Pascal's Mugging: Tiny Probabilities of Vast Utilities

1. There aren't 3^^^3 people and there is no machine that can simulate even one person, let alone that many people.

2. Nobody has such magic powers.

3. Even if you don't accept 1 and 2 above, there's no reason to expect that the person is telling the truth. He might kill the people even if you give him the $5, or conversely he might not kill them even if you don't give him the $5.

I also don't understand the appeal of Occam's razor, so I'm pretty sure I'm not part of the target audience for this paradox.

Why would not giving him $5 make it more likely that people would die, as opposed to less likely? The two would seem to cancel out. It's the same old "what if we are living in a simulation?" argument- it is, at least, possible that me hitting the sequence of letters "QWERTYUIOP" leads to a near-infinity of death and suffering in the "real world", due to AGI overlords with wacky programming. Yet I do not refrain from hitting those letters, because there's no entanglement which drives the probabilities in that direction as opposed to some other random direction; my actions do not alter the expected future state of the universe. You could just as easily wind up saving lives as killing people.

Very interesting thought experiment!

One place where it might fall down is that our disutility for causing deaths is probably not linear in the number of deaths, just as our utility for money flattens out as the amount gets large. In fact, I could imagine that its value is connected to our ability to intuitively grasp the numbers involved. The disutility might flatten out really quickly so that the disutility of causing the death of 3^^^^3 people, while large, is still small enough that the small probabilities from the induction are not overwhelmed by it.

People say the fact that there are many gods neutralizes Pascal’s wager - but I don't understand that at all. It seems to be a total non sequetor. Sure, it opens the door to other wagers being valid, but that is a different issue.

Lets say I have a simple game against you where, if I choose 1 I win a lotto ticket and if I choose 0 I loose. There is also a number of other games tables around the room with people winning or not winning lotto tickets. If I want to win the lotto, what number should I pick?

Also I don't tink there is a fundimental issue with having favour with Allah, Christ and Zeus simultaniously. (so you could actualy win, then get up and go play at another table - although there would be a time cost to that).

Now there is the more detailed argument where you argue that a god who desired you disbelieve in him and oppose his will is equally likely to one that desires that you believe in him and supports his will. But as long as there is any imperfection in the mirror then there is a Pascal’s wager to be had.

What if a philosopher tries Pascal's Mugging on the AI for a joke, and the tiny probabilities of 3^^^^3 lives being at stake, override everything else in the AI's calculations?

Suppose that depends on how he calculates the probability of the threat of the mugger. The very act of giving a specific probability to a threat like that opens one up to an infinite risk (i.e. that they will demand infinite things in exchange for infinity x 3^^^^3 lives). So this is a bit like comparing what I might call naive utilitarianism (where one doesn’t consider the wider effects of one’s acts and rules) with pure utilitarianism (where one takes everything into account).

Whether that neutralizes Pascal’s wager relates to how one resolves the mirror issue I mentioned. If that produces a tidy result then the problem above doesn’t occur.

This is an instance of the general problem of attaching a probability to matrix scenarios. And you can pascal-mug yourself, without anyone showing up to assert or demand anything - just think: what if things are set up so that whether I do, or do not do, something, determines whether those 3^^^^3 people will be created and destroyed? It's just as possible as the situation in which a messenger from Outside shows up and tells you so.

The obvious way to attach probabilities to matrix scenarios is to have a unified notion of possible world capacious enough to encompass both matrix worlds and worlds in which your current experiences are veridical; and then you look at relative frequencies or portions of world-measure for the two classes of possibility. For example, you could assume the correctness of our current physics across all possible worlds, and then make a Drake/Bostrom-like guesstimate of the frequency of matrix construction across all those universes, and of the "demography" and "political character" of those simulations. Garbage in, garbage out; but you really can get an answer if you make enough assumptions. In that regard, it is not too different to any other complicated decision made against a background of profound uncertainty.

Tom and Andrew, it seems very implausible that someone saying "I will kill 3^^^^3 people unless X" is literally zero Bayesian evidence that they will kill 3^^^^3 people unless X. Though I guess it could plausibly be weak enough to take much of the force out of the problem.

Andrew, if we're in a simulation, the world containing the simulation could be able to support 3^^^^3 people. If you knew (magically) that it couldn't, you could substitute something on the order of 10^50, which is vastly less forceful but may still lead to the same problem.

Andrew and Steve, you could replace "kill 3^^^^3 people" with "create 3^^^^3 units of disutility according to your utility function". (I respectfully suggest that we all start using this form of the problem.)

Michael Vassar has suggested that we should consider any number of identical lives to have the same utility as one life. That could be a solution, as it's impossible to create 3^^^^3 distinct humans. But, this also is irrelevant to the create-3^^^^3-disutility-units form.

IIRC, Peter de Blanc told me that any consistent utility function must have an upper bound (meaning that we must discount lives like Steve suggests). The problem disappears if your upper bound is low enough. Hopefully any realistic utility function has such a low upper bound, but it'd still be a good idea to solve the general problem.

I see a similarity to the police chief example. Adopting a policy of paying attention to any Pascalian muggings would encourage others to manipulate you using them. At first it doesn't seem like this would have nearly enough disutility to justify ignoring muggings, but it might when you consider that it would interfere with responding to any real threat (unlikely as it is) of 3^^^^3 deaths.

Mitchell, it doesn't seem to me like any sort of accurate many-worlds probability calculation would give you a probability anywhere near low enough to cancel out 3^^^^3. Would you disagree? It seems like there's something else going on in our intuitions. (Specifically, our intuitions that an good FAI would need to agree with us on this problem.)

Sorry, the first link was supposed to be to Absence of Evidence is Evidence of Absence.

Mitchell, I don't see how you can Pascal-mug yourself. Tom is right that the possibility that typing QWERTYUIOP will destroy the universe can be safely ignored; there is no evidence either way, so the probability equals the prior, and the Solomonoff prior that typing QWERTYUIOP will save the universe is, as far as we know, exactly the same. But the mugger's threat is a shred of Bayesian evidence that you have to take into account, and when you do, it massively tips the expected utility balance. Your suggested solution does seem right but utterly intractable.

Tom and Andrew, it seems very implausible that someone saying "I will kill 3^^^^3 people unless X" is literally zero Bayesian evidence that they will kill 3^^^^3 people unless X. Though I guess it could plausibly be weak enough to take much of the force out of the problem.

Nothing could possibly be that weak.

Tom is right that the possibility that typing QWERTYUIOP will destroy the universe can be safely ignored; there is no evidence either way, so the probability equals the prior, and the Solomonoff prior that typing QWERTYUIOP will save the universe is, as far as we know, exactly the same.

Exactly the same? These are different scenarios. What happens if an AI actually calculates the prior probabilities, using a Solomonoff technique, without any a priori desire that things should exactly cancel out?

OK, let's try this one more time:

1. Even if you don't accept 1 and 2 above, there's no reason to expect that the person is telling the truth. He might kill the people even if you give him the $5, or conversely he might not kill them even if you don't give him the $5.

To put it another way, conditional on this nonexistent person having these nonexistent powers, why should you be so sure that he's telling the truth? Perhaps you'll only get what you want by not giving him the $5. To put it mathematically, you're computing pX, where p is the probability and X is the outcome, and you're saying that if X is huge, then just about any nonzero p will make pX be large. But you're forgetting two things: first, if you have the imagination to imagine X to be super-huge, you should be able to have the imagination to imagine p to be super-small. (I.e., if you can talk about 3^^^^3, you can talk about 1/3^^^^3.) Second, once you allow these hypothetical super-large X's, you have to acknowledge the possibility that you got the sign wrong.

I have to go with Tom McGabe on this one; This is just a restatement of the core problem of epistemology. It's not unique to AI, either.

3. Even if you don't accept 1 and 2 above, there's no reason to expect that the person is telling the truth. He might kill the people even if you give him the $5, or conversely he might not kill them even if you don't give him the $5.

But if a Bayesian AI actually calculates these probabilities by assessing their Kolmogorov complexity - or any other technique you like, for that matter - without desiring that they come out exactly equal, can you rely on them coming out exactly equal? If not, an expected utility differential of 2 to the negative googolplex times 3^^^^3 still equals 3^^^^3, so whatever tiny probability differences exist will dominate all calculations based on what we think of as the "real world" (the mainline of probability with no wizards).

if you have the imagination to imagine X to be super-huge, you should be able to have the imagination to imagine p to be super-small

But we can't just set the probability to anything we like. We have to calculate it, and Kolmogorov complexity, the standard accepted method, will not be anywhere near that super-small.

Addendum: In computational terms, you can't avoid using a 'hack'. Maybe not the hack you described, but something, somewhere has to be hard-coded. How else would you avoid solipsism?

This case seems to suggest the existence of new interesting rationality constraints, which would go into choosing rational probabilities and utilities. It might be worth working out what constraints one would have to impose to make an agent immune to such a mugging.

Eliezer,

OK, one more try. First, you're picking 3^^^^3 out of the air, so I don't see why you can't pick 1/3^^^^3 out of the air also. You're saying that your priors have to come from some rigorous procedure but your utility comes from simply transcribing what some dude says to you. Second, even if for some reason you really want to work with the utility of 3^^^^3, there's no good reason for you not to consider the possibility that it's really -3^^^^3, and so you should be doing the opposite. The issue is not that two huge numbers will exactly cancel out; the point is that you're making up all the numbers here but are artificially constraining the expected utility differential to be positive.

If I really wanted to consider this example realistically, I'd say that this guy has no magic powers, so I wouldn't worry about him killing 3^^^^3 people or whatever. A slightly more realistic scenario would be something like a guy with a bomb in a school, in which case I'd defer to the experts (presumably whoever in the police force deals with people like that) on their judgment of how best to calm him down. There I could see an (approximate) probability calculation being relevant, but, again, they key thing would be whether giving him $5 (or whatever) would make him more or less likely to set the fuse. It wouldn't be appropriate to say a priori that it could only help.

pdf23ds, under certain straightforward physical assumptions, 3^^^^3 people wouldn't even fit in anyone's future light-cone, in which case the probability is literally zero. So the assumption that our apparent physics is the physics of the real world too, really could serve to decide this question. The only problem is that that assumption itself is not very reasonable.

