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Thanks. Is there a particular source whose notation yours most aligns with?

When you write  I understand that to mean that  for all . But when I look up definitions of conditional probability it seems that that notation would usually mean  for all  

Am I confused or are you just using non-standard notation?

Fair enough, but in that example making irreversible decisions is unavoidable. What if we consider a modified tree such that one and only one branch is traversible in both directions, and utility can be anywhere? 

I expect we get that the reversible brach is the most popular across the distribution of utility functions (but not necessarily that most utility functions prefer it). That sounds like cause for optimism—'optimal policies tend to avoid irreversible changes'.

I've been thinking about whether these results could be interpeted pretty differently under different branding.

The current framing, if I understand it correctly, is something like, 'Powerseeking is not desirable. We can prove that keeping your options open tends to be optimal and tends to meet a plausible definition of powerseeking. Therefore we should expect RL agents to seek power, which is bad.'

An alternative framing would be, 'Making irreversible changes is not desirable. We can prove that keeping your options open tends to be optimal. Therefore we should not expect RL agents to make irreversible changes, which is good.'

I don't think that the second framing is better than the first, but I do think that if you had run with it instead then lots of people would be nodding their heads and feeling reassured about corrigibility, instead of feeling like their views about instrumental convergence had been confirmed. That makes me feel like we shouldn't update our views too much based on formal results that leave so much room for interpretation. If I showed a bunch of theorems about MDPs, with no exposition, to two people with different opinions about alignment, I expect they might come to pretty different conclusions about what they meant. 

What do you think?

(To be clear I think this is a great post and paper, I just worry that there are pitfalls when it comes to interpretation.)