That is: in this case at least it seems like there's concrete reason to believe we can have some cake and eat some too.

I disagree with this framing. Sure, if you have 5 different cakes, you can eat some and have some. But for any particular cake, you can't do both. Similarly, if you face 5 (or infinitely many) identical decision problems, you can choose to be updateful in some of them (thus obtaining useful Value of Information, that increases your utility in some worlds), and updateless in others (thus obtaining useful strategic coherence, that increases your utility in other worlds). The fundamental dichotomy remains as sharp, and it's misleading to imply we can surmount it. It's great to discuss, given this dichotomy, which trade-offs we humans are more comfortable making. But I've felt this was obscured in many relevant conversations.

This content-work seems primarily aimed at discovering and navigating actual problems similar to the decision-theoretic examples I'm using in my arguments. I'm more interested in gaining insights about what sorts of AI designs humans should implement. IE, the specific decision problem I'm interested in doing work to help navigate is the tiling problem.

My point is that the theoretical work you are shooting for is so general that it's closer to "what sorts of AI designs (priors and decision theories) should always be implemented", rather than "what sorts of AI designs should humans in particular, in this particular environment, implement".
And I think we won't gain insights on the former, because there are no general solutions, due to fundamental trade-offs ("no-free-lunchs").
I think we could gain many insights on the former, but that the methods better fit for that are less formal/theoretical and way messier/"eye-balling"/iterating.

Excellent explanation, congratulations! Sad I'll have to miss the discussion.

Interlocutor: Neither option is plausible. If you update, you're not dynamically consistent, and you face an incentive to modify into updatelessness. If you bound cross-branch entanglements in the prior, you need to explain why reality itself also bounds such entanglements, or else you're simply advising people to be delusional.

You found yourself a very nice interlocutor. I think we truly cannot have our cake and eat it: either you update, making you susceptible to infohazards=traps (if they exist, and they might exist), or you don't, making you entrenched forever. I think we need to stop dancing around this fact, recognize that a fully-general solution in the formalism is not possible, and instead look into the details of our particular case. Sure, our environment might be adversarially bad, traps might be everywhere. But under this uncertainty, which ways do we think are best to recognize and prevent traps (while updating on other things). This is kind of studying and predicting generalization: given my past observations, where do I think I will suddenly fall out of distribution (into a trap)?

Me: I'm not sure if that's exactly the condition, but at least it motivates the idea that there's some condition differentiating when we should be updateful vs updateless. I think uncertainty about "our own beliefs" is subtly wrong; it seems more like uncertainty about which beliefs we endorse.

This was very though-provoking, but unfortunately I still think this crashes head-on with the realization that, a priori and in full generality, we can't differentiate between safe and unsafe updates. Indeed, why would we expect that no one will punish us by updating on "our own beliefs" or "which beliefs I endorse"? After all, that's just one more part of reality (without a clear boundary separating it).

It sounds like you are correctly explaining that our choice of prior will be, in some important sense, arbitrary: we can't know the correct one in advance, we always have to rely on extrapolating contingent past observations.
But then, it seems like your reaction is still hoping that we can have our cake and eat it: "I will remain uncertain about which beliefs I endorse, and only later will I update on the fact that I am in this or that reality. If I'm in the Infinite Counterlogical Mugging... then I will just eventually change my prior because I noticed I'm in the bad world!". But then again, why would we think this update is safe? That's just not being updateless, and losing out on the strategic gains from not updating.

Since a solution doesn't exist in full generality, I think we should pivot to more concrete work related to the "content" (our particular human priors and our particular environment) instead of the "formalism". For example:

• Conceptual or empirical work on which are the robust and safe ways to extract information from humans (Suddenly LLM pre-training becomes safety work)
• Conceptual or empirical work on which actions or reasoning are more likely to unearth traps under different assumptions (although this work could unearth traps)
• Compilation or observation of properties of our environment (our physical reality) that could have some weak signal on which kinds of moves are safe
  • Unavoidably, this will involve some philosophical / almost-ethical reflection about which worlds we care about and which ones we are willing to give up.

(I will not try to prove transitivity here, since my goal is to get the overall picture across; I have not checked it, although I expect it to hold.)

Transitivity doesn't hold, here's a counterexample.

The intuitive story is: X's action tells you whether Z failed, Y fails sometimes, and Z fails more rarely.

The full counterexample (all of the following is according to your beliefs $P_1$): Say available actions are 0 and 1. There is a hidden fair coin, and your utility is high if you manage to match the coin, and low if you don't. Y peeks at the coin, and takes the correct action, except when it fails, which has a 1/4 chance. Z does the same, but it only fails with a 1/100 chance. X plays 1 iff Z has failed.
Given X's and Y's action, you always go with Y's action, since X tells you nothing about the coin, and Y gives you some information. Given Z's and Y's actions, you always go with Z's, because it's less likely to have failed (even when they disagree). But given Z's and X's, there will be some times (1/100), in which you see X played 1, and then you will not play the same as Z.

The same counterexample works for beliefs (or continuous actions) instead of discrete actions (where you will choose a probability $p\in [0,1]$ to believe, instead of an action $a\in \{0,1\}$), but needs a couple small changes. Now both Z and Y fail with 1/4 probability (independently). Also, Y outputs its guess as 0.75 or 0.25 (instead of 1 or 0), because YOU (that is, $P_1$) will be taking into account the possibility that it has failed (and Y better output whatever you will want to guess after seeing it). Instead of Z, consider A as the third expert, which outputs 0.5 if Z and Y disagree, 15/16 if they agree on yes, and 1/16 if they agree on no. X still tells you whether Z failed. Seeing Y and X, you always go with Y's guess. Seeing A and Y, you always go with A's guess. But if you see A = 15/16 and X = 1, you know both failed, and guess 0. (In fact, even when you see X = 0, you will guess 1 instead of 15/16.)