Lacking for the moment a rational way to delimit the range of possible worlds, one can utilize what I'll call a Chalmers prior, which simply specifies directly how much time you will spend thinking about matrix scenarios. (I name it for David Chalmers because I once heard him give an estimate of the odds that we are in a matrix; I think it was about 10%.) The rationality of having a Chalmers prior can be justified by observing one's own cognitive resource-boundedness, and the apparently endless amount of time one could spend thinking about matrix scenarios. (Is there a name for this sort of scheduling meta-heuristic, in which one limits the processing time available for potentially nonterminating lines of thought?)

Well... I think we act diffrently from the AI because we not only know Pascals Mugging, we know that it is known. I don't see why an AI could not know the knowledge of it, though, but you do not seem to consider that, which might simply show that it is not relevant, as you, er, seem to have given this some thought...

But maybe an AI cannot in fact know the knowledge of something.

Konrad: In computational terms, you can't avoid using a 'hack'. Maybe not the hack you described, but something, somewhere has to be hard-coded.

Well, yes. The alternative to code is not solipsism, but a rock, and even a rock can be viewed as being hard-coded as a rock. But we would prefer that the code be elegant and make sense, rather than using a local patch to fix specific problems as they come to mind, because the latter approach is guaranteed to fail if the AI becomes more powerful than you and refuses to be patched.

Andrew: You're saying that your priors have to come from some rigorous procedure

The priors have to come from some computable procedure. We would prefer it to be a good one, as agents with nonsense priors will not attain sensible posteriors.

but your utility comes from simply transcribing what some dude says to you.

No. Certain hypothetical scenarios, which we describe using the formalism of Turing machines, have fixed utilities - that is, if some description of the universe is true, it has a certain utility.

The problem with this scenario is not that we believe everything the dude tells us. The problem is that the description of a certain very large universe with a very large utility, does not have a correspondingly tiny prior probability if we use Solmonoff's prior. And then as soon as we see any evidence, no matter how tiny, anything whose entanglement is not as tiny as the very large universe is large, that expected utility differential instantly wipes out all other factors in our decision process.

Second, even if for some reason you really want to work with the utility of 3^^^^3, there's no good reason for you not to consider the possibility that it's really -3^^^^3, and so you should be doing the opposite.

A Solomonoff inductor might indeed consider it, though there's the problem of any bounded rationalist not being able to consider all computations. It seems "reasonable" for a bounded mind to consider it here; you did, after all.

The issue is not that two huge numbers will exactly cancel out; the point is that you're making up all the numbers here but are artificially constraining the expected utility differential to be positive.

Let the differential be negative. Same problem. If the differential is not zero, the AI will exhibit unreasonable behavior. If the AI literally thinks in Solomonoff induction (as I have described), it won't want the differential to be zero, it will just compute it.

Well, are you going to give us your answer?

To solve this problem, the AI would need to calculate the probability of the claim being true, for which it would need to calculate the probability of 3^^^^3 people even existing. Given what it knows about the origins and rate of reproduction of humans, wouldn't the probability of 3^^^^3 people even existing be approximately 1/3^^^^3? It's as you said, multiply or divide it by the number of characters in the bible, it's still nearly the same damned incomprehensably large number. Unless you are willing to argue that there are some bizarre properties of the other universe that would allow so many people to spontaneously arise from nothing- but this is yet another explanatory assumption, and one that I see no way of assigning a probability to.

Here's one for you: Lets assume for arguement's sake that "humans" could include human cosciousnesses, not just breathing humans. Then, if a universe with 3^^^^3 "humans" actually existed, what would be the odds that they were NOT all copies of the same parasitic consciousness?

Pascal's wager type arguments fail due to their symmetry (which is preserved in finite cases).

Eliezer Sorry to say (because it makes me sound callous), but if someone can and is willing to create and then destroy 3^^^3 people for less than $5, then there is no value in life, and definitely no moral structure to the universe. The creation and destruction of 3^^^3 people (or more) is probably happening all the time. Therefore the AI is safe declining the wager on purely selfish grounds.

Eliezer, I'd like to take a stab at the internal criterion question. One differerence between me and the program you describe is that I have a hoped for future. Say "I'd like to play golf on Wednesday." Now, I could calculate the odds of Wednesday not actually arriving (nuclear war,asteroid impact...), or me not being alive to see it (sudden heartattack...), and I would get an answer greater than zero. Why don't I operate on those non-zero probabilities? (The other difference between me and the program you describe) I think it has to do with faith. That is I have faith that my hoped for future will occur, or at least some semblance of it. I seem to have this faith despite previous losses. Take the field of AI. There is a hoped for future, a computer will demonstrate intelligence, some hope the machine will become conscious. There is a faith that "we can solve these problems" I'm not sure the machine you describe would have either characteristic. I don't know how to formalize this, but it seems an important aspect of the situation.

IIRC, Peter de Blanc told me that any consistent utility function must have an upper bound (meaning that we must discount lives like Steve suggests). The problem disappears if your upper bound is low enough. Hopefully any realistic utility function has such a low upper bound, but it'd still be a good idea to solve the general problem.

Nick, please see my blog (just click on my name). I have a post about this.

"Let the differential be negative. Same problem. If the differential is not zero, the AI will exhibit unreasonable behavior. If the AI literally thinks in Solomonoff induction (as I have described), it won't want the differential to be zero, it will just compute it."

How can a computation arrive at a nonzero differential, starting with zero data? If I ask a rational AI to calculate the probability of me typing "QWERTYUIOP" saving 3^^^^3 human lives, it knows literally nothing about the causal interactions between me and those lives, because they are totally unobservable.

GeniusNZ, you have to consider not only all proposed gods, but all possible gods and reward/punishment structures. Since the number and range of conceivable divine rewards and punishments is infinite for each action, the incentives are all equally balanced, and thus give you no reason to prefer one action over another.

Ultimately, I think Tom McCabe is right -- the truth of a proposition depends in part on its meaningfulness.

What is the probability that the sun will rise tomorrow? Nearly 1, if you're thinking of dawns. Nearly 0, if you're thinking of Copernicus. Bayesian reasoning can evaluate propositions, but at the limit, one must already have a rational vocabulary in which to express hypotheses.

When someone threatens to kill 3^^^^3 people, this calls into question

1) whether the assertion is meaningful at all 2) whether the lives in question are equivalent to "human lives" already observed, or are unlike in kind -- in other words, whether they should be valued similarly.

After all, analogously to the original Wager's problem, these 3^^^^3 people could be of a negative moral value -- it could be good to kill them. And no, Pascal's Mugger cannot just respond that he means people like you and me, because they are obviously not exactly analogous, since they are unobservable.

I generally share Tom McCabe's conclusion, that is, that they exactly cancel out because a symmetry has not been broken. The reversed hypothesis has the same complexity as the original hypothesis, and the same evidence supporting it. No differential entanglement. However, I think that this problem is worth attention because a) so many people who normally agree disagree here, and b) I suspect that the problem is related to normal utilitarianism with no discounting and an unbounded future. Of course, we already have some solutions in that case and we should try them and see what we get. Our realistic AGI is boundedly rational in some respect to another. How does it's limitation in predicting the mundane consequences of any given action relate to its limitation in predicting the probabilities in Pascal's Mugging.

Benquo, replace "kill 3^^^^3 people" with "create 3^^^^3 disutility units" and the problem reappears.

Michael, do you really think the mugger's statement is zero evidence?

It seems to me that the cancellation is an artifact of the particular example, and that it would be easy to come up with an example in which the cancellation does not occur. For example, maybe you have previous experience with the mugger. He has mugged you before about minor things and sometimes you have paid him and sometimes not. In all cases he has been true to his word. This would seem to tip the probabilities at least slightly in favor of him being truthful about his current much larger threat.

However clever your algorithm, at that level, something's bound to confuse it. Gimme FAI with checks and balances every time.

You could always just give up being a consequentialist and ontologically refuse to give in to the demands of anyone taking part in a Pascal mugging because consistently doing so would lead to the breakdown of society.

Re: "However clever your algorithm, at that level, something's bound to confuse it. Gimme FAI with checks and balances every time."

I agree that a mature Friendly Artificial Intelligence should defer to something like humanity's volition.

However, before it can figure out what humanity's volition is and how to accomplish it, an FAI first needs to:

1. self-improve into trans-human intelligence while retaining humanity's core goals
2. avoid UnFriendly Behavior (for example, murdering people to free up their resources) in the process of doing step (1)

If the AI falls prey to a paradoxes early on in the process of self-improvement, the FAI has failed and has to be shut down or patched.

Why is that a problem? Because if the AI falls prey to a paradox later on in the process of self-improvement, when the computer can outsmart human beings, the result could be catastrophic. (As Eliezer keeps pointing out: a rational AI might not agree to be patched, just as Gandhi would not agree to have his brain modified into becoming a psychopath, and Hitler would not agree to have his brain modified to become an egalitarian. All things equal, rational agents will try to block any actions that would prevent them from accomplishing their current goals.)

So you want to create an elegant (to the point, ideally, of being "provably correct") structure that doesn't need patches or hacks. If you have to constantly patch or hack early on in the process, that increases the chances that you've missed something fundamental, and that the AI will fail later on, when it's too late to patch.

Rolf: I agree with everything you just said, especially the bit about patches and hacks. I just wouldn't be happy having a FAI's sanity dependent on any single part of it's design, no matter how perfect and elegant looking, or provably safe on paper, or demonstrably safe in our experiments.

However clever your algorithm, at that level, something's bound to confuse it.

