Pascal's Muggle: Infinitesimal Priors and Strong Evidence

I don't like to be a bearer of bad news here, but it ought to be stated. This whole leverage ratio idea is very obviously an intelligent kludge / patch / work around because you have two base level theories that either don't work together or don't work individually.

You already know that something doesn't work. That's what the original post was about and that's what this post tries to address. But this is a clunky inelegant patch, that's fine for a project or a website, but given belief in the rest of your writings on AI, this is high stakes. At those stakes saying "we know it doesn't work, but we patched the bugs we found" is not acceptable.

The combination of your best guess at picking the rigtht decision theory and your best guess at epistemology produces absurd conclusions. Note that you allready know this. This knowledge which you already have motivated this post.

The next step is to identify which is wrong, the decision theory or the epistemology. After that you need to find something that's not wrong to replace it. That sucks, it's probably extreamly hard, and it probably sets you back to square one on multiple points. But you can't know that one of your foundations is wrong and just keep going. Once you know you are wrong you need to act consistently with that.

Mugger: Give me five dollars, and I'll save 3↑↑↑3 lives using my Matrix Powers.

Me: I'm not sure about that.

Mugger: So then, you think the probability I'm telling the truth is on the order of 1/3↑↑↑3?

Me: Actually no. I'm just not sure I care as much about your 3↑↑↑3 simulated people as much as you think I do.

Mugger: "This should be good."

Me: There's only something like n=10^10 neurons in a human brain, and the number of possible states of a human brain exponential in n. This is stupidly tiny compared to 3↑↑↑3, so most of the lives you're saving will be heavily duplicated. I'm not really sure that I care about duplicates that much.

Mugger: Well I didn't say they would all be humans. Haven't you read enough Sci-Fi to know that you should care about all possible sentient life?

Me: Of course. But the same sort of reasoning implies that, either there are a lot of duplicates, or else most of the people you are talking about are incomprehensibly large, since there aren't that many small Turing machines to go around. And it's not at all obvious to me that you can describe arbitrarily large minds whose existence I should care about without using up a lot of complexity. More generally, I can't see any way to describe worlds which I care about to a degree that vastly outgrows their complexity. My values are complicated.

This post has not at all misunderstood my suggestion from long ago, though I don't think I thought about it very much at the time. I agree with the thrust of the post that a leverage factor seems to deal with the basic problem, though of course I'm also somewhat expecting more scenarios to be proposed to upset the apparent resolution soon.

Hm, a linear "leverage penalty" sounds an awful lot like adding the complexity of locating you of the pool of possibilities to the total complexity.

Thing 2: consider the case of the other people on that street when the Pascal's Muggle-ing happens. Suppose they could overhear what is being said. Since they have no leverage of their own, are they free to assign a high probability to the muggle helping 3^^^3 people? Do a few of them start forward to interfere, only to be held back by the cooler heads who realize that all who interfere will suddenly have the probability of success reduced by a factor of 3^^^3?

A simplified version of the argument here:

• The utility function isn't up for grabs.
• Therefore, we need unbounded utility.
• Oops! If we allow unbounded utility, we can get non-convergence in our expectation.
• Since we've already established that the utility function is not up for grabs, let's try and modify the probability to fix this!

My response to this is that the probability distribution is even less up for grabs. The utility, at least, is explicitly there to reflect our preferences. If we see that a utility function is causing our agent to take the wrong actions, then it makes sense to change it to better reflect the actions we wish our agent to take.

The probability distribution, on the other hand, is a map that should reflect the territory as well as possible! It should not be modified on account of badly-behaved utility computations.

This may be taken as an argument in favor of modifying the utility function; Sniffnoy makes a case for bounded utility in another comment.

It could alternatively be taken as a case for modifying the decision procedure. Perhaps neither the probability nor the utility are "up for grabs", but how we use them should be modified.

One (somewhat crazy) option is to take the median expectation rather than the mean expectation: we judge actions by computing the lowest utility score that we have 50% chance of making or beating, rather than by computing the average. This makes the computation insensitive to extreme (high or low) outcomes with small probabilities. Unfortunately, it also makes the computation insensitive to extreme (high or low) options with 49% probabilities: it would prefer a gamble with a 49% probability of utility -3^^^3 and 51% probability of utility +1, to a gamble with 51% probability of utility 0, and 49% probability of +3^^^3.

But perhaps there are more well-motivated alternatives.

I have a problem with calling this a "semi-open FAI problem", because even if Eliezer's proposed solution turns out to be correct, it's still a wide open problem to develop arguments that can allow us to be confident enough in it to incorporate it into an FAI design. This would be true even if nobody can see any holes in it or have any better ideas, and doubly true given that some FAI researchers consider a different approach (which assumes that there is no such thing as "reality-fluid", that everything in the multiverse just exists and as a matter of preference we do not / can not care about all parts of it in equal measure, #4 in this post) to be at least as plausible as Eliezer's current approach.

(As always, the term "magical reality fluid" reflects an attempt to demarcate a philosophical area where I feel quite confused, and try to use correspondingly blatantly wrong terminology so that I do not mistake my reasoning about my confusion for a solution.)

This seems like a really useful strategy!

Just thought of something:

How sure are we that P(there are N people) is not at least as small as 1/N for sufficiently large N, even without a leverage penalty? The OP seems to be arguing that the complexity penalty on the prior is insufficient to generate this low probability, since it doesn't take much additional complexity to generate scenarios with arbitrarily more people. Yet it seems to me that after some sufficiently large number, P(there are N people) must drop faster than 1/N. This is because our prior must be normalized. That is:

Sum(all non-negative integers N) of P(there are N people) = 1.

If there was some integer M such that for all n > M, P(there are n people) >= 1/n, the above sum would not converge. If we are to have a normalized prior, there must be a faster-than-1/N falloff to the function P(there are N people).

In fact, if one demands that my priors indicate that my expected average number of people in the universe/multiverse is finite, then my priors must diminish faster than 1/N^2. (So that that the sum of N*P(there are N people) converges).

TL:DR If your priors are such that the probability of there being 3^^^3 people is not smaller than 1/(3^^^3), then you don't have a normalized distribution of priors. If your priors are such that the probability of there being 3^^^3 people is not smaller than 1/((3^^^3)^2) then your expected number of people in the multiverse is divergent/infinite.

How does this style of reasoning work on something more like the original Pascal's Wager problem?

Suppose a (to all appearances) perfectly ordinary person goes on TV and says "I am an avatar of the Dark Lords of the Matrix. Please send me $5. When I shut down the simulation in a few months, I will subject those who send me the money to [LARGE NUMBER] years of happiness, and those who do not to [LARGE NUMBER] years of pain".

Here you can't solve the problem by pointing out the very large numbers of people involved, because there aren't very high numbers of people involved. Your probability should depend only on your probability that this is a simulation, your probability that the simulators would make a weird request like this, and your probability that this person's specific weird request is likely to be it. None of these numbers help you get down to a 1/[LARGE NUMBER] level.

I've avoided saying 3^^^3, because maybe there's some fundamental constraint on computing power that makes it impossible for simulators to simulate 3^^^3 years of happiness in any amount of time they might conceivably be willing to dedicate to the problem. But they might be able to simulate some number of years large enough to outweigh our prior against any given weird request coming from the Dark Lords of the Matrix.

(also, it seems less than 3^^^3-level certain that there's no clever trick to get effectively infinite computing power or effectively infinite computing time, like the substrateless computation in Permutation City)

This is an awful lot of words to expend to notice that

(1) Social interactions need to be modeled in a game-theoretic setting, not straightforward expected payoff

(2) Distributions of expected values matter. (Hint: p(N) = 1/N is a really bad model as it doesn't converge).

(3) Utility functions are neither linear nor symmetric. (Hint: extinction is not symmetric with doubling the population.)

(4) We don't actually have an agreed-upon utility function anyway; big numbers plus a not-well-agreed-on fuzzy notion is a great way to produce counterintuitive results. The details don't really matter; as fuzzy approaches infinity, you get nonintuitiveness.

It's much more valuable to address some of these imperfections in the setup of the problem than continuing to wade through the logic with bad assumptions in hand.

Friendly neighborhood Matrix Lord checking in!

I'd like to apologize for the behavior of my friend in the hypothetical. He likes to make illusory promises. You should realize that regardless of what he may tell you, his choice of whether to hit the green button is independent of your choice of what to do with your $5. He may hit the green button and save 3↑↑↑3 lives, or he may not, at his whim. Your $5 can not be reliably expected to influence his decision in any way you can predict.

You are no doubt accustomed to thinking about enforceable contracts between parties, since those are a staple of your game theoretic literature as well as your storytelling traditions. Often, your literature omits the requisite preconditions for a binding contract since they are implicit or taken for granted in typical cases. Matrix Lords are highly atypical counterparties, however, and it would be a mistake to carry over those assumptions merely because his statements resemble the syntactic form of an offer between humans.

