I think that the criticism sees it the second way and so sees the arguments as not establishing what they are supposed to establish, and I see it the first way—there might be a further fact that says why OT and IC don’t apply to AGI like they theoretically should, but the burden is on you to prove it. Rather than saying that we need evidence OT and IC will apply to AGI.
I agree with that burden of proof. However, we do have evidence that IC will apply, if you think we might get AGI through RL.
I think that hypothesized AI catastrophe is usually due to power-seeking behavior and instrumental drives. Iprovedthat that optimal policies are generally power-seeking in MDPs. This is a measure-based argument, and it is formally correct under broad classes of situations, like “optimal farsighted agents tend to preserve their access to terminal states” (Optimal Farsighted Agents Tend to Seek Power, §6.2 Theorem 19) and “optimal agents generally choose paths through the future that afford strictly more options” (Generalizing the Power-Seeking Theorems, Theorem 2).
The theorems aren’t conclusive evidence:
maybe we don’t get AGI through RL
learned policies are not going to be optimal
the results don’t prove how hard it is tweak the reward function distribution, to avoid instrumental convergence (perhaps a simple approval penalty suffices! IMO: doubtful, but technically possible)
perhaps the agents inherit different mesa objectives during training
The optimality theorems + mesa optimization suggest that not only might alignment be hard because of Complexity of Value, it might also be hard for agents with very simple goals! Most final goals involve instrumental goals; agents trained through ML may stumble upon mesa optimizers, which are generalizing over these instrumental goals; the mesa optimizers are unaligned and seek power, even though the outer alignment objective was dirt-easy to specify.
But the theorems are evidence that RL leads to catastrophe at optimum, at least. We’re not just talking about “the space of all possible minds and desires” anymore.
We know there are many possible AI systems (including “powerful” ones) that are not inclined toward omnicide
Any possible (at least deterministic) policy is uniquely optimal with regard to some utility function. And many possible policies do not involve omnicide.
On its own, this point is weak; reading part of his 80K talk, I do not think it is a key part of his argument. Nonetheless, here’s why I think it’s weak:
“All states have self-loops, left hidden to reduce clutter.
In AI: A Modern Approach (3e), the agent starts at 1 and receives reward for reaching 3. The optimal policy for this reward function avoids 2, and one might suspect that avoiding 2 is instrumentally convergent. However, a skeptic might provide a reward function for which navigating to 2 is optimal, and then argue that “instrumental convergence″ is subjective and that there is no reasonable basis for concluding that 2 is generally avoided.
We can do better… for any way of independently and identically distributing reward over states, 1011 of reward functions have farsighted optimal policies which avoid 2. If we complicate the MDP with additional terminal states, this number further approaches 1.
If we suppose that the agent will be forced into 2 unless it takes preventative action, then preventative policies are optimal for 1011 of farsighted agents – no matter how complex the preventative action. Taking 2 to represent shutdown, we see that avoiding shutdown is instrumentally convergent in any MDP representing a real-world task and containing a shutdown state. We argue that this is a special case of a more general phenomenon: optimal farsighted agents tend to seek power.”
I agree that your paper strengthens the IC (and is also, in general, very cool!). One possible objection to the ICT, as traditionally formulated, has been that it’s too vague: there are lots of different ways you could define a subset of possible minds, and then a measure over that subset, and not all of these ways actually imply that “most” minds in the subset have dangerous properties. Your paper definitely makes the ICT crisper, more clearly true, and more closely/concretely linked to AI development practices.
I still think, though, that the ICT only gets us a relatively small portion of the way to believing that extinction-level alignment failures are likely. A couple of thoughts I have are:
It may be useful to distinguish between “power-seeking behavior” and omnicide (or equivalently harmful behavior). We do want AI systems to pursue power-seeking behaviors, to some extent. Making sure not to lock yourself in the bathroom, for example, qualifies as a power-seeking behavior—it’s akin to avoiding “State 2″ in your diagram—but it is something that we’d want any good house-cleaning robot to do. It’s only a particular subset of power-seeking behavior that we badly want to avoid (e.g. killing people so they can’t shut you off.)
This being said, I imagine that, if we represented the physical universe as an MDP, and defined a reward function over states, and used a sufficiently low discount rate, then the optimal policy for most reward functions probably would involve omnicide. So the result probably does port over to this special case. Still, I think that keeping in mind the distinction between omnicide and “power-seeking behavior” (in the context of some particular MDP) does reduce the ominousness of the result to some degree.
