invariant” distribution over utility functions. Reading between the lines shows roughly what it would look like.
I also moved from Sd to any G≤Sd - not sure if you looked at that, but at least the parts I was using all seem to work just as well with any subgroup. This gives preferences shared by a larger set of distributions, e.g. for an MDP you could in some cases have s1 preferred to s2 for all priors on U that are merely invariant to permuting U(s1) and U(s2) (rather than requiring them to be invariant to all permutations of utilities).
FWIW—here (finally) is the related post I mentioned, which motivated this observation: Natural Abstraction: Convergent Preferences Over Information Structures The context is a power-seeking-style analysis of the naturality of abstractions, where I was determined to have transitive preferences.
It had quite a bit of scope creep already, so I ended up not including a general treatment of the (transitive) ‘sum over orbits’ version of retargetability (and some parts I considered only optimality—sorry! still think it makes sense to start there first and then generalize in this case). The full translation also isn’t necessarily as easy as I thought—it turns out that ≥nmost is transitive specifically for binary functions, so the other cases may not translate as easily as IsOptimal. After noticing that I decided to leave the general case for later.
I did use the sum-over-orbits form, though; which turns out to describe the preferences shared by every “G-invariant” distribution over utility functions. Reading between the lines shows roughly what it would look like.
I also moved from Sd to any G≤Sd - not sure if you looked at that, but at least the parts I was using all seem to work just as well with any subgroup. This gives preferences shared by a larger set of distributions, e.g. for an MDP you could in some cases have s1 preferred to s2 for all priors on U that are merely invariant to permuting U(s1) and U(s2) (rather than requiring them to be invariant to all permutations of utilities).