Did my Matrix Lord friend (who you just met a few minutes ago!) volunteer to have his green save-the-multitudes button and your $5 placed under the control of a mutually trustworthy third party escrow agent who will reliably uphold the stated bargain?

Alternately, if my Matrix Lord friend breaches his contract with you, is someone Even More Powerful standing by to forcibly remedy the non-performance?

Absent either of the above conditions, is my Matrix Lord friend participating in an iterated trading game wherein cheating on today's deal will subject him to less attractive terms on future deals, such that the net present value of his future earnings would be diminished by more than the amount he can steal from you today?

Since none of these three criteria seem to apply, there is no deal to be made here. The power asymmetry enables him to do whatever he feels like regardless of your actions, and he is just toying with you! Do you really think your $5 means anything to him? He'll spend it making 3↑↑↑3 paperclips for all you know.

Your $5 will not exert any predictable causal influence on the fate of the hypothetical 3↑↑↑3 Matrix Lord hostages. Decision theory doesn't even begin to apply.

You should stick to taking boxes from Omega; at least she has an established reputation for paying out as promised.

Related: Would an AI conclude it's likely to be a Boltzmann brain? ;)

I don't at all think that this is central to the problem, but I do think you're equating "bits" of sensory data with "bits" of evidence far too easily. There is no law of probability theory that forbids you from assigning probability 1/3^^^3 to the next bit in your input stream being a zero -- so as far as probability theory is concerned, there is nothing wrong with receiving only one input bit and as a result ending up believing a hypothesis that you assigned probability 1/3^^^3 before.

Similarly, probability theory allows you to assign prior probability 1/3^^^3 to seeing the blue hole in the sky, and therefore believing the mugger after seeing it happen anyway. This may not be a good thing to do on other principles, but probability theory does not forbid it. ETA: In particular, if you feel between a rock and a bad place in terms of possible solutions to Pascal's Muggle, then you can at least consider assigning probabilities this way even if it doesn't normally seem like a good idea.

Two quick thoughts:

• Any two theories can be made compatible if allowing for some additional correction factor (e.g. a "leverage penalty") designed to make them compatible. As such, all the work rests with "is the leverage penalty justified?"

• For said justification, there has to some sort of justifiable territory-level reasoning, including "does it carve reality at its joints?" and such, "is this the world we live in?".

The problem I see with the leverage penalty is that there is no Bayesian updating way that will get you to such a low prior. It's the mirror from "can never process enough bits to get away from such a low prior", namely "can never process enough bits to get to assigning such low priors" (the blade cuts both ways).

The reason for that is in part that your entire level of confidence you have in the governing laws of physics, and the causal structure and dependency graphs and such is predicated on the sensory bitstream of your previous life - no more, it's a strictly upper bound. You can gain confidence that a prior to affect a googleplex people is that low only by using that lifetime bitstream you have accumulated - but then the trap shuts, just as you can't get out of such a low prior, you cannot use any confidence you gained in the current system by ways of your lifetime sensory input to get to such a low prior. You can be very sure you can't affect that many, based on your understanding of how causal nodes are interconnected, but you can't be that sure (since you base your understanding on a comparatively much smaller number of bits of evidence):

It's a prior ex machina, with little more justification than just saying "I don't deal with numbers that large/small in my decision making".

As near as I can figure, the corresponding state of affairs to a complexity+leverage prior improbability would be a Tegmark Level IV multiverse in which each reality got an amount of magical-reality-fluid corresponding to the complexity of its program (1/2 to the power of its Kolmogorov complexity) and then this magical-reality-fluid had to be divided among all the causal elements within that universe - if you contain 3↑↑↑3 causal nodes, then each node can only get 1/3↑↑↑3 of the total realness of that universe.

This reminds me a lot of Levin's universal search algorithm, and the associated Levin complexity.

To formalize, I think you will want to assign each program p, of length #p, a prior weight 2^-#p (as in usual Solomonoff induction), and then divide that weight among the execution steps of the program (each execution step corresponding to some sort of causal node). So if program p executes for t steps before stopping, then each individual step gets a prior weight 2^-#p/t. The connection to universal search is as follows: Imagine dovetailing all possible programs on one big computer, giving each program p a share 2^-#'p of all the execution steps. (If the program stops, then start it again, so that the computer doesn't have idle steps). In the limit, the computer will spend a proportion 2^-#p/t of its resources executing each particular step of p, so this is an intuitive sense of the step's prior "weight".

You'll then want to condition on your evidence to get a posterior distribution. Most steps of most programs won't in any sense correspond to an intelligent observer (or AI program) having your evidence, E, but some of them will. Let nE(p) be the number of steps in a program p which so-correspond (for a lot of programs nE(p) will be zero) and then program p will get posterior weight proportional to 2^-#p x (nE(p) / t). Normalize, and that gives you the posterior probability you are in a universe executed by a program p.

You asked if there are any anthropic problems with this measure. I can think of a few:

1. Should "giant" observers (corresponding to lots of execution steps) count for more weight than "midget" observers (corresponding to fewer steps)? They do in this measure, which seems a bit counter-intuitive.

2. The posterior will tend to focus weight on programs which have a high proportion (nE(p) / t) of their execution steps corresponding to observers like you. If you take your observations at face value (i.e. you are not in a simulation), then this leads to the same sort of "Great Filter" issues that Katja Grace noticed with the SIA. There is a shift towards universes which have a high density of habitable planets, occupied by observers like us, but where very few or none of those observers ever expand off their home worlds to become super-advanced civilizations, since if they did they would take the executions steps away from observers like us.

3. There also seems to be a good reason in this measure NOT to take your observations at face value. The term nE(p) / t will tend to be maximized in universes very unlike ours: ones which are built of dense "computronium" running lots of different observer simulations, and you're one of them. Our own universe is very "sparse" in comparison (very few execution steps corresponding to observers).

4. Even if you deal with simulations, there appears to be a "cyclic history" problem. The density nE(p)/t will tend to be is maximized if civilizations last for a long time (large number of observers), but go through periodic "resets", wiping out all traces of the prior cycles (so leading to lots of observers in a state like us). Maybe there is some sort of AI guardian in the universe which interrupts civilizations before they create their own (rival) AIs, but is not so unfriendly as to wipe them out altogether. So it just knocks them back to the stone age from time to time. That seems highly unlikely a priori, but it does get magnified a lot in posterior probability.

On the plus side, note that there is no particular reason in this measure to expect you are in a very big universe or multiverse, so this defuses the "presumptuous philosopher" objection (as well as some technical problems if the weight is dominated by infinite universes). Large universes will tend to correspond to many copies of you (high nE(p)) but also to a large number of execution steps t. What matters is the density of observers (hence the computronium problem) rather than the total size.

You probably shouldn't let super-exponentials into your probability assignments, but you also shouldn't let super-exponentials into the range of your utility function. I'm really not a fan of having a discontinuous bound anywhere, but I think it's important to acknowledge that when you throw a trip-up (^^^) into the mix, important assumptions start breaking down all over the place. The VNM independence assumption no longer looks convincing, or straightforward. Normally my preferences in a Tegmark-style multiverse would reflect a linear combination of my preferences for its subcomponents; but throw a 3^^^3 in the mix, and this is no longer the case, so suddenly you have to introduce new distinctions between logical uncertainty and at least one type of reality fluid.

My short-term hack for Pascal's Muggle is to recognize that my consequentialism module is just throwing exceptions, and fall back on math-free pattern matching, including low-weighted deontological and virtue-ethical values that I've kept around for just such an occasion. I am very unhappy with this answer, but the long-term solution seems to require fully figuring out how I value different kinds of reality fluid.

Is it just me, or is everyone here overly concerned with coming up with patches for this specific case and not the more general problem? If utilities can grow vastly larger than the prior probability of the situation that contains them, then an expected utility system will become almost useless. Acting on situations with probabilities as tiny as can possibly be represented in that system, since the math would vastly outweigh the expected utility from acting on anything else.

I've heard people come up with apparent resolutions to this problem. Like counter balancing every possible situation with an equally low probability situation that has vast negative utility. There are a lot of problems with this though. What if the utilities don't exactly counterbalance? An extra bit to represent a negative utility for example, might add to the complexity and therefore the prior probability. Or even a tiny amount of evidence for one scenario over the other would completely upset it.

And even if that isn't the case, your utility might not have negative. Maybe you only value the number of paperclips in the universe. The worst that can happen is you end up in a universe with no paperclips. You can't have negative paperclips, so the lowest utility you can have is 0. Or maybe your positive and negative values don't exactly match up. Fear is a better motivator than reward, for example. The fear of having people suffer may have more negative utility than the opposite scenario of just as many people living happy lives or something (and since they are both different scenarios with more differences than a single number, they would have different prior probabilities to begin with.)

