Diffractor

Agreed. The bargaining solution for the entire game can be very different from adding up the bargaining solutions for the subgames. If there's a subgame where Alice cares very much about victory in that subgame (interior decorating choices) and Bob doesn't care much, and another subgame where Bob cares very much about it (food choice) and Alice doesn't care much, then the bargaining solution of the entire relationship game will end up being something like "Alice and Bob get some relative weights on how important their preferences are, and in all the subgames, the weighted sum of their utilities is maximized. Thus, Alice will be given Alice-favoring outcomes in the subgames where she cares the most about winning, and Bob will be given Bob-favoring outcomes in the subgames where he cares the most about winning"

And in particular, since it's a sequential game, Alice can notice if Bob isn't being fair, and enforce the bargaining solution by going "if you're not aiming for something sorta like this, I'll break off the relationship". So, from Bob's point of view, aiming for any outcome that's too Bob-favoring has really low utility since Alice will inevitably catch on. (this is the time-extended version of "give up on achieving any outcome that drives the opponent below their BATNA") Basically, in terms of raw utility, it's still a bargaining game deep down, but once both sides take into account how the other will react, the payoff matrix for the restaurant game (taking the future interactions into account) will look like "it's a really bad idea to aim for an outcome the other party would regard as unfair"

Actually, they apply anyways in *all* circumstances, not *just* after the rescaling and shifting is done! Scale-and-shift invariance means that no matter how you stretch and shift the two axes, the bargaining solution always hits the same probability-distribution over outcomes, so monotonicity means "if you increase the payoff numbers you assign for some or all of the outcomes, the Pareto frontier point you hit will give you an increased number for your utility score over what it'd be otherwise" (no matter how you scale-and-shift). And independence of irrelevant alternatives says "you can remove any option that you have 0 probability of taking and you'll still get the same probability-distribution over outcomes as you would in the original game" (no matter how you scale-and-shift)

If you're looking for curriculum materials, I believe that the most useful reference would probably be my "Infra-exercises", a sequence of posts containing all the math exercises you need to reinvent a good chunk of the theory yourself. Basically, it's the textbook's exercise section, and working through interesting math problems and proofs on one's own has a *much* better learning feedback loop and retention of material than slogging through the old posts. The exercises are short on motivation and philosophy compared to the posts, though, much like how a functional analysis textbook takes for granted that you want to learn functional analysis and doesn't bother motivating it.

The primary problem is that the exercises aren't particularly calibrated in terms of difficulty, and in order for me to get useful feedback, someone has to actually work through all of them, so feedback has been a bit sparse. So I'm stuck in a situation where I keep having to link *everyone* to the infra-exercises over and over and it'd be really good to just get them out and publicly available, but if they're as important as I think, then the best move is something like "release them one at a time and have a bunch of people work through them as a group" like the fixpoint exercises, instead of "just dump them all as public documents".

I'll ask around about speeding up the public - ation of the exercises and see what can be done there.

I'd strongly endorse linking this introduction even if the exercises are linked as well, because this introduction serves as the table of contents to all the other applicable posts.

So, if you make Nirvana infinite utility, yes, the fairness criterion becomes "if you're mispredicted, you have any probability at all of entering the situation where you're mispredicted" instead of "have a significant probability of entering the situation where you're mispredicted", so a lot more decision-theory problems can be captured if you take Nirvana as infinite utility. But, I talk in another post in this sequence (I think it was "the many faces of infra-beliefs") about why you want to do Nirvana as 1 utility instead of infinite utility.

Parfit's Hitchiker with a perfect predictor is a perfectly fine acausal decision problem, we can still represent it, it just cannot be represented as an infra-POMDP/causal decision problem.

Yes, the fairness criterion is tightly linked to the pseudocausality condition. Basically, the acausal->pseudocausal translation is the part where the accuracy of the translation might break down, and once you've got something in pseudocausal form, translating it to causal form from there by adding in Nirvana won't change the utilities much.

