This arbitrary choice effectively unrolls the state graph into a tree with a constant branching factor (+ self-loops in the terminal states) and we get that the POWER of all the states is equal.
Not necessarily true—you’re still considering the IID case.
I think using a well-chosen reward distribution is necessary, otherwise POWER depends on arbitrary choices in the design of the MDP’s state graph. E.g. suppose the student in the above example writes about every action they take in a blog that no one reads, and we choose to include the content of the blog as part of the MDP state.
Yes, if you insist in making really weird modelling choices (and pretending the graph still well-models the original situation, even though it doesn’t), you can make POWER say weird things. But again, I can prove that up to a large range of perturbation, most distributions will agree that some obvious states have more POWER than other obvious states.
Your original claim was that POWER isn’t a good formalization of intuitive-power/influence. You seem to be arguing that because there exists a situation “modelled” by an adversarially chosen environment grounding such that POWER returns “counterintuitive” outputs (are they really counterintuitive, given the information available to the formalism?), therefore POWER is inappropriately sensitive to the reward function distribution. Therefore, it’s not a good formalization of intuitive-power.
I deny both of the ‘therefores.’
The right thing to do is just note that there is some dependence on modelling choices, which is another consideration to weigh (especially as we move towards more sophisticated application of the theorems to e.g. distributions over mesa objectives and their attendant world models). But you should sure that the POWER-seeking conclusions hold under plausible modelling choices, and not just the specific one you might have in mind. I think that my theorems show that they do in many reasonable situations (this is a bit argumentatively unfair of me, since the theorems aren’t public yet, but I’m happy to give you access).
If this doesn’t resolve your concern and you want to talk more about this, I’d appreciate taking this to video, since I feel like we may be talking past each other.
Just to summarize my current view: For MDP problems in which the state representation is very complex, and different action sequences always yield different states, POWER-defined-over-an-IID-reward-distribution is equal for all states, and thus does not match the intuitive concept of power.
At some level of complexity such problems become relevant (when dealing with problems with real-world-like environments). These are not just problems that show up when one adverserially constructs an MDP problem to game POWER, or when one makes “really weird modelling choices”. Consider a real-world inspired MDP problem where a state specifies the location of every atom. What makes POWER-defined-over-IID problematic in such an environment is the sheer complexity of the state, which makes it so that different action sequences always yield different states. It’s not “weird modeling decisions” causing the problem.
I also (now) think that for some MDP problems (including many grid-world problems), POWER-defined-over-IID may indeed match the intuitive concept of power well, and that publications about such problems (and theorems about POWER-defined-over-IID) may be very useful for the field. Also, I see that the abstract of the paper no longer makes the claim “We prove that, with respect to a wide class of reward function distributions, optimal policies tend to seek power over the environment”, which is great (I was concerned about that claim).
Not necessarily true—you’re still considering the IID case.
Yes, if you insist in making really weird modelling choices (and pretending the graph still well-models the original situation, even though it doesn’t), you can make POWER say weird things. But again, I can prove that up to a large range of perturbation, most distributions will agree that some obvious states have more POWER than other obvious states.
Your original claim was that POWER isn’t a good formalization of intuitive-power/influence. You seem to be arguing that because there exists a situation “modelled” by an adversarially chosen environment grounding such that POWER returns “counterintuitive” outputs (are they really counterintuitive, given the information available to the formalism?), therefore POWER is inappropriately sensitive to the reward function distribution. Therefore, it’s not a good formalization of intuitive-power.
I deny both of the ‘therefores.’
The right thing to do is just note that there is some dependence on modelling choices, which is another consideration to weigh (especially as we move towards more sophisticated application of the theorems to e.g. distributions over mesa objectives and their attendant world models). But you should sure that the POWER-seeking conclusions hold under plausible modelling choices, and not just the specific one you might have in mind. I think that my theorems show that they do in many reasonable situations (this is a bit argumentatively unfair of me, since the theorems aren’t public yet, but I’m happy to give you access).
If this doesn’t resolve your concern and you want to talk more about this, I’d appreciate taking this to video, since I feel like we may be talking past each other.
EDIT: Removed a distracting analogy.
Just to summarize my current view: For MDP problems in which the state representation is very complex, and different action sequences always yield different states, POWER-defined-over-an-IID-reward-distribution is equal for all states, and thus does not match the intuitive concept of power.
At some level of complexity such problems become relevant (when dealing with problems with real-world-like environments). These are not just problems that show up when one adverserially constructs an MDP problem to game POWER, or when one makes “really weird modelling choices”. Consider a real-world inspired MDP problem where a state specifies the location of every atom. What makes POWER-defined-over-IID problematic in such an environment is the sheer complexity of the state, which makes it so that different action sequences always yield different states. It’s not “weird modeling decisions” causing the problem.
I also (now) think that for some MDP problems (including many grid-world problems), POWER-defined-over-IID may indeed match the intuitive concept of power well, and that publications about such problems (and theorems about POWER-defined-over-IID) may be very useful for the field. Also, I see that the abstract of the paper no longer makes the claim “We prove that, with respect to a wide class of reward function distributions, optimal policies tend to seek power over the environment”, which is great (I was concerned about that claim).