Ultimately, for most real-world tasks, I think it’s unlikely that people will develop RL systems using hand-coded reward functions (and then deploy them). I buy the framing in (e.g.) the DM “scalable agent alignment” paper, Rohin’s “narrow value learning” sequence, and elsewhere: that, over time, the RL development process will necessarily look less-and-less like “pick a reward function and then let an RL algorithm run until you get a policy that optimizes the reward function sufficiently well.” There’s seemingly just not that much that you can do using hand-written reward functions. I think that these more sophisticated training processes will probably be pretty strongly attracted toward non-omnicidal policies. At a higher level, engineers will also be attracted toward using training processes that produce benign/useful policies. They should have at least some ability to notice or foresee issues with classes of training processes, before any of them are used to produce systems that are willing and able to commit omnicide. Ultimately, in other words, I think it’s reasonable to be optimistic that we’ll do much better than random when producing the policies of advanced AI systems.
I do still think that the ICT is true, though, and I do still think that it matters: it’s (basically) necessary for establishing a high level of misalignment risk. I just don’t think it’s sufficient to establish a high level of risk (and am skeptical of certain other premises that would be sufficient to establish this).
But the theorems are evidence that RL leads to catastrophe at optimum, at least.
RL with a randomly chosen reward leads to catastrophe at optimum.
Iprovedthat that optimal policies are generally power-seeking in MDPs.
The proof is for randomly distributed rewards.
Ben’s main critique is that the goals evolve in tandem with capabilities, and goals will be determined by what humans care about. These are specific reasons to deny the conclusion of analysis of random rewards.
(A random Python program will error with near-certainty, yet somehow I still manage to write Python programs that don’t error.)
I do agree that this isn’t enough reason to say “there is no risk”, but it surely is important for determining absolute levels of risk. (See also this comment by Ben.)
Right, it’s for randomly distributed rewards. But if I show a property holds for reward functions generically, then it isn’t necessarily enough to say “we’re going to try to try to provide goals without that property”. Can we provide reward functions without that property?
Every specific attempt so far has been seemingly unsuccessful (unless you want the AI to choose a policy at random or shut down immediately). The hope might be that future goals/capability research will help, but I’m not personally convinced that researchers will receive good Bayesian evidence via their subhuman-AI experimental results.
I agree it’s relevant that we will try to build helpful agents, and might naturally get better at that. I don’t know that it makes me feel much better about future objectives being outer aligned.
ETA: also, i was referring to the point you made when i said
“the results don’t prove how hard it is tweak the reward function distribution, to avoid instrumental convergence”
Every specific attempt so far has been seemingly unsuccessful
Idk, I could say that every specific attempt made by the safety community to demonstrate risk has been seemingly unsuccessful, therefore systems must not be risky. This pretty quickly becomes an argument about priors and reference classes and such.
But I don’t really think I disagree with you here. I think this paper is good, provides support for the point “we should have good reason to believe an AI system is safe, and not assume it by default”, and responds to an in-fact incorrect argument of “but why would any AI want to kill us all, that’s just anthropomorphizing”.
But when someone says “These arguments depend on some concept of a ‘random mind’, but in reality it won’t be random, AI researchers will fix issues and goals and capabilities will evolve together towards what we want, seems like IC may or may not apply”, it seems like a response of the form “we have support for IC, not just in random minds, but also for random reward functions” has not responded to the critique and should not be expected to be convincing to that person.
Aside:
I don’t know that it makes me feel much better about future objectives being outer aligned.
I am legitimately unconvinced that it matters whether you are outer aligned at optimum. Not just being a devil’s advocate here. (I am also not convinced of the negation.)
it seems like a response of the form “we have support for IC, not just in random minds, but also for random reward functions” has not responded to the critique and should not be expected to be convincing to that person.
I agree that the paper should not be viewed as anything but slight Bayesian evidence for the difficulty of real objective distributions. IIRC I was trying to reply to the point of “but how do we know IC even exists?” with “well, now we can say formal things about it and show that it exists generically, but (among other limitations) we don’t (formally) know how hard it is to avoid if you try”.
I find myself agreeing with the idea that an agent unaware of it’s task will seek power, but also conclude that an agent aware of it’s task will give-up power.