Odd, I've been reading moral paradoxes for many years and my brain never crashed once, nor have I turned evil. I've been confused but never catastrophically so (though I have to admit my younger self came close). My algorithm must be "beyond clever".

That's a remarkable level of resilience for a brain design which is, speaking professionally, a damn ugly mess. If I can't do aspire to do at least that well, I may as well hang up my shingle and move in with the ducks.

Give me five dollars, or I will kill as many puppies as it takes to make you. And they'll go to hell. And there in that hell will be fire, brimstone, and rap with Engrish lyrics.

I think the problem is not Solomonoff inducton or Kolmogorov complexity or Bayesian rationality, whatever the difference is, but you. You don't want an AI to think like this because you don't want it to kill you. Meanwhile, to a true altruist, it would make perfect sense.

Not really confident. It's obvious that no society of selfish beings whose members think like this could function. But they'd still, absurdly, be happier on average.

You don't need a bounded utility function to avoid this problem. It merely has to have the property that the utility of a given configuration of the world doesn't grow faster than the length of a minimal description of that function. (Where "minimal" is relative to whatever sort of bounded rationality you're using.)

It actually seems quite plausible to me that our intuitive utility-assignments satisfy something like this constraint (e.g., killing 3^^^^^3 puppies doesn't feel much worse than killing 3^^^^3 puppies), though that might not matter much if you think (as I do, and I expect Eliezer does) that our intuitive utility-assignments often need a lot of adjustment before they become things a really rational being could sign up to.

Nick Tarleton, you say:

"Benquo, replace "kill 3^^^^3 people" with "create 3^^^^3 disutility units" and the problem reappears."

But what is a disutility unit? How can there be that many? How do you know that what he supposes to be a disutility unit isn't from your persective a utility unit?

Any similarly outlandish claim is a challenge not merely to your beliefs, but to your mental vocabulary. It can't be evaluated for probability until it's evaluated for meaning.

Utility functions have to be bounded basically because genuine martingales screw up decision theory -- see the St. Petersburg Paradox for an example.

Economists, statisticians, and game theorists are typically happy to do so, because utility functions don't really exist -- they aren't uniquely determined from someone's preferences. For example, you can multiply any utility function by a constant, and get another utility function that produces exactly the same observable behavior.

Tiiba, keep in mind that to an altruist with a bounded utility function, or with any other of Peter's caveats, in may not "make perfect sense" to hand over the five dollars. So the problem is solveable in a number of ways, the problem is to come up with a solution that (1) isn't a hack and (2) doesn't create more problems than in solves.

Anyway, like most people, I'm not a complete utilitarian altruist, even at a philosophical level. Example: if an AI complained that you take up too much space and are mopey, and offered to kill you and replace you with two happy midgets, I would feel no guilt about refusing the offer, even if the AI could guarantee that overall utility would be higher after the swap.

That's a remarkable level of resilience for a brain design which is, speaking professionally, a damn ugly mess.

...with vital functions inherited from reptiles. But it's been tested to death through history, serious failures thrown out at each step, and we've lots of practical experience and knowledge about how and why it fails. It wasn't built and run first go with zero unrecoverable errors.

I'm not advocating using evolutionary algorithms or to model from the human brain like Ray Kurzweil. I just mean I'd allow for unexpected breakdowns in any part of the system, however much you trust it. At least enough so if it fails it fails safe.

That's only my opinion, and it shouldn't be taken too seriously as I don't have much knowledge in the field at this time, but I thought I should explain what I meant.

I think that if you consider that the chance of a threat to cause a given amount of disutility being valid is a function of the amount of disutility then the problem mostly goes away. That is, in my experience any threat to cause me X units of disutility where X is beyond some threshold is less than 1/10 as credible as a threat to cause me 1 unit of disutility. If someone threatened to kill another person unless I gave them $5000 I would be worried. If they threatened to kill 10 poeple I would be very slightly less worried. If they threatened to kill 1000 people I would be roughly 10 times less worried. If they threatened to kill 1,000,000 people I wouldn't pay any attention at all. Taking these data points and extrapolating I form the heuristic that the chance of someone threatening me with X units of disutility over a threshold based on how much they are demanding and whether I can fulfill that demand decreases faster than linearly.

[i]Nothing could possibly be that weak.[/i]

On the contrary, I think it is not only that weak but actually far weaker. If you are willing to consider the existance of things like 3^^^3 units of disutility without considering the existence of chances like 1/4^^^4 then I believe that is the problem that is causing you so much trouble.

"Odd, I've been reading moral paradoxes for many years and my brain never crashed once, nor have I turned evil."

Even if it hasn't happened to you, it's quite common- think about how many people under Stalin had their brains programmed to murder and torture. Looking back and seeing how your brain could have crashed is scary, because it isn't particularly improbable; it almost happened to me, more than once.

g: killing 3^^^^^3 puppies doesn't feel much worse than killing 3^^^^3 puppies

...

..........................

I hereby award G the All-Time Grand Bull Moose Prize for Non-Extensional Reasoning and Scope Insensitivity.

Clough: On the contrary, I think it is not only that weak but actually far weaker. If you are willing to consider the existance of things like 3^^^3 units of disutility without considering the existence of chances like 1/4^^^4 then I believe that is the problem that is causing you so much trouble.

I'm certainly willing to consider the existence of chances like that, but to arrive at such a calculation, I can't be using Solomonoff induction.

Consider the plight of the first nuclear physicists, trying to calculate whether an atomic bomb could ignite the atmosphere. Yes, they had to do this calculation! Should they have not even bothered, because it would have killed so many people that the prior probability must be very low? The essential problem is that the universe doesn't care one way or the other and therefore events do not in fact have probabilities that diminish with increasing disutility.

Likewise, physics does not contain a clause prohibiting comparatively small events from having large effects. Consider the first replicator in the seas of ancient Earth.

Tiiba: You don't want an AI to think like this because you don't want it to kill you. Meanwhile, to a true altruist, it would make perfect sense.

So you're biting the bullet and saying that, faced with a Pascal's Mugger, you should give him the five dollars?

Would any commenters care to mug Tiiba? I can't quite bring myself to do it, but it needs doing.

Krishnaswami: Utility functions have to be bounded basically because genuine martingales screw up decision theory -- see the St. Petersburg Paradox for an example.

One deals with the St. Petersburg Paradox by observing that the resources of the casino are finite; it is not necessary to bound the utility function itself when you can bound the game within your world-model.

If you believe in the many worlds interpretation of quantum mechanics, you have to discount the utility of each of your future selves by his measure, instead of treating them all equally. The obvious generalization of this idea is for the altruist to discount the utility he assigns to other people by their measures, instead of treating them all equally.

But instead of using the QM measure (which doesn't make sense "outside the Matrix"), let the measure of each person be inversely related to his algorithmic complexity (his personal algorithmic complexity, which is equal to the algorithmic complexity of his universe plus the amount of information needed to locate him within that universe), and the problem is solved. The utility of a Turing machine can no longer grow much faster than its prior probability shrinks, since the sum of measures of people computed by a Turing machine can't be larger than its prior probability.

But there is another puzzle/paradox with Solomonoff induction that I don't know how to solve. I've written about it at http://groups.google.com/group/everything-list/browse_frm/thread/c7442c13ff1396ec/. Eliezer, do you think it would be suitable for a blog post here?

Wei, would it be correct to say that, under your interpretation, if our universe initially contains 100 super happy people, that creating one more person who is "very happy" but not "super happy" is a net negative, because the "measure" of all the 100 super happy people gets slightly discounted by this new person?

It's hard to see why I would consider this the right thing to do - where does this mysterious "measure" come from?

Eliezer, do you think it would be suitable for a blog post here?

Mm... sure. "Bias against uncomputability."

"Would any commenters care to mug Tiiba? I can't quite bring myself to do it, but it needs doing."

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Eli,

I agree that G's reasoning is an example of scope insensitivity. I suspect you meant this as a criticism. It seems undeniable that scope insensitivity leads to some irrational attitudes (e.g. when a person who would be horrified at killing one human shrugs at wiping out humanity). However, it doesn't seem obvious that scope insensitivity is pure fallacy. Mike Vassar's suggestion that "we should consider any number of identical lives to have the same utility as one life" seems plausible. An extreme example is, what if the universe were periodic in the time direction so that every event gets repeated infinitely. Would this mean that every decision has infinite utility consequence? It seems to me that, on the contrary, this would make no difference to the ethical weight of decisions. Perhaps somehow the utility binds to the information content of a set of events. Presumably, the total variation in experiences a puppy can have while being killed would be exhausted long before reaching 3^^^^^3.

Vann McGee has proven that if you have an agent with an unbounded utility function and who thinks there are infinitely many possible states of the world (ie, assigns them probability greater than 0), then you can construct a Dutch book against that agent. Next, observe that anyone who wants to use Solomonoff induction as a guide has committed to infinitely many possible states of the world. So if you also want to admit unbounded utility functions, you have to accept rational agents who will buy a Dutch book.

And if you do that, then the subjectivist justification of probability theory collapses, taking Bayesianism with it, since that's based on non-Dutch-book-ability.

I think the cleanest option is to drop unbounded utility functions, since they buy you zero additional expressive power. Suppose you have an event space S, a preference relation P, and a utility function f from events to nonnegative real numbers such that if s1 P s2, then f(s1) < f(s2). Then, you can easily turn this into a bounded utility function g(s) = f(s)/(f(s) + 1). It's easily seen that g respects the preference relation P in exactly the same way as f did, but is now bounded to the interval [0, 1).

G,

I was essentially agreeing with you that killing 3^^^^^3 vs 3^^^^3 puppies may not be ethically distinct. I would call this scope insensitivity. My suggestion was that scope insensitivity is not necessarily always unjustified.