Resolutions that involve tweaking the probability of different events is just cheating since the probability shouldn't change if the universe hasn't. It's how you act on those probabilities that we should be concerned about. And changing the utility function is pretty much cheating too. You can make all sorts of arbitrary tweaks that would solve the problem, like having a maximum utility or something. But if you really found out you lived in a universe where 3^^^3 lives existed (perhaps aliens have been breeding extensively, or we really do live in a simulation, etc), are you just supposed to stop caring about all life since it exceeds your maximum amount of caring?

I apologize if I'm only reiterating arguments that have already been gone over. But it's concerning to me that people are focusing on extremely sketchy patches to a specific case of this problem, and not the more general problem, that any expected utility function becomes apparently worthless in a probabilistic universe like ours.

EDIT: I think I might have a solution to the problem and posted it here.

Nick Beckstead's finished but as-yet unpublished dissertation has much to say on this topic. Here is Beckstead's summary of chapters 6 and 7 of his dissertation:

[My argument for the overwhelming importance of shaping the far future] asks us to be happy with having a very small probability of averting an existential catastrophe [or bringing about some other large, positive "trajectory change"], on the grounds that the expected value of doing so is extremely enormous, even though there are more conventional ways of doing good which have a high probability of producing very good, but much less impressive, outcomes. Essentially, we're asked to choose a long shot over a high probability of something very good. In extreme cases, this can seem irrational on the grounds that it's in the same ballpark as accepting a version of Pascal's Wager.

In chapter 6, I make this worry more precise and consider the costs and benefits of trying to avoid the problem. When making decisions under risk, we make trade-offs between how good outcomes might be and how likely it is that we get good outcomes. There are three general kinds of ways to make these tradeoffs. On two of these approaches, we try to maximize expected value. On one of the two approaches, we hold that there are limits to how good (or bad) outcomes can be. On this view, no matter how bad an outcome is, it could always get substantially worse, and no matter how good an outcome is, it could always get substantially better. On the other approach, there are no such limits, at least in one of these directions. Either outcomes could get arbitrarily good, or they could get arbitrarily bad. On the third approach, we give up on ranking outcomes in terms of their expected value.

The main conclusion of chapter 6 is that all of these approaches have extremely unpalatable implications. On the approach where there are upper and lower limits, we have to be timid — unwilling to accept extremely small risks in order to enormously increase potential positive payoffs. Implausibly, this requires extreme risk aversion when certain extremely good outcomes are possible, and extreme risk seeking when certain extremely bad outcomes are possible, and it requires making one's ranking of prospects dependent on how well things go in remote regions of space and time.

In the second case, we have to be reckless — preferring very low probabilities of extremely good outcomes to very high probabilities of less good, but still excellent, outcomes — or rank prospects non-transitively. I then show that, if a theory is reckless, what it would be best to do, according to that theory, depends almost entirely upon what would be best in terms of considerations involving infinite value, no matter how implausible it is that we can bring about any infinitely good or bad outcomes, provided it is not certain. In this sense, there really is something deeply Pascalian about the reckless approach.

Some might view this as a reductio of expected utility theory. However, I show that the only way to avoid being both reckless and timid is to rank outcomes in a circle, claiming that A is better than B, which is better than C,. . . , which is better than Z, which is better than A. Thus, if we want to avoid these two other problems, we have to give up not only on expected utility theory, but we also have to give up on some very basic assumptions about how we should rank alternatives. This makes it much less clear that we can simply treat these problems as a failure of expected utility theory.

What does that have to do with the rough future-shaping argument? The problem is that my formalization of the rough future-shaping argument commits us to being reckless. Why? By Period Independence [the assumption that "By and large, how well history goes as a whole is a function of how well things go during each period of history"], additional good periods of history are always good, how good it is to have additional periods does not depend on how many you've already had, and there is no upper limit (in principle) to how many good periods of history there could be. Therefore, there is no upper limit to how good outcomes can be. And that leaves us with recklessness, and all the attendant theoretical difficulties.

At this point, we are left with a challenging situation. On one hand, my formalization of the rough future-shaping argument seemed plausible. However, we have an argument that if its assumptions are true, then what it is best to do depends almost entirely on infinite considerations. That's a very implausible conclusion. At the same time, the conclusion does not appear to be easy to avoid, since the alternatives are the so-called timid approach and ranking alternatives non-transitively.

In chapter 7, I discuss how important it would be to shape the far future given these three different possibilities (recklessness, timidity, and non-transitive rankings of alternatives). As we have already said, in the case of recklessness, the best decision will be the decision that is best in terms of infinite considerations. In the first part of the chapter, I highlight some difficulties for saying what would be best with respect to infinite considerations, and explain how what is best with respect to infinite considerations may depend on whether our universe is infinitely large, and whether it makes sense to say that one of two infinitely good outcomes is better than the other.

In the second part of the chapter, I examine how a timid approach to assessing the value of prospects bears on the value of shaping the far future. The answer to this question depends on many complicated issues, such as whether we want to accept something similar to Period Independence in general even if Period Independence must fail in extreme cases, whether the universe is infinitely large, whether we should include events far outside of our causal control when aggregating value across space and time, and what the upper limit for the value of outcomes is.

In the third part of the chapter, I consider the possibility of using the reckless approach in contexts where it seems plausible and using the timid approach in the contexts where it seems plausible. This approach, I argue, is more plausible in practice than the alternatives. I do not argue that this mixed strategy is ultimately correct, but instead argue that it is the best available option in light of our cognitive limitations in effectively formalizing and improving our processes for thinking about infinite ethics and long shots.

If an AI's overall architecture is such as to enable it to carry out the "You turned into a cat" effect - where if the AI actually ends up with strong evidence for a scenario it assigned super-exponential improbability, the AI reconsiders its priors and the apparent strength of evidence rather than executing a blind Bayesian update, though this part is formally a tad underspecified - then at the moment I can't think of anything else to add in.

Ex ante, when the AI assigns infinitesimal probability to the real thing, and meaningful probability to "hallucination/my sensors are being fed false information," why doesn't it self-modify/self-bind to treat future apparent cat transformations as hallucinations?

"Now, in this scenario we've just imagined, you were taking my case seriously, right? But the evidence there couldn't have had a likelihood ratio of more than 10^10^26 to 1, and probably much less. So by the method of imaginary updates, you must assign probability at least 10^-10^26 to my scenario, which when multiplied by a benefit on the order of 3↑↑↑3, yields an unimaginable bonanza in exchange for just five dollars -"

Me: "Nope."

I don't buy this. Consider the following combination of features of the world and account of anthropic reasoning (brought up by various commenters in previous discussions), which is at least very improbable in light of its specific features and what we know about physics and cosmology, but not cosmically so.

• A world small enough not to contain ludicrous numbers of Boltzmann brains (or Boltzmann machinery)
• Where it is possible to create hypercomputers through complex artificial means
• Where hypercomputers are used to compute arbitrarily many happy life-years of animals, or humanlike beings with epistemic environments clearly distinct from our own (YOU ARE IN A HYPERCOMPUTER SIMULATION tags floating in front of their eyes)
• And the hypercomputed beings are not less real or valuable because of their numbers and long addresses

Treating this as infinitesimally likely, and then jumping to measurable probability on receipt of (what?) evidence about hypercomputers being possible, etc, seems pretty unreasonable to me.

The behavior you want could be approximated with a bounded utility function that assigned some weight to achieving big payoffs/achieving a significant portion (on one of several scales) of possible big payoffs/etc. In the absence of evidence that the big payoffs are possible, the bounded utility gain is multiplied by low probability and you won't make big sacrifices for it, but in the face of lots of evidence, and if you have satisfied other terms in your utility function pretty well, big payoffs could become a larger focus.

Basically, I think such a bounded utility function could better track the emotional responses driving your intuitions about what an AI should do in various situations than jury-rigging the prior. And if you don't want to track those responses then be careful of those intuitions and look to empirical stabilizing assumptions.

Just gonna jot down some thoughts here. First a layout of the problem.

1. Expected utility is a product of two numbers, probability of the event times utility generated by the event.
2. Traditionally speaking, when the event is claimed to affect 3^^^3 people, the utility generated is on the order of 3^^^3
3. Traditionally speaking, there's nothing about the 3^^^3 people that requires a super-exponentially large extension to the complexity of the system (the univers/multivers/etc). So the probability of the event does not scale like 1/(3^^^3)
4. Thus Expected Payoff becomes enormous, and you should pay the dude $5.
5. If you actually follow this, you'll be mugged by random strangers offerring to save 3^^^3 people or whatever super-exponential numbers they can come up with.

In order to avoid being mugged, your suggestion is to apply a scale penalty (leverage penalty) to the probability. You then notice that this has some very strange effects on your epistemology - you become incapable of ever believing the 5$ will actually help no matter how much evidence you're given, even though evidence can make the expected payoff large. You then respond to this problem with what appears to be an excuse to be illogical and/or non-bayesian at times (due to finite computing power).