I think this is a slight misunderstanding of the theory in the paper. I’d translate the theory of the paper to English as:
If we do not know an agent’s goal, but we know that the agent knows its goal and is optimal w.r.t it, then from our perspective the agent is more likely to go to higher-power states. (From the agent’s perspective, there is no probability, it always executes the deterministic perfect policy for its reward function.)
Any time the paper talks about “distributions” over reward functions, it’s talking from our perspective. The way the theory does this is by saying that first a reward function is drawn from the distribution, then it is given to the agent, then the agent thinks really hard, and then the agent executes the optimal policy. All of the theoretical analysis in the paper is done “before” the reward function is drawn, but there is no step where the agent is doing optimization but doesn’t know its reward.
In your paper, theorem 19 suggests that given a choice between two sets of 1-cycles C1 and C2 the agent is more likely to select the larger set.
I’d rewrite this as:
Theorem 19 suggests that, if an agent that knows its reward is about to choose between C1 and C2, but we don’t know the reward and our prior is that it is uniformly distributed, then we will assign higher probability to the agent going to the larger set.
I do not see how the agent ‘seeks’ out powerful states because, as you say, the agent is fixed.
I do think this is mostly a matter of translation of math to English being hard. Like, when Alex says “optimal agents seek power”, I think you should translate it as “when we don’t know what goal an optimal agent has, we should assign higher probability that it will go to states that have higher power”, even though the agent itself is not thinking “ah, this state is powerful, I’ll go there”.
Great observation. Similarly, a hypothesis called “Maximum Causal Entropy” once claimed that physical systems involving intelligent actors tended tended towards states where the future could be specialized towards many different final states, and that maybe this was even part of what intelligence was. However, people objected: (monogamous) individuals don’t perpetually maximize their potential partners—they actually pick a partner, eventually.
My position on the issue is: most agents steer towards states which afford them greater power, and sometimes most agents give up that power to achieve their specialized goals. The point, however, is that they end up in the high-power states at some point in time along their optimal trajectory. I imagine that this is sufficient for the catastrophic power-stealing incentives: the AI only has to disempower us once for things to go irreversibly wrong.
If there’s a collection of ‘turned-off’ terminal states where the agent receives no further reward for all time then every optimized policy will try to avoid such a state.
To clarify, I don’t assume that. The terminal states, even those representing the off-switch, also have their reward drawn from the same distribution. When you distribute reward IID over states, the off-state is in fact optimal for some low-measure subset of reward functions.
But, maybe you’re saying “for realistic distributions, the agent won’t get any reward for being shut off and therefore π∗ won’t ever let itself be shut off”. I agree, and this kind of reasoning is captured by Theorem 3 of Generalizing the Power-Seeking Theorems. The problem is that this is just a narrow example of the more general phenomenon. What if we add transient “obedience” rewards, what then? For some level of farsightedness (γ close enough to 1), the agent will still disobey, and simultaneously disobedience gives it more control over the future.
The paper doesn’t draw the causal diagram “Power → instrumental convergence”, it gives sufficient conditions for power-seeking being instrumentally convergent. Cycle reachability preservation is one of those conditions.
In general, I’d suspect that there are goals we could give the agent that significantly reduce our gain. However, I’d also suspect the opposite.
Yes, right. The point isn’t that alignment is impossible, but that you have to hit a low-measure set of goals which will give you aligned or non-power-seeking behavior. The paper helps motivate why alignment is generically hard and catastrophic if you fail.
It seems reasonable to argue that we would if we could guarantee r=h.
Yes, if r=h, introduce the agent. You can formalize a kind of “alignment capability” by introducing a joint distribution over the human’s goals and the induced agent goals (preliminary Overleaf notes). So, if we had goal X, we’d implement an agent with goal X’, and so on. You then take our expected optimal value under this distribution and find whether you’re good at alignment, or whether you’re bad and you’ll build agents whose optimal policies tend to obstruct you.
There might be a way to argue over randomness and say this would double our gain.
The doubling depends on the environment structure. There are game trees and reward functions where this holds, and some where it doesn’t.
More speculatively, what if |r−h|<ϵ?
If the rewards are ϵ-close in sup-norm, then you can get nice regret bounds, sure.
Great question. One thing you could say is that an action is power-seeking compared to another, if your expected (non-dominated subgraph; see Figure 19) power is greater for that action than for the other.