So, the flaw in your reasoning is after updating we're in the city, doesn't go "logically impossible, infinite utility". We just go "alright, off-history measure gets converted to 0 utility", a perfectly standard update. So updates to (0,0) (ie, there's 0 probability I'm in this situation in the first place, and my expected utility for not getting into this situation in the first place is 0, because of probably dying in the desert)

As for the proper way to do this analysis, it's a bit finicky. There's something called "acausal form", which is the fully general way of representing decision-theory problems. Basically, you just give an infrakernel that tells you your uncertainty over which history will result, for each of your policies.

So, you'd have

Ie, if you pay, 99 percent chance of ending up alive but paying and 1 percent chance of dying in the desert, if you don't pay, 99 percent chance of dying in the desert and 1 percent chance of cheating them, no extra utility juice on either one.

You update on the event "I'm alive". The off-event utility function is like "being dead would suck, 0 utility". So, your infrakernel updates to (leaving off the scale-and-shift factors, which doesn't affect anything)

Because, the probability mass on "die in desert" got burned and turned into utility juice, 0 of it since it's the worst thing. Let's say your utility function assigns 0.5 utility to being alive and rich, and 0.4 utility to being alive and poor. So the utility of the first policy is , and the utility of the second policy is , so it returns the same answer of paying up. It's basically thinking "if I don't pay, I'm probably not in this situation in the first place, and the utility of "I'm not in this situation in the first place" is also about as low as possible."

BUT

There's a very mathematically natural way to translate any decision theory to "causal form", and as it turns out, the process which falls directly out of the math is that thing where you go "hard-code in all possible policies, go to Nirvana if I behave differently from the hard-coded policy". This has an advantage and a disadvantage. The advantage is that now your decision-theory problem is in the form of an infra-POMDP, a much more restrictive form, so you've got a much better shot at actually developing a practical algorithm for it. The disadvantage is that not all decision-theory problems survive the translation process unchanged. Speaking informally the "fairness criterion" to translate a decision-theory problem into causal form without too much loss in fidelity is something like "if I was mispredicted, would I actually have a good shot at entering the situation where I was mispredicted to prove the prediction wrong".

Counterfactual mugging fits this. If Omega flubs its prediction, you've got a 50 percent chance of being able to prove it wrong.

XOR blackmail fits this. If the blackmailer flubs its prediction and thinks you'll pay up, you've got like a 90 percent chance of being able to prove it wrong.

Newcomb's problem fits this. If Omega flubs its prediction and thinks you'll 2-box, you'll definitely be able to prove it wrong.

Transparent Newcomb and Parfait's Hitchiker don't fit this "fairness property" (especially for 100 percent accuracy), and so when you translate them to a causal problem, it ruins things. If the predictor screws up and thinks you'll 2-box on seeing a filled transparent box/won't pay up on seeing you got saved, then the transparent box is empty/you die in the desert, and you don't have a significant shot at proving them wrong.

Let's see what's going wrong. Our two a-environments are

Update on the event "I didn't die in the desert". Then, neglecting scale-and-shift, our two a-environments are

Letting N be the utility of Nirvana,

If you pay up, then the expected utilities of these are and

If you don't pay up, then the expected utilities of these are and

Now, if N is something big like 100, then the worst-case utilities of the policies are 0.396 vs 0.005, as expected, and you pay up. But if N is something like 1, then the worst-case utilities of the policies are 0.01 vs 0.005, which... well, it technicallygets the right answer, but those numbers are suspiciously close to each other, the agent isn't thinking properly. And so, without too much extra effort tweaking the problem setup, it's possible to generate decision-theory problems where the agent just straight-up makes the wrong decision after changing things to the causal setting.

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 , and a bunch of hypotheses , 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 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.

Yeah, "transferrable utility games" are those where there is a resource, and the utilities of all players are linear in that resource (in order to redenominate everyone's utilities as being denominated in that resource modulo a shift factor). I believe the post mentioned this.