It seems to me that an alternative would be to rescale the untility value, instead of the probability. This way, you wouldn't run into any epistemic issues anywhere because you aren't messing with the epistemics.

I'm not proposing we rescale Utility(save X people) by a factor 1/X, as that would make Utility(save X people) = Utility(save 1 person) all the time, which is obviously problematic. Rather, my idea is to make Utility a per capita quantity. That way, when the random hobo tells you he'll save 3^^^3 people, he's making a claim that requires there to be at least 3^^^3 people to save. If this does turn out to be true, keeping your Utility as a per capita quantity will require a rescaling on the order of 1/(3^^^3) to account for the now-much-larger population. This gives you a small expected payoff without requiring problematically small prior probabilities.

It seems we humans may already do a rescaling of this kind anyway. We tend to value rare things more than we would if they were common, tend to protect an endangered species more than we would if it weren't endangered, and so on. But I'll be honest and say that I haven't really thought the consequences of this utility re-scaling through very much. It just seems that if you need to rescale a product of two numbers and rescaling one of the numbers causes problems, we may as well try rescaling the other and see where it leads.

Any thoughts?

I think the simpler solution is just to use a bounded utility function. There are several things suggesting we do this, and I really don't see any reason to not do so, instead of going through contortions to make unbounded utility work.

Consider the paper of Peter de Blanc that you link -- it doesn't say a computable utility function won't have convergent utilities, but rather that it will iff said function is bounded. (At least, in the restricted context defined there, though it seems fairly general.) You could try to escape the conditions of the theorem, or you could just conclude that utility functions should be bounded.

Let's go back and ask the question of why we're using probabilities and utilities in the first place. Is it because of Savage's Theorem? But the utility function output by Savage's Theorem is always bounded.

OK, maybe we don't accept Savage's axiom 7, which is what forces utility functions to be bounded. But then we can only be sure that comparing expected utilities is the right thing to do for finite gambles, not for infinite ones, so talking about sums converging or not -- well, it's something that shouldn't even come up. Or alternatively, if we do encounter a situation with infinitely many choices, each of differing utility, we simply don't know what to do.

Maybe we're not basing this on Savage's theorem at all -- maybe we simply take probability for granted (or just take for granted that it should be a real number and ground it in something like Cox's theorem -- after all, like Savage's theorem, Cox's theorem only requires that probability be finitely additive) and are then deriving utility from the VNM theorem. The VNM theorem doesn't prohibit unbounded utilities. But the VNM theorem once again only tells us how to handle finite gambles -- it doesn't tell us that infinite gambles should also be handled via expected utility.

OK, well, maybe we don't care about the particular grounding -- we're just going to use probability and utility because it's the best framework we know, and we'll make the probability countably additive and use expected utility in all cases hey, why not, seems natural, right? (In that case, the AI may want to eventually reconsider whether probability and utility really is the best framework to use, if it is capable of doing so.) But even if we throw all that out, we still have the problem de Blanc raises. And, um, all the other problems that have been raised with unbounded utility. (And if we're just using probability and utility to make things nice, well, we should probably use bounded utility to make things nicer.)

I really don't see any particular reason utility has to be unbounded either. Eliezer Yudkowsky seems to keep using this assumption that utility should be unbounded, or just not necessarily bounded, but I've yet to see any justification for this. I can find one discussion where, when the question of bounded utility functions came up, Eliezer responded, "[To avert a certain problem] the bound would also have to be substantially less than 3^^^^3." -- but this indicates a misunderstanding of the idea of utility, because utility functions can be arbitrarily (positively) rescaled or recentered. Individual utility "numbers" are not meaningful; only ratios of utility differences. If a utility function is bounded, you can assume the bounds are 0 and 1. Talk about the value of the bound is as meaningless as anything else using absolute utility numbers; they're not amounts of fun or something.

Sure, if you're taking a total-utilitarian viewpoint, then your (decision-theoretic) utility function has to be unbounded, because you're summing a quantity over an arbitrarily large set. (I mean, I guess physical limitations impose a bound, but they're not logical limitations, so we want to be able to assign values to situations where they don't hold.) (As opposed to the individual "utility" functions that your'e summing, which is a different sort of "utility" that isn't actually well-defined at present.) But total utilitarianism -- or utilitarianism in general -- is on much shakier ground than decision-theoretic utility functions and what we can do with them or prove about them. To insist that utility be unbounded based on total utilitarianism (or any form of utilitarianism) while ignoring the solid things we can say seems backwards.

Not everything has to scale linearly, after all. There seems to be this idea out there that utility must be unbounded because there are constants C_1 and C_2 such that adding to the world of person of "utility" (in the utilitarian sense) C_1 must increase your utility (in the decision-theoretic sense) by C_2, but this doesn't need to be so. This to me seems a lot like insisting "Well, no matter how fast I'm going, I can always toss a baseball forward in my direction at 1 foot per second relative to me; so it will be going 1 foot per second faster than me, so the set of possible speeds is unbounded." As it turns out, the set of possible speeds is bounded, velocities don't add linearly, and if you toss a baseball forward in your direction at 1 foot per second relative to you, it will not be going 1 foot per second faster.

My own intuition is more in line with earthwormchuck163's comment -- I doubt I would be that joyous about making that many more people when so many are going to be duplicates or near-duplicates of one another. But even if you don't agree with this, things don't have to add linearly, and utilities don't have to be unbounded.

I get the sense you're starting from the position that rejecting the Mugging is correct, and then looking for reasons to support that predetermined conclusion. Doesn't this attitude seem dangerous? I mean, in the hypothetical world where accepting the Mugging is actually the right thing to do, wouldn't this sort of analysis reject it anyway? (This is a feature of debates about Pascal's Mugging in general, not just this post in particular.)

It seems to me like the whistler is saying that the probability of saving knuth people for $5 is exactly 1/knuth after updating for the Matrix Lord's claim, not before the claim, which seems surprising. Also, it's not clear that we need to make an FAI resistant to very very unlikely scenarios.

I enjoyed this really a lot, and while I don't have anything insightful to add, I gave five bucks to MIRI to encourage more of this sort of thing.

(By "this sort of thing" I mean detailed descriptions of the actual problems you are working on as regards FAI research. I gather that you consider a lot of it too dangerous to describe in public, but then I don't get to enjoy reading about it. So I would like to encourage you sharing some of the fun problems sometimes. This one was fun.)

If the AI actually ends up with strong evidence for a scenario it assigned super-exponential improbability, the AI reconsiders its priors and the apparent strength of evidence rather than executing a blind Bayesian update, though this part is formally a tad underspecified.

I would love to have a conversation about this. Is the "tad" here hyperbole or do you actually have something mostly worked out that you just don't want to post? On a first reading (and admittedly without much serious thought -- it's been a long day), it seems to me that this is where the real heavy lifting has to be done. I'm always worried that I'm missing something, but I don't see how to evaluate the proposal without knowing how the super-updates are carried out.

Really interesting, though.

If someone suggests to me that they have the ability to save 3^^^3 lives, and I assign this a 1/3^^^3 probability, and then they open a gap in the sky at billions to one odds, I would conclude that it is still extremely unlikely that they can save 3^^^3 lives. However, it is possible that their original statement is false and yet it would be worth giving them five dollars because they would save a billion lives. Of course, this would require further assumptions on whether people are likely to do things that they have not said they would do, but are weaker versions of things they did say they would do but are not capable of.

Also, I would assign lower probabilities when they claim they could save more people, for reasons that have nothing to do with complexity. For instance, "the more powerful a being is, the less likely he would be interested in five dollars" or :"a fraudster would wish to specify a large number to increase the chance that his fraud succeeds when used on ordinary utility maximizers, so the larger the number, the greater the comparative likelihood that the person is fraudulent".

the phrase "Pascal's Mugging" has been completely bastardized to refer to an emotional feeling of being mugged that some people apparently get when a high-stakes charitable proposition is presented to them, regardless of whether it's supposed to have a low probability.

1) Sometimes what you may actually be seeing is disagreement on whether the hypothesis has a low probability.

2) Some of the arguments against Pascal's Wager and Pascal's Mugging don't depend on the probability. For instance, Pascal's Wager has the "worshipping the wrong god" problem--what if there's a god who prefers that he not be worshipped and damns worshippers to Hell? Even if there's a 99% chance of a god existing, this is still a legitimate objection (unless you want to say there's a 99% chance specifically of one type of god).

3) In some cases, it may be technically true that there is no low probability involved but there may be some other small number that the size of the benefit is multiplied by. For instance, most people discount events that happen far in the future. A highly beneficial event that happens far in the future would have the benefit multiplied by a very small number when considering discounting.

Of course in cases 2 and 3 that is not technically Pascal's mugging by the original definition, but I would suggest the definition should be extended to include such cases. Even if not, they should at least be called something that acknowledges the similarity, like "Pascal-like muggings".

