# Pascal's Muggle: Infinitesimal Priors and Strong Evidence

**Followup to:** Pascal's Mugging: Tiny Probabilities of Vast Utilities, The Pascal's Wager Fallacy Fallacy, Being Half-Rational About Pascal's Wager Is Even Worse

**Short form: **Pascal's Muggle

*tl;dr: If you assign superexponentially infinitesimal probability to claims of large impacts, then apparently you should ignore the possibility of a large impact even after seeing huge amounts of evidence. 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. For the same reason, any evidence you encounter showing that the human species could create a sufficiently large number of descendants - no matter how normal the corresponding laws of physics appear to be, or how well-designed the experiments which told you about them - must be rejected out of hand. There is a possible reply to this objection using Robin Hanson's anthropic adjustment against the probability of large impacts, and in this case you will treat a Pascal's Mugger as having decision-theoretic importance exactly proportional to the Bayesian strength of evidence they present you, without quantitative dependence on the number of lives they claim to save. This however corresponds to an odd mental state which some, such as myself, would find unsatisfactory. In the end, however, I cannot see any better candidate for a prior than having a leverage penalty plus a complexity penalty on the prior probability of scenarios.*

In late 2007 I coined the term "Pascal's Mugging" to describe a problem which seemed to me to arise when combining conventional decision theory and conventional epistemology in the obvious way. On conventional epistemology, the prior probability of hypotheses diminishes exponentially with their complexity; if it would take 20 bits to specify a hypothesis, then its prior probability receives a 2^{-20} penalty factor and it will require evidence with a likelihood ratio of 1,048,576:1 - evidence which we are 1048576 times more likely to see if the theory is true, than if it is false - to make us assign it around 50-50 credibility. (This isn't as hard as it sounds. Flip a coin 20 times and note down the exact sequence of heads and tails. You now believe in a state of affairs you would have assigned a million-to-one probability beforehand - namely, that the coin would produce the exact sequence HTHHHHTHTTH... or whatever - after experiencing sensory data which are more than a million times more probable if that fact is true than if it is false.) The problem is that although this kind of prior probability penalty may seem very strict at first, it's easy to construct physical scenarios that grow in size vastly faster than they grow in complexity.

I originally illustrated this using Pascal's Mugger: A poorly dressed street person says "I'm actually a Matrix Lord running this world as a computer simulation, along with many others - the universe above this one has laws of physics which allow me easy access to vast amounts of computing power. Just for fun, I'll make you an offer - you give me five dollars, and I'll use my Matrix Lord powers to save 3↑↑↑↑3 people inside my simulations from dying and let them live long and happy lives" where ↑ is Knuth's up-arrow notation. This was originally posted in 2007, when I was a bit more naive about what kind of mathematical notation you can throw into a random blog post without creating a stumbling block. (E.g.: On several occasions now, I've seen someone on the Internet approximate the number of dust specks from this scenario as being a "billion", since any incomprehensibly large number equals a billion.) Let's try an easier (and *way *smaller) number instead, and suppose that Pascal's Mugger offers to save a googolplex lives, where a googol is 10^{100} (a 1 followed by a hundred zeroes) and a googolplex is 10 to the googol power, so 10^{10100} or 10^{10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000} lives saved if you pay Pascal's Mugger five dollars, if the offer is honest.

If Pascal's Mugger had only offered to save a mere *googol *lives (10^{100}), we could perhaps reply that although the notion of a Matrix Lord may sound simple to say in English, if we actually try to imagine all the machinery involved, it works out to a substantial amount of computational complexity. (Similarly, Thor is a worse explanation for lightning bolts than the laws of physics because, among other points, an anthropomorphic deity is more complex than calculus in *formal* terms - it would take a larger computer program to simulate Thor as a complete mind, than to simulate Maxwell's Equations - even though in mere human words Thor sounds much easier to explain.) To imagine this scenario in formal detail, we might have to write out the laws of the higher universe the Mugger supposedly comes from, the Matrix Lord's state of mind leading them to make that offer, and so on. And so (we reply) when mere verbal English has been translated into a formal hypothesis, the Kolmogorov complexity of this hypothesis is more than 332 bits - it would take more than 332 ones and zeroes to specify - where 2^{-332} ~ 10^{-100}. Therefore (we conclude) the net expected value of the Mugger's offer is still tiny, once its prior improbability is taken into account.

But once Pascal's Mugger offers to save a *googolplex* lives - offers us a scenario whose value is constructed by twice-repeated exponentiation - we seem to run into some difficulty using this answer. Can we really claim that the complexity of this scenario is on the order of a googol bits - that to formally write out the hypothesis would take one hundred billion billion times more bits than there are atoms in the observable universe?

And a tiny, paltry number like a googolplex is only the beginning of computationally simple numbers that are unimaginably huge. Exponentiation is defined as repeated multiplication: If you see a number like 3^{5}, it tells you to multiply five 3s together: 3×3×3×3×3 = 243. Suppose we write 3^{5} as 3↑5, so that a single arrow ↑ stands for exponentiation, and let the double arrow ↑↑ stand for repeated exponentation, or *tetration*. Thus 3↑↑3 would stand for 3↑(3↑3) or 3^{33} = 3^{27} = 7,625,597,484,987. Tetration is also written as follows: ^{3}3 = 3↑↑3. Thus ^{4}2 = 2^{222} = 2^{24} = 2^{16} = 65,536. Then pentation, or repeated tetration, would be written with 3↑↑↑3 = ^{33}3 = ^{7,625,597,484,987}3 = 3^{3...3} where the ... summarizes an exponential tower of 3s seven trillion layers high.

But 3↑↑↑3 is still quite simple computationally - we could describe a small Turing machine which computes it - so a hypothesis involving 3↑↑↑3 should not therefore get a large complexity penalty, if we're penalizing hypotheses by algorithmic complexity.

I had originally intended the scenario of Pascal's Mugging to point up what seemed like a basic problem with combining conventional epistemology with conventional decision theory: Conventional epistemology says to penalize hypotheses by an exponential factor of computational complexity. This seems pretty strict in everyday life: "What? for a mere 20 bits I am to be called a million times less probable?" But for stranger hypotheses about things like Matrix Lords, the size of the hypothetical universe can blow up enormously faster than the exponential of its complexity. 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...

This problem is worse than just giving five dollars to Pascal's Mugger - our expected utilities don't converge at all! Conventional epistemology tells us to sum over the predictions of all hypotheses weighted by their computational complexity and evidential fit. This works fine with *epistemic* probabilities and sensory predictions because no hypothesis can predict more than probability 1 or less than probability 0 for a sensory experience. As hypotheses get more and more complex, their contributed predictions have tinier and tinier weights, and the sum converges quickly. But decision theory tells us to calculate expected *utility* by summing the utility of each possible outcome, times the probability of that outcome conditional on our action. If hypothetical utilities can grow faster than hypothetical probability diminishes, the contribution of an average term in the series will keep increasing, and this sum will never converge - not if we try to do it the same way we got our epistemic predictions, by summing over complexity-weighted possibilities. (See also this similar-but-different paper by Peter de Blanc.)

Unfortunately I failed to make it clear in my original writeup that this was where the problem came from, and that it was general to situations beyond the Mugger. Nick Bostrom's writeup of Pascal's Mugging for a philosophy journal used a Mugger offering a quintillion days of happiness, where a quintillion is merely 1,000,000,000,000,000,000 = 10^{18}. It takes at least two exponentiations to outrun a singly-exponential complexity penalty. I would be willing to assign a probability of less than 1 in 10^{18} to a random person being a Matrix Lord. You may not have to invoke 3↑↑↑3 to cause problems, but you've got to use something like 10^{10100} - double exponentiation or better. Manipulating ordinary hypotheses about the ordinary physical universe taken at face value, which just contains 10^{80} atoms within range of our telescopes, should not lead us into such difficulties.

(And then the phrase "Pascal's Mugging" got *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.* This is enough to make me regret having ever invented the term "Pascal's Mugging" in the first place; and for further thoughts on this see The Pascal's Wager Fallacy Fallacy (just because the stakes are high does not mean the probabilities are low, and Pascal's Wager is fallacious because of the low probability, not the high stakes!) and Being Half-Rational About Pascal's Wager Is Even Worse. Again, when dealing with issues the mere size of the apparent universe, on the order of 10^{80} - for *small* large numbers - we do *not* run into the sort of decision-theoretic problems I originally meant to single out by the concept of "Pascal's Mugging". My rough intuitive stance on x-risk charity is that if you are one of the tiny fraction of all sentient beings who happened to be born here on Earth before the intelligence explosion, when the existence of the whole vast intergalactic future depends on what we do now, you should expect to find yourself surrounded by a *smorgasbord *of opportunities to affect small large numbers of sentient beings. There is then no reason to worry about tiny probabilities of having a large impact when we can expect to find medium-sized opportunities of having a large impact, so long as we restrict ourselves to impacts no larger than the size of the known universe.)

One proposal which has been floated for dealing with Pascal's Mugger in the decision-theoretic sense is to penalize hypotheses that let you affect a large number of people, in proportion to the number of people affected - what we could call perhaps a "leverage penalty" instead of a "complexity penalty".

Unfortunately this potentially leads us into a different problem, that of *Pascal's Muggle.*

Suppose a poorly-dressed street person asks you for five dollars in exchange for doing a googolplex's worth of good using his Matrix Lord powers.

"Well," you reply, "I think it very improbable that I would be able to affect so many people through my own, personal actions - who am I to have such a great impact upon events? Indeed, I think the probability is somewhere around one over googolplex, maybe a bit less. So no, I won't pay five dollars - it is unthinkably improbable that I could do so much good!"

"I see," says the Mugger.

A wind begins to blow about the alley, whipping the Mugger's loose clothes about him as they shift from ill-fitting shirt and jeans into robes of infinite blackness, within whose depths tiny galaxies and stranger things seem to twinkle. In the sky above, a gap edged by blue fire opens with a horrendous tearing sound - you can hear people on the nearby street yelling in sudden shock and terror, implying that they can see it too - and displays the image of the Mugger himself, wearing the same robes that now adorn his body, seated before a keyboard and a monitor.

"That's not actually me," the Mugger says, "just a conceptual representation, but I don't want to drive you insane. Now give me those five dollars, and I'll save a googolplex lives, just as promised. It's easy enough for me, given the computing power my home universe offers. As for why I'm doing this, there's an ancient debate in philosophy among my people - something about how we ought to sum our expected utilities - and I mean to use the video of this event to make a point at the next decision theory conference I attend. Now will you give me the five dollars, or not?"

"Mm... no," you reply.

"*No?*" says the Mugger. "I understood earlier when you didn't want to give a random street person five dollars based on a wild story with no evidence behind it. But now I've offered you evidence."

"Unfortunately, you haven't offered me *enough* evidence," you explain.

"Really?" says the Mugger. "I've opened up a fiery portal in the sky, and that's not enough to persuade you? What do I have to do, then? Rearrange the planets in your solar system, and wait for the observatories to confirm the fact? I suppose I could also explain the true laws of physics in the higher universe in more detail, and let you play around a bit with the computer program that encodes all the universes containing the googolplex people I would save if you gave me the five dollars -"

"Sorry," you say, shaking your head firmly, "there's just no *way* you can convince me that I'm in a position to affect a googolplex people, because the prior probability of that is one over googolplex. If you wanted to convince me of some fact of merely 2^{-100 }prior probability, a mere decillion to one - like that a coin would come up heads and tails in some particular pattern of a hundred coinflips - then you could just show me 100 bits of evidence, which is within easy reach of my brain's sensory bandwidth. I mean, you could just flip the coin a hundred times, and my eyes, which send my brain a hundred megabits a second or so - though that gets processed down to one megabit or so by the time it goes through the lateral geniculate nucleus - would easily give me enough data to conclude that this decillion-to-one possibility was true. But to conclude something whose prior probability is on the order of one over googolplex, I need on the order of a googol bits of evidence, and you can't present me with a sensory experience containing a googol bits. 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."