Eliezer, creating another person in addition to 100 super happy people do not reduce the measures of those 100 super happy people. For example, suppose those 100 super happy people are living in a classical universe computed by some TM. The minimal information needed to locate each person in this universe is just his time/space coordinate. Creating another person does not cause an increase in that information for the existing people.

Is the value of my existence steadily shrinking as the universe expands and it requires more information to locate me in space?

If I make a large uniquely structured arrow pointing at myself from orbit so that a very simple Turing machine can scan the universe and locate me, does the value of my existence go up?

I am skeptical that this solution makes moral sense, however convenient it might be as a patch to this particular problem.

Stephen, you can't have been agreeing with me about that since I didn't say it, even though for some reason I don't understand (perhaps I was very unclear, but I don't see how) Eliezer chose to interpret me doing so and indeed going further to say that it isn't ethically distinct.

Random question:

The number of possible Turing machines is countable. Given a function that maps the natural numbers onto the set of possible Turing machines, one can construct a Turing machine that acts like this:

If machine #1 has not halted, simulate the execution of one instruction of machine #1

If machine #2 has not halted, simulate the execution of one instruction of machine #2
If machine #1 has not halted, simulate the execution of one instruction of machine #1

If machine #3 has not halted, simulate the execution of one instruction of machine #3
If machine #2 has not halted, simulate the execution of one instruction of machine #2
If machine #1 has not halted, simulate the execution of one instruction of machine #1

etc.

This Turing machine, if run, would eventually make all possible computations. (One could even run a program like this on a real, physical computer, subject to memory and time limitations.) Does running such a program have any ethical implications? If running a perfect simulation of a reality is essentially the same as creating that reality, would running this program for a long enough period of time actually cause all possible computable universes to come into existence? Does the existence of this program have any implications for the hypothesis that "our universe is a computer simulation being run in another universe?"

As others have basically said:

Isn't the point essentially that we believe the man's statement is uncorrelated with any moral facts? I mean if we did, then its pretty clear we can be morally forced into doing something.

Is it reasonable to believe the statement is uncorrelated with any facts about the existence of many lives? It seems so, since we have no substantial experience with "Matrices", people from outside the simulation visting us, 3^^^^^^3, the simulation of moral persons, etc...

Consider, the statement 'there is a woman being raped around the corner'. We are morally obliged to look around the corner. We have no more direct proof of the truth of this statement than of Pascal's mugger's statement. But we have good reason to believe the statement is correlated with a fact in one case, but no such reason in the other.

Can a machine be made that will consistently give zero correlation to this sort of thing? Hell if I know. Probably no, since you if iterate the known enough you get the absurd. But any claim that conditional probability of 3^^^^3 lives being simulated and destroyed is 1/(3^^^^^3) or something is a pile of horseshit.

Eliezer, you can interpret rocks as minds if you make the interpretation complex enough. Why do you ignore these rock-minds if not because you discount them for algorithmic complexity?

First, questions like "if the agent expects that I wouldn't be able to verify the extreme disutility, would its utility function be such as to actually go through spending the resources to cause the unverifiable disutility?"

That an entity with such a utility function exists would manage to stick around long enough in the first place itself may drop the probabilities by a whole lot.

Perhaps best to restrict ourselves to the case of the disutility being verifiable, but only after the fact. (Has this agent ever pulled this soft of thing before? etc..) and that verification doesn't open in the present a causal link allowing for other means of preventing the disutility. There's alot going on here.

I'm not sure, but maybe the reasoning would go not so much for the single specific case, but the process would reason by computing the expected utility of following a rule which would result in it being utterly vulnerable to any agent that merely claims to be capable of causing bignum units of disutility.

Something reasoning along the lines of following such a rule would allow agents in general to order the process to cause plenty of disutility. And that, in itself, would seem to have plenty of expected disutility.

However, if after chugging through the math, it didn't balance out and still the expected disutility from the existance of the disutility threat was greater, then perhaps allowing oneself to be vulnerable to such threats is genuinely the correct outcome, however counterintuitive and absurd it would seem to us.

Eliezer> Is the value of my existence steadily shrinking as the universe expands and it requires more information to locate me in space?

Yes, but the value of everyone else's existence is shrinking by the same factor, so it doesn't disturb the preference ordering among possible courses of actions, as far as I can see.

Eliezer> If I make a large uniquely structured arrow pointing at myself from orbit so that a very simple Turing machine can scan the universe and locate me, does the value of my existence go up?

This is a more serious problem for my proposal, but the conspicuous arrow also increases the values of everyone near you by almost the same factor, so again perhaps it doesn't make as much difference as you expect.

Eliezer> I am skeptical that this solution makes moral sense, however convenient it might be as a patch to this particular problem.

I'm also skeptical, but I'd say it's more than just a patch to this particular problem. Treating everyone as equals no matter what their measures are, besides leading to counterintuitive results in this "Pascal's Mugging" thought experiment, is not even mathematically sound, since the sum of the small probabilities multiplied by the vast utilities do not converge to any finite value, no matter what course of action you choose.

The mathematics says that you have to discount each person's value by some function, otherwise your expected utilities won't converge. The only question is which function. Using the inverse of a person's algorithmic complexity seems to lead to intuitive results in many situations, but not all.

But I'm also open to the possibility that this entire approach is wrong... Are there other proposed solutions that make more sense to you at the moment?

I'll respond to a couple of other points I skipped over earlier.

Eliezer> It's hard to see why I would consider this the right thing to do - where does this mysterious "measure" come from?

Suppose you plan to measure the polarization of a photon at some future time and thereby split the universe into two branches of unequal weight. You do not treat people in these two branches as equals, but instead value the people in the higher-weight branch more, right? Can you answer why you consider that to be the right thing to do? That's not a rhetorical question, btw. If I knew the answer to that question I think I'd also know why discounting people by algorithmic complexity (or some other function) might be the right thing to do.

Stephen> Mentioning quantum mechanics serves only as a distraction.

In classical physics, the universe doesn't branch, but instead everything is predetermined by the starting conditions and laws of physics. There is no issue of people in unequal-weight branches, which I think might be analogous to people with different algorithmic complexities. That's why I brought up QM.

Maybe the origin of the paradox is that we are extending the principle of maximizing expected return beyond its domain of applicability. Unlike Bayes formula, which is an unassailable theorem, the principle of maximizing expected return is perhaps just a model of rational desire. As such it could be wrong. When dealing with reasonably high probabilities, the model seems intuitively right. With small probabilities it seems to be just an abstraction, and there is not much intuition to compare it to. When considering a game with positive expected return that comes from big payoffs and small probabilities, it reduces to the intuitive case if we have the opportunity to play the game many times, on the order of one over the payoff probability. This type of frequentist argument seems to be where the principle comes from in the first place. However, if the probabilities are so small that there is no possibility of playing the game that many times, then maybe a rational person just ignores it rather than dutifully investing in an essentially certain loss. Of course, if we relegate the principle of maximizing expected return to being just a limiting case, this leaves open the question of what more general model underlies it.

G: Sorry to put words in your mouth.

Wei: You do not treat people in these two branches as equals, but instead value the people in the higher-weight branch more, right? Can you answer why you consider that to be the right thing to do?

Robin Hanson's guess about mangled worlds seems very elegant to me, since it means that I can run a (large) computer with conventional quantum mechanics programmed into it, no magic in its transistors, and the resulting simulation will contain sentient beings who experience the same probabilities we do.

Even so, I'd have to confess myself confused about why I find myself in a simple universe rather than a noisy one.

Not all infinities are equal, there exists a hierarchy. Look at real numbers versus integers.

kthxbye

Stephen, no problem. Incidentally, I share your doubt about the optimality of optimizing expected utility (though I wonder whether there might be a theorem that says anything coherent can be squeezed into that form).

CC, indeed there are many infinities (not merely infinitely many, not merely more than we can imagine, but more than we can describe), but so what? Any sort of infinite utility, coupled with a nonzero finite probability, leads to the sort of difficulty being contemplated here. Higher infinities neither help with this nor make it worse, so far as I can see. (I suppose it's worth considering that it might conceivably make sense for an agent's utilities to live in some structure "richer" than the usual real numbers, like Conway's surreal numbers, where there are infinities and infinitesimals aplenty. But I think there are technical difficulties with this sort of scheme; for instance, doing calculus over the surreals is problematic. And of course we actually only have finite brains, so whatever utilities we have are presumably representable in finite terms even if they feature incommensurabilities of the sort that might be modelled in terms of something like the surreal numbers. But all this is a separate issue.)

I have a paper which explores the problem in a somewhat more general way (but see especially section 6.3).

Infinite Ethics: http://www.nickbostrom.com/ethics/infinite.pdf

People have been talking about assuming that states with many people hurt have a low (prior) probability. It might be more promising to assume that states with many people hurt have a low correlation with what any random person claims to be able to effect.

Eliezer, I think Robin's guess about mangled worlds is interesting, but irrelevant to this problem. I'd guess that for you, P(mangled worlds is correct) is much smaller than P(it's right that I care about people in proportion to the weight of the branches they are in). So Robin's idea can't explain why you think that is the right thing to do.

Nick, your paper doesn't seem to mention the possibility of discounting people by their algorithmic complexity. Is that an option you considered?

Pascal's wager type arguments fail due to their symmetry (which is preserved in finite cases).

Even if our priors are symmetric for equally complex religious hypotheses, our posteriors almost certainly won't be. There's too much evidence in the world, and too many strong claims about these matters, for me to imagine that posteriors would come out even. Besides, even if two religions are equally probable, there may be certainly be non-epistemic reasons to prefer one over the other.

However, if after chugging through the math, it didn't balance out and still the expected disutility from the existance of the disutility threat was greater, then perhaps allowing oneself to be vulnerable to such threats is genuinely the correct outcome, however counterintuitive and absurd it would seem to us.