Has the following reply to Pascal's Mugging been discussed on LessWrong?

1. Almost any ordinary good thing you could do has some positive expected downstream effects.
2. These positive expected downstream effects include lots of things like, "Humanity has slightly higher probability of doing awesome thing X in the far future." Possible values of X include: create 3^^^^3 great lives or create infinite value through some presently unknown method, and stuff like, in a scenario where the future would have been really awesome, it's one part in 10^30 better.
3. Given all the possible values of X whose probability is raised by doing ordinary good things, the expected value of doing any ordinary good thing is higher than the expected value of paying the mugger.
4. Therefore, almost any ordinary good thing you could do is better than paying the mugger. [I take it this is the conclusion we want.]

The most obvious complaint I can think of for this response is that it doesn't solve selfish versions of Pascal's Mugging very well, and may need to be combined with other tools in that case. But I don't remember people talking about this and I don't currently see what's wrong with this as a response to the altruistic version of Pascal's Mugging. (I don't mean to suggest I would be very surprised if someone quickly and convincingly shoots this down.)

One point I don't see mentioned here that may be important is that someone is saying this to you.

I encounter lots of people. Each of them has lots of thoughts. Most of those thoughts, they do not express to me (for which I am grateful). How do they decide which thoughts to express? To a first approximation, they express thoughts which are likely, important and/or amusing. Therefore, when I hear a thought that is highly important or amusing, I expect it had less of a likelihood barrier to being expressed, and assign it a proportionally lower probability.

Note that this doesn't apply to arguments in general -- only to ones that other people say to me.

This is probably obvious, but if this problem persisted, a Pascal-Mugging-vulnerable AI would immediately get mugged even without external offers or influence. The possibility alone, however remote, of a certain sequence of characters unlocking a hypothetical control console which could potentially access an above Turing computing model which could influence (insert sufficiently high number) amounts of matter/energy, would suffice. If an AI had to decide "until what length do I utter strange tentative passcodes in the hope of unlocking some higher level of physics", it would get mugged by the shadow of a matrix lord every time.

It sounds like what you're describing is something that Iain Banks calls an "Out of Context Problem" - it doesn't seem like a 'leverage penalty' is the proper way to conceptualize what you're applying, as much as a 'privilege penalty'.

In other words, when the sky suddenly opens up and blue fire pours out, the entire context for your previous set of priors needs to be re-evaluated - and the very question of "should I give this man $5" exists on a foundation of those now-devaluated priors.

Is there a formalized tree or mesh model for Bayesian probabilities? Because I think that might be fruitful.

There's something very counterintuitive about the notion that Pascal's Muggle is perfectly rational. But I think we need to do a lot more intuition-pump research before we'll have finished picking apart where that counterintuitiveness comes from. I take it your suggestion is that Pascal's Muggle seems unreasonable because he's overly confident in his own logical consistency and ability to construct priors that accurately reflect his credence levels. But he also seems unreasonable because he doesn't take into account that the likeliest explanations for the Hole In The Sky datum either trivialize the loss from forking over $5 (e.g., 'It's All A Dream') or provide much more credible generalized reasons to fork over the $5 (e.g., 'He Really Is A Matrix Lord, So You Should Do What He Seems To Want You To Do Even If Not For The Reasons He Suggests'). Your response to the Holy In The Sky seems more safe and pragmatic because it leaves open that the decision might be made for those reasons, whereas the other two muggees were explicitly concerned only with whether the Lord's claims were generically right or generically wrong.

Noting these complications doesn't help solve the underlying problem, but it does suggest that the intuitively right answer may be overdetermined, complicating the task of isolating our relevant intuitions from our irrelevant ones.

One scheme with the properties you want is Wei Dai's UDASSA, e.g. see here. I think UDASSA is by far the best formal theory we have to date, although I'm under no delusions about how well it captures all of our intuitions (I'm also under no delusions about how consistent our intuitions are, so I'm resigned to accepting a scheme that doesn't capture them).

I think it would be more fair to call this allocation of measure part of my preferences, instead of "magical reality fluid." Thinking that your preferences are objective facts about the world seems like one of the oldest errors in the book, which is only possibly justified in this case because we are still confused about the hard problem of consciousness.

As other commenters have observed, it seems clear that you should never actually believe that the mugger can influence the lives of 3^^^^3 other folks and will do so at your suggestion, whether or not you've made any special "leverage adjustment." Nevertheless, even though you never believe that you have such influence, you would still need to pass to some bounded utility function if you want to use the normal framework of expected utility maximization, since you need to compare the goodness of whole worlds. Either that, or you would need to make quite significant modifications to your decision theory.

What if the mugger says he will give you a single moment of pleasure that is 3^^^3 times more intense than a standard good experience? Wouldn't the leverage penalty not apply and thus make the probability of the mugger telling the truth much higher?

I think the real reason the mugger shouldn't be given money is that people are more likely to be able to attain 3^^^3 utils by donating the five dollars to an existential risk-reducing charity. Even though the current universe presumably couldn't support 3^^^3 utils, there is a chance of being able to create or travel to vast numbers of other universes, and I think this chance is greater than the chance of the mugger being honest.

Am I missing something? These points seem too obvious to miss, so I'm assigning a fairly large probability to me either being confused or that these were already mentioned.

Indeed, you can't ever present a mortal like me with evidence that has a likelihood ratio of a googolplex to one - evidence I'm a googolplex times more likely to encounter if the hypothesis is true, than if it's false - because the chance of all my neurons spontaneously rearranging themselves to fake the same evidence would always be higher than one over googolplex. You know the old saying about how once you assign something probability one, or probability zero, you can never change your mind regardless of what evidence you see? Well, odds of a googolplex to one, or one to a googolplex, work pretty much the same way."

On the other hand, if I am dreaming, or drugged, or crazy, then it DOESN'T MATTER what I decide to do in this situation. I will still be trapped in my dream or delusion, and I won't actually be five dollars poorer because you and I aren't really here. So I may as well discount all probability lines in which the evidence I'm seeing isn't a valid representation of an underlying reality. Here's your $5.

Random thoughts here, not highly confident in their correctness.

Why is the leverage penalty seen as something that needs to be added, isn't it just the obviously correct way to do probability.

Suppose I want to calculate the probability that a race of aliens will descend from the skies and randomly declare me Overlord of Earth some time in the next year. To do this, I naturally go to Delphi to talk to the Oracle of Perfect Priors, and she tells me that the chance of aliens descending from the skies and declaring an Overlord of Earth in the next year is 0.0000007%.

If I then declare this to be my probability of become Overlord of Earth in an alien-backed coup, this is obviously wrong. Clearly I should multiply it by the probability that the aliens pick me, given that the aliens are doing this. There are about 7-billion people on earth, and updating on the existence of Overlord Declaring aliens doesn't have much effect on that estimate, so my probability of being picked is about 1 in 7 billion, meaning my probability of being overlorded is about 0.0000000000000001%. Taking the former estimate rather than the latter is simply wrong.

Pascal's mugging is a similar situation, only this time when we update on the mugger telling the truth, we radically change our estimate of the number of people who were 'in the lottery', all the way up to 3^^^^3. We then multiply 1/3^^^^3 by the probability that we live in a universe where Pascal's muggings occur (which should be very small but not super-exponentially small). This gives you the leverage penalty straight away, no need to think about Tegmark multiverses. We were simply mistaken to not include it in the first place.

Okay, that makes sense. In that case, though, where's the problem? Claims in the form of "not only is X a true event, with details A, B, C, ..., but also it's the greatest event by metric M that has ever happened" should have low enough probability that a human writing it down specifically in advance as a hypothesis to consider, without being prompted by some specific evidence, is doing really badly epistemologically.

Also, I'm confused about the relationship to MWI.

Many of the conspiracy theories generated have some significant overlap (i.e. are not mutually exclusive), so one shouldn't expect the sum of their probabilities to be less than 1. It's permitted for P(Cube A is red) + P(Sphere X is blue) to be greater than 1.

Edit: formatting fixed. Thanks, wedrifid.

My response to the mugger:

• You claim to be able to simulate 3^^^^3 unique minds.
• It takes log(3^^^^3) bits just to count that many things, so my absolute upper bound on the prior for an agent capable of doing this is 1/3^^^^3.
• My brain is unable to process enough evidence to overcome this, so unless you can use your matrix powers to give me access to sufficient computing power to change my mind, get lost.

My response to the scientist:

• Why yes, you do have sufficient evidence to overturn our current model of the universe, and if your model is sufficiently accurate, the computational capacity of the universe is vastly larger than we thought.
• Let's try building a computer based on your model and see if it works.

This system does seem to lead to the odd effect that you would probably be more willing to pay Pascal's Mugger to save 10^10^100 people than you would be willing to pay to save 10^10^101 people, since the leverage penalties make them about equal, but the latter has a higher complexity cost. In fact the leverage penalty effectively means that you cannot distinguish between events providing more utility than you can provide an appropriate amount of evidence to match.