I think this has a fix-point selection problem: If one or both of them start with a different prior under which the other player punishes them for not racing / doesn't reward them enough (maybe because they have very little faith in the other's rationality, or because they think it's not within their power to decide that, and also there's not enough evidential correlation in their decisions), then they'll race.

Of course, considerations about whether the other player normatively endorses something LDT-like also enter the picture. And even if individual humans would endorse it (and that's already a medium-big if), I worry our usual decision structures (for example in AI labs) don't incentivize it (and what's the probability some convincing decision theorist cuts through them? not sure).

we have only said that P2B is the convergent instrumental goal. Whenever there are obvious actions that directly lead towards the goal, a planner should take them instead.

Hmm, given your general definition of planning, shouldn't it include realizations (and their corresponding guided actions) of the form "further thinking about this plan is worse than already acquiring some value now", so that P2B itself already includes acquiring the terminal goal (and optimizing solely for P2B is thus optimal)?

I guess your idea is "plan to P2B better" means "plan with the sole goal of improving P2B", so that it's a "non-value-laden" instrumental goal.

Since this hypothesis makes distinct predictions, it is possible for the confidence to rise above 50% after finitely many observations.

I was confused about why this is the case. I now think I've got an answer (please anyone confirm):
The description length of the Turing Machine enumerating theorems of PA is constant. The description length of any Turing Machine that enumerates theorems of PA up until time-step n and the does something else grows with n (for big enough n). Since any probability prior over Turing Machines has an implicit simplicity bias, no matter what prior we have, for big enough n the latter Turing Machines will (jointly) get arbitrarily low probability relative to the first one. Thus, after enough time-steps, given all observations are PA theorems, our listener will assign arbitrarily higher probability to the first one than all the rest, and thus the first one will be over 50%.

Edit: Okay, I now saw you mention the "getting over 50%" problem further down:

I don't know if the argument works out exactly as I sketched; it's possible that the rich hypothesis assumption needs to be "and also positive weight on a particular enumeration". Given that, we can argue: take one such enumeration; as we continue getting observations consistent with that observation, the hypothesis which predicts it loses no weight, and hypotheses which (eventually) predict other things must (eventually) lose weight; so, the updated probability eventually believes that particular enumeration will continue with probability > 1/2.

But I think the argument goes through already with the rich hypothesis assumption as initially stated. If the listener has non-zero prior probability on the speaker enumerating theorems of PA, it must have non-zero probability on it doing so in a particular enumeration. (unless our specification of the listener structure doesn't even consider different enumerations? but I was just thinking of their hypothesis space as different Turing Machines the whole time) And then my argument above goes through, which I think is just your argument + explicitly mentioning the additional required detail about the simplicity prior.

Nice!

⊬a

Should be $⊢\neg a$, right?

In particular, this theorem shows that players $i$ with very low $w_i$ (little capital/influence on $q$) will accurately predict $\hat{p}={\sum }_j{w}_j{p}_j$

You mean $f\left(\hat{p}\right)$?

Solution: Black box the whole setup and remove it from the simulation to avoid circularity.

Addendum: I now notice this amounts to brute-forcing a solution to certain particular counterfactuals.

Hi Vanessa! Thanks again for your previous answers. I've got one further concern.

        Are all mesa-optimizers really only acausal attackers?

I think mesa-optimizers don't need to be purely contained in a hypothesis (rendering them acausal attackers), but can be made up of a part of the hypotheses-updating procedures (maybe this is obvious and you already considered it).

Of course, since the only way to change the AGI's actions is by changing its hypotheses, even these mesa-optimizers will have to alter hypothesis selection. But their whole running program doesn't need to be captured inside any hypothesis (which would be easier for classifying acausal attackers away).

That is, if we don't think about how the AGI updates its hypotheses, and just consider them magically updating (without any intermediate computations), then of course, the only mesa-optimizers will be inside hypotheses. If we actually think about these computations and consider a brute-force search over all hypotheses, then again they will only be found inside hypotheses, since the search algorithm itself is too simple and provides no further room for storing a subagent (even if the mesa-optimizer somehow takes advantage of the details of the search). But if more realistically our AGI employs more complex heuristics to ever-better approximate optimal hypotheses update, mesa-optimizers can be partially or completely encoded in those (put another way, those non-optimal methods can fail / be exploited). This failure could be seen as a capabilities failure (in the trivial sense that it failed to correctly approximate perfect search), but I think it's better understood as an alignment failure.

The way I see PreDCA (and this might be where I'm wrong) is as an "outer top-level protocol" which we can fit around any superintelligence of arbitrary architecture. That is, the superintelligence will only have to carry out the hypotheses update (plus some trivial calculations over hypotheses to find the best action), and given it does that correctly, since the outer objective we've provided is clearly aligned, we're safe. That is, PreDCA is an outer objective that solves outer alignment. But we still need to ensure the hypotheses update is carried out correctly (and that's everything our AGI is really doing).

I don't think this realization rules out your Agreement solution, since if truly no hypothesis can steer the resulting actions in undesirable ways (maybe because every hypothesis with a user has the human as the user), then obviously not even optimizers in hypothesis update can find malign hypotheses (although they can still causally attack hacking the computer they're running on etc.). But I think your Agreement solution doesn't completely rule out any undesirable hypothesis, but only makes it harder for an acausal attacker to have the user not be the human. And in this situation, an optimizer in hypothesis update could still select for malign hypotheses in which the human is subtly incorrectly modelled in such a precise way that has relevant consequences for the actions chosen. This can again be seen as a capabilities failure (not modelling the human well enough), but it will always be present to some degree, and it could be exploited by mesa-optimizers.