"So no matter what evidence I show you," the Mugger says - as the blue fire goes on crackling in the torn sky above, and screams and desperate prayers continue from the street beyond - "you can't ever notice that you're in a position to help a googolplex people."

"Right!" you say. "I can believe that you're a Matrix Lord. I mean, I'm not a *total* Muggle, I'm psychologically capable of responding in *some* fashion to that giant hole in the sky. But it's just completely forbidden for me to assign any significant probability whatsoever that you will actually save a googolplex people after I give you five dollars. You're lying, and I am absolutely, absolutely, absolutely confident of that."

"So you weren't *just* invoking the leverage penalty as a plausible-sounding way of getting out of paying me the five dollars earlier," the Mugger says thoughtfully. "I mean, I'd understand if that was just a rationalization of your discomfort at forking over five dollars for what seemed like a tiny probability, when I hadn't done my duty to present you with a corresponding amount of evidence before demanding payment. But you... you're acting like an AI would if it was actually programmed with a leverage penalty on hypotheses!"

"Exactly," you say. "I'm forbidden *a priori* to believe I can ever do that much good."

"Why?" the Mugger says curiously. "I mean, all I have to do is press this button here and a googolplex lives will be saved." The figure within the blazing portal above points to a green button on the console before it.

"Like I said," you explain again, "the prior probability is just too infinitesimal for the massive evidence you're showing me to overcome it -"

The Mugger shrugs, and vanishes in a puff of purple mist.

The portal in the sky above closes, taking with the console and the green button.

(The screams go on from the street outside.)

A few days later, you're sitting in your office at the physics institute where you work, when one of your colleagues bursts in through your door, seeming highly excited. "I've got it!" she cries. "I've figured out that whole dark energy thing! Look, these simple equations retrodict it exactly, there's no way that could be a coincidence!"

At first you're also excited, but as you pore over the equations, your face configures itself into a frown. "No..." you say slowly. "These equations may look extremely simple so far as computational complexity goes - and they do exactly fit the petabytes of evidence our telescopes have gathered so far - but I'm afraid they're far too improbable to ever believe."

"What?" she says. "Why?"

"Well," you say reasonably, "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. But that would mean that we, here on Earth, are in a position to affect a googolplex people - since, if we blow ourselves up via a nanotechnological war or *(cough) *make certain other errors, those googolplex people will never come into existence. 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."

"Hmm..." she says. "I hadn't thought of that. But what if these equations are right, and yet somehow, everything I do is exactly balanced, down to the googolth decimal point or so, with respect to how it impacts the chance of modern-day Earth participating in a chain of events that leads to creating an intergalactic civilization?"

"How would *that* work?" you say. "There's only seven billion people on today's Earth - there's probably been only a hundred billion people who ever existed total, or will exist before we go through the intelligence explosion or whatever - so even before analyzing your exact position, it seems like your leverage on future affairs couldn't reasonably be less than a one in ten trillion part of the future or so."

"But then given this physical theory which seems obviously true, my acts might imply expected utility differentials on the order of 10^{10100}^{-13}," she explains, "and I'm not allowed to believe that no matter how much evidence you show me."

This problem may not be as bad as it looks; with some further reasoning, the leverage penalty may lead to more sensible behavior than depicted above.

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 solely_{10} responsible for any given event. Or if "fate" is taken to be a sufficiently ultimate fate that there's at most 10 other decisions of similar magnitude that could cumulate to determine someone's outcome utility to within ±50%, then any given person could have their fate_{10} determined on at most 10 occasions. We would surely agree, while assigning priors at the dawn of reasoning, that an agent randomly selected from the pool of all agents in Reality has at most a 100/X chance of being able to be solely_{10} responsible for the fate_{10} of X people. Any reasoning we do about universes, their complexity, sensory experiences, and so on, should maintain this net balance. You can even strip out the part about agents and carry out the reasoning on pure causal nodes; the chance of a randomly selected causal node being in a unique_{100} position on a causal graph with respect to 3↑↑↑3 other nodes ought to be at most 100/3↑↑↑3 for finite causal graphs. (As for infinite causal graphs, well, if problems arise *only* when introducing infinity, maybe it's infinity that has the problem.)

Suppose we apply the Hansonian leverage penalty to the face-value scenario of our own universe, in which there are apparently no aliens and the galaxies we can reach in the future contain on the order of 10^{80} atoms; which, if the intelligence explosion goes well, might be transformed into on the very loose order of... let's ignore a lot of intermediate calculations and just call it the equivalent of 10^{80} centuries of life. (The neurons in your brain perform lots of operations; you don't get only one computing operation per element, because you're powered by the Sun over time. The universe contains a lot more negentropy than just 10^{80} bits due to things like the gravitational potential energy that can be extracted from mass. Plus we should take into account reversible computing. But of course it also takes more than one computing operation to implement a century of life. So I'm just going to xerox the number 10^{80} for use in these calculations, since it's not supposed to be the main focus.)

Wouldn't it be terribly odd to find ourselves - where by 'ourselves' I mean the hundred billion humans who have ever lived on Earth, for no more than a century or so apiece - solely_{100,000,000,000} responsible for the fate_{10} of around 10^{80} units of life? Isn't the prior probability of this somewhere around 10^{-68}?

Yes, according to the leverage penalty. But a prior probability of 10^{-68} is not an insurmountable epistemological barrier. If you're taking things at face value, 10^{-68} is just 226 bits of evidence or thereabouts, and your eyes are sending you a megabit per second. Becoming convinced that *you,* yes *you* are an Earthling is epistemically doable; you just need to see a stream of sensory experiences which is 10^{68} times more probable if you are an Earthling than if you are someone else. If we take everything at face value, then there could be around 10^{80} centuries of life over the history of the universe, and only 10^{11} of those centuries will be lived by creatures who discover themselves occupying organic bodies. Taking everything at face value, the sensory experiences of your life are unique to Earthlings and should immediately convince you that you're an Earthling - just looking around the room you occupy will provide you with sensory experiences that plausibly belong to only 10^{11} out of 10^{80} life-centuries.

If we *don't* take everything at face value, then there might be such things as ancestor simulations, and it might be that your experience of looking around the room is something that happens in 10^{20} ancestor simulations for every time that it happens in 'base level' reality. In this case your probable leverage on the future is diluted (though it may be large even post-dilution). But this is not something that the Hansonian leverage penalty *forces *you to believe - not when the putative stakes are still as small as 10^{80}. Conceptually, the Hansonian leverage penalty doesn't interact much with the Simulation Hypothesis (SH) at all. If you don't believe SH, then you think that the experiences of creatures like yours are rare in the universe and hence present strong, convincing evidence for you occupying the leverage-privileged position of an Earthling - much stronger evidence than its prior improbability. (There's some separate anthropic issues here about whether or not this is *itself* evidence for SH, but I don't think that question is intrinsic to leverage penalties per se.)

A key point here is that even if you accept a Hanson-style leverage penalty, it doesn't have to manifest as an inescapable commandment of modesty. You need not refuse to believe (in your deep and irrevocable humility) that you could be someone as special as an Ancient Earthling. Even if Earthlings matter in the universe - even if we occupy a unique position to affect the future of galaxies - it is still possible to encounter pretty convincing evidence that you're an Earthling. Universes the size of 10^{80} do not pose problems to conventional decision-theoretic reasoning, or to conventional epistemology.

Things play out similarly if - still taking everything at face value - you're wondering about the chance that you could be special even for an Earthling, because you might be one of say 10^{4} people in the history of the universe who contribute a major amount to an x-risk reduction project which ends up actually saving the galaxies. The vast majority of the improbability here is just in being an Earthling in the first place! Thus most of the clever arguments for not taking this high-impact possibility at face value would also tell you not to take being an Earthling at face value, since Earthlings as a whole are much more unique within the total temporal history of the universe than you are supposing yourself to be unique among Earthlings. But given ¬SH, the prior improbability of being an Earthling can be overcome by a few megabits of sensory experience from looking around the room and querying your memories - it's not like 10^{80} is enough future beings that the number of agents randomly hallucinating similar experiences outweighs the number of real Earthlings. Similarly, if you don't think lots of Earthlings are hallucinating the experience of going to a donation page and clicking on the Paypal button for an x-risk charity, that sensory experience can easily serve to distinguish you as one of 10^{4} people donating to an x-risk philanthropy.

Yes, there are various clever-sounding lines of argument which involve *not* taking things at face value - "Ah, but maybe you should consider yourself as an indistinguishable part of this here large reference class of deluded people who think they're important." Which I consider to be a bad idea because it renders you a permanent Muggle by putting you into an inescapable reference class of self-deluded people and then dismissing all your further thoughts as insufficient evidence because you *could* just be deluding yourself further about whether these are good arguments. Nor do I believe the world can only be saved by good people who are incapable of distinguishing themselves from a large class of crackpots, all of whom have no choice but to continue based on the tiny probability that they are not crackpots. (For more on this see Being Half-Rational About Pascal's Wager Is Even Worse.) In this case you are a Pascal's Muggle not because you've explicitly assigned a probability like one over googolplex, but because you took an improbability like 10^{-6} at unquestioning face value and then cleverly questioned all the evidence which could've overcome that prior improbability, and so, in practice, you can never climb out of the epistemological sinkhole. By the same token, you should conclude that you are just self-deluded about being an Earthling since real Earthlings are so rare and privileged in their leverage.

In general, leverage penalties don't translate into advice about modesty or that you're just deluding yourself - they just say that to be rationally coherent, your picture of the universe has to imply that your sensory experiences are at least as rare as the corresponding magnitude of your leverage.

Which brings us back to Pascal's Mugger, in the original alleyway version. The Hansonian leverage penalty seems to imply that to be coherent, *either* you believe that your sensory experiences are *really actually* 1 in a googolplex - that only 1 in a googolplex beings experiences what you're experiencing - or else you really *can't *take the situation at face value.

Suppose the Mugger is telling the truth, and a googolplex other people are being simulated. Then there are at least a googolplex people in the universe. Perhaps some of them are hallucinating a situation similar to this one by sheer chance? Rather than telling you flatly that you can't have a large impact, the Hansonian leverage penalty implies a coherence requirement on how uniquely you think your sensory experiences identify the position you believe yourself to occupy. When it comes to believing you're one of 10^{11} Earthlings who can impact 10^{80} other life-centuries, you need to think your sensory experiences are unique to Earthlings - identify Earthlings with a likelihood ratio on the order of 10^{69}. This is quite achievable, if we take the evidence at face value. But when it comes to improbability on the order of 1/3↑↑↑3, the prior improbability *is *inescapable - your sensory experiences *can't *possibly be that unique - which is assumed to be appropriate because almost-everyone who ever believes they'll be in a position to help 3↑↑↑3 people *will in fact* be hallucinating. Boltzmann brains should be much more common than people in a unique position to affect 3↑↑↑3 others, at least if the causal graphs are finite.

Furthermore - although I didn't realize this part until recently - applying Bayesian updates from that starting point may partially avert the Pascal's Muggle effect:

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

You: "Nope."

Mugger: "Why not? It's a really large impact."

You: "Yes, and I assign a probability on the order of 1 in 3↑↑↑3 that I would be in a unique position to affect 3↑↑↑3 people."

Mugger: "Oh, is that really the probability that you assign? Behold!"

*(A gap opens in the sky, edged with blue fire.)*

Mugger: "Now what do you think, eh?"