Is there any particular reason an AI wouldn't be able to self-modify with regards to its prior/algorithm for deciding prior probabilities? A basic Solomonoff prior should include a non-negligible chance that it itself isn't perfect for finding priors, if I'm not mistaken. That doesn't answer the question as such, but it isn't obvious to me that it's necessary to answer this one to develop a Friendly AI.

As near as I can figure, the corresponding state of affairs to a complexity+leverage prior improbability would be a Tegmark Level IV multiverse in which each reality got an amount of magical-reality-fluid corresponding to the complexity of its program (1/2 to the power of its Kolmogorov complexity) and then this magical-reality-fluid had to be divided among all the causal elements within that universe - if you contain 3↑↑↑3 causal nodes, then each node can only get 1/3↑↑↑3 of the total realness of that universe.

The difference between this and average utilitarianism is that we divide the probability by the hypothesis size, rather than dividing the utility by that size. The closeness of the two seems a bit surprising.

Robin Hanson has suggested that the logic of a leverage penalty should stem from the general improbability of individuals being in a unique position to affect many others (which is why I called it a leverage penalty). At most 10 out of 3↑↑↑3 people can ever be in a position to be "solely responsible" for the fate of 3↑↑↑3 people if "solely responsible" is taken to imply a causal chain that goes through no more than 10 people's decisions; i.e. at most 10 people can ever be solely10 responsible for any given event.

This bothers me because it seems like frequentist anthropic reasoning similar to the Doomsday argument. I'm not saying I know what the correct version should be, but assuming that we can use a uniform distribution and get nice results feels like the same mistake as the principle of indifference (and more sophisticated variations that often worked surprisingly well as an epistemic theory for finite cases). Things like Solomonoff distributions are more flexible...

(As for infinite causal graphs, well, if problems arise only when introducing infinity, maybe it's infinity that has the problem.)

The problem goes away of we try to employ a universal distribution for the reality fluid, rather than a uniform one. (This does not make that a good idea, necessarily.)

This setup is not entirely implausible because the Born probabilities in our own universe look like they might behave like this sort of magical-reality-fluid - quantum amplitude flowing between configurations in a way that preserves the total amount of realness while dividing it between worlds - and perhaps every other part of the multiverse must necessarily work the same way for some reason.

If we try to use universal-distribution reality-fluid instead, we would expect to continue to see the same sort of distribution we had seen in the past: we would believe that we went down a path where the reality fluid concentrated into the Born probabilities, but other quantum paths which would be very improbable according to the Born probabilities may get high probability from some other rule.

There is likely a broader-scoped discussion on this topic that I haven't read, so please point me to such a thread if my comment is addressed -- but it seems to me that there is a simpler resolution to this issue (as well as an obvious limitation to this way of thinking), namely that there's an almost immediate stage (in the context of highly-abstract hypotheticals) where probability assessment breaks down completely.

For example, there are an uncountably-infinite number of different parent universes we could have. There are even an uncountably-infinite number of possible laws of physics that could govern our universe. And it's literally impossible to have all these scenarios "possible" in the sense of a well-defined measure, simply because if you want an uncountable sum of real numbers to add up to 1, only countably many terms can be nonzero.

This is highly related to the axiomatic problem of cause and effect, a famous example being the question "why is there something rather than nothing" -- you have to have an axiomatic foundation before you can make calculations, but the sheer act of adopting that foundation excludes a lot of very interesting material. In this case, if you want to make probabilistic expectations, you need a solid axiomatic framework to stipulate how calculations are made.

Just like with the laws of physics, this framework should agree with empirically-derived probabilities, but just like physics there will be seemingly-well-formulated questions that the current laws cannot address. In cases like hobos who make claims to special powers, the framework may be ill-equipped to make a definitive prediction. More generally, it will have a scope that is limited of mathematical necessity, and many hypotheses about spirituality, religion, and other universes, where we would want to assign positive but marginal probabilities, will likely be completely outside its light cone.

A few thoughts:

I haven't strongly considered my prior on being able to save 3^^^3 people (more on this to follow). But regardless of what that prior is, if approached by somebody claiming to be a Matrix Lord who claims he can save 3^^^3 people, I'm not only faced with the problem of whether I ought to pay him the $5 - I'm also faced with the question of whether I ought to walk over to the next beggar on the street, and pay him $0.01 to save 3^^^3 people. Is this person 500 times more likely to be able to save 3^^^3 people? From the outset, not really. And giving money to random people has no prior probability of being more likely to save lives than anything else.

Now suppose that the said "Matrix Lord" opens the sky, splits the Red Sea, demonstrates his duplicator box on some fish and, sure, creates a humanoid Patronus. Now do I have more reason to believe that he is a Time Lord? Perhaps. Do I have reason to think that he will save 3^^^3 lives if I give him $5? I don't see convincing reason to believe so, but I don't see either view as problematic.

Obviously, once you're not taking Hanson's approach, there's no problem with believing you've made a major discovery that can save an arbitrarily large number of lives.

But here's where I noticed a bit of a problem in your analogy: In the dark matter case you say ""if these equations are actually true, then our descendants will be able to exploit dark energy to do computations, and according to my back-of-the-envelope calculations here, we'd be able to create around a googolplex people that way."

Well, obviously the odds here of creating exactly a googolplex people is no greater than one in a googolplex. Why? Because those back of the hand calculations are going to get us (at best say) an interval from 0.5 x 10^(10^100) to 2 x 10^(10^100) - an interval containing more than a googolplex distinct integers. Hence, the odds of any specific one will be very low, but the sum might be very high. (This is simply worth contrasting with your single integer saved of the above case, where presumably your probabilities of saving 3^^^3 + 1 people are no higher than they were before.)

Here's the main problem I have with your solution:

"But if I actually see strong evidence for something I previously thought was super-improbable, I don't just do a Bayesian update, I should also question whether I was right to assign such a tiny probability in the first place - whether it was really as complex, or unnatural, as I thought. In real life, you are not ever supposed to have a prior improbability of 10^-100 for some fact distinguished enough to be written down, and yet encounter strong evidence, say 10^10 to 1, that the thing has actually happened."

Sure you do. As you pointed out, dice rolls. The sequence of rolls in a game of Risk will do this for you, and you have strong reason to believe that you played a game of Risk and the dice landed as they did.

We do probability estimates because we lack information. Your example of a mathematical theorem is a good one: The Theorem X is true or false from the get-go. But whenever you give me new information, even if that information is framed in the form of a question, it makes sense for me to do a Bayesian update. That's why a lot of so-called knowledge paradoxes are silly: If you ask me if I know who the president is, I can answer with 99%+ probability that it's Obama, if you ask me whether Obama is still breathing, I have to do an update based on my consideration of what prompted the question. I'm not committing a fallacy by saying 95%, I'm doing a Bayesian update, as I should.

You'll often find yourself updating your probabilities based on the knowledge that you were completely incorrect about something (even something mathematical) to begin with. That doesn't mean you were wrong to assign the initial probabilities: You were assigning them based on your knowledge at the time. That's how you assign probabilities.

In your case, you're not even updating on an "unknown unknown" - that is, something you failed to consider even as a possibility - though that's the reason you put all probabilities at less than 100%, because your knowledge is limited. You're updating on something you considered before. And I see absolutely no reason to label this a special non-Bayesian type of update that somehow dodges the problem. I could be missing something, but I don't see a coherent argument there.

As an aside, the repeated references to how people misunderstood previous posts are distracting to say the least. Couldn't you just include a single link to Aaronson's Large Numbers paper (or anything on up-arrow notation, I mention Aaronson's paper because it's fun)? After all, if you can't understand tetration (and up), you're not going to understand the article to begin with.

(edit: wide-open)

This link seems to no longer work.

Is it reasonable to take this as evidence that we shouldn't use expected utility computations, or not only expected utility computations, to guide our decisions?

If I understand the context, the reason we believed an entity, either a human or an AI, ought to use expected utility as a practical decision making strategy, is because it would yield good results (a simple, general architecture for decision making). If there are fully general attacks (muggings) on all entities that use expected utility as a practical decision making strategy, then perhaps we should revise the original hypothesis.

Utility as a theoretical construct is charming, but it does have to pay its way, just like anything else.

P.S. I think the reasoning from "bounded rationality exists" to "non-Bayesian mind changes exist" is good stuff. Perhaps we could call this "on seeing this, I become willing to revise my model" phenomenon something like "surprise", and distinguish it from merely new information.

This would mean that all our decisions were dominated by tiny-seeming probabilities (on the order of 2-100 and less) of scenarios where our lightest action affected 3↑↑4 people... which would in turn be dominated by even more remote probabilities of affecting 3↑↑5 people...

I'm pretty ignorant of quantum mechanics, but I gather there was a similar problem, in that the probability function for some path appeared to be dominated by an infinite number of infinitessimally-unlikely paths, and Feynman solved the problem by showing that those paths cancelled each other out.