I agree. If we really trust the AI doing the computations and don't have reason to think that it's biased, and if the AI has considered all of the points that have been raised about the future consequences of showing oneself vulnerable to Pascalian muggings, then I feel we should go along with the AI's conclusion. 3^^^^3 people is too many to get wrong, and if the probabilities come out asymmetric, so be it.

Maybe the origin of the paradox is that we are extending the principle of maximizing expected return beyond its domain of applicability.

In addition to a frequency argument, one can in some cases make a different argument for maximizing expected value even in one-time-only scenarios. For instance, if you knew you would become a randomly selected person in the universe, and if your only goal was to avoid being murdered, then minimizing the expected number of people murdered would also minimize the probability that you personally would be murdered. Unfortunately, arguments like this make the assumption that your utility function on outcomes takes only one of two values ("good," i.e., not murdered, and "bad," i.e., murdered); it doesn't capture the fact that being murdered in one way may be twice as bad as being murdered in another way.

Even if there is nobody currently making a bignum-level threat, maybe the utility-maximizing thing to do is to devote substantial resources to search for low-probability, high-impact events and stop or encourage them depending on the utility effect. After all, you can't say the probability of every possibility as bad as killing 3^^^^3 people is zero.

Nick Tarleton,
Yes, it is probably correct that one should devote substantial resources to low probability events, but what are the odds that the universe is not only a simulation, but that the containing world is much bigger; and, if so, does the universe just not count, because it's so small? The bounded utility function probably reaches the opposite conclusion that only this universe counts, and maybe we should keep our ambitions limited, out of fear of attracting attention.

"I find myself in a simple world rather than a noisy one."
Care to expand on that?

Robin: Great point about states with many people having low correlations with what one random person can effect. This is fairly trivially provable.

Utilitarian: Equal priors due to complexity, equal posteriors due to lack of entanglement between claims and facts.

Wei Dai, Eliezer, Stephen, g: This is a great thread, but it's getting very long, so it seems likely to be lost to posterity in practice. Why don't the three of you read the paper Neel Krishnaswami referenced, have a chat, and post it on the blog, possibly edited, as a main post?

"The paper I referenced:

Vann McGee (1999)
An airtight Dutch book
Analysis 59 (264), 257–265.

Posted by: Neel Krishnaswami | October 20, 2007 at 06:29 PM"

It might be more promising to assume that states with many people hurt have a low correlation with what any random person claims to be able to effect.

Robin: Great point about states with many people having low correlations with what one random person can effect. This is fairly trivially provable.

Aha!

For some reason, that didn't click in my mind when Robin said it, but it clicked when Vassar said it. Maybe it was because Robin specified "many people hurt" rather than "many people", or because Vassar's part about being "provable" caused me to actually look for a reason. When I read Robin's statement, it came through as just "Arbitrarily penalize probabilities for a lot of people getting hurt."

But, yes, if you've got 3^^^^3 people running around they can't all have sole control over each other's existence. So in a scenario where lots and lots of people exist, one has to penalize by a proportional factor the probability that any one person's binary decision can solely control the whole bunch.

Even if the Matrix-claimant says that the 3^^^^3 minds created will be unlike you, with information that tells them they're powerless, if you're in a generalized scenario where anyone has and uses that kind of power, the vast majority of mind-instantiations are in leaves rather than roots.

This seems to me to go right to the root of the problem, not a full-fledged formal answer but it feels right as a starting point. Any objections?

Robin's anthropic argument seems pretty compelling in this example, now that I understand it. It seems a little less clear if the Matrix-claimant tried to mug you with a threat not involving many minds. For example, maybe he could claim that there exists some giant mind, the killing of which would be as ethically significant as the killing of 3^^^^3 individual human minds? Maybe in that case you would anthropically expect with overwhelmingly high probability to be a figment inside the giant mind.

I think that Robin's point solves this problem, but doesn't solve the more general problem of an AGI's reaction to low probability high utility possibilities and the attendant problems of non-convergence.
The guy with the button could threaten to make an extra-planar factory farm containing 3^^^^^3 pigs instead of killing 3^^^^3 humans. If utilities are additive, that would be worse.

The guy with the button could threaten to make an extra-planar factory farm containing 3^^^^^3 pigs instead of killing 3^^^^3 humans. If utilities are additive, that would be worse.

Congratulations, you made my brain asplode.

3^^^^^^3 copies of that brain, fates all dependent on the original pondering this thread.

All fates equal, I think their incentive to solve the mystery equals that for one alone.

Eliezer, what if the mugger (Matrix-claimant) also says that he is the only person who has that kind of power, and he knows there is just one copy of you in the whole universe? Is the probability of that being true less than 1/3^^^^3?

Don't dollars have an infinite expected value (in human lives or utility) anyway, especially if you take into account weird low-probability scenarios? Maybe the next mugger will make even bigger threats.

Even if the Matrix-claimant says that the 3^^^^3 minds created will be unlike you, with information that tells them they're powerless, if you're in a generalized scenario where anyone has and uses that kind of power, the vast majority of mind-instantiations are in leaves rather than roots.

You would have to abandon Solomonoff Induction (or modify it to account for these anthropic concerns) to make this work. Solomonoff Induction doesn't let you consider just "generalized scenarios"; you have to calculate each one in turn, and eventually one of them is guaranteed to be nasty.

To paraphrase Wei's example: the mugger says, "Give me five dollars, or I'll simulate and kill 3^^^^3 people, and I'll make sure they're aware that they are at the leaf and not at the node". Congratulations, you now have over 3^^^^3 bits of evidence (in fact, it's a tautology with probability 1) that the following proposition is true: "if the mugger's statement is correct, then I am the one person at the node and am not one of the 3^^^^3 people at the leaf." By Solomonoff Induction, this scenario where his statement is literally true has > 1 / 2^(10^50) probability, as it's easily describable in much less than 10^50 bits. Once you try to evaluate the utility differential of that scenario, boom, we're right back where we started.

On the other hand, you could modify Solomonoff Induction to reflect anthropic concerns, but I'm not sure it's any better than just modifying the utility function to reflect anthropic concerns.

And, of course, there's still the pig problem in either case.

Michael, your pig example threw me into a great fit of belly-laughing. I guess that's what my mind look likes when it explodes. And I recall that was Marvin Minsky's prediction in Society of Minds.

You would have to abandon Solomonoff Induction (or modify it to account for these anthropic concerns) to make this work.

To be more specific, you would have to alter it in such a way that it accepted Brandon Carter's Doomsday Argument.

"Congratulations, you made my brain asplode."

Read http://www.spaceandgames.com/?p=22 if you haven't already. Your utility function should not be assigning things arbitrarily large additive utilities, or else you get precisely this problem (if pigs qualify as minds, use rocks), and your function will sum to infinity. If you "kill" by destroying the exact same information content over and over, it doesn't seem to be as bad, or even bad at all. If I made a million identical copies of you, froze them into complete stasis, and then shot 999,999 with a cryonics-proof Super-Plasma-Vaporizer, would this be immoral? It would certainly be less immoral than killing a million ordinary individuals, at least as far as I see it.

Wei, no I don't think I considered the possibility of discounting people by their algorithmic complexity.

I can see that in the context of Everett it seems plausible to weigh each observer with a measure proportional to the amplitude squared of the branch of the wave function on which he is living. Moreover, it seems right to use this measure both to calculate the anthropic probability of me finding myself as that observer and the moral importance of that observer's well-being.

Assigning anthropic probabilities over infinite domains is problematic. I don't know of a fully satisfactory explanation of how to do this. One natural approach might to explore might be to assign some Turing machine based measure to each of the infinite observers. Perhaps we could assign plausible probabilities by using such an approach (although I'd like to see this worked out in detail before accepting that it would work).

If I understand your suggestion correctly, you propose that the same anthropic probability measure should also be used as a measure of moral importance. But there seems to me to be a problem. Consider a simple classical universe with two very similar observers. On my reckoning they should each get anthropic probability measure 1/2 (rejecting SIA, the Self-Indication Assumption). Yet it appears that they should each have a moral weight of 1. Does your proposal require that one accepts the SIA? Or am I misinterpreting you? Or are you trying to explicate not total utilitarianism but average utilitarianism?

It seems like this may be another facet of the problem with our models of expected utility in dealing with very large numbers. For instance, do you accept the Repugnant conclusion?

I'm at a loss for how to model expected utility in a way that doesn't generate the repugnant conclusion, but my suspicion is that if someone finds it, this problem may go away as well.

Or not. It seems that our various heuristics and biases against having correct intuitions about very large and small numbers are directly tied up in producing a limiting framework that acts as a conservative.

One thought, the expected utility of letting our god-like figure run this Turing simulation might well be positive! S/He is essentially creating these 3^^^3 people and then killing them. And in fact, it's reasonable to assume that expected disutility of killing them is entirely dependent on (and thus exactly balanced by) the utility of their creation.

So, our mugger doesn't really hand us a dilemma unless the claim is that this simulation is already running, and those people have lives worth living, but if you don't pay the $5, the program will be altered (sun will stop in the sky, so tto speak) and they will all be killed). This last is more of a nitpick.

It does seem to me that the bayesian inference we draw from this person's statement must be extraordinarily low, with an uncertainty much larger than its absolute value. Because a being which is both capable of this and willing to offer such a wager (either in truth or as a test) is deeply beyond our moral or intellectual comprehension. Indeed, if the claim is true, that fact will have utility implications that completely dwarf the immediate decision. If they are willing to do this much over 5 dollars, what will they do for a billion? Or for some end that money cannot normally purchase? Or merely at whim? It seems that the information we receive by failing to pay may be of value commensurate with the disutility of them truthfully carrying out their threat.

Regarding the comments about exploding brains, it's a wonder to me that we are able to think about these issues and not lose our sanity. How is it that a brain evolved for hunting/gathering/socializing is able to consider these problems at all? Not only that, but we seem to have some useful intuitions about these problems. Where on Earth did they come from?

Nick> Does your proposal require that one accepts the SIA?