You: "Well... I can't actually say this observation has a likelihood ratio of 3↑↑↑3 to 1. No stream of evidence that can enter a human brain over the course of a century is ever going to have a likelihood ratio larger than, say, 10^{1026} to 1 at the *absurdly most, *assuming one megabit per second of sensory data, for a century, each bit of which has at least a 1-in-a-trillion error probability. I'd probably start to be dominated by Boltzmann brains or other exotic minds well before then."

Mugger: "So you're not convinced."

You: "Indeed not. The probability that you're telling the truth is so tiny that God couldn't find it with an electron microscope. Here's the five dollars."

Mugger: "Done! You've saved 3↑↑↑3 lives! Congratulations, you're never going to top that, your peak life accomplishment will now always lie in your past. But why'd you give me the five dollars if you think I'm lying?"

You: "Well, because the evidence you *did* present me with had a likelihood ratio of at least a billion to one - I would've assigned less than 10^{-9} prior probability of seeing this when I woke up this morning - so in accordance with Bayes's Theorem I promoted the probability from 1/3↑↑↑3 to at least 10^{9}/3↑↑↑3, which when multiplied by an impact of 3↑↑↑3, yields an expected value of at least a billion lives saved for giving you five dollars."

I confess that I find this line of reasoning a bit suspicious - it seems overly clever. But on the level of intuitive virtues of rationality, it does seem less stupid than the original Pascal's Muggle; this muggee is at least *behaviorally *reacting to the evidence. In fact, they're reacting in a way exactly proportional to the evidence - they would've assigned the same net importance to handing over the five dollars if the Mugger had offered 3↑↑↑4 lives, so long as the strength of the evidence seemed the same.

(Anyone who tries to apply the lessons here to actual x-risk reduction charities (which I think is probably a bad idea), keep in mind that 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, and that sensory evidence for being an Earthling is what gives you most of your belief that your actions can have an outsized impact.)

So why not just run with this - why not just declare the decision-theoretic problem resolved, if we have a rule that seems to give reasonable behavioral answers in practice? Why not just go ahead and program that rule into an AI?

Well... I still feel a bit nervous about the idea that Pascal's Muggee, after the sky splits open, is handing over five dollars while claiming to assign probability on the order of 10^{9}/3↑↑↑3 that it's doing any good.

I think that my own reaction in a similar situation would be along these lines instead:

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

Me: "Nope."

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

Me: "Yeah... that probably *has *to follow. I don't see any way around that revealed belief, given that I'm not actually giving you the five dollars. 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, if there was no *a priori* mathematical requirement that they balance. 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. So if I assigned probabilities much greater than 1/3↑↑↑3 to your doing something that affected 3↑↑↑3 people, my actions would be overwhelmingly dominated by even a tiny difference in likelihood ratio elevating the probability that you saved 3↑↑↑3 people over the probability that you did something bad to them. The only way this hypothesis can't dominate my actions - really, the only way my expected utility sums can converge at all - is if I assign probability on the order of 1/3↑↑↑3 or less. I don't see any way of escaping that part."

Mugger: "But can you, in your mortal uncertainty, truly assign a probability as low as 1 in 3↑↑↑3 to any proposition whatever? Can you truly believe, with your error-prone neural brain, that you could make 3↑↑↑3 statements *of any kind *one after another, and be wrong, on average, about once?"

Me: "Nope."

Mugger: "So give me five dollars!"

Me: "Nope."

Mugger: "Why not?"

Me: "Because even though I, in my mortal uncertainty, will eventually be wrong about all sorts of things if I make enough statements one after another, this fact can't be used to increase the probability of arbitrary statements beyond what my prior says they should be, because then my prior would sum to more than 1. There must be some kind of required condition for taking a hypothesis seriously enough to worry that I might be overconfident about it -"

Mugger: "Then behold!"

*(A gap opens in the sky, edged with blue fire.)*

Mugger: "Now what do you think, eh?"

Me *(staring up at the sky):* "...whoa." *(Pause.)* "You turned into a cat."

Mugger: "What?"

Me: "Private joke. Okay, I think I'm going to have to rethink a *lot *of things. But if you want to tell me about how I was wrong to assign a prior probability on the order of 1/3↑↑↑3 to your scenario, I will shut up and listen very carefully to what you have to say about it. Oh, and here's the five dollars, can I pay an extra twenty and make some other requests?"

*(The thought bubble pops, and we return to two people standing in an alley, the sky above perfectly normal.)*

Mugger: "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^{1026} to 1, and probably much less. So by the method of imaginary updates, you must assign probability at least 10^{-1026} 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."

Mugger: "How can you possibly say that? You're not being logically coherent!"

Me: "I agree that I'm not being logically coherent, but I think that's acceptable in this case."

Mugger: "This ought to be good. Since when are rationalists allowed to deliberately be logically incoherent?"

Me: "Since we don't have infinite computing power -"

Mugger: "That sounds like a fully general excuse if I ever heard one."

Me: "No, this is a *specific* consequence of bounded computing power. Let me start with a simpler example. Suppose I believe in a set of mathematical axioms. Since I don't have infinite computing power, I won't be able to know all the deductive consequences of those axioms. And *that* means I will necessarily fall prey to the conjunction fallacy, in the sense that you'll present me with a theorem X that is a deductive consequence of my axioms, but which I don't know to be a deductive consequence of my axioms, and you'll ask me to assign a probability to X, and I'll assign it 50% probability or something. 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. So I assign P(X&Y) = 90%, or something like that. Well, that's the conjunction fallacy - I assigned P(X&Y) > P(X). The thing is, if you *then* ask me P(X), after I've seen Y, I'll reply that P(X) is 91% or at any rate something higher than P(X&Y). I'll have changed my mind about what my prior beliefs logically imply, because I'm not logically omniscient, even if that looks like assigning probabilities *over time* which are incoherent in the Bayesian sense."

Mugger: "And how does this work out to my not getting five dollars?"

Me: "In the scenario you're asking me to imagine, you present me with evidence which I currently think Just Plain Shouldn't Happen. And if that actually *does* happen, the sensible way for me to react is by questioning my prior assumptions and the reasoning which led me assign such low probability. One way that I handle my lack of logical omniscience - my finite, error-prone reasoning capabilities - is by being willing to assign infinitesimal probabilities to non-privileged hypotheses so that my prior over all possibilities can sum to 1. 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 in advance, and yet encounter strong evidence, say 10^{10} to 1, that the thing has actually happened. If something like that happens, you don't do a Bayesian update to a posterior of 10^{-90}. Instead you question both whether the evidence might be weaker than it seems, *and* whether your estimate of prior improbability might have been poorly calibrated, because rational agents who actually have well-calibrated priors should not encounter situations like that until they are ten billion days old. Now, this may mean that I end up doing some non-Bayesian updates: I say some hypothesis has a prior probability of a quadrillion to one, you show me evidence with a likelihood ratio of a billion to one, and I say 'Guess I was wrong about that quadrillion to one thing' rather than being a Muggle about it. And then I shut up and listen to what *you* have to say about how to estimate probabilities, because on my worldview, I wasn't *expecting* to see you turn into a cat. But for me to make a super-update like that - reflecting a posterior belief that I was logically incorrect about the prior probability - you have to really actually show me the evidence, you can't just ask me to imagine it. This is something that only logically incoherent agents ever say, but that's all right because I'm not logically omniscient."

At some point, we're going to have to build some sort of actual prior into, you know, some sort of actual self-improving AI.

(Scary thought, right?)

So far as I can presently see, the logic requiring some sort of leverage penalty - not just so that we don't pay $5 to Pascal's Mugger, but also so that our expected utility sums converge at all - seems clear enough that I can't yet see a good alternative to it (feel welcome to suggest one), and Robin Hanson's rationale is by far the best I've heard.

In fact, what we actually need is more like a combined leverage-and-complexity penalty, to avoid scenarios like this:

Mugger: "Give me $5 and I'll save 3↑↑↑3 people."

You: "I assign probability exactly 1/3↑↑↑3 to that."

Mugger: "So that's one life saved for $5, on average. That's a pretty good bargain, right?"

You: "Not by comparison with x-risk reduction charities. But I also like to do good on a smaller scale now and then. How about a penny? Would you be willing to save 3↑↑↑3/500 lives for a penny?"

Mugger: "Eh, fine."

You: "Well, the probability of that is 500/3↑↑↑3, so here's a penny!" *(Goes on way, whistling cheerfully.)*

Adding a complexity penalty *and* a leverage penalty is necessary, not just to avert this exact scenario, but so that we don't get an infinite expected utility sum over a 1/3↑↑↑3 probability of saving 3↑↑↑3 lives, 1/(3↑↑↑3 + 1) probability of saving 3↑↑↑3 + 1 lives, and so on. If we combine the standard complexity penalty with a leverage penalty, the whole thing should converge.

Probability penalties are epistemic features - they affect what we believe, not just what we do. Maps, ideally, correspond to territories. Is there any territory that this complexity+leverage penalty can correspond to - any state of a single reality which would make these the true frequencies? Or is it only interpretable as pure uncertainty over realities, with there being no single reality that could correspond to it? To put it another way, the complexity penalty and the leverage penalty seem unrelated, so perhaps they're mutually inconsistent; can we show that the union of these two theories has a model?

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. (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 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. It seems worth noting that part of what's motivating this version of the 'territory' is that our sum over all real things, weighted by reality-fluid, can then converge. In other words, the reason why complexity+leverage works in decision theory is that the union of the two theories has a model in which the total multiverse contains an amount of reality-fluid that can sum to 1 rather than being infinite. (Though we need to suppose that either (a) only programs with a finite number of causal nodes exist, or (2) programs can divide finite reality-fluid among an infinite number of nodes via some measure that gives every experience-moment a well-defined relative amount of reality-fluid. Again see caveats about basic philosophical confusion - perhaps our map needs this property over its uncertainty but the territory doesn't have to work the same way, etcetera.)

If an AI's overall architecture is also 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.

In other words: This is my best current idea for how a prior, e.g. as used in an AI, could yield decision-theoretic convergence over explosively large possible worlds.

However, I would still call this a semi-open FAI problem (edit: wide-open) because it seems quite plausible that somebody is going to kick holes in the overall view I've just presented, or come up with a better solution, possibly within an hour of my posting this - the proposal is both recent and weak even by my standards. I'm also worried about whether it turns out to imply anything crazy on anthropic problems. Over to you, readers.

## Comments (404)

TopI 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.

I'm not sure that the kludge works anyway, since there are still some "high impact" scenarios which don't get kludged out. Let's imagine the mugger's pitch is as follows. "I am the Lord of the Matrix, and guess what - you're in it! I'm in the process of running a huge number of simulations of human civilization, in series, and in each run of the simulation I am making a very special offer to some carefully selected people within it. If you are prepared to hand over $5 to me, I will kindly prevent one dust speck from entering the eye of one person in each of the next googleplex simulations that I run! Doesn't that sound like a great offer?"

Now, rather naturally, you're going to tell him to get lost. And in the worlds where there really is a Matrix Lord, and he's telling the truth, the approached subjects almost always tell him to get lost as well (the Lord is careful in whom he approaches), which means that googleplexes of preventable dust specks hit googleplexes of eyes. Each rejection of the offer causes a lower total utility than would be obtained from accepting it. And if those worlds have a measure > 1/googleplex, there is on the face of it a net loss in expected utility. More likely, we're just going to get non-convergent expected utilities again.

The general issue is that the causal structure of the hypothetical world is highly linear. A reasonable proportion of nodes (perhaps 1 in a billion) do indeed have the ability to affect a colossal number of other nodes in such a world. So the high utility outcome doesn't get suppressed by a locational penalty.

I'd be more worried about that if I couldn't (apparently) visualize what a corresponding Tegmark Level IV universe looks like. If the union of two theories has a model, they can't be mutually inconsistent. Whether this corresponding multiverse is plausible is a different problem.

