paulom comments on Parametrically retargetable decision-makers tend to seek power

[paulom]

I appreciate this generalization of the results—I think it’s a good step towards showing the underlying structure involved here.

One point I want to comment on is transitivity of ${\ge }_{most}^n$, as a relation on induced functions $f:\Theta \to R$. Namely, it isn’t, and can even contain cycles of non-equivalent elements. (This came up when I was trying to apply a version of these results, and hoping that ${\ge }_{most}^n$ would be the preference relation I was looking for out of the box.) Quite possibly you noticed this since you give ‘limited transitivity’ in Lemma B.1 rather than full transitivity, but to give a concrete example:

Let $$V=\left(\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\\ 2& 3& 1\end{array}\right)$$ and $f\left(i|j\right)=V_{ij}$. The permutations are $\sigma \in S_3$ with the usual action on $\{1,2,3\}$. Then we have ^[1] $f_1{\ge }_{most}^2{f}_2{\ge }_{most}^2{f}_3{\ge }_{most}^2{f}_1$ (and $f_2{\ngeqq }_{most}^2{f}_1$). This also works on retargetability directly, with $f$ being $A\stackrel{2}{\to }B$, $B\stackrel{2}{\to }C$, $C\stackrel{2}{\to }A$ 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.

A narrow point is that for a non-transitive relation, I think the notation should be something other than $\ge$ (maybe $\succcurlyeq$).

But more importantly, I think we would really rather a transitive (at least acyclic) relation, if we want to interpret this is ‘most $\theta$ prefer’ or any kind of preference /​ aggregation of preferences. If our theorem gives us only an intransitive relation as our conclusion, then we should tweak it.

One way you can do this: aim for a stronger relation like ${\ge }_{\text{o-m}}^n$:

Definition (Orbit-mean dominance?): Let $O_{f,A\ne B}\left(\theta \right)=\{{\theta }^{\prime }\in \text{Orbit}{|}_{\Theta }(\theta ):f(A|{\theta }^{\prime })\ne f(B|{\theta }^{\prime }\left)\right\}$. Write $f\left(B|\theta \right){\ge }_{\text{o-m}}^{n}f\left(A|\theta \right)$ if $\forall \theta :{\sum }_{O_{f,A\ne B}\left(\theta \right)}f\left(B|{\theta }^{\prime }\right)\ge n{\sum }_{O_{f,A\ne B}\left(\theta \right)}f\left(A|{\theta }^{\prime }\right)$.

Since the orbits are under $S_d$ i.e. finite, it’s easy to just sum over them. More generally, you could parameterize this with an arbitrary aggregator $g:{\text{Orbits}}_{\Theta }\left(f\right)\to R$ in place of summation; I’m not sure whether this general form or the $\sum$ case should be the focus.

This is transitive for $n=1$ and acyclic for^[2] $n>1$ (consider $\theta$ by $\theta$); and possibly any orbit-based transitive relation is representable in basically this form^[3] (with some $g$), since I’d guess any partial order on sets with cardinality $\le c$ can be represented as a pointwise inequality of functions, but I haven’t thought about this too carefully.

With this notion of ${\ge }_{\text{o-m}}^n$, we also need a stronger version of retargetability for the main theorem to hold. For the $\sum$ version, this could be

Definition (scalar-retargetability): Write $f$ is $A\underset{scalar}{\overset{B}{\to }}$ if there exists $\sigma \in S_d$ such that for all $\theta$ with $f\left(A|{\theta }^A\right)-f\left(B|{\theta }^A\right)=c>0$ we have $f\left(B|\sigma {\theta }^A\right)-f\left(A|\sigma {\theta }^A\right)\ge c$ (and likewise multiply scalar-retargetable).

Then scalar-retargetability from $A$ to $B$ will imply $f\left(B|\theta \right){\ge }_{\text{o-m}}^{n}f\left(A|\theta \right)$.

And: I think many (all?) of the main power-seeking results are already secretly in this form. For example, $\theta$-wise comparison of ${\sum }_{{\theta }^{\prime }\in \text{Orbit}{|}_{\Theta }\left(\theta \right)}\text{IsOptimal}(X|C,{\theta }^{\prime })$ gives a preference relation ${\ge }_{\text{o-m}}^n$ identical to the relation ${\ge }_{most}^n$. Assuming this also works for the other rationalities, then the cases we care about were transitive all along exactly because the relations can be expressed in this way.

What do you think?

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1. ↩︎

   We get the same single orbit $\{1,2,3\}$ for all $\theta$ a.k.a. $j$; the orbit elements $j$ with $f\left(i|j\right)>f\left(i^{\prime }|j\right)$ are the columns where row $i$ $>$ row $i^{\prime }$. There are always two such columns when comparing row $i$ and row $i+1$ (mod 3). For example, $$\begin{array}{c}f(1,1)=1<3=f(2,1)\\ f(1,2)=2>1=f(2,2)\\ f(1,3)=3>2=f(2,3)\end{array}$$

2. ↩︎

   We exclude $\theta$ s.t.$f\left(A|{\theta }^{\prime }\right)=f\left(B|{\theta }^{\prime }\right)$ in this version of the definition to match the behaviour of ${\ge }_{most}^n$ with $n>1$, and allow $n$-scalar-retargetability to imply ${\ge }_{\text{o-m}}^n$. There’s a case that you should include them, in which case you do get transitivity, and even the stronger property: if $x{\le }^{n}y{\le }^{m}z$, then $x{\le }^{nm}z$. I think this corresponds to looking at likelihood ratios of $P(A\wedge \neg B)::P(B\wedge \neg A)$ vs. $P\left(A\right)::P\left(B\right)$.

3. ↩︎

   Compare also what would give you a total order (instead of partial order): aggregating over all of $\Theta$ at once, like ${\int }_{\Theta }f\left(A|\theta \right)d\mu \left(\theta \right)$, instead of aggregating orbitwise at each $\theta$.

[TurnTrout]

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:

This also works on retargetability directly, with $f$ being $A\stackrel{2}{\to }B$, $B\stackrel{2}{\to }C$, $C\stackrel{2}{\to }A$ 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.

What are $A,B,C$ here?

[paulom]

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 ${\ge }_{most}^n$ is transitive specifically for binary functions, so the other cases may not translate as easily as $\text{IsOptimal}$. 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 $S_d$ to any $G\le S_d$ - 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 $s_1$ preferred to $s_2$ for all priors on $U$ that are merely invariant to permuting $U\left(s_1\right)$ and $U\left(s_2\right)$ (rather than requiring them to be invariant to all permutations of utilities).

[paulom]

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 $1\stackrel{2}{\to }2$, $2\stackrel{2}{\to }3$, $3\stackrel{2}{\to }1$ retargetable (as arguments $i$ to $f\left(i|j\right)$).