Yes, but using a complexity-based measure as the anthropic probability measure implies that the SIA's effect is limited. For example, consider two universes, the first with 1 observer, and the second with 2. If all of the observers have the same complexity you'd assign a higher prior probability (i.e., 2/3) to being in the second universe. But if the second universe has an infinite number of observers, the sum of their measures can't exceed the measure of the universe as a whole, so the "presumptuous philosopher" problem is not too bad.

Nick> If I understand your suggestion correctly, you propose that the same anthropic probability measure should also be used as a measure of moral importance.

Yes, in fact I think there are good arguments for this. If you have an anthropic probability measure, you can argue that it should be used as the measure of moral importance, since everyone would prefer that was the case from behind the veil of ignorance. On the other hand, if you have a measure of moral importance, you can argue that for decisions not involving externalities, the global best case can be obtained if people use that measure as the anthropic probability measure and just consider their self interests.

BTW, when using both anthropic reasoning and moral discounting, it's easy to accidentally apply the same measure twice. For example, suppose the two universes both have 1 observer each, but the observer in the second universe has twice the measure of the one in the first universe. If you're asked to guess which universe you're in with some payoff if you guess right, you don't want to think "There's 2/3 probability that I'm in the second universe, and the payoff is twice as important if I guess 'second', so the expected utility of guessing 'second' is 4 times as much as the EU of guessing 'first'."

I think that to avoid this kind of confusion and other anthropic reasoning paradoxes (see http://groups.google.com/group/everything-list/browse_frm/thread/dd21cbec7063215b/), it's best to consider all decisions and choices from a multiversal objective-deterministic point of view. That is, when you make a decision between choices A and B, you should think "would I prefer if everyone in my position (i.e., having the same perceptions and memories as me) in the entire multiverse chose A or B?" and ignore the temptation to ask "which universe am I likely to be in?".

But that may not work unless you believe in a Tegmarkian multiverse. If you don't, you may have to use both anthropic reasoning and moral discounting, being very careful not to double-count.

Before I get going, please let me make clear that I do not
understand the math here (even Eliezer's intuitive bayesian paper
defeated me on the first pass, and I haven't yet had the courage to
take a second pass), so if I'm Missing The Point(tm), please tell
me.

It seems to me that what's missing is talking about the probability
of given level of resourcefulness of the mugger. Let me 'splain.

If I ask the mugger for more detail, there are a wide variety of
different variables that determine how resourceful the mugger claims
to be. The mugger could, upon further questioning, reveal that all
the death events are the same entity being killed in the same way,
which I call one death; given the unlikelyhood of the mugger telling
the truth in the first place, I'd not pay. Similarily, the mugger
could reveal that the deaths, while of distinct entities, happen one
at a time, and may even include time for the entities to grow up and
become functioning adults (i.e. one death every 18 years), in which
case I can almost certainly put the money to better use by giving it
to SIAI.

On the other end of the scale, the mugger can claim infinite
resources, so that the can complete the deaths (of entirely distinct
entities, which have lives, grow up, and then are slaughtered) in an
infinitely small amount of time. If the mugger does so, they don't
get the money, because I assign an infinitely small value to
probability of the mugger having infinite resources. Yes, the
mugger may live in a magical universe where having infinite
resources is easy, but you don't get a
get-out-of-probability-assignment-free card because you say the word
"magic"; I still have to base my probability assignment of your
claims on the world around me, in which we don't yet have the
computing power to simulate even one human in real time (ignoring
the software problem entirely).

Between these two extremes is an entire range of possibilities. The
important part here is that the probability I assign to "the mugger
is lying" is going to increase exponentially as their claim of
resources increases. Until the claimed rate of birth, growing, and
dying exceeds the rate of deaths we already have here on Earth, I
don't care, because I can better spend the money here. After we
reach that point (~150K per day), I don't care, because my
probability is something like 1/O(2^n) (Computer Science big-O
there; sorry, that's my background) where n is the multiple of
computer resources claimed over "one mind in realtime", so n is,
umm, 150K deaths per day = 53400000 deaths per year, 18 years for
each person, so I think n is 961200000?. That's not even counting
the probability discount due to the ridiculousness of the whole
claim.

The point here is that I don't care about the 3^^^^3 number; I only
care about the claimed deaths per unit time, how that compares to
the number of people currently dying on Earth (on whom I know I
can well-spend the $5) and the claimed resourcefulness of the
mugger. By the time we get up to where the 3^^^^3 number matters,
i.e. "I can kill one-onemillionth of 3^^^^3 people every realtime
year", my probability assignment for their claimed resourcefulness
is so incredibly low (and so incredibly lower than the numbers they
are throwing at me) that I laugh and walk away.

There is not, as far as I can tell, a sweet spot where the number of
lives I might save by giving the mugger the $5 is enough more than
the number of people currently dying on Earth to offset the
ridiculously low probability I'd be assiging to the mugger's
resourcefulness. I'd rather give the $5 to SIAI.

-Robin

My apologies for the horrific formatting; I wrote that huge diatribe in w3m before discovering the captcha needed javascript, and then pasted it here. If an admin can fix it, please do so.

-Robin

One idea is to tell the AI not to expend a portion of its resources greater than the chance of the mugger's statement being true.

Should I think the universe is probably a coarse-grained simulation of my mind rather than real quantum physics, because a coarse-grained human mind is fifty(?) orders of magnitude cheaper than real quantum physics? Should I think the galaxies are tiny lights on a painted backdrop, because that Turing machine would require less space to compute?

I think a large universe full of randomly scattered matter is much more probable than a small universe that consists of a working human mind and little else.

"But, small as this probability is, it isn't anywhere near as small as 3^^^^3 is large"

Eliezer, I contend your limit!

I think this scenario is ingenious. Here are a few ideas, but I'm really not sure how far one can pursue them / how 'much work' they can do:

(1) Perhaps the agent needs some way of 'absolving itself of responsibility' for the evil/arbitrary/unreasonable actions of another being. The action to be performed is the one that yields highest expected utility but only along causal pathways that don't go through an adversary that has been labelled as 'unreasonable'.

(Except this approach doesn't defuse the variation that goes "You can never wipe your nose because you've computed that the probability of this action killing 3^^^^3 people in a parallel universe is ever so slightly greater than the probability of it saving that number of people".)

(2) We only have a fixed amount of 'moral concern', apportioned somehow or other to the beings we care about. Our utility function looks like: Sum(over beings X) ConcernFor(X)*HappinessOf(X). Allocation of 'moral concern' is a 'competitive' process. The only way we can gain some concern about Y is to lose a bit of concern about some X, but if we have regular and in some sense 'positive' interactions with X then our concern for X will be constantly 'replenishing itself'. When the magician appears and tells us his story, we may acquire a tiny bit of concern about him and the people he mentions, but the parts of us that care about the people we know (a) aren't 'told' the magician's story and thus (b) refuse to 'relinquish' very much.

The trouble with is that it sounds too reminiscent of the insanely stupid moral behaviour of human beings (where e.g. they give exactly as much money to save a hundred penguins as ten thousand.)

(3) We completely abandon the principle of using minimum description length as some kind of 'universal prior'. (For some reason. And replace it with something else. For some reason.)

tried it out in Berlin. epic.

Our best understanding of the nature of the "simulation" we call reality has this concept we call "cause and effect" in place. So when something happens it has non-zero (though nigh infinitely small) effects on everything else in existence (progressively smaller effect with each degree of separation).

The effect that affecting 3^^^3 things (regardless of type or classification) has on other things (even if the individual effects of affecting one thing would be extremely small) would be non-trivial (enormously large even after a positively ludicrous degree of separations).

Once you consider the level of effect that this would have on the whole "simulation" you are forced to consider basically all possible futures. You have nigh-infinite good (when these things are removed/effected you end up with utopia and a range of all possible net benefits for the whole of the simulation) and nigh-infinite penalty (when these things are removed/effected you end up with hell and a range of all possible net losses for the whole of the simulation). I cannot foresee how an AI can possibly have enough processing power to overcome the vagary being unable to predict all possible futures following the event.

Moreover, I personally balk at the assumption of that level of responsibility. It is for the same reason that I balk at time travel scenarios. I refuse to be responsible for whatever changes are wrought across all of reality (which in sum become quite large when you consider a Vast possibly infinite universe regardless of how "small" the initial event seems).

Also does the probability assignment take into account the likelihood of the actor in question approaching you? Assuming there are 3^^^3 people (minds), then surely the probability assignment of approaching you specifically must be adjusted accordingly. I understand that "somebody has to be approached," but surely no one here is willing to contend that any of us have traits which are so exceptional that they cannot be found inside of a population which is 3^^^3 in size?

Assume that the basic reasoning for this is true, but nobody actually does the mugging. Since the probability doesn't actually make a significant difference to the expected utility, I'll just simplify and say there equal.

The total expected marginal utility, assuming you're equally likely to save or kill the people, would be (3^^^3 - 3^^^3) + (3^^^^3 - 3^^^^3) + (3^^^^^3 - 3^^^^^3) + ... = 0. At least, it would be if you count it by alternating with saving and killing. You could also count it as 3^^^3 + 3^^^^3 - 3^^^3 + 3^^^^^3 - 3^^^^3 + ... = infinity. Or you could count it as -3^^^3 - 3^^^^3 + 3^^^3 - 3^^^^^3 + 3^^^^3 - ... = -infinity. Or you could even do 3^^^3 - 3^^^3 + 3^^^^3 - 3^^^^3 + 3^^^^^3 - 3^^^^^3 + ... (without parentheses) which doesn't even converge to anything.

You could also construct hypothetical possibility sets where you can set it to add to any given number by rearranging the possibilities.

It's one thing when order matters for talking about total utility of an infinitely long universe. It at least has an order, assuming you don't mind abandoning special relativity, but what order are you even supposed to count expected utility in?