*4 points [-]Why is decision/probability theory allowed to constrain the space of "physical" models? It seems that the proper theory should not depend on metaphysical assumptions.

If they are starting to require uncertain metaphysical assumptions, I think that counts as "not working together".

Metaphysical assumptions are one thing: this one involves normative assumptions. There is zero reason to think we evolved values that can make any sense at all of saving 3^^^3 people. The software we shipped with cannot take numbers like that in it's domain. That we can think up thought experiments that confuse our ethical intuitions is already incredibly likely. Coming up with kludgey methods to make decisions that give intuitively correct answers to the thought experiments while preserving normal normative reasoning and then--- from there--- concluding something about what the universe must be like is a really odd epistemic position to take.

*25 points [-]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.

Bostrom would probably try to argue that you do. See Bostrom (2006).

Am I crazy, or does Bostrom's argument in that paper fall flat almost immediately, based on a bad moral argument?

His first, and seemingly most compelling, argument for Duplication over Unification is that, assuming an infinite universe, it's certain (with probability 1) that there is already an identical portion of the universe where you're torturing the person in front of you. Given Unification, it's meaningless to distinguish between that portion and this portion, given their physical identicalness, so torturing the person is morally blameless, as you're not increasing the number of unique observers being tortured. Duplication makes the two instances of the person distinct due to their differing spatial locations, even if every other physical and mental aspect is identical, so torturing is still adding to the suffering in the universe.

However, you can flip this over trivially and come to a terrible conclusion. If Duplication is true, you merely have to simulate a person until they experience a moment of pure hedonic bliss, in some ethically correct manner that everyone agrees is morally good to experience and enjoy. Then, copy the fragment of the simulation covering the experiencing of that emotion, and duplicate it endlessly. Each duplicate is distinct, and so you're increasing the amount of joy in the universe every time you make a copy. It would be a net win, in fact, if you killed every human and replaced the earth with a computer doing nothing but running copies of that one person experiencing a moment of bliss. Unification takes care of this, by noting that duplicating someone adds, at most, a single bit of information to the universe, so spamming the universe with copies of the happy moment counts either the same as the single experience, or at most a trivial amount more.

Am I thinking wrong here?

True just if your

summum bonumis exactly an aggregate of moments of happiness experienced.I take the position that it is not.

I don't think one even has to resort to a position like "only one copy counts".

True, but that's then striking more at the heart of Bostrom's argument, rather than my counter-argument, which was just flipping Bostrom around. (Unless your summum malum is significantly different, such that duplicate tortures and duplicate good-things-equivalent-to-torture-in-emotional-effect still sum differently?)

I'd argue that the torture portion is not identical to the not-torture portion and that the difference is caused by at least one event in the common prior history of both portions of the universe where they diverged. Unification only makes counterfactual worlds real; it does not cause every agent to experience every counterfactual world. Agents are differentiated by the choices they make and agents who perform torture are not the same agents as those who abstain from torture. The difference can be made arbitrarily small, for instance by choosing an agent with a 50% probability of committing torture based on the outcome of a quantum coin flip, but the moral question in that case is why an agent would choose to become 50% likely to commit torture in the first place. Some counterfactual agents will choose to become 50% likely to commit torture, but they will be very different than the agents who are 1% likely to commit torture.

I think you're interpreting Bostrom slightly wrong. You seem to be reading his argument (or perhaps just my short distillation of it) as arguing that you're not currently torturing someone, but there's an identical section of the universe elsewhere where you

aretorturing someone, so you might as well start torturing now.As you note, that's contradictory - if you're not currently torturing, then your section of the universe must not be identical to the section where the you-copy

istorturing.Instead, assume that you are currently torturing someone. Bostrom's argument is that you're not making the universe worse, because there's a you-copy which is torturing an identical person elsewhere in the universe. At most one of your copies is capable of taking blame for this; the rest are just running the same calculations "a second time", so to say. (Or at least, that's what he's arguing that Unification would say, and using this as a reason to reject it and turn to Duplication, so each copy is morally culpable for causing new suffering.)

I think it not unlikely that if we have a successful intelligence explosion and subsequently discover a way to build something 4^^^^4-sized, then we will figure out a way to grow into it, one step at a time. This 4^^^^4-sized supertranshuman mind then should be able to discriminate "interesting" from "boring" 3^^^3-sized things. If you could convince the 4^^^^4-sized thing to write down a list of all nonboring 3^^^3-sized things in its spare time, then you would have a formal way to say what an "interesting 3^^^3-sized thing" is, with description length (the description length of humanity = the description length of our actual universe) + (the additional description length to give humanity access to a 4^^^^4-sized computer -- which isn't much because access to a universal Turing machine would do the job and more).

Thus, I don't think that it needs a 3^^^3-sized description length to pick out interesting 3^^^3-sized minds.

*3 points [-]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: So then, you think the probability that you should care as much about my 3↑↑↑3 simulated people as I thought you did is on the order of 1/3↑↑↑3?

After thinking about it a bit more I decided that I actually do care about simulated people almost exactly as the mugger thought I did.

*2 points [-]Didn't you feel sad when Yoona-939 was terminated, or wish all happiness for Sonmi-451?

*4 points [-]All the other Yoona-939s were fine, right? And that Yoona-939 was terminated quickly enough to prevent divergence, wasn't she?

(my point is, you're making it seem like you're breaking the degeneracy by labeling them. But their being identical is deep)

But now she's... you know... now she's...

(wipes away tears)slightly less real.You hit pretty strong diminishing returns on existence once you've hit the 'at least one copy' point.

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.

*16 points [-]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

otherpeople 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?This is indeed a good argument for viewing the leverage penalty as a special case of a locational penalty (which I think is more or less what Hanson proposed to begin with).

*10 points [-]Suppose we had a planet of 3^^^3 people (their universe has novel physical laws). There is a planet-wide lottery. Catherine wins. There was a 1/3^^^3 chance of this happening. The lotto representative comes up to her and asks her to hand over her ID card for verification.

All over the planet, as a fun prank, a small proportion of people have been dressing up as lotto representatives and running away with peoples' ID cards. This is very rare - only one person in 3^^3 does this today.

If the lottery prize is 3^^3 times better than getting your ID card stolen, should Catherine trust the lotto official? No, because there are 3^^^3/3^^3 pranksters, and only 1 real official, and 3^^^3/3^^3 is 3^^(3^^3 - 3), which is a whole lot of pranksters. She hangs on to her card, and doesn't get the prize. Maybe if the reward were 3^^^3 times greater than the penalty, we could finally get some lottery winners to actually collect their winnings.

All of which is to say, I don't think there's any locational penalty - the crowd near the muggle should have exactly the same probability assignments as her, just as the crowd near Catherine has the same probability assignments as her about whether this is a prankster or the real official. I think the penalty is the ratio of lotto officials to pranksters (conditional on a hypothesis like "the lottery has taken place"). If the hypothesis is clever, though, it could probably evade this penalty (hypothesize a smaller population with a reward of 3^^^3 years of utility-satisfaction, maybe, or 3^^^3

newpeople created), and so what intuitively seems like a defense against pascal's mugging may not be.*15 points [-]A simplified version of the argument here:

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.

If the agent defines its utility indirectly in terms of designer's preference, a disagreement in evaluation of a decision by agent's utility function and designer's preference doesn't easily indicate that designer's evaluation is more accurate, and if it's not, then the designer should defer to the agent's judgment instead of adjusting its utility.

Similarly, if the agent is good at building its map, it might have a better map than the designer, so a disagreement is not easily resolved in favor of the designer. On the other hand, there can be a bug in agent's world modeling code in which case it should be fixed! And similarly, if there is a bug in agent's indirect utility definition, it too should be fixed. The arguments seem analogous to me, so why would preference be more easily debugged than world model?

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.

You're right. Edited.

*1 point [-]In my view, we could make act-based agents without answering this or any similar questions. So I'm much less interested in answering them then I used to be. (There are possible approaches that do have to answer all of these questions, but at this point they seem very much less promising to me.)

We've briefly discussed this issue in the abstract, but I'm curious to get your take in a concrete case. Does that seem right to you? Do you think that we need to understand issues like this one, and have confidence in that understanding, prior to building powerful AI systems?

*1 point [-]FAI designs that require high confidence solutions to many philosophical problems also do not seem very promising to me at this point. I endorse looking for alternative approaches.

I agree that act-based agents seem to require fewer high confidence solutions to philosophical problems. My main concern with act-based agents is that these designs will be in competition with fully autonomous AGIs (either alternative designs, or act-based agents that evolve into full autonomy due to inadequate care of their owners/users) to colonize the universe. The dependence on humans and lack of full autonomy in act-based agents seem likely to cause a significant weakness in at least one crucial area of this competition, such as general speed/efficiency/creativity, warfare (conventional, cyber, psychological, biological, nano, etc.), cooperation/coordination, self-improvement, and space travel. So even if these agents turn out to be "safe", I'm not optimistic that we "win" in the long run.

My own idea is to aim for FAI designs that can correct their philosophical errors, autonomously, the same way that we humans can. Ideally, we'd fully understand how humans reason about philosophical problems and how philosophy normatively

oughtto be done before programming or teaching that to an AI. But realistically, due to time pressure, we might have to settle for something suboptimal like teaching through examples of human philosophical reasoning. Of course there's lots of ways for this kind of AI to go wrong as well, so I also consider it to be a long shot.Let me ask you a related question. Suppose act-based designs are as successful as you expect them to be. We still need to understand issues like the one described in Eliezer's post (or solve the meta-problem of understanding philosophical reasoning)

at some point, right? When do you think that will be? In other words, how much time do you think successfully creating act-based agents buys us?It's not so much that I have confidence in these approaches, but that I think (1) they are the most natural to explore at the moment, and (2) issues that seem like they can be cleanly avoided for these approaches seem less likely to be fundamental obstructions in general.

Whenever such issues bear directly on our decision-making in such a way that making errors would be really bad. For example, when we encounter a situation where we face a small probability of a very large payoff, then it matters how well we understand the particular tradeoff at hand. The goal / best case is that the development of AI doesn't depend on sorting out these kinds of considerations for its own sake, only insofar as the AI has to actually make critical choices that depend on these considerations.

I wrote a little bit about efficiency here. I don't see why an approval-directed agent would be at a serious disadvantage compared to an RL agent (though I do see why an imitation learner would be at a disadvantage by default, and why an approval-directed agent may be unsatisfying from a safety perspective for non-philosophical reasons).

Ideally you would synthesize data in advance in order to operate without access to counterfactual human feedback at runtime---it's not clear if this is possible, but it seems at least plausible. But it's also not clear to me it is necessary, as long as we can tolerate very modest (<1%) overhead from oversight.

Of course if such a period goes on long enough then it will be a problem, but that is a slow-burning problem that a superintelligent civilization can address at its leisure. In terms of technical solutions, anything we can think of now will easily be thought of in this future scenario. It seems like the only thing we really lose is the option of technological relinquishment or serious slow-down, which don't look very attractive/feasible at the moment.

Isn't a crucial consideration here how soon after the development of AI they will be faced with such choices? If the answer is "soon" then it seems that we should try to solve the problems ahead of time or try to delay AI. What's your estimate? And what do you think the first such choices will be?

I think that we are facing some issues all of the time (e.g. some of these questions probably bear on "how much should we prioritize fast technological development?" or "how concerned should we be with physics disasters?" or so on), but that it will be a long time before we face really big expected costs from getting these wrong. My best guess is that we will get to do many-centuries-of-current-humanity worth of thinking before we really need to get any of these questions right.

I don't have a clear sense of what the first choices will be. My view is largely coming from not seeing any serious candidates for critical choices.

