Thanks for taking the time to write this out. 

Regarding the theorems (in the POWER paper; I've now spent some time on the current version): The abstract of the paper says: "With respect to a class of neutral reward function distributions, we provide sufficient conditions for when optimal policies tend to seek power over the environment." I didn't find a description of those sufficient conditions (maybe I just missed it?).

I'm sorry - although I think I mentioned it in passing, I did not draw sufficient attention to the fact that I've been talking about a drastically broadened version of the paper, compared to what was on arxiv when you read it. The new version should be up in a few days. I feel really bad about this - especially since you took such care in reading the arxiv version!

The theorems hold for all finite MDPs in which the formal sufficient conditions are satisfied (i.e. the required environmental symmetries exist; see proposition 6.9, theorem 6.13, corollary 6.14). For practical advice, see subsection 6.3 and beginning of section 7. 

(I shared the Overleaf with Ofer; if other lesswrong readers want to read without waiting for arxiv to update, message me! ETA: The updated version is now on arxiv.)

I further argue that we can take any MDP environment and "unroll" its state graph into a tree-with-constant-branching-factor (e.g. by adding an "action log" to the state representation) such that we get a "functionally equivalent" MDP in which the POWER (IID) of all the states are equal. My best guess is that you don't agree with this point, or think that the instrumental convergence thesis doesn't apply in a meaningful sense to such MDPs (but I don't yet understand why).

I agree that you can do that. I also think that instrumental convergence doesn't apply in such MDPs (as in, "most" goals over the environment won't incentivize any particular kind of optimal action), unless you restrict to certain kinds of reward functions. 

Fix a reward function distribution $D_{\text{iid}}^M$ in the original MDP $M$. For simplicity, let's suppose $D_{\text{iid}}^M$ is max-ent (and thus IID). Let's suppose we agree that optimal policies under $D_{\text{iid}}^M$ tend to avoid getting shut off.

Translated to the rolled-out MDP $M^{\prime }$, $D_{\text{iid}}^M$ no longer distributes reward uniformly over states. In fact, in its support, each reward function has the rather unusual property that its reward is only dependent on the current state, and not on the action log's contents. When translated into $M^{\prime }$, $D_{\text{iid}}^M$ imposes heavy structural assumptions on its reward functions, and it's not max-ent over the states of $M^{\prime }$. By the "functional equivalence", it still gives you the same optimality probabilities as before, and so it still tends to incentivize shutdown avoidance. 

However, if you take a max-ent over the rolled-out states of $M^{\prime }$, then this max-ent won't incentivize shutdown avoidance.

To see why, consider how absurdly expressive utility functions are when their domains are entire state-action histories. In Coherence arguments do not imply goal-directed behavior [LW · GW], Rohin Shah wrote:

Actually, no matter what the policy is, we can view the agent as an EU maximizer. The construction is simple: the agent can be thought as optimizing the utility function U, where U(h, a) = 1 if the policy would take action a given history h, else 0.

...

Consider the following examples:

• A robot that constantly twitches
• The agent that always chooses the action that starts with the letter “A”
• The agent that follows the policy <policy> where for every history the corresponding action in <policy> is generated randomly.

These are not goal-directed by my “definition”. However, they can all be modeled as expected utility maximizers

When defined over state-action histories, it's dead easy to write down objectives which don't pursue instrumental subgoals. 

However, how easy is it to write down state-based utility functions which do the same? I guess there's the one that maximally values dying. What else? While more examples probably exist, it seems clear that they're much harder to come by.

And so when your reward depends on your action history, this is strictly more expressive than state-based reward - so expressive that it becomes easy to directly incentivize any sequence of actions via the reward function. And thus, instrumental convergence disappears for "most objectives."

However, from our perspective, we still have a distribution over goals we might want to give the agent. And these goals are generally very structured - they aren't just randomly selected preferences over action-histories+current state. So we should still expect instrumental convergence to exist empirically (at a first approximation, perhaps via a simplicity prior over reward functions/utility functions). It just doesn't exist for most "unstructured" distributions in the unrolled environment.

The first state has the largest POWER (IID), but for most reward functions the optimal policy is to immediately transition to a lower-POWER state (even in the limit as $\gamma$ approaches 1). 

Note that the RSD optimality probability theorem (Theorem 6.13) applies here, and it correctly predicts that when $\gamma \approx 1$, most reward functions incentivize navigating to the larger set of 1-cycles (the 4 below the high-POWER state). As I explain in section 6.3, section 7, and appendix B of the new paper, you have to be careful in applying Thm 6.13, because 

The paper says: "Theorem 6.6 shows it’s always robustly instrumental and power-seeking to take actions which allow strictly more control over the future (in a graphical sense)." I don't yet understand the theorem, but is there somewhere a description of the set/distribution of MDP transition functions for which that statement applies? (Specifically, the "always robustly instrumental" part, which doesn't seem to hold in the example above.)

Yeah, I'm aware of this kind of situation. I think that that sentence from the paper was poorly worded. In the new version, I'm more careful to emphasize the environmental symmetries which are sufficient to conclude power-seeking: 

Some researchers speculate that intelligent reinforcement learning agents would be incentivized to seek resources and power in pursuit of their objectives. Other researchers are skeptical, because human-like power-seeking instincts need not be present in RL agents. To clarify this debate, we develop the first formal theory of the statistical tendencies of optimal policies in reinforcement learning. In the context of Markov decision processes, we prove that certain environmental symmetries are sufficient for optimal policies to tend to seek power over the environment. These symmetries exist in many environments in which the agent can be shut down or destroyed. We prove that for most prior beliefs one might have about the agent’s reward function (including as a special case the situations where the reward function is known), one should expect optimal policies to seek power in these environments. These policies seek power by keeping a range of options available and, when the discount rate is sufficiently close to 1, by navigating towards larger sets of potential terminal states.

(emphasis added)

See appendix B of the new paper for an example similar to yours, referenced by subsection 6.3 ("how to reason about other environments").

Are you referring here to POWER when it is defined over a reward distribution that corresponds to some simplicity prior?

Yup! POWER depends on the reward distribution; if you want to reason formally about a simplicity prior, plug it into POWER.

My argument is just that in MDPs where the state graph is a tree-with-a-constant-branching-factor—which is plausible in very complex environments—POWR (IID) is equal in all states. The argument doesn't mention description length (the description length concept arose in this thread in the context of discussing what reward function distribution should be used for defining instrumental convergence).

Right, okay, I agree with that. I think we agree about how POWER works here, but disagree about the link between optimality probability-wrt-a-distribution, and instrumental convergence.

If so, I argue that claim doesn't make sense: you can take any formal environment, however large and complex, and just add to it a simple "action logger" (that doesn't influence anything, other than effectively adding to the state representation a log of all the actions so far). If the action space is constant, the state graph of the modified MDP is a tree-with-a-constant-branching-factor; which would imply that adding that action logger somehow destroyed the applicability of the instrumental convergence thesis to that MDP; which doesn't make sense to me.

Yeah, I think that wrt the action-logger-MDP, instrumental convergence doesn't exist for goals over the new action-logger-MDP. See the earlier part of this comment.