This paper—accepted as a poster to NeurIPS 2022— is the sequel to Optimal Policies Tend to Seek Power. The new theoretical results are extremely broad, discarding the requirements of full observability, optimal policies, or even requiring a finite number of options.

Abstract:

If capable AI agents are generally incentivized to seek power in service of the objectives we specify for them, then these systems will pose enormous risks, in addition to enormous benefits. In fully observable environments, most reward functions have an optimal policy which seeks power by keeping options open and staying alive. However, the real world is neither fully observable, nor must trained agents be even approximately reward-optimal.

We consider a range of models of AI decision-making, from optimal, to random, to choices informed by learning and interacting with an environment. We discover that many decision-making functions are retargetable, and that retargetability is sufficient to cause power-seeking tendencies. Our functional criterion is simple and broad.

We show that a range of qualitatively dissimilar decision-making procedures incentivize agents to seek power. We demonstrate the flexibility of our results by reasoning about learned policy incentives in Montezuma's Revenge. These results suggest a safety risk: Eventually, retargetable training procedures may train real-world agents which seek power over humans.

Examples of agent designs the power-seeking theorems now apply to:

- Boltzmann-rational agents,
- Expected utility maximizers and minimizers,
- Even if they uniformly randomly sample a few plans and then choose the best sampled

- Satisficers (as I formalized them),
- Quantilizing with a uniform prior over plans, and
- RL-trained agents under certain modeling assumptions.

The key insight is that the original results hinge not on optimality per se, but on the retargetability of the policy-generation process via a reward or utility function or some other parameter. See Satisficers Tend To Seek Power: Instrumental Convergence Via Retargetability for intuitions and illustrations.

# Why am I only now posting this?

First, I've been way more excited about shard theory. I still think these theorems are really cool, though.

Second, I think the results in this paper are informative about the default incentives for decision-makers which "care about things." IE, make decisions on the basis of e.g. how many diamonds that decision leads to, or how many paperclips, and so on. However, I think that conventional accounts and worries around "utility maximization" are subtly misguided. Whenever I imagined posting this paper, I felt like "ugh sharing this result will just make it worse." I'm not looking to litigate that concern right now, but I do want to flag it.

Third, Optimal Policies Tend to Seek Power makes the "reward is the optimization target" mistake *super strongly*. Parametrically retargetable decision-makers tend to seek power makes the mistake less hard, both because it discusses utility functions and learned policies instead of optimal policies, and also thanks to edits I've made since realizing my optimization-target mistake.

# Conclusion

This paper isolates the key mechanism—retargetability—which enables the results in Optimal Policies Tend to Seek Power. This paper also takes healthy steps away from the optimal policy regime (which I consider to be a red herring for alignment) and lays out a bunch of theory I found—and still find—beautiful.

This paper is both published in a top-tier conference and, unlike the previous paper, actually has a shot of being applicable to realistic agents and training processes. Therefore, compared to the original^{[1]} optimal policy paper, I think this paper is better for communicating concerns about power-seeking to the broader ML world.

^{^}I've since updated the optimal policy paper with disclaimers about Reward is not the optimization target, so the updated version is at least passable in this regard. I still like the first paper, am proud of it, and think it was well-written within its scope. It also takes a more doomy tone about AGI risk, which seems good to me.

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 ≥nmost, as a relation on induced functions f:Θ→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 ≥nmost 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=⎛⎜⎝123312231⎞⎟⎠ and f(i|j)=Vij. The permutations are σ∈S3 with the usual action on {1,2,3}. Then we have

^{[1]}f1≥2mostf2≥2mostf3≥2mostf1 (and f2≱2mostf1). 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.A narrow point is that for a non-transitive relation, I think the notation should be something other than ≥ (maybe ≽).

But more importantly, I think we would really rather a transitive (at least acyclic) relation, if we want to interpret this is 'most θ 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 ≥no-m:

Definition (Orbit-mean dominance?): Let Of,A≠B(θ)={θ′∈Orbit|Θ(θ):f(A|θ′)≠f(B|θ′)}. Write f(B|θ)≥no-mf(A|θ) if ∀θ:∑Of,A≠B(θ)f(B|θ′)≥n∑Of,A≠B(θ)f(A|θ′).

Since the orbits are under Sd i.e. finite, it's easy to just sum over them. More generally, you could parameterize this with an arbitrary aggregator g:OrbitsΘ(f)→R in place of summation; I'm not sure whether this general form or the ∑ case should be the focus.

This is transitive for n=1 and acyclic for

^{[2]}n>1 (consider θ by θ); 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 ≤c can be represented as a pointwise inequality of functions, but I haven't thought about this too carefully.With this notion of ≥no-m, we also need a stronger version of retargetability for the main theorem to hold. For the ∑ version, this could be

Definition (scalar-retargetability): Write f is AB−−−→scalar if there exists σ∈Sd such that for all θ with f(A|θA)−f(B|θA)=c>0 we have f(B|σθA)−f(A|σθA)≥c (and likewise multiply scalar-retargetable).

Then scalar-retargetability from A to B will imply f(B|θ)≥no-mf(A|θ).

And: I think many (all?) of the main power-seeking results are already secretly in this form. For example, θ-wise comparison of ∑θ′∈Orbit|Θ(θ)IsOptimal(X|C,θ′) gives a preference relation ≥no-m identical to the relation ≥nmost. 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?

We get the same single orbit {1,2,3} for all θ a.k.a. j; the orbit elements j with f(i|j)>f(i′|j) are the columns where row i > row i′. There are always two such columns when comparing row i and row i+1 (mod 3). For example, f(1,1)=1<3=f(2,1)f(1,2)=2>1=f(2,2)f(1,3)=3>2=f(2,3) ↩︎

We exclude θ s.t. f(A|θ′)=f(B|θ′) in this version of the definition to match the behaviour of ≥nmost with n>1, and allow n-scalar-retargetability to imply ≥no-m. There's a case that you

shouldinclude them, in which case you do get transitivity, and even the stronger property: if x≤ny≤mz, then x≤nmz. I think this corresponds to looking at likelihood ratios of P(A∧¬B)::P(B∧¬A) vs. P(A)::P(B). ↩︎Compare also what would give you a total order (instead of partial order): aggregating over all of Θ at once, like ∫Θf(A|θ)dμ(θ), instead of aggregating orbitwise at each θ. ↩︎

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?

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)).