Anything to do with expansion into space looks like it will be very far away in subjective time (though perhaps not far in calendar time). Maybe there is some stuff with simulations, or value drift, but neither of those look very big in expectation for now. Maybe all of these issues together make 5% difference in expectation over the next few hundred subjective issues? (Though this is a pretty unstable estimate.)

How did you arrive at the conclusion that we're not facing big expected costs with these questions? It seems to me that for example the construction of large nuclear arsenals and lack of sufficient safeguards against nuclear war has already caused a large expected cost, and may have been based on one or more incorrect philosophical understandings (e.g., to the question of, what is the right amount of concern for distant strangers and future people). Similarly with "how much should we prioritize fast technological development?" But this is just from intuition since I don't really know how to compute expected costs when the uncertainties involved have a large moral or normative component.

Do you expect technological development to have plateaued by then (i.e., AIs will have invented essentially all technologies feasible in this universe)? If so, do you think there won't be any technologies among them that would let some group of people/AIs unilaterally alter the future of the universe according to their understanding of what is normative? (For example, intentionally or accidentally destroy civilization, or win a decisive war against the rest of the world.) Or do you think something like a world government will have been created to control the use of such technologies?

There are lots of things we don't know, and my default presumption is for errors to be non-astronomically-costly, until there are arguments otherwise.

I agree that philosophical problems have some stronger claim to causing astronomical damage, and so I am more scared of philosophical errors than e.g. our lack of effective public policy, our weak coordination mechanisms, global warming, the dismal state of computer security.

But I don't see really strong arguments for philosophical errors causing great damage, and so I'm skeptical that we are facing big expected costs (big compared to the biggest costs we can identify and intervene on, amongst them AI safety).

That is, there seems to be a pretty good case that AI may be built soon, and that we lack the understanding to build AI systems that do what we want, that we will nevertheless build AI systems to help us get what we want in the short term, and that in the long run this will radically reduce the value of the universe. The cases for philosophical errors causing damage are overall much more speculative, have lower stakes, and are less urgent.

I agree that philosophical progress would very slightly decrease the probability of nuclear trouble, but this looks like a very small effect. (Orders of magnitude smaller than the effects from say increased global peace and stability, which I'd probably list as a higher priority right now than resolving philosophical uncertainty.) It's possible we disagree about the mechanics of this particular situation.

No. I think that 200 years of subjective time probably amounts 5-10 more doublings of the economy, and that technological change is a plausible reason that philosophical error would eventually become catastrophic.

I said "best guess" but this really is a pretty wild guess about the relevant timescales.

As with the special case of nuclear weapons, I think that philosophical error is a relatively small input into world-destruction.

I don't expect this to cause philosophical errors to become catastrophic. I guess the concern is that the war will be won by someone who doesn't much care about the future, thereby increasing the probability that resources are controlled by someone who prefers not undergo any further reflection? I'm willing to talk about this scenario more, but at face value the prospect of a decisive military victory wouldn't bump philosophical error above AI risk as a concern for me.

I'm open to ending up with a more pessimistic view about the consequences of philosophical error, either by thinking through more possible scenarios in which it causes damage or by considering more abstract arguments.

But if I end up with a view more like yours, I don't know if it would change my view on AI safety. It still feels like the AI control problem is a different issue which can be considered separately.

*11 points [-]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)

When we jump to the version involving causal nodes having Large leverage over other nodes in a graph, there aren't Large numbers of distinct people involved, but there's Large numbers of life-centuries involved and those moments of thought and life have to be instantiated by causal nodes.

Infinity makes my calculations break down and cry, at least at the moment.

*14 points [-]Imagine someone makes the following claims:

Then they threaten, unless you give them $5, to kidnap you, give you the immortality drug, stick you in the spaceship, launch it at near-light speed, and have you stuck (presumably bound in an uncomfortable position) in the spaceship for the 3^^^3 years the universe will last.

(okay, there are lots of contingent features of the universe that will make this not work, but imagine something better. Pocket dimension, maybe?)

If their claims are true, then their threat seems credible even though it involves a large amount of suffering. Can you explain what you mean by life-centuries being instantiated by causal nodes, and how that makes the madman's threat less credible?

*2 points [-]Are you sure it wouldn't be rational to pay up? I mean, if the guy looks like he could do that for $5, I'd rather not take chances. If you pay, and it turns out he didn't have all that equipment for torture, you could just sue him and get that $5 back, since he defrauded you. If he starts making up rules about how you can never ever tell anyone else about this, or later check validity of his claim or he'll kidnap you, you should, for game-theoretical reasons not abide, since being the kinda agent that accepts those terms makes you valid target for such frauds. Reasons for not abiding being the same as for single-boxing.

That requires a MTTF of 3^^^3 years, or a per-year probability of failure of roughly 1/3^^^3.

This implies that physical properties like the cosmological constant and the half-life of protons can be measured to a precision of roughly 1/3^^^3 relative error.

To me it seems like both of those claims have prior probability ~ 1/3^^^3. (How many spaceships would you have to build and how long would you have to test them to get an MTTF estimate as large as 3^^^3? How many measurements do you have to make to get the standard deviation below 1/3^^^3?)

If what he says is true, then there will be 3^^^3 years of life in the universe. Then, assuming this anthropic framework is correct, it's very unlikely to find yourself at the beginning rather than at any other point in time, so this provides 3^^^3-sized evidence against this scenario.

I'm not entirely sure that the doomsday argument also applies to different time slices of the same person, given that Eliezer in 2013 remembers being Eliezer in 2012 but not vice versa.

Say the being that suffers for 3^^^3 seconds is morally relevant but not in the same observer moment reference class as humans for some reason. (IIRC putting all possible observers in the same reference class leads to bizarre conclusions...? I can't immediately re-derive why that would be.) But anyway it really seems that the magical causal juice is the important thing here, not the anthropic/experiential nature or lack thereof of the highly-causal nodes, in which case the anthropic solution isn't quite hugging the real query.

The only reason that I have ever thought of is that our reference class should intuitively consist of only sentient beings, but that nonsentient beings should still be able to reason. Is this what you were thinking of? Whether it applies in a given context may depend on what exactly you mean by a reference class in that context.

*3 points [-]If it can reason but isn't sentient then it maybe doesn't have "observer" moments, and maybe isn't itself morally relevant—Eliezer seems to think that way anyway. I've been trying something like, maybe messing with the non-sentient observer has a 3^^^3 utilon effect on human utility somehow, but that seems psychologically-architecturally impossible for humans in a way that might end up being fundamental. (Like, you either have to make 3^^^3 humans, which defeats the purpose of the argument, or make a single human have a 3^^^3 times better life without lengthening it, which seems impossible.) Overall I'm having a really surprising amount of difficulty thinking up an example where you have a lot of causal importance but no anthropic counter-evidence.

Anyway, does "anthropic" even really have anything to do with qualia? The way people talk about it it clearly does, but I'm not sure it even shows up in the definition—a non-sentient optimizer could totally make anthropic updates. (That said I guess Hofstadter and other strange loop functionalists would disagree.) Have I just been wrongly assuming that everyone else was including "qualia" as fundamental to anthropics?

Yeah, this whole line of reasoning fails if you can get to 3^^^3 utilons without creating ~3^^^3 sentients to distribute them among.

I'm not sure what you mean. If you use an anthropic theory like what Eliezer is using here (e.g. SSA, UDASSA) then an amount of causal importance that is large compared to the rest of your reference class implies few similar members of the reference class, which is anthropic counter-evidence, so of course it would be impossible to think of an example. Even if nonsentients can contribute to utility, if I can create 3^^^3 utilons using nonsentients, than some other people probably can to, so I don't have a lot of causal importance compared to them.

This is the contrapositive of the grandparent. I was saying that if we assume that the reference class is sentients, then nonsentients need to reason using different rules i.e. a different reference class. You are saying that if nonsentients should reason using the same rules, then the reference class cannot comprise only sentients. I actually agree with the latter much more strongly, and I only brought up the former because it seemed similar to the argument you were trying to remember.

There are really two separate questions here, that of how to reason anthropically and that of how magic reality-fluid is distributed. Confusing these is common, since the same sort of considerations affect both of them and since they are both badly understood, though I would say that due to UDT/ADT, we now understand the former much better, while acknowledging the possibility of unknown unknowns. (Our current state of knowledge where we confuse these actually feels a lot like people who have never learnt to separate the descriptive and the normative.)

The way Eliezer presented things in the post, it is not entirely clear which of the two he meant to be responsible for the leverage penalty. It seems like he meant for it to be an epistemic consideration due to anthropic reasoning, but this seems obviously wrong given UDT. In the Tegmark IV model that he describes, the leverage penalty is caused by reality-fluid, but it seems like he only intended that as an analogy. It seems a lot more probable to me though, and it is possible that Eliezer would express uncertainty as to whether the leverage penalty is actually caused by reality-fluid, so that it is a bit more than an analogy. There is also a third mathematically equivalent possibility where the leverage penalty is about values, and we just care less about individual people when there are more of them, but Eliezer obviously does not hold that view.

*10 points [-]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 theorythat forbids you from assigning probability 1/3^^^3 to the next bit in your input stream being a zero -- so as far asprobability theoryis 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 leastconsiderassigning probabilities this way even if it doesn't normally seem like a good idea.True, but it seems crazy to be that certain about what you'll see. It doesn't seem that unlikely to hallucinate that happening. It doesn't seem that unlikely for all the photons and phonons to just happen to converge in some pattern that makes it look and sound exactly like a Matrix Lord.

You're basically assuming that your sensory equipment is vastly more reliable than you have evidence to believe, just because you want to make sure that if you get a positive, you won't just assume it's a false positive.

Actually, there is such a law. You cannot reasonably start, when you are born into this world, naked, without any sensory experiences, expecting that the next bit you experience is much more likely to be 1 rather than 0. If you encounter one hundred zillion bits and they all are 1, you still wouldn't assign 1/3^^^3 probability to next bit you see being 0, if you're rational enough.

Of course, this is mudded by the fact that you're not born into this world without priors and all kinds of stuff that weights on your shoulders. Evolution has done billions of years worth of R&D on your priors, to get them straight. However, the gap these evolution-set priors would have to cross to get even close to that absurd 1/3^^^3... It's a theoretical possibility that's by no stretch a realistic one.

This seems like a

really usefulstrategy!Agreed - placeholders and kludges should

looklike placeholders and kludges. I became a happier programmer when I realised this, because up until then I was always conflicted about how much time I should spend making some unsatisfying piece of code look beautiful.Related: Would an AI conclude it's likely to be a Boltzmann brain? ;)

Everyone's a Boltzmann brain to some degree.

*5 points [-]Or even if the AI experienced an intelligence explosion the danger is that it would not believe it had really become so important because the prior odds of you being the most important thing that will probably ever exist is so low.

Edit: The AI could note that it uses a lot more computing power than any other sentient and so give itself an anothropic weight much greater than 1.

With respect to this being a "danger," don't Boltzmann brains have a decision-theoretic weight of zero?

Why zero? If you came to believe there was a 99.99999% chance you are currently dreaming wouldn't it effect your choices?

*7 points [-]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)

mustdrop 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.

Hm. Technically for EU

differentialsto converge we only need that the number of peoplewe expectedly affectsums to something finite, but having a finite expected number of people existing in the multiverse would certainly accomplish that.The problem is that the Solomonoff prior picks out 3^^^3 as much more likely than most of the numbers of the same magnitude because it has much lower Kolmogorov complexity.

I'm not familiar with Kolmogorov complexity, but isn't the aparent simplicity of 3^^^3 just an artifact of what notation we happen to have invented? I mean, "^^^" is not really a basic operation in arithmetic. We have a nice compact way of describing what steps are needed to get from a number we intuitively grok, 3, to 3^^^3, but I'm not sure it's safe to say that makes it simple in any significant way. For one thing, what would make 3 a simple number in the first place?

