Said actions or lack thereof cause a fairly low utility differential compared to the actions in other, non-doomy hypotheses. Also I want to draw a critical distinction between "full knightian uncertainty over meteor presence or absence", where your analysis is correct, and "ordinary probabilistic uncertainty between a high-knightian-uncertainty hypotheses, and a low-knightian uncertainty one that says the meteor almost certainly won't happen" (where the meteor hypothesis will be ignored unless there's a meteor-inspired modification to what you do that's also very cheap in the "ordinary uncertainty" world, like calling your parents, because the meteor hypothesis is suppressed in decision-making by the low expected utility differentials, and we're maximin-ing expected utility)
Something analogous to what you are suggesting occurs. Specifically, let's say you assign 95% probability to the bandit game behaving as normal, and 5% to "oh no, anything could happen, including the meteor". As it turns out, this behaves similarly to the ordinary bandit game being guaranteed, as the "maybe meteor" hypothesis assigns all your possible actions a score of "you're dead" so it drops out of consideration.
The important aspect which a hypothesis needs, in order for you to ignore it, is that no matter what you do you get the same outcome, whether it be good or bad. A "meteor of bliss hits the earth and everything is awesome forever" hypothesis would also drop out of consideration because it doesn't really matter what you do in that scenario.
To be a wee bit more mathy, probabilistic mix of inframeasures works like this. We've got a probability distribution ζ∈ΔN, and a bunch of hypotheses ψi∈□X, things that take functions as input, and return expectation values. So, your prior, your probabilistic mixture of hypotheses according to your probability distribution, would be the function
It gets very slightly more complicated when you're dealing with environments, instead of static probability distributions, but it's basically the same thing. And so, if you vary your actions/vary your choice of function f, and one of the hypotheses ψi is assigning all these functions/choices of actions the same expectation value, then it can be ignored completely when you're trying to figure out the best function/choice of actions to plug in.
So, hypotheses that are like "you're doomed no matter what you do" drop out of consideration, an infra-Bayes agent will always focus on the remaining hypotheses that say that what it does matters.
Well, taking worst-case uncertainty is what infradistributions do. Did you have anything in mind that can be done with Knightian uncertainty besides taking the worst-case (or best-case)?
And if you were dealing with best-case uncertainty instead, then the corresponding analogue would be assuming that you go to hell if you're mispredicted (and then, since best-case things happen to you, the predictor must accurately predict you).
This post is still endorsed, it still feels like a continually fruitful line of research. A notable aspect of it is that, as time goes on, I keep finding more connections and crisper ways of viewing things which means that for many of the further linked posts about inframeasure theory, I think I could explain them from scratch better than the existing work does. One striking example is that the "Nirvana trick" stated in this intro (to encode nonstandard decision-theory problems), has transitioned from "weird hack that happens to work" to "pops straight out when you make all the math as elegant as possible". Accordingly, I'm working on a "living textbook" (like a textbook, but continually being updated with whatever cool new things we find) where I try to explain everything from scratch in the crispest way possible, to quickly catch up on the frontier of what we're working on. That's my current project.
I still do think that this is a large and tractable vein of research to work on, and the conclusion hasn't changed much.
Availability: Almost all times between 10 AM and PM, California time, regardless of day. Highly flexible hours. Text over voice is preferred, I'm easiest to reach on Discord. The LW Walled Garden can also be nice.
A note to clarify for confused readers of the proof. We started out by assuming □(cross→U=−10), and cross. We conclude □(cross→U=10)∨□(cross→U=0) by how the agent works. But the step from there to □⊥ (ie, inconsistency of PA) isn't entirely spelled out in this post.Pretty much, that follows from a proof by contradiction. Assume con(PA) ie ¬□⊥, and it happens to be a con(PA) theorem that the agent can't prove in advance what it will do, ie, ¬□(¬cross). (I can spell this out in more detail if anyone wants) However, combining □(cross→U=−10) and □(cross→U=10) (or the other option) gets you □(¬cross), which, along with ¬□(¬cross), gets you ⊥. So PA isn't consistent, ie, □⊥.
In the proof of Lemma 3, it should be "Finally, since χFC(z,z)=z, we have that polyFC(z)⋅polyFB∖C(z)=QFz.
Thus, QFz⋅QFx∩y∩z and QFx∩z⋅QFy∩z are both equal to polyFC(x∩z)⋅polyFB∖C(y∩z)⋅polyFC(z)⋅polyFB∖C(z).instead.
Any idea of how well this would generalize to stuff like Chicken or games with more than 2-players, 2-moves?
I don't know, we're hunting for it, relaxations of dynamic consistency would be extremely interesting if found, and I'll let you know if we turn up with anything nifty.
Looks good. Re: the dispute over normal bayesianism: For me, "environment" denotes "thingy that can freely interact with any policy in order to produce a probability distribution over histories". This is a different type signature than a probability distribution over histories, which doesn't have a degree of freedom corresponding to which policy you pick.But for infra-bayes, we can associate a classical environment with the set of probability distributions over histories (for various possible choices of policy), and then the two distinct notions become the same sort of thing (set of probability distributions over histories, some of which can be made to be inconsistent by how you act), so you can compare them.