My understanding of figure 7 of your paper indicates that cycle reachability cannot be a sufficient condition.
Shortly after Theorem 19, the paper says: “In appendix C.6.2, we extend this reasoning to k-cycles (k >1) via theorem 53 and explain how theorem19 correctly handles fig. 7”. In particular, see Figure 19.
The key insight is that Theorem 19 talks about how many agents end up in a set of terminal states, not how many go through a state to get there. If you have two states with disjoint reachable terminal state sets, you can reason about the phenomenon pretty easily. Practically speaking, this should often suffice: for example, the off-switch state is disjoint from everything else.
If not, you can sometimes consider the non-dominated subgraph in order to regain disjointness. This isn’t in the main part of the paper, but basically you toss out transitions which aren’t part of a trajectory which is strictly optimal for some reward function. Figure 19 gives an example of this.
The main idea, though, is that you’re reasoning about what the agent’s end goals tend to be, and then say “it’s going to pursue some way of getting there with much higher probability, compared to this small set of terminal states (ie shutdown)”. Theorem 17 tells us that in the limit, cycle reachability totally controls POWER.
I think I still haven’t clearly communicated all my mental models here, but I figured I’d write a reply now while I update the paper.
Thank you for these comments, by the way. You’re pointing out important underspecifications. :)
My philosophy is that aligned/general is OK based on a shared (?) premise that,
I think one problem is that power-seeking agents are generally not that corrigible, which means outcomes are extremely sensitive to the initial specification.
I agree with that burden of proof. However, we do have evidence that IC will apply, if you think we might get AGI through RL.
I think that hypothesized AI catastrophe is usually due to power-seeking behavior and instrumental drives. I proved that that optimal policies are generally power-seeking in MDPs. This is a measure-based argument, and it is formally correct under broad classes of situations, like “optimal farsighted agents tend to preserve their access to terminal states” (Optimal Farsighted Agents Tend to Seek Power, §6.2 Theorem 19) and “optimal agents generally choose paths through the future that afford strictly more options” (Generalizing the Power-Seeking Theorems, Theorem 2).
The theorems aren’t conclusive evidence:
maybe we don’t get AGI through RL
learned policies are not going to be optimal
the results don’t prove how hard it is tweak the reward function distribution, to avoid instrumental convergence (perhaps a simple approval penalty suffices! IMO: doubtful, but technically possible)
perhaps the agents inherit different mesa objectives during training
The optimality theorems + mesa optimization suggest that not only might alignment be hard because of Complexity of Value, it might also be hard for agents with very simple goals! Most final goals involve instrumental goals; agents trained through ML may stumble upon mesa optimizers, which are generalizing over these instrumental goals; the mesa optimizers are unaligned and seek power, even though the outer alignment objective was dirt-easy to specify.
But the theorems are evidence that RL leads to catastrophe at optimum, at least. We’re not just talking about “the space of all possible minds and desires” anymore.
Also
In the linked slides, the following point is made in slide 43:
On its own, this point is weak; reading part of his 80K talk, I do not think it is a key part of his argument. Nonetheless, here’s why I think it’s weak:
I agree that your paper strengthens the IC (and is also, in general, very cool!). One possible objection to the ICT, as traditionally formulated, has been that it’s too vague: there are lots of different ways you could define a subset of possible minds, and then a measure over that subset, and not all of these ways actually imply that “most” minds in the subset have dangerous properties. Your paper definitely makes the ICT crisper, more clearly true, and more closely/concretely linked to AI development practices.
I still think, though, that the ICT only gets us a relatively small portion of the way to believing that extinction-level alignment failures are likely. A couple of thoughts I have are:
It may be useful to distinguish between “power-seeking behavior” and omnicide (or equivalently harmful behavior). We do want AI systems to pursue power-seeking behaviors, to some extent. Making sure not to lock yourself in the bathroom, for example, qualifies as a power-seeking behavior—it’s akin to avoiding “State 2″ in your diagram—but it is something that we’d want any good house-cleaning robot to do. It’s only a particular subset of power-seeking behavior that we badly want to avoid (e.g. killing people so they can’t shut you off.)
This being said, I imagine that, if we represented the physical universe as an MDP, and defined a reward function over states, and used a sufficiently low discount rate, then the optimal policy for most reward functions probably would involve omnicide. So the result probably does port over to this special case. Still, I think that keeping in mind the distinction between omnicide and “power-seeking behavior” (in the context of some particular MDP) does reduce the ominousness of the result to some degree.