*6 points [-]In the nicest possible way, shouldn't you have stopped right there? Shouldn't the appearance of this unfamiliar and formidable-looking word have told you that I wasn't appealing to some intuitive notion of complexity, but to a particular formalisation that you would need to be familiar with to challenge? If instead of commenting you'd Googled that term, you would have found the Wikipedia article that answered this and your next question.

*1 point [-]You can as a rough estimate of the complexity of a number take the amount of lines of the shortest program that would compute the number from basic operations. More formally, substitute lines of a program with states of a Turing Machine.

But what numbers are you allowed to start with on the computation? Why can't I say that, for example, 12,345,346,437,682,315,436 is one of the numbers I can do computation from (as a starting point), and thus it has extremely small complexity?

You could say this -- doing so would be like describing your own language in which things involving 12,345,346,437,682,315,436 can be expressed concisely.

So Kolmogorov complexity is somewhat language-dependent. However, given two languages in which you can describe numbers, you can compute a constant such that the complexity of any number is off by at most that constant between the two languages. (The constant is more or less the complexity of describing one language in the other). So things aren't actually too bad.

But if we're just talking about Turing machines, we presumably express numbers in binary, in which case writing "3" can be done very easily, and all you need to do to specify 3^^^3 is to make a Turing machine computing ^^^.

But can't this constant itself be arbitrarily large when talking about arbitrary numbers? (Of course, for any

specificnumber, it is limited in size.)Well... Given any number

N, you can in principle invent a programming language where the program`do_it`

outputsN.The constant depends on the two languages, but not on the number. As army1987 points out, if you pick the number first, and then make up languages, then the difference can be arbitrarily large. (You could go in the other direction as well: if your language specifies that no number less than 3^^^3 can be entered as a constant, then it would probably take approximately log(3^^^3) bits to specify even small numbers like 1 or 2.)

But if you pick the languages first, then you can compute a constant based on the languages, such that for

allnumbers, the optimal description lengths in the two languages differ by at most a constant.The context this in which this comes up here generally requires something like "there's a way to compare the complexity of numbers which always produces the same results independent of language, except in a finite set of cases. Since that set is finite and my argument doesn't depend on any specific number, I can always base my argument on a case that's not in that set."

If that's how you're using it, then you don't get to pick the languages first.

You do get to pick the languages first because there is a large but finite (say no more than 10^6) set of reasonable languages-modulo-trivial-details that could form the basis for such a measurement.

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.

Caveat emptor, the boxes she gave me always were empty!

*6 points [-]Two quick thoughts:

Anytwo theories can be made compatible if allowing for some additional correction factor (e.g. a "leverage penalty") designed tomakethem 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

tosuch a low prior. It's the mirror from "can never process enough bits to getawayfrom such a low prior", namely "can never process enough bits to gettoassigning 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

tosuch a low prior. You can beverysure you can't affect that many, based on your understanding of how causal nodes are interconnected, but you can't bethatsure (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".

*6 points [-]Is it just me, or is everyone here overly concerned with coming up with patches for

this specific caseand 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

exactlycounterbalance? 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.

*1 point [-]The idea is that it'd be great to have a formalism where they do by construction.

Also, when there's no third party, it's not distinct enough from Pascal's Wager as to demand extra terminology that focusses on the third party, such as "Pascal's Mugging". If it is just agent doing contemplations by itself, that's the agent making a wager on it's hypotheses, not getting mugged by someone.

I'll just go ahead and use "Pascal Scam" to describe a situation where an in-distinguished agent promises unusually huge pay off, and the mark erroneously gives in due to some combination of bad priors and bad utility evaluation. The common errors seem to be 1: omit the consequence of keeping the money for a more distinguished agent, 2: assign too high prior, 3: and, when picking between approaches, ignore the huge cost of acting in a manner which encourages disinformation. All those errors act in favour of the scammer (and some are optional), while non-erroneous processing would assign huge negative utility to paying up even given high priors.

There is no real way of doing that without changing your probability function or your utility function. However you can't change those. The real problem is with the expected utility function and I don't see any way of fixing it, though perhaps I missed something.

Any agent subject to Pascal's Mugging would fall pray to this problem first, and it would be far worse. While the mugger is giving his scenario, the agent could imagine an even more unlikely scenario. Say one where the mugger actually gives him 3^^^^^^3 units of utility if he does some arbitrary task, instead of 3^^^3. This possibility immediately gets so much utility that it far outweighs anything the mugger has to say after that. Then the agent may imagine

an even more unlikely scenariowhere it gets 3^^^^^^^^^^3 units of utility, and so on.I don't really know what an agent would do if the expected utility of any action approached infinity. Perhaps it would generally work out as some things would approach infinity faster than others. I admit I didn't consider that. But I don't know if that would necessarily be the case. Even if it is it seems "wrong" for expected utilities of everything to be infinite and only tiny probabilities to matter for anything. And if so then it would work out for the pascal's mugging scenario too I think.

*2 points [-]Last time I checked, priors were fairly subjective even here. We don't know what is the best way to assign priors. Things like "Solomonoff induction" depend to arbitrary choice of machine.

Nope, people who end up 419-scammed or waste a lot of money investing into someone like Randel L Mills or Andrea Rossi live through their life ok until they read a harmful string in a harmful set of circumstances (bunch of other believers around for example).

*0 points [-]Priors are indeed up for grabs, but a set of priors about the universe ought be consistent with itself, no? A set of priors based only on complexity may indeed not be the best set of priors -- that's what all the discussions about "leverage penalties" and the like are about, enhancing Solomonoff induction with something extra. But what you seem to suggest is a set of priors about the universe that are designed for the express purposes of making

human utilitycalculations balance out? Wouldn't such a set of priors require the anthroporphization of the universe, and effectively mean sacrificing all sense of epistemic rationality?*1 point [-]The best "priors" about the universe are 1 for what that universe right around you is, and 0 for everything else. Other priors are a compromise, an engineering decision.

What I am thinking is that

there is a considerably better way to assign priors which we do not know of yet - the way which will assign equal probabilities to each side of a die if it has no reason to prefer one over the other - the way that does correspond to symmetries in the evidence.

We don't know that there will still be same problem when we have a non-stupid way to assign priors (especially as the non-stupid way ought to be considerably more symmetric). And it may be that some value systems are intrinsically incoherent. Suppose you wanted to maximize blerg without knowing what blerg even really is. That wouldn't be possible, you can't maximize something without having a measure of it. But I still can tell you i'd give you 3^^^^3 blergs for a dollar, without either of us knowing what blerg is supposed to be or whenever 3^^^^3 blergs even make sense (if blerg is an unique good book of up to 1000 page length, it doesn't because duplicates aren't blerg).

True, but the goal of a probability function is to represent the actual probability of an event happening as closely as possible. The map should correspond to the territory. If your map is good, you shouldn't change it unless you observe actual changes in the territory.

I don't know if those things have such extremes in low probability vs high utility to be called pascal's mugging. But even so, the human brain doesn't operate on anything like Solomonoff induction, Bayesian probability theory, or expected utility maximization.

*2 points [-]The actual probability is either 0 or 1 (either happens or doesn't happen). Values in-between quantify ignorance and partial knowledge (e.g. when you have no reason to prefer one side of the die to the other), or, at times, are chosen very arbitrarily (what is the probability that a physics theory is "correct").

New names for same things are kind of annoying, to be honest, especially ill chosen... if it happens by your own contemplation, I'd call it Pascal's Wager. Mugging implies someone making threats, scam is more general and can involve promises of reward. Either way the key is the high payoff proposition wrecking some havoc, either through it's prior probability being too high, other propositions having been omitted, or the like.

People are still agents, though.

Yes but the goal is to assign whatever outcome that will actually happen with the highest probability as possible, using whatever information we have. The fact that some outcomes result in ridiculously huge utility gains does not imply anything about how likely they are to happen, so there is no reason that should be taken into account (unless it actually does, in which case it should.)

Pascal's mugging was an absurd scenario with absurd rewards that approach infinity. What you are talking about is just normal everyday scams. Most scams do not promise such huge rewards or have such low probabilities (if you didn't know any better it is feasible that someone could have an awesome invention or need your help with transaction fees.)

And the problem with scams is that people overestimate their probability. If they were to consider how many emails in the world are actually from Nigerian Princes vs scammers, or how many people promise awesome inventions without any proof they will actually work, they would reconsider. In pascal's mugging, you fall for it even after having considered the probability of it happening in detail.

Your probability estimation could be absolutely correct. Maybe 1 out of a trillion times a person meets someone claiming to be a matrix lord, they are actually telling the truth. And they still end up getting scammed, so that the 1 in a trillionth counter-factual of themselves gets infinite reward.

They are agents, but they aren't subject to this specific problem because we don't really use expected utility maximization. At best maybe some kind of poor approximation of it. But it is a problem for building AIs or any kind of computer system that makes decisions based on probabilities.

I think you're considering a different problem than Pascal's Mugging, if you're taking it as a given that the probabilities are indeed 1 in a trillion (or for that matter 1 in 10). The original problem doesn't make such an assumption.

What you have in mind, the case of definitely known probabilities, seems to me more like The LifeSpan dilemma where e.g. "an unbounded utility on lifespan implies willingness to trade an 80% probability of living some large number of years for a 1/(3^^^3) probability of living some sufficiently longer lifespan"

The wiki page on it seems to suggest that this is the problem.

Also this

which is pretty concerning.

I'm curious what you think the problem with Pascal's Mugging is though. That you can't easily estimate the probability of such a situation? Well that is true of anything and isn't really unique to Pascal's Mugging. But we can still approximate probabilities. A necessary evil to live in a probabilistic world without the ability to do perfect Bayesian updates on all available information, or unbiased priors.

*0 points [-]I abhor using unnecessary novel jargon.

Bad math being internally bad, that's the problem. Nothing to do with any worlds, real or imaginary, just a case of internally bad math - utilities are undefined, it is undefined if you pay up or not, the actions chosen are undefined. Akin to maximizing blerg without any definition of what blerg even is - maximizing "expected utility" without having defined it.

Speed prior works, for example (it breaks some assumptions of Blanc. Namely, the probability is not bounded from below by any computable function of length of the hypothesis).

*0 points [-]There is no evidence for the actual existence of neatly walled-of and unupdateable utility functions or probability functions, any more than there is for a

luz'.Utility and probability functions are not perfect or neatly walled off. But that doesn't mean you should change them to fix a problem with your expected utility function. The goal of a probability function is to represent the actual probability of an event happening as closely as possible. And the goal of a utility function is to represent what you states you would prefer the universe to be in. This also shouldn't change unless you've actually changed your preferences.

There's plenty of evidence of people changing their preferences over significant periods of time: it would be weird not to. And I am well aware that the theory of stable utility functions is standardly patched up with a further theory of terminal values, for which there is also no direct evidence.

Of course people can change their preferences. But if your preferences are not consistent you will likely end up in situations that are less preferable than if you had the same preferences the entire time. It also makes you a potential money pump.

What? Terminal values are not a patch for utility functions. It's basically another word that means the same thing, what state you would prefer the world to end up in. And how can there be evidence for a decision theory?

Well, how is this different from changing our preferences to utility functions to fix problems with our naive preferences?

I don't know what you mean. All I'm saying is that you shouldn't change your preferences because of a problem with your expected utility function. Your preferences are just what you want. Utility functions are just a mathematical way of expressing that.

Human preferences don't naturally satisfy the VNM axioms, thus by expressing them as a utility function you've already changed them.

*0 points [-]I don't see why our preferences can't be expressed by a utility function even as they are. The only reason it wouldn't work out is if there were circular preferences, and I don't think most peoples preferences would work out to be truly circular if they were to think about the specific occurrence and decide what they really preferred.