Why are you not a science fiction writer?

How confident are you of "Probability penalties are epistemic features - they affect what we believe, not just what we do. Maps, ideally, correspond to territories."? That seems to me to be a strong heuristic, even a very very strong heuristic, but I don't think it's strong enough to carry the weight you're placing on it here. I mean, more technically, the map corresponds to some relationship between the territory and the map-maker's utility function, and nodes on a causal graph, which are, after all, probabilistic, and thus are features of maps, not of territories, are features of the map-maker's utility function, not just summaries of evidence about the territory.
I suspect that this formalism mixes elements of division of magical reality fluid between maps with elements of division of magical reality fluid between territories.

...at what point are you overthinking this?

The link labelled "the prior probability of hypotheses diminishes exponentially with their complexity" is malformed.

Is there any justification for the leverage penalty? I understand that it would apply if there were a finite number of agents, but if there's an infinite number of agents, couldn't all agents have an effect on an arbitrarily larger number of other agents? Shouldn't the prior probability instead be P(event A | n agents will be effected) = (1 / n) + P(there being infinite entities)? If this is the case, then it seems the leverage penalty won't stop one from being mugged.

This setup is not entirely implausible because the Born probabilities in our own universe look like they might behave like this sort of magical-reality-fluid - quantum amplitude flowing between configurations in a way that preserves the total amount of realness while dividing it between worlds - and perhaps every other part of the multiverse must necessarily work the same way for some reason.

I should like to point out that if realness were not preserved, i.e., if some worlds at time t were more real than others, their inhabitants would have no way of discerning that fact.

Just a digression that has no bearing on the main point of the post:

I would be willing to assign a probability of less than 1 in 10^18 to a random person being a Matrix Lord.

The probability that we're in a simulation, times the number of expected Matrix Lords at one moment per simulation, divided by population, should be a lower bound on that probability. I would think it would be at least 1 / population.

The usual analyses of Pascal's Wager, like many lab experiments, privileges the hypothesis and doesn't look for alternative hypotheses.

Why would anyone assume that the Mugger will do as he says? What do we know about the character of all powerful beings? Why should they be truthful to us? If he knows he could save that many people, but refrains from doing so because you won't give him five dollars, he is by human standards a psycho. If he's a psycho, maybe he'll kill all those people if I give him 5 dollars. That actually seems more likely behavior from such a dick.

The situation you are in isn't the experimental hypothetical of knowing what the mugger will do depending on what your actions are. It's a situation where you observer X,Y, and Z, and are free to make inferences from them. If he has the power, I infer the mugger is a sadistic dick who likes toying with creatures. I expect him to renege on the bet, and likely invert it. "Ha Ha! Yes, I saved those beings, knowing that each would go on to torture a zillion zillion others."

This is a mistake theists make all the time. They think hypothesizing an all powerful being allows them to account for all mysteries, and assume that once the power is there, the privileged hypothesis will be fulfilled. But you get no increased probability of any event from hypothesizing power unless you also establish a prior on behavior. From the little I've seen of the mugger, if he has the power to do what he claims, he is malevolent. If he doesn't have the power, he is impotent to deliver and deluded or dishonest besides. Either way, I have no expectation of gain by appealing to such a person.

Then you present me with a brilliant lemma Y, which clearly seems like a likely consequence of my mathematical axioms, and which also seems to imply X - once I see Y, the connection from my axioms to X, via Y, becomes obvious.

Seems a lot like learning a proof of X. It shouldn't surprise us that learning a proof of X increases your confidence in X. The mugger genie has little ground to accuse you of inconsistency for believing X more after learning a proof of it.

Granted the analogy isn't exact; what is learned may fall well short of rigorous proof. You may have only learned a good argument for X. Since you assign only 90% posterior likelihood I presume that's intended in your narrative.

Nevertheless, analogous reasoning seems to apply. The mugger genie has little ground to accuse you of inconsistency for believing X more after learning a good argument for it.

Continuing from what I said in my last comment about the more general problem with Expected Utility Maximizing, I think I might have a solution. I may be entirely wrong, so any criticism is welcome.

Instead of calculating Expected Utility, calculate the probability that an action will result in a higher utility than another action. Choose the one that is more likely to end up with a higher utility. For example, if giving Pascal's mugger the money only has a one out of a trillionth chance of ending up with a higher utility than not giving him your money, you wouldn't give it.

Now there is an apparent inconsistency with this system. If there is a lottery, and you have a 1/100 chance of winning, you would never buy a ticket. Even if the reward is $200 and the cost of a ticket only $1. Or even regardless how big the reward is. However if you are offered the chance to buy a lot of tickets all at once, you would do so, since the chance of winning becomes large enough to outgrow the chance of not winning.

However I don't think that this is a problem. If you expect to play the lottery a bunch of times in a row, then you will choose to buy the ticket, because making that choice in this one instance also means that you will make the same choice in every other instance. Then the probability of ending up with more money at the end of the day is higher.

So if you expect to play the lottery a lot, or do other things that have low chances of ending up with high utilities, you might participate in them. Then when all is done, you are more likely to end up with a higher utility than if you had not done so. However if you get in a situation with an absurdly low chance of winning, it doesn't matter how large the reward is. You wouldn't participate, unless you expect to end up in the same situation an absurdly large number of times.

This method is consistent, it seems to "work" in that most agents that follow it will end up with higher utilities than agents that don't follow it, and Expected Utility is just a special case of it that only happens when you expect to end up in similar situations a lot. It also seems closer to how humans actually make decisions. So can anyone find something wrong with this?

Here's a question, if we had the ability to input a sensory event with a likelyhoodratio of 3^^^^3:1 this whole problem would be solved?

This seems like an exercise in scaling laws.

The odds of being a hero who save 100 lives are less 1% of the odds of being a hero who saves 1 life. So in the absence of good data about being a hero who saves 10^100 lives, we should assume that the odds are much, much less than 1/(10^100).

In other words, for certain claims, the size of the claim itself lowers the probability.

More pedestrian example: ISTR your odds of becoming a musician earning over $1 million a year are much, much less than 1% of your odds of becoming a musician who earns over $10,000 a year.

I don't know of any set of axioms that imply that you should take expected utilities when considering infinite sets of possible outcomes that do not also imply that the utility function is bounded. If we think that our utility functions are unbounded and we want to use the Solomonoff prior, why are we still taking expectations?

(I suppose because we don't know how else to aggregate the utilities over possible worlds. Last week, I tried to see how far I could get if I weakened a few of the usual assumptions. I couldn't really get anywhere interesting because my axioms weren't strong enough to tell you how to decide in many cases, even when the generalized probabilities and generalized utilities are known.)

Isn't this more of social recognition of a scam?

While there are decision-theoretic issues with the Original Pascal's Wager, one of the main problems is that it is a scam ("You can't afford not to do it! It's an offer you can't refuse!"). It seems to me that you can construct plenty of arguments like you just did, and many people wouldn't take you up on the offer because they'd recognize it as a scam. Once something has a high chance of being a scam (like taking the form of Pascal's Wager), it won't get much more of your attention until you lower the likelihood that it's a scam. Is that a weird form of Confirmation Bias?

But nonetheless, couldn't the AI just function in the same way as that? I would think it would need to learn how to identify what is a trick and what isn't a trick. I would just try to think of it as a Bad Guy AI who is trying to manipulate the decision making algorithms of the Good Guy AI.

I think this comes down to bounded computation

I've heard some people try to claim silly things like, the probability that you're telling the truth is counterbalanced by the probability that you'll kill 3↑↑↑3 people instead, or something else with a conveniently equal and opposite utility. But there's no way that things would balance out exactly in practice

With a human's bounded computational resources, maybe assuming that it balances out is the best you can do. You have to make simplifying assumptions if you want to reason about numbers as large as 3↑↑↑3.

I considered this, and I'm not sure if I am considering the mugging from the right perspective.

For instance, in the case of a mugger who is willing to talk with you, even if the actual amount of evidence was mathematically indeterminate (Say the amount is defined as 'It's a finite number higher than any number that could fit in your brain.' and the probability is defined as 'closer to 0 then any positive number you can fit in your brain that isn't 0') you might still attempt to figure out the direction that talking about evidence made the evidence about the mugger go and use that for decision making

If as you talk to him, the mugger provides more and more evidence that he is a matrix lord, you could say "Sure, Here's 5 dollars."

Or If as you talk to him, the mugger provides more and more evidence that he is a mugger, you could say "No, go away."

(Note: I'm NOT saying the above is correct or incorrect yet! Among other things, you could also use the SPEED at which the mugger was giving you evidence as an aid to decision making. You might say yes to a Mugger who offers a million bits of evidence all at once, and no to a Mugger who offers evidence one bit at a time.)