Ultimately, for most real-world tasks, I think it’s unlikely that people will develop RL systems using hand-coded reward functions (and then deploy them). I buy the framing in (e.g.) the DM “scalable agent alignment” paper, Rohin’s “narrow value learning” sequence, and elsewhere: that, over time, the RL development process will necessarily look less-and-less like “pick a reward function and then let an RL algorithm run until you get a policy that optimizes the reward function sufficiently well.” There’s seemingly just not that much that you can do using hand-written reward functions. I think that these more sophisticated training processes will probably be pretty strongly attracted toward non-omnicidal policies. At a higher level, engineers will also be attracted toward using training processes that produce benign/useful policies. They should have at least some ability to notice or foresee issues with classes of training processes, before any of them are used to produce systems that are willing and able to commit omnicide. Ultimately, in other words, I think it’s reasonable to be optimistic that we’ll do much better than random when producing the policies of advanced AI systems.
I do still think that the ICT is true, though, and I do still think that it matters: it’s (basically) necessary for establishing a high level of misalignment risk. I just don’t think it’s sufficient to establish a high level of risk (and am skeptical of certain other premises that would be sufficient to establish this).
RL with a randomly chosen reward leads to catastrophe at optimum.
The proof is for randomly distributed rewards.
Ben’s main critique is that the goals evolve in tandem with capabilities, and goals will be determined by what humans care about. These are specific reasons to deny the conclusion of analysis of random rewards.
(A random Python program will error with near-certainty, yet somehow I still manage to write Python programs that don’t error.)
I do agree that this isn’t enough reason to say “there is no risk”, but it surely is important for determining absolute levels of risk. (See also this comment by Ben.)
Right, it’s for randomly distributed rewards. But if I show a property holds for reward functions generically, then it isn’t necessarily enough to say “we’re going to try to try to provide goals without that property”. Can we provide reward functions without that property?
Every specific attempt so far has been seemingly unsuccessful (unless you want the AI to choose a policy at random or shut down immediately). The hope might be that future goals/capability research will help, but I’m not personally convinced that researchers will receive good Bayesian evidence via their subhuman-AI experimental results.
I agree it’s relevant that we will try to build helpful agents, and might naturally get better at that. I don’t know that it makes me feel much better about future objectives being outer aligned.
ETA: also, i was referring to the point you made when i said
“the results don’t prove how hard it is tweak the reward function distribution, to avoid instrumental convergence”
Idk, I could say that every specific attempt made by the safety community to demonstrate risk has been seemingly unsuccessful, therefore systems must not be risky. This pretty quickly becomes an argument about priors and reference classes and such.
But I don’t really think I disagree with you here. I think this paper is good, provides support for the point “we should have good reason to believe an AI system is safe, and not assume it by default”, and responds to an in-fact incorrect argument of “but why would any AI want to kill us all, that’s just anthropomorphizing”.
But when someone says “These arguments depend on some concept of a ‘random mind’, but in reality it won’t be random, AI researchers will fix issues and goals and capabilities will evolve together towards what we want, seems like IC may or may not apply”, it seems like a response of the form “we have support for IC, not just in random minds, but also for random reward functions” has not responded to the critique and should not be expected to be convincing to that person.
Aside:
I am legitimately unconvinced that it matters whether you are outer aligned at optimum. Not just being a devil’s advocate here. (I am also not convinced of the negation.)
I agree that the paper should not be viewed as anything but slight Bayesian evidence for the difficulty of real objective distributions. IIRC I was trying to reply to the point of “but how do we know IC even exists?” with “well, now we can say formal things about it and show that it exists generically, but (among other limitations) we don’t (formally) know how hard it is to avoid if you try”.
I think I agree with most of what you’re arguing.
[Deleted]
I think this is a slight misunderstanding of the theory in the paper. I’d translate the theory of the paper to English as:
Any time the paper talks about “distributions” over reward functions, it’s talking from our perspective. The way the theory does this is by saying that first a reward function is drawn from the distribution, then it is given to the agent, then the agent thinks really hard, and then the agent executes the optimal policy. All of the theoretical analysis in the paper is done “before” the reward function is drawn, but there is no step where the agent is doing optimization but doesn’t know its reward.