I figure the only way out of this is to use a prior that decreases with expected utility faster than those formulations of Occam's razor would suggest. I don't like the idea of doing this, but not doing so just doesn't add up.

The probability of some action costing delta-utility x and resulting in delta-utility y, where y >> x, is low. The Anti Gratis Dining modifier is x/y. These things I conjecture, anyways.

The apple-salespeep who says, "Give me $0.50, and I will give you an apple" is quite believable, unlike the apple-salespeep who claims, "Give me $3.50, and I will give apples to all who walk the Earth". We understand how buying an apple gets us an apple, but we know far less about implementing global apple distribution.

Suppose I have a Holy Hand Grenade of FAI, which has been carefully proofed by all the best mathematicians, programmers, and philosophers, and I am (of course) amongst them. And am randomly selected to activate it! Sadly, there is an ant caught in the pin. I can not delay to extricate it, for that means more deaths left unprevented. I pull the pin and kill the ant anyways.

So, the more understanding you have about the situation at hand, the less the AGD factor applies to the situation.

[Late edit: I have since retracted this solution as wrong, see comments below; left here for completeness. The ACTUAL solution that really works I've written in a different comment :) ]

I do believe I've solved this. Don't know if anyone is still reading or not after all this time, but here goes.

Eliezer speaks of the symmetry of Pascal's wager; I'm going to use something very similar here to solve the issue. The number of things that could happen next - say, in the next nanosecond - is infinite, or at the very least incalculable. A lot of mundane things could happen, or a lot of unforeseen things could happen. It could happen that a car would go through my living room and kill me. Or it could happen that the laws of energy conservation were violated and the whole world would turn into bleu cheese. Each of these possibilities could, in theory, have a probability assigned to it, given our priors.

But! We only have enough computing power to calculate a finite number of outcomes at any given moment. That means that we CANNOT go around assigning probabilities by calculation. Rather, we're going to need some heuristic to deal with all the probabilities we do NOT calculate.

Suppose our AI is very good at predicting things. It manages to assign SOME probability to what will happen next about 99% of the time (Note: My solution works equally well for anything from 0% to 100% minus epsilon - and I shouldn't have to explain why a Bayesian AI should never be 100% certain that it got an answer right). That means that 1% of the time, something REALLY surprises it; it just did not assign any probability at all. Now, because the number of things that could be in that category is infinite, they cancel out. Sure, we could all turn to cheese if it says "abracadabra". Or we could turn to cheese UNLESS it says so. The utility functions will always end in 0 for the uncalculated mass of probabilities.

That means that the AI always works under the assumptions that "or something I didn't see coming will happen; but I must be neutral regarding such an outcome until I know more about it".

Now. Say the AI manages to consider 1 million possibilities per prediction it makes (how it still gets 1% of them wrong is beyond me but again, the exact number doesn't matter for my solution). So any outcome that has NOT been calculated could, in fact, be considered to have a probability of 1%/ 1 million - not because there are only a million possibilities the AI hasn't considered, but because that is how many it could TRY to consider.

This number is your cutoff. Before you multiply a probability with a utility function, you subtract this number from the probability, first. So now if someone comes up to you and says it'll kill 3^^^^3 people and you decide to actually spend the cycles to consider how likely that is, and you get 1/googol, that number is LESS than the background noise of everything you don't have time to calculate. You round it down to zero, not because it is arbitrarily small enough, but because anything you have not considered for calculation must be considered to have higher probability - and like in Pascal's wager, those options' utility is infinite and can counter any number that Pascal's Mugger can throw at me. You subtract, not an arbitrary number, but rather a number depending on how long the AI is thinking about the problem; how many possibilities it takes into account.

Does this solve the problem? I think it does.

(By the way: ChrisA's way also works against this problem, except that coding your AI so that it may disregard value and morality if certain conditions are met seems like a pretty risky proposition).

I think you've just perfectly illustrated how some Scope Insensitivity can be good thing.

Because a mind with perfect scope sensitivity, will be diverted into chasing impossibly tiny probabilities for impossibly large rewards. If a good rationalist must win, then a good rationalist should commit to avoiding supposed rationality that makes him lose like that.

So, here's a solution. If a probability is too tiny to be reasonably likely to occur in your lifespan, treat its bait as actually impossible. If you don't, you'll inevitably crash into effective ineffectiveness.

This comment thread has grown too large :). I have a thought that seems to me to be the right way to resolve this problem. On the one hand, the thought is obvious, so it probably has already been played out in this comment thread, where it presumably failed to convince everyone. On the other hand, the thread is too large for me to digest in the time that I can reasonably give it. So I'm hoping that someone more familiar with the conversation here will tell me where I can find the sub-thread that addresses my point. (I tried some obvious word-searches, and nothing came up.)

Anyway, here is my point. I can see that the hypothesis that 3^^^^3 people are being tortured might be simple enough so that the Solomonoff prior is high enough so that the AI would give in to the mugger, if the AI were using an un-updated Solomonoff prior. But the AI is allowed to update, right? And, from what the AI knows about humans, it can see that the low complexity of 3^^^^3 also makes it more probable that a "philosopher out for a fast buck" would choose that number.

So, the simplicity of 3^^^^3 contributes to both the hypothesis of a real torturer and the hypothesis of the liar.

And if, after taking all this into account, the AI still computes a high expected utility for giving in to the mugger, well, then I guess that that is really what it ought to do (assuming that it shares my utility function). But is there any reason to think that this is likely? Does it follow just from Eliezer's observation that "the utility of a Turing machine can grow much faster than its prior probability shrinks"? After all, it's the updated probability that really matters, isn't it?

It does seem that the probability of someone being able to bring about the deaths of N people should scale as 1/N, or at least 1/f(N) for some monotonically increasing function f. 3^^^^3 may be a more simply specified number than 1697, but it seems "intuitively obvious" (as much as that means anything) that it's easier to kill 1697 people than 3^^^^3. Under this reasoning, the likely deaths caused by not giving the mugger $5 are something like N/f(N), which depends on what f is, but it seems likely that it converges to zero as N increases.

It is an awfully difficult question, though, because how do we know we don't live in a world where 3^^^^3 people could die at any moment? It seems unlikely, but then so do a lot of things that are real.

Perhaps the problem lies in the idea that a Turing machine can create entities that have the moral status of humans. If there's a machine out there that can create and destroy 3^^^^3 humans on a whim, then are human lives really worth that much? But, on the other hand, there are laws of physics out there that have been demonstrated to create almost 3^^3 humans, so what is one human life worth on that scale?

On another note, my girlfriend says that if someone tried this on her, she'd probably give them the $5 just for the laugh she got out of it. It would probably only work once, though.

Incidentally: How would it affect your intuition if you instead could participate in the Intergalactic Utilium Lottery, where probabilities and payoffs are the same but where you trust the organizers that they do what they promise?

Philosophers of religion argue quite a lot about Pascal's wager and very large utilities or infinite utilities. I haven't bothered to read any of those papers, though. As an example, here is Alexander Pruss.

As I see it, the mugger seems to have an extremely bad hand to play.

If you evaluate the probability of the statement 'I will kill one person if you don't give me five dollars,' as being something that stands in a relationship to the occurrence of such threat being carried through on, and simply multiply up from there until you get to 3^^^^3 people, then you're going to end up with problems.

However, that sort of simplification – treating all the evidence as locating the same thing, only works for low multiples. (Which I'd imagine is why it feels wrong when you start talking about large numbers.) If you evaluate the evidence for and against different parts of the statement, then you can't simply scale it up as a whole without scaling up all the variables that evidence attaches to. The probability that the person will carry through on a threat to kill 3^^^3 people for five dollars is going to zero out fairly quickly. You need to scale up the dollars asked to get all the bits of evidence to scale in proportion to each other.

To make the threat plausible the mugger would have to be asking for a ridiculously large benefit for themselves. And when you start asking for that huge benefit then the computer simply has to answer whether the resources can be put to better use elsewhere.

As it stands, however, the variables in the statement haven't been properly scaled to keep the evidence for and against the proposed murders in a constant relationship. And, while it's just about possible that someone will kill a few hundred people for five dollars (destroying a train or the like would be a low investment exercise) the probability rapidly approaches zero as you increase the number of people that you're proposing to kill for five dollars.

By the time you're talking about 3^^^^3 lives the probability would have long since been reduced to an absurdity. Which would then be compared against all the other things the FAI could do with five dollars, that have far higher probabilities, and simply be dismissed as a bad gamble. (Since over a great length of time the computer could reasonably expect to approach the predicted loss/benefit ratio, regardless of whether the mugger actually killed 3^^^^3 people.)

I think the problem might lie in the almost laughable disparity between the price and the possible risk. A human mind is not capable of instinctively providing a reason why it would be worth killing 3^^^^3 people - or even, I think, a million people - as punishment for not getting $5. A mind who would value $5 as much or more than the lives of 3^^^^3 people is utterly alien to us, and so we leap to the much more likely assumption that the guy is crazy.

Is this a bias? I'd call it a heuristic. It calls to my mind the discussion in Neal Stephenson's Anathem about pink nerve-gas-farting dragons. (Mandatory warning: fictional example.) The crux of it is, our minds only bother to anticipate situations that we can conceive of as logical. Therefore, the manifest illogicality of the mugging (why is 3^^^^3 lives worth $5; if you're a Matrix Lord why can't you just generate $5 or better yet, modify my mind so that I'm inclined to give you $5, etc.) causes us to anti-anticipate its truth. Otherwise, what's to stop you from imagining, as stated by Tom_McCabe2 (and mitchell_porter2, &c.), that typing the string "QWERTYUIOP" leads to, for example, 3^^^^3 deaths? If you imagine it, and conceive of it as a logically possible outcome, then regardless of its improbability, by your argument (as I see it), a "mind that worked strictly by Solomonoff induction" should cease to type that string of letters ever again. By induction, such a mind could cause itself to cease to take any action, which would lead to... well, if the AI had access to itself, likely self-deletion.