Though mapping out which outcomes are more preferred than others is not enough to assign them an actual utility, you'd somehow have to guess how

muchmore preferable one outcome is to another quantitatively.But even then I think most people could if they thought about it enough. The problem is that our utility functions are complex and we don't really know what they are, not that they don't exist.Or they might violate the independence axiom, but in any case what do you mean by " think about the specific occurrence and decide what they really preferred", since the result of such thinking is likely to depend on the exact order they thought about things in.

*6 points [-]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:

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 think he was assuming a natural scale. After all, you can just pick some everyday-sized utility difference to use as your unit, and measure everytihng on that scale. It wouldn't really matter what utility difference you pick as long as it is a natural size, since multiplying by 3^^^3 is easily enough for the argument to go through.

*5 points [-]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?

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.

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.

It seems reasonable to me because on the stated assumptions - the floating tags seen by vast numbers of other beings but not yourself - you've managed to generate sensory data with a vast likelihood ratio. The vast update is as reasonable as this vast ratio, no more, no less.

The problem is that you seem to be introducing one dubious piece to deal with another. Why is the hypothesis that those bullet points hold infinitesimally unlikely rather than very unlikely in the first place?

*5 points [-]Has the following reply to Pascal's Mugging been discussed on LessWrong?

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.)

*5 points [-]The obvious problem with this is that your utility is

not definedif you are willing to accept muggings, so you can't use the framework of expected utility maximization at all. The point of the mugger is just to illustrate this, I don't think anyone thinks you should actually pay them (after all, you might encounter a more generous mugger tomorrow, or any number of more realistic opportunities to do astronomical amounts of good...)Part of the issue is that I am coming at this problem from a different perspective than maybe you or Eliezer is. I believe that paying the mugger is basically worthless in the sense that doing almost any old good thing is better than paying the mugger. I would like to have a satisfying explanation of this. In contrast, Eliezer is interested in reconciling a view about complexity priors with a view about utility functions, and the mugger is an illustration of the conflict.

I do not have a proposed reconciliation of complexity priors and unbounded utility functions. Instead, the above comment is a recommended as an explanation of why paying the mugger is basically worthless in comparison with ordinary things you could do. So this hypothesis would say that if you set up your priors and your utility function in a reasonable way, the expected utility of downstream effects of ordinary good actions would greatly exceed the expected utility of paying the mugger.

Even if you decided that the expected utility framework somehow breaks down in cases like this, I think various related claims would still be plausible. E.g., rather than saying that doing ordinary good things has higher expected utility, it would be plausible that doing ordinary good things is "better relative to your uncertainty" than paying the mugger.

On a different note, another thing I find unsatisfying about the downstream effects reply is that it doesn't seem to match up with why ordinary people think it is dumb to pay the mugger. The ultimate reason I think it is dumb to pay the mugger is strongly related to why ordinary people think it is dumb to pay the mugger, and I would like to be able to thoroughly understand the most plausible common-sense explanation of why paying the mugger is dumb. The proposed relationship between ordinary actions and their distant effects seems too far off from why common sense would say that paying the mugger is dumb. I guess this is ultimately pretty close to one of Nick Bostrom's complaints about empirical stabilizing assumptions.

*1 point [-]I think we are all in agreement with this (modulo the fact that all of the expected values end up being infinite and so we can't compare in the normal way; if you e.g. proposed a cap of 3^^^^^^^3 on utilities, then you certainly wouldn't pay the mugger).

It seems very likely to me that ordinary people are best modeled as having bounded utility functions, which would explain the puzzle.

So it seems like there are two issues:

And then there is the third issue Eliezer is dealing with, where he wants to be able to have an unbounded utility function even if that doesn't describe anyone's preferences (since it seems like boundedness is an unfortunate restriction to randomly impose on your preferences for technical reasons), and formally it's not clear how to do that. At the end of the post he seems to suggest giving up on that though.

*2 points [-]Obviously to really put the idea of people having bounded utility functions to the test, you have to forget about it solving problems of small probabilities and incredibly good outcomes and focus on the most unintuitive consequences of it. For one, having a bounded utility function means caring arbitrarily little about differences between the goodness of different sufficiently good outcomes. And all the outcomes could be certain too. You could come up with all kinds of thought experiments involving purchasing huge numbers of years happy life or some other good for a few cents. You know all of this so I wonder why you don't talk about it.

Also I believe that Eliezer thinks that an unbounded utility function describes at least his preferences. I remember he made a comment about caring about new happy years of life no matter how many he'd already been granted.

(I haven't read most of the discussion in this thread or might just be missing something so this might be irrelevant.)

As far as I know the strongest version of this argument is Benja's, here (which incidentally seems to deserve many more upvotes than it got).

Benja's scenario isn't a problem for normal people though, who are not reflectively consistent and whose preferences manifestly change over time.

Beyond that, it seems like people's preferences regarding the lifespan dilemma are somewhat confusing and probably inconsistent, much like their preferences regarding the repugnant conclusion. But that seems mostly orthogonal to pascal's mugging, and the basic point---having unbounded utility

by definitionmeans you are willing to accept negligible chances of sufficiently good outcomes against probability nearly 1 of any fixed bad outcome, so if you object to the latter you are just objecting to unbounded utility.I agree I was being uncharitable towards Eliezer. But it is true that at the end of this post he was suggesting giving up on unbounded utility, and that everyone in this crowd seems to ultimately take that route.

It's in Nick Bostrom's Infinite Ethics paper, which has been discussed repeatedly here, and has been floating around in various versions since 2003. He uses the term "empirical stabilizing assumption."

I bring this up routinely in such discussions because of the misleading intuitions you elicit by using an example like a mugging that sets off many "no-go heuristics" that track chances of payoffs, large or small. But just because ordinary things may have a higher chance of producing huge payoffs than paying off a Pascal's Mugger (who doesn't do demonstrations), doesn't mean your activities will be completely unchanged by taking huge payoffs into account.

*3 points [-]Maybe the answer to this reply is that if there is a downstream multiplier for ordinary good accomplished, there is also a downstream multiplier for good accomplished by the mugger in the scenario where he is telling the truth. And multiplying each by a constant doesn't change the bottom line.

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:

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.

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.

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).

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.

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 itanyway? (This is a feature of debates about Pascal's Mugging in general, not just this post in particular.)That's just how it is when you reason about reason; Neurath's boat must be repaired while on the open sea. In this case, our instincts strongly suggest that what the decision theory seems to say we should do must be wrong, and we have to turn to the rest of our abilities and beliefs to adjudicate between them.

Well, besides that thing about wanting expected utilities to converge, from a rationalist-virtue perspective it seems relatively less dangerous to start from a position of someone rejecting something with no priors or evidence in favor of it, and relatively more dangerous to start from a position of rejecting something that has strong priors or evidence.

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'm a lot more worried about making an FAI behave correctly if it encounters a scenario which

we thoughtwas very very unlikely.*3 points [-]Also, if the AI spreads widely and is around for a long time, it will eventually run into very unlikely scenarios. Not 1/3^^^3 unlikely, but pretty unlikely.

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.

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.

Reallyinteresting, though.That hyperbole one. I wasn't intending the primary focus of this post to be on the notion of a super-update - I'm not sure if

thatpart needs to make it into AIs, though it seems to me to be partially responsible for my humanlike foibles in the Horrible LHC Inconsistency. I agree that this notion is actuallyveryunderspecified but so is almost all of bounded logical uncertainty.Using "a tad" to mean "very" is understatement, not hyperbole.

*9 points [-]One could call it

hypobole.Specifically, litotes.

*4 points [-]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".

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

specificallyof 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".

1) It's been applied to

cryonic preservation, fer crying out loud. It's reasonable to suspect that the probability of that working is low, but anyone who says with current evidence that the probability isbeyond astronomically lowis being too silly to take seriously.The benefit of cryonic preservation isn't astronomically high, though, so you don't need a probability that is beyond astronomically low. First of all,even an infinitely long life after being revived only has a finite present value, and possibly a very low one, because of discounting. Second, the benefit from cryonics is the benefit you'd gain from being revived after being cryonically preserved,

minus the benefit that you'd gain from being revived after not cryonically preserved.(A really advanced society might be able to simulate us. If simulations count as us, simulating us counts as reviving us without the need for cryonic preservation.)I do not think that you have gotten Luke's point. He was addressing your point #1, not trying to make a substantive argument in favor of cryonics.

*2 points [-]That argument is isomorphic to the one discussed in the post here:

Essentially, it's hard to argue that the probabilities you assign should be balanced so exactly, and thus (if you're an altruist) Pascal's Wager exhorts you either to devote your entire existence to proselytizing for some god, or proselytizing for atheism, depending on which type of deity seems to you to have the slightest edge in probability (maybe with some weighting for the awesomeness of their heavens and awfulness of their hells).

So that's why you still need a mathematical/epistemic/decision-theoretic reason to reject Pascal's Wager and Mugger.

What you have is a divergent sum whose sign will depend to the order of summation, so maybe some sort of re-normalization can be applied to make it balance itself out in absence of evidence.

*4 points [-]Actually, there is no order of summation in which the sum will converge, since the terms get arbitrary large. The theorem you are thinking of applies to conditionally convergent series, not all divergent series.

*2 points [-]Strictly speaking, you don't always need the sums to converge. To choose between two actions you merely need the sign of difference between utilities of two actions, which you can represent with divergent sum. The issue is that it is not clear how to order such sum or if it's sign is even meaningful in any way.

Any similarities are arguments for giving them a maximally

differentname to avoid confusion, not a similar one. Would the English language really be better if rubies were called diyermands?Chemistry would not be improved by providing completely different names to chlorate and perchlorate (e.g. chlorate and sneblobs). Also, I think English might be better if rubies were called diyermands. If all of the gemstones were named something that followed a scheme similar to diamonds, that might be an improvement.

*3 points [-]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.

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

notscale like 1/(3^^^3)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 tothisproblem 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 capitaquantity. 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?

*3 points [-]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.

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.)

Not 'a lot' and present-day non-sharing imperatives are driven by an (obvious) strategy to accumulate a long-term advantage for FAI projects over AGI projects which is impossible if all lines of research are shared at all points when they are not yet imminently dangerous. No present-day knowledge is imminently dangerous AFAIK.

I can't help but remember HPJEV talk about plausible deniability and how that relates to you telling people whether there is dangerous knowledge out there.

Do you believe this to be possible? In modern times with high mobility of information

and peopleI have strong doubts a gnostic approach would work. You can hide small, specific, contained "trade secrets", you can't hide a large body of knowledge that needs to be actively developed.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.

Are you sure? I would expect that it's possible to recover from that, and some actions would make you more likely to recover than others.

If all of my experiences are dreaming/drugged/crazy/etc. experiences then what decision I make only matters if I value having one set of dreaming/drugged/crazy experiences over a different set of such experiences.

The thing is, I sure do seem to value having one set of experiences over another. So if all of my experiences are dreaming/drugged/crazy/etc. experiences then it seems I do value having one set of such experiences over a different set of such experiences.

So, given that, do I choose the dreaming/drugged/crazy/etc. experience of giving you $5 (and whatever consequences that has?). Or of refusing to give you $5 (and whatever consequences

thathas)? Or something else?But that would destroy your ability to deal with optical illusions and misdirection.

Perhaps I should say ...in which I can't reasonably expect to GET evidence entangled with an underlying reality.

*2 points [-]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.

*2 points [-]Edit: formatting fixed. Thanks, wedrifid.

My response to the mugger:

My response to the scientist:

Try an additional linebreak before the first bullet point.

Why does that prior follow from the counting difficulty?