I’d rewrite this as:
[Deleted]
I do think this is mostly a matter of translation of math to English being hard. Like, when Alex says “optimal agents seek power”, I think you should translate it as “when we don’t know what goal an optimal agent has, we should assign higher probability that it will go to states that have higher power”, even though the agent itself is not thinking “ah, this state is powerful, I’ll go there”.
Great observation. Similarly, a hypothesis called “Maximum Causal Entropy” once claimed that physical systems involving intelligent actors tended tended towards states where the future could be specialized towards many different final states, and that maybe this was even part of what intelligence was. However, people objected: (monogamous) individuals don’t perpetually maximize their potential partners—they actually pick a partner, eventually.
My position on the issue is: most agents steer towards states which afford them greater power, and sometimes most agents give up that power to achieve their specialized goals. The point, however, is that they end up in the high-power states at some point in time along their optimal trajectory. I imagine that this is sufficient for the catastrophic power-stealing incentives: the AI only has to disempower us once for things to go irreversibly wrong.
[Deleted]
To clarify, I don’t assume that. The terminal states, even those representing the off-switch, also have their reward drawn from the same distribution. When you distribute reward IID over states, the off-state is in fact optimal for some low-measure subset of reward functions.
But, maybe you’re saying “for realistic distributions, the agent won’t get any reward for being shut off and therefore π∗ won’t ever let itself be shut off”. I agree, and this kind of reasoning is captured by Theorem 3 of Generalizing the Power-Seeking Theorems. The problem is that this is just a narrow example of the more general phenomenon. What if we add transient “obedience” rewards, what then? For some level of farsightedness (γ close enough to 1), the agent will still disobey, and simultaneously disobedience gives it more control over the future.
The paper doesn’t draw the causal diagram “Power → instrumental convergence”, it gives sufficient conditions for power-seeking being instrumentally convergent. Cycle reachability preservation is one of those conditions.
Yes, right. The point isn’t that alignment is impossible, but that you have to hit a low-measure set of goals which will give you aligned or non-power-seeking behavior. The paper helps motivate why alignment is generically hard and catastrophic if you fail.
Yes, if r=h, introduce the agent. You can formalize a kind of “alignment capability” by introducing a joint distribution over the human’s goals and the induced agent goals (preliminary Overleaf notes). So, if we had goal X, we’d implement an agent with goal X’, and so on. You then take our expected optimal value under this distribution and find whether you’re good at alignment, or whether you’re bad and you’ll build agents whose optimal policies tend to obstruct you.
The doubling depends on the environment structure. There are game trees and reward functions where this holds, and some where it doesn’t.
If the rewards are ϵ-close in sup-norm, then you can get nice regret bounds, sure.
[Deleted]
The freshly updated paper answers this question in great detail; see section 6 and also appendix B.
Great question. One thing you could say is that an action is power-seeking compared to another, if your expected (non-dominated subgraph; see Figure 19) power is greater for that action than for the other.
Power is kinda weird when defined for optimal agents, as you say—when γ=1, POWER can only decrease. See Power as Easily Exploitable Opportunities for more on this.
Shortly after Theorem 19, the paper says: “In appendix C.6.2, we extend this reasoning to k-cycles (k >1) via theorem 53 and explain how theorem19 correctly handles fig. 7”. In particular, see Figure 19.
The key insight is that Theorem 19 talks about how many agents end up in a set of terminal states, not how many go through a state to get there. If you have two states with disjoint reachable terminal state sets, you can reason about the phenomenon pretty easily. Practically speaking, this should often suffice: for example, the off-switch state is disjoint from everything else.
If not, you can sometimes consider the non-dominated subgraph in order to regain disjointness. This isn’t in the main part of the paper, but basically you toss out transitions which aren’t part of a trajectory which is strictly optimal for some reward function. Figure 19 gives an example of this.
The main idea, though, is that you’re reasoning about what the agent’s end goals tend to be, and then say “it’s going to pursue some way of getting there with much higher probability, compared to this small set of terminal states (ie shutdown)”. Theorem 17 tells us that in the limit, cycle reachability totally controls POWER.
I think I still haven’t clearly communicated all my mental models here, but I figured I’d write a reply now while I update the paper.
Thank you for these comments, by the way. You’re pointing out important underspecifications. :)
I think one problem is that power-seeking agents are generally not that corrigible, which means outcomes are extremely sensitive to the initial specification.