However, in the case below, you can't even do that - Or you could attempt to, but with the worry that even talking about it itself makes a decision:

Cruel Mugger: "Give me 5 dollars and I use my powers to save a shitload of lives. Do anything else, like talking about evidence or walking away, and they die."

So, to consider the problem from the right perspective, should I be attempting to solve the Mugging, the Cruel Mugging, both separately, or both as if they are the same problem?

Even if the prior probability of your saving 3↑↑↑3 people and killing 3↑↑↑3 people, conditional on my giving you five dollars, exactly balanced down to the log(3↑↑↑3) decimal place, the likelihood ratio for your telling me that you would "save" 3↑↑↑3 people would not be exactly 1:1 for the two hypotheses down to the log(3↑↑↑3) decimal place.

The scenario is already so outlandish that it seems unwarranted to assume that the mugger is saying the truth with more than 0.5 certainty. The motives of such a being to engage in this kind of prank, if truly in such a powerful position, would have to be very convulted. Isn't it at least as likely that the opposite will happen if I hand over the five dollars?

Okay, I guess if that's my answer, I'll have to hand over the money if the mugger says "don't give me five dollars!" Or do I?

Suppose you could conceive of what the future will be like if it were explained to you.

Are there more or less than a googleplex differentiable futures which are conceivable to you? If there are more, then selecting a specific one of those conceivable futures is more bits than posited as possible. If fewer, then...?

the vast majority of the improbable-position-of-leverage in any x-risk reduction effort comes from being an Earthling in a position to affect the future of a hundred billion galaxies,

Why does "Earthling" imply sufficient evidence for the rest of this (given a leverage adjustment)? Don't we have independent reason to think otherwise, eg the Great Filter argument?

Mind you, the recent MIRI math paper and follow-up seem (on their face) to disprove some clever reasons for calling seed AGI actually impossible and thereby rejecting a scenario in which Earth will "affect the future of a hundred billion galaxies". There may be a lesson there.

Typo:

opportunities to affect small large numbers of sentient beings

So it looks like the Pascal's mugger problem can be reduced to two problems that need to be solved anyway for an FAI: how to be optimally rational given a finite amount of computing resources, and how to assign probabilities for mathematical statements in a reasonable way.

Does that sound right?

TL:DR

If a poorly-dressed street person offers to save 10(10^100) lives (googolplex lives) for $5 using their Matrix Lord powers, and you claim to assign this scenario less than 10-(10^100) probability, then apparently you should continue to believe absolutely that their offer is bogus even after they snap their fingers and cause a giant silhouette of themselves to appear in the sky.

I don't see why this is necessarily the case. What am I missing here?

Here is a Summary of what I understandso far

A "correct" epistemology would satisfy our intuition that we should ignore the Pascal's Mugger who doesn't show any evidence, and pay the Matrix Lord, who snaps his fingers and shows his power.

The problem is that no matter how low a probability we assign to the mugger telling the truth, the mugger can name an arbitrarily large number of people to save, and thus make it worth it to pay him anyway. If we weigh the mugger's claim at infinitesimally small, however, we won't be sufficiently convinced by the Matrix Lord's evidence.

The matter is further complicated by the fact that the number of people Matrix Lord claims to save suggests a universe which is so complex that it gets a major complexity penalty.

Here is my Attempt at solution

Here is the set of all possible universes

Each possible universe has a probability. They all add up to one. Since there are infinite possible universes, many of these universes have infinitesimally low probability. Bayes theorem adjusts the probability of each.

The Matrix Lord / person turning into a cat scenario is such that a universe which previously had an infinitesimally low probability now has a rather large likelihood.

What happens when a person turns into a cat?

All of the most likely hypothesis are suddenly eliminated, and everything changes.

Working through some examples to demonstrate that this is a solution

You have models U1, U2, U3...and so on. P(Un) is the probability that you live in Universe n. Your current priors:

P(U1) = 60%

P(U2) = 30%

P(U3) = epsilon

P(U4) = delta

...and so on.

Mr. Matrix turns into a cat or something. Now our hypothesis space is as follows:

P(U1) = 0

P(U2) = 0

P(U3) = 5% (previously Epsilon)

P(U4) = delta

In essence, the utter elimination of all the remotely likely hypothesis suddenly makes several universes which were previously epsilon/delta/arbitrarily small in probability much more convincing.

Basically, if the scenario with the Time Lord happened to us, we aught to act in approximately the same way that the idealized "rational agent" would act if it were given no information whatsoever (so all prior probabilities are assigned using complexity alone), and then a voice from the sky suddenly specifies a hypothesis of arbitrarily high complexity from the space of possible universes and claims that it is true.

Come to think of it, you might even think of your current memories as playing the role of the "voice from the sky". There is no meta-prior saying you should trust your memories, but you have nothing else. Similarly, when Mr. Matrix turned into a cat, he eliminated all your non-extremely-unlikely hypotheses, so you have nothing to go on but his word.

"Robin Hanson has suggested that the logic of a leverage penalty should stem from the general improbability of individuals being in a unique position to affect many others (which is why I called it a leverage penalty)."

As I mentioned in a recent discussion post, I have difficulty accepting Robin's solution as valid -- for starters it has the semblance of possibly working in the case of people who care about people, because that's a case that seems as it should be symmetrical, but how would it e.g. work for a Clippy who is tempted with the creation of paperclips? There's no symmetry here because paperclips don't think and Clippy knows paperclips don't think.

And how would it work if the AI in question in asked to evaluate whether such a hypothetical offer should be accepted by a random individual or not? Robin's anthropic solution says that the AI should judge that someone else ought hypothetically take the offer, but it would judge the probabilities differently if it had to judge things in actual life. That sounds as if it ought violate basic principles of rationality?

My effort to steelman Robin's argument attempted to effectively replace "lives" with "structures of type X that the observer cares about and will be impacted", and "unique position to affect" with "unique position of not directly observing" -- hence Law of Visible Impact.

The prior probability of us being in a position to impact a googolplex people is on the order of one over googolplex, so your equations must be wrong

That's not at all how validity of physical theories is evaluated. Not even a little bit.

By that logic, you would have to reject most current theories. For example, Relativity restricted the maximum speed of travel, thus revealing that countless future generations will not be able to reach the stars. Archimedes's discovery of the buoyancy laws enabled future naval battles and ocean faring, impacting billions so far (which is not a googolplex, but the day is still young). The discovery of fission and fusion still has the potential to destroy all those potential future lives. Same with computer research.

The only thing that matters in physics is the old mundane "fits current data, makes valid predictions". Or at least has the potential to make testable predictions some time down the road. The only time you might want to bleed (mis)anthropic considerations into physics is when you have no way of evaluating the predictive power of various models and need to decide which one is worth pursuing. But that is not physics, it's decision theory.

Once you have a testable working theory, your anthropic considerations are irrelevant for evaluating its validity.

Informally speaking, it seems like superexponential numbers of people shouldn't be possible. If a person is some particular type of computation, and exactly identical copies of a person should only count once, then number of people is bounded by number of unique computations (exponential). It does not seem like the raw Kolmogorov complexity of the number will be the right complexity penalty if each person has to be a different computation.

I find it truly bizarre that nobody here seems to be taking MWI seriously. That is, it's not 1 person handing over $5 or not, it's all the branching possible futures of those possibilities. In other words, I hand over $5, then depending how my head radiates heat for the next second there are now many copies of me experiencing $5-less-ness.

How many? Well, answering that question may require a theory of magical reality fluid (or "measure"), but naively speaking it seems that it should be something more akin to 3^^^3 (or googolplex) than to 3^^^^3. So the problem may still exist; but this MWI issue certainly deserves consideration, and the fact that Eliezer didn't apparently consider it makes me suspicious that he hasn't thought as deeply about this as he claims. Even if throwing this additional factor of 3^^^3 into the mix doesn't dissolve the problem entirely, it may well put it into the range where further arguments, such as earthwormchuck163's "there aren't 3^^^^3 different people", could solve it.

Someone who reacts to gap in the sky with "its most likely a hallucination" may, with incredibly low probability, encounter the described hypothetical where it is not a hallucination, and lose out. Yet this person would perform much more optimally when their drink got spiced with LSD or if they naturally developed an equivalent fault.

And of course the issue is that maximum or even typical impact of faulty belief processing which is described here could be far larger than $5 - the hypothesis could have required you to give away everything, to work harder than you normally would and give away income, or worse, to kill someone. And if it is processed with disregard for probability of a fault, such dangerous failure modes are rendered more likely.

I question whether keeping probabilities summing to one is a valid justification for acting as if the mugger being honest has a probability of roughly 1/3^^^3. Since we know that due to our imperfect reasoning, the probability is greater than 1/3^^^3, we know that the expected value of giving the mugger $5 is unimaginably large. Of course, acknowledging this fact causes our probabilities to sum to above one, but this seems like a small price to pay.

Edit: Could someone explain why I've lost points for this?