I was thinking that using (length of program) + (memory required to run program) as a penalty makes more sense to me than (length of program) + (size of impact). I am assuming that any program that can simulate X minds must be able to handle numbers the size of X, so it would need more than log(X) bits of memory, which makes the prior less than 2^-log(X).

I wouldn't be overly surprised if there were some other situation that breaks this idea too, but I was just posting the first thing that came to mind when I read this.

You're trying to italicize those long statements? It's possible that you need to get rid of the spaces around the asterisks.

But you're probably better off just using quote boxes with ">" instead.

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.

It's not that odd. If someone asked to borrow ten dollars, and said he'd pay you back tomorrow, would you believe him? What if he said he'd pay back $20? $100? $1000000? All the money in the world?

At some point, the probability goes down faster than the price goes up. That's why you can't just get a loan and keep raising the interest to make up for the fact that you probably won't ever pay it back.

Comment deleted06 May 2013 05:59:01PM*[-]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.

Comment deleted07 May 2013 04:54:16AM*[-]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.

*2 points [-]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

possiblyjustified 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.

*1 point [-]A note - it looks like what Eliezer is suggesting here is

notthe same as UDASSA. See my analysis here - and endoself's reply - and here.The big difference is that UDASSA won't impose the same locational penalty on nodes in extreme situations, since the measure is shared unequally between nodes. There are programs q with relatively short length that can select out such extreme nodes (parties getting genuine offers from Matrix Lords with the power of 3^^^3) and so give them much higher relative weight than 1/3^^^3. Combine this with an unbounded utility, and the mugger problem is still there (as is the divergence in expected utility).

*1 point [-]Yeah, I like this solution too. It doesn't have to be based on the universal distribution, any distribution will work. You must have some way of distributing your single unit of care across all creatures in the multiverse. What matters is not the large number of creatures affected by the mugger, but their total weight according to your care function, which is less than 1 no matter what outlandish numbers the mugger comes up with. The "leverage penalty" is just the measure of your care for not losing $5, which is probably more than 1/3^^^^3.

*2 points [-]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.

You are mistaken. A prior isn't something that can be mistaken per se. The closest it can get is assigning a low probability to something that is true. However, any prior system will say that the probability it gives of something being true is exactly equal to the probability of it being true, therefore it is well-calibrated. It will occasionally give low probabilities for things that are true, but only to the extent that unlikely things sometimes happen.

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.

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...

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.)

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

wewent 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.Just to jump in here - the solution to the doomsday argument is that it is a low-information argument in a high-information situation. Basically, once you know you're the 10 billionth zorblax, your prior should indeed put you in the middle of the group of zorblaxes, for 20 billion total, no matter what a zorblax is. This is correct and makes sense. The trouble comes if you open your eyes, collect additional data, like population growth patterns, and then

never use any of that to update the prior.When people put population growth patterns and the doomsday prior together in the same calculation for the "doomsday date," that's just blatantly having data but not updating on it.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.

*2 points [-]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.

Honestly, at this point, I would strongly update in the direction that I am being deceived in some manner. Possibly I am dreaming, or drugged, of the person in front of me has some sort of perception-control device. I do not see any reason why someone who could open the sky, split the Red Sea, and so on, would need $5; and if he did, why not make it himself? Or sell the fish?

The only reasons I can imagine for a genuine Matrix Lord pulling this on me are very bad for me. Either he's a sadist who likes people to suffer - in which case I'm doomed no matter what I do - or there's something that he's not telling me (perhaps doing what he says once surrenders my free will, allowing him to control me forever?), which implies that he believes that I would reject his demand if I knew the truth behind it, which strongly prompts me to reject his demand.

Or he's insane, following no discernable rules, in which case the only thing to do is to try to evade notice (something I've clearly already failed at).

*3 points [-]That your universe is controlled by a sadist doesn't suggest that every possible action you could do is equivalent. Maybe all your possible fates are miserable, but some are far more miserable than others. More importantly, a being might be sadistic in some respects/situations but not in others.

I also have to assign a very, very low prior to anyone's being able to figure out in 5 minutes what the Matrix Lord's exact motivations are. Your options are too simplistic even to describe minds of human-level complexity, much less ones of the complexity required to design or oversee physics-breakingly large simulations.

I think indifference to our preferences (except as incidental to some other goal, e.g., paperclipping) is more likely than either sadism or beneficence. Only very small portions of the space of values focus on human-style suffering or joy. Even in hypotheticals that seem designed to play with human moral intuitions. Eliezer's decision theory conference explanation makes as much sense as any.

Well, if I'm going to free-form speculate about the scenario, rather than use it to explore the question it was introduced to explore, the most likely explanation that occurs to me is that the entity is doing the Matrix Lord equivalent of free-form speculating... that is, it's wondering "what would humans do, given this choice and that information?" And, it being a Matrix Lord, its act of wondering creates a human mind (in this case, mine) and gives it that choice and information.

Which makes it likely that I haven't actually lived through most of the life I remember, and that I won't continue to exist much longer than this interaction, and that most of what I think is in the world around me doesn't actually exist.

That said, I'm not sure what use free-form speculating about such bizarre and underspecified scenarios really is, though I'll admit it's kind of fun.

It's kind of fun. Isn't that reason enough?

Looking at the original question - i.e. how to handle very large utilities with very small probability - I find that I have a mental safety net there. The safety net says that the situation is a lie. It does not matter how much utility is claimed, because anyone can state any arbitrarily large number, and a number has been chosen (in this case, by the Matrix Lord) in a specific attempt to overwhelm my utility function. The small probability is chosen (a) because I would not believe a larger probability and (b) so that I have no recourse when it fails to happen.

I am reluctant to fiddle with my mental safety nets because, well, they're

safetynets - they're there for a reason. And in this case, the reason is that such a fantastically unlikely event is unlikely enough that it's not likely to happenever, toanyone. Not even once in the whole history of the universe. If I (out of all the hundreds of billions of people in all of history) do ever run across such a situation, then it's so incredibly overwhelmingly more likely that I am being deceived that I'm far more likely to gain by immediately jumping to the conclusion of 'deceit' than by assuming that there's any chance of this being true.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.

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.

I don't think you can give me a moment of pleasure that intense without using 3^^^3 worth of atoms on which to run my brain, and I think the leverage penalty still applies then. You definitely can't give me a moment of worthwhile happiness that intense without 3^^^3 units of background computation.

The article said the leverage penalty "[penalizes] hypotheses that let you affect a large number of people, in proportion to the number of people affected." If this is

allthe leverage penalty does, then it doesn't matter if it takes 3^^^3 atoms or units of computation, because atoms and computations aren't people.That said, the article doesn't precisely define what the leverage penalty is, so there could be something I'm missing. So, what exactly

isthe leverage penalty? Does it penalize how many units of computation, rather than people, you can affect? This sounds much less arbitrary than the vague definition of "person" and sounds much easier to define: simply divide the prior of a hypothesis by the number of bits flipped by your actions in it and then normalize.You're absolutely right. I'm not sure how I missed or forgot about reading that.

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.

Relevant math, similar features in classical optics and Quantum Mechanics.

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.

How does this work with Clippy (the only paperclipper in known existence) being tempted with 3^^^^3 paperclips?

That's part of why I dislike Robin Hanson's original solution. That the tempting/blackmailing offer involves 3^^^^3 other people, and that you are also a person should be merely incidental to one

particular illustrationof the problem of Pascal's Mugging -- and as such it can't be part of a solution to the core problem.To replace this with something like "causal nodes", as Eliezer mentions, might perhaps solve the problem. But I wish that we started talking about Clippy and his paperclips instead, so that the original illustration of the problem which involves incidental symmetries doesn't mislead us into a "solution" overreliant on symmetries.

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.)

*1 point [-]"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

evaluatewhether such a hypothetical offer should be accepted by a random individual or not? Robin's anthropic solution says that the AI should judge that someoneelseought 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.

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.

That's perfectly credible since it implies a lack of leverage.

10^10 is not a significant factor compared to the sensory experience of seeing something float in a bathtub.

To build an AI one must be a tad more formal than this, and once you start trying to be formal, you will soon find that you need a prior.

Oh, I assumed that negative leverage is still leverage. Given that it might amount to an equivalent of killing a googolplex of people, assuming you equate never being born with killing.

I see. I cannot comment on anything AI-related with any confidence. I thought we were talking about evaluating the likelihood of a certain model in physics to be accurate. In that latter case anthropic considerations seem irrelevant.

It's likely that anything around today has a huge impact on the state of the future universe. As I understood the article, the leverage penalty requires considering how unique your opportunity to have the impact would be too, so Archimedes had a massive impact, but there have also been a massive number of people through history who would have had the chance to come up with the same theories had they not already been discovered, so you have to offset Archimedes leverage penalty by the fact that he wasn't uniquely capable of having that leverage.

Neither was any other scientist in history ever, including the the one in the Eliezer's dark energy example. Personally, I take a very dim view of applying anthropics to calculating probabilities of future events, and this is what Eliezer is doing.

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...?

...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.

If our math has to handle infinities we have bigger problems. Unless we use measures, and then we have the same issue and seemingly forced solution as before. If we don't use measures, things fail to add up the moment you imagine "infinity".

Then this solution just assumes the possibility of infinite people is 0. If this solution is based on premises that are probably false, then how is it a solution at all? I understand that infinity makes even bigger problems, so we should instead just call your solution a pseudo-solution-that's-probably-false-but-is--still-the-best-one-we-have, and dedicate more efforts to finding a real solution.

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:

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.

I expect far less than 1 Matrix Lord per simulated population. I expect the vast majority of simulations are within UFAIs trying to gain certain types of information through veridical simulation, no Matrix Lords there.

*0 points [-]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

morelikely 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.

*1 point [-]Yes, privileging a hypothesis isn't discussed in great detail, but the alternatives you mention in your post don't resolve the dilemma. Even if you think that that the probabilities of a "good" and "bad" alternatives balance each other out to the quadrillionth decimal point, the utilities you get in your calculation are astronomical. If you think there's a 0.0000

quadrillion zeros1 greater chance that the beggar will do good than harm, the expected utility of your $5 donation is inconceivably greater than than a trillion years of happiness. If you think there's at least a 0.0000quadrillion zeros1 chance that $5 will cause the mugger to act malevolently, your $5 donation is inconceivably worse than a trillion years of torture. Both of theses expectations seem off.You can't just say "the probabilities balance out". You have to explain why the probabilities balance out to a bignum number of decimal points.

<blockquote> 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. </blockquote>

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.

*0 points [-]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?

So if I'm getting what you're saying correctly, it would not sacrifice a single cent for a 49% chance to save a human life?

And on the other hand it could be tempted to a game where it'd have 51% chance of winning a cent, and 49% chance of being destroyed?

If the solution for the problem of infinitesmal probabilities, is to effectively ignore every probability under 50%, that's a solution that's worse than the problem...

I stupidly didn't consider that kind of situation for some reason... Back to the drawing board I guess.

Though to be fair it would still come out ahead 51% of the time, and in a real world application it would probably choose to spend the penny, since it would expect to make choices similarly in the future, and that would help it come out ahead an even higher percent of the time.

But yes, a 51% chance of losing a penny for nothing probably shouldn't be worth more than a 49% chance at saving a life for a penny. However allowing a large enough reward to outweigh a small enough probability means the system will get stuck in situations where it is pretty much guaranteed to lose, on the slim, slim chance that it could get a huge reward.

Caring only about the percent of the time you "win" seemed like a more rational solution but I guess not.

Though another benefit of this system could be that you could have weird utility functions. Like a rule that says any outcome where one life is saved is worth more than any amount of money lost. Or Asimov's three laws of robotics, which wouldn't work under an Expected Utility function since it would only care about the first law. This is allowed because in the end all that matters is which outcomes you prefer to which other outcomes. You don't have to turn utilities into numbers and do math on them.