I agree with the parts I could verify within about 10 minutes of staring (it’s been a while). The scalar-retargetability is nice, and I like the delineation of what definitions yield what properties. Seems like an additional hour of work would yield a good AF post, where I’d expect most of the useful additional work to come from fleshing out the example more and justifying the claims in a bit more detail.
To clarify:
This also works on retargetability directly, with f being A2→B, B2→C, C2→A retargetable. Notice also that f is invariant under joint permutations (constant diagonals), and I think can be represented as EU-determined, so neither of these save it.
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).
Thanks for the reply. I’ll clean this up into a standalone post and/or cover this in a related larger post I’m working on, depending on how some details turn out.
What are A,B,C here?
Variables I forgot to rename, when I changed how I was labelling the arguments of f in my example. This should be 12→2, 22→3, 32→1 retargetable (as arguments i to f(i|j)).
This is a nice contribution, thank you!
I agree with the parts I could verify within about 10 minutes of staring (it’s been a while). The scalar-retargetability is nice, and I like the delineation of what definitions yield what properties. Seems like an additional hour of work would yield a good AF post, where I’d expect most of the useful additional work to come from fleshing out the example more and justifying the claims in a bit more detail.
To clarify:
What are A,B,C here?
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).
Thanks for the reply. I’ll clean this up into a standalone post and/or cover this in a related larger post I’m working on, depending on how some details turn out.
Variables I forgot to rename, when I changed how I was labelling the arguments of f in my example. This should be 12→2, 22→3, 32→1 retargetable (as arguments i to f(i|j)).