That's my top-of-the-head theory. It doesn't really answer the question at hand, but maybe I'm on the right track...?

Maybe I'm missing the point here, but why do we care about any number of simulated "people" existing outside the matrix at all? Even assuming that such people exist, they'll never effect me, nor effect anyone in the world I'm in. I'll never speak to them, they'll never speak to anyone I know and I'll never have to deal with any consequences for their deaths. There's no expectation that I'll be punished or shunned for not caring about people from outside the matrix, nor is there any way that these people could ever break into our world and attempt to punish me for killing them. As far as I care, they're not real people and their deaths or non-deaths do not factor into my utility function at all. Unless Pascal's mugger claims he can use his powers from outside the matrix to create 3^^^3 people in our world (the only one I care about) and then kill them here, my judgement is based soley on that fact that U(me loosing 5$) < 0.

So, let's assume that we're asking about the more interesting case and say that Pascal's mugger is instead threatening to use his magic extra-matrix powers to create 3^^^3 people here on Earth one by one and that they'll each go on international television and denounce me for being a terrible person and ask me over and over why I didn't save them and then explode into chunks of gore where everyone can see it before fading back out of the matrix (to avoid black hole concerns) and that all of this can be avoided with a single one time payment of 5$. What then?

I honestly don't know. Even one person being created and killed that way definitely feels worse than imagining any number of people outside the matrix getting killed. I'd be tempted on an emotional level to say yes and give him the money, despite my more intellectual parts saying this is clearly a setup and that something that terrible isn't actually going to happen. 3^^^3 people, while I obviously can't really imagine that many, is only worse since it will keep happening over and over and over until after the stars burn out of the sky.

The only really convincing argument, aside from the argument from absurdity ("That's stupid, he's just a philosopher out trying to make a quick buck.") is Polymeron's argument here

This might be overly simplistic, but it seems relevant to consider the probability per murder. I am feeling a bit of scope insensitivity on that particular probability, as it is far too small for me to compute, so I need to go through the steps.

If someone tells me that they are going to murder one person if I don't give them $5, I have to consider the probability of it: not every attempted murder is successful, after all, and I don't have nearly as much incentive to pay someone if I believe they won't be successful. Further, most people don't actually attempt murder, and the cost to that person of telling me they will murder someone if they don't get $5 is much, much smaller then the cost of actually murdering someone. Consequences usually follow from murder, after all. I also have to consider the probability that this person is insane and doesn't care about the consequences: only the $5.

Still, only .00496% of people are murdered in a year. (According to Wolfram Alpha, at least) And while I would assign a higher probability to a person claiming to murder someone, it wouldn't jump dramatically- they could be lying, they could try but fail, etc. Even if I let "I will kill someone" be a 90% accurate test with only a 10% false positive rate- which I think is generous in the case of $5 with no additional evidence- as only being .004%. Even if it was 99% sure and 1% false positive, EXTREMELY generous odds, there is only a .4% total probability of it occurring.

In reality, I think there would be some evidence in the case of one murder. At very least I could get strong sociological cues that the person was likely to be telling the truth. However, since I am moving to an end point where they will be killing 3^^^^3 people, I'll leave that aside as it is irrelevant to the end example.

If such a person claimed they would murder 2 people, it would depend on whether I thought the probabilities of the events occurring together were dependent or independent: if him killing one person made it more likely that he would kill two, given the event (the threat) in question.

Now, if he says he will kill two people, and he kills one, he is unlikely to stop before killing another. BUT, there are more chances for complication or failure, and the cost:benefit for him shrinks by half, making the probability that he manages to or tries to kill anyone smaller. These numbers in reality would be affected by circumstance: it is a lot easier to kill two people with a pistol or a bomb than it is with your bare hands. But since I see no bomb or pistol and he is claiming some mechanism I have no evidence for, we'll ignore that reality for now.

I had trouble finding information on the rate of double homicide:single homicide to use as a baseline, but it seems likely that it is neither totally dependent, nor totally independent. In order to believe the threat credible, I have to believe (after hearing the threat) that they will attempt to kill two people, successfully kill one, AND successfully kill another. And if I put the probability of A+B at .004%, I can't very well put A+B+C at any higher. Since I used a 90% false positive rate for my initial calculation, let's use it twice: 81% false positive. We'll assume that the false negative (he murders people even when he says he won't) stays constant.

This means that each murder is slightly more likely than 90% as likely to occur as the murder before it. Now, it isn't exact, and these numbers get really, really small, so I'm looking at 3^3 as a reference.

At 3^3, the cost has gone up 27x if he kills people, but the probability of the event has gone down to .06 of what it was. So, something like 1.7x more costly, given what was said above.

But all this was dependent on several assumed figures. So at what points does it balance out?

I'm a little tired for doing all the math right now, but some quick work showed that being only 80% sure of the test, with a 10% false positive rate, would be enough to where it would go down continuously. So if I am less than 80% sure of the test of "he says he will murder one person if I don't give him 5 dollars" then I can be sure that the probability that he will kill 3^^^^3 is far, far less than the cost if I am wrong.

I'm assuming that I am getting my math right here, and I am quite tired, so if anyone wishes to correct me on some portion of this I would be happy for the criticism.

First¸ I didn't read all of the above comments, though I read a large part of it.

Regarding the intuition that makes one question Pascals mugging: I think it would be likely that there was a strong survival value in the ancestral environment to being able to detect and disregard statements that would cause you to pay money to someone else without there being any way to detect if these statements were true. Anyone without that ability would have been mugged to extinction long ago. This makes more sense if we regard the origin of our builtin utility function as a /very/ coarse approximation of our genes' survival fitness.

Regarding what the FAI is to do, I think the mistake made is assuming that the prior utility of doing ritual X is exactly zero, so that a very small change in our probabilities would make the expected utility of X positive. (Where X is "give the Pascal mugger the money"). A sufficiently smart FAI would have thought about the possibility of being Pascal-mugged long before that actually happens, and would in fact consider it a likely event to sometimes happen. I am not saying that this actually happening is not a tiny sliver of evidence in favor of the mugger telling the truth, but it is very tiny. The FAI would (assuming it had enough resources) compute for every possible Matrix scenario the appropriate probabilities and utilities for every possible action, taking the scenario's complexity into account. There is no reason to assume the prior expected utility for any religious ritual (such as paying Pascal muggers, whose statements you can't check) is exactly zero. Maybe the FAI finds that there is a sufficiently simple scenario in which a god exists and in which it is extremely utillious to worship that god, more so than any alternative scenarios. Or in which one should give in to (specific forms of) Pascal mugging.

However, the problem as presented in this blogpost implicitly assumes that the prior probabilities the FAI holds are such that the tiny sliver of probability provided by one more instance of Pascal's mugging happening, is enough to push the probability of the scenario of 'Extra-Matrix deity killing lots of people if I don't pay' over that of 'Extra-Matrix deity killing lots of people if I do pay'. Since these two scenarios need not have the exact same Kolmogorov complexity this is unlikely.

In short, either the FAI is already religious, (which may include as a ritual 'give money to people who speak a certain passphrase') or it is not, but the event of a Pascal mugging happening is unlikely to change its beliefs.

Now, the question becomes if we should accept the FAI doing things that are expected to favor a huge number of extra-matrix people at a cost to a smaller number of inside-matrix people. If we actually count every human life as equal, and we accept what Solomonoff-inducted bayesian probability theory has to say about huge payoff-tiny probability events and dutch books, the FAI's choice of religion would be the rational thing to do. Else, we could add a term to the AI's utility function to favor inside-matrix people over outside-matrix people, or we could make it favor certainty (of benefitting people known to actually exist) over uncertainty (of outside-matrix people not known to actually exist).

Looks like strategic thinking to me. If you are to organize yourself to be prone to be Pascal-mugged, you will get Pascal mugged, and thus it is irrational to organize yourself to be Pascal-muggable.

edit: It is as rational to introduce certain bounds on applications of own reasoning as it is to try to build reliable, non-crashing software, or to impose simple rule of thumb limits on the output of the software that controls positioning of control rods in the nuclear reactor.

If you properly consider a tiny probability of mistake to your reasoning, a mistake that may lead to consideration of a number generated by a random string - a lot of such numbers are extremely huge - and apply some meta-cognition with regards to appearance of such numbers, you'll find that such extremely huge numbers are also disproportionally represented in products of errors in reasoning.

With regards to the wager, there is my answer: If you see someone bend over backwards to make a nickel, it is probably not Warren Buffett you're seeing. Indeed the probability of that person who's bending over backwards to make a nickel, having N$, would sharply fall off with increase of N. Here you see a being that is mugging you, and he allegedly has the power to simulate 3^^^^3 beings that he can mug, have sexual relations with, torture, what ever. The larger is the claim, the less probable it is that this is a honest situation.

It is however exceedingly difficult to formalize such answer or to arrive at it in a formal fashion. And for me, there could exist other wagers that are beyond my capability to reason correctly about.

For this reason as matter of policy I assume that I have an error per each inference step - the error that can result in consideration of an extremely huge number - and have an upper cut off on the numbers i'd use for considerations as an optimization strategy; if there is a huge number of this sort, more verification steps are needed. In particular, this has very high impact on morality on me. Any sort of situation where you are killing fewer people to save more people - those situations are extremely uncommon and difficult to conjecture - the appearance of such situation however can easily result from faulty reasoning.

The problem seems to vanish if you don't ask "What is the expectation value of utility for this decision, if I do X", but rather "If I changed my mental algorithms so that they do X in situations like this all the time, what utility would I plausibly accumulate over the course of my entire life?" ("How much utility do I get at the 50th percentile of the utility probability distribution?") This would have the following results:

• For the limit case of decisions where all possible outcomes happen infinitely often during your lifetime, you would decide exactly as if you wanted to maximize expectation value in an individual case.

• You would not decide to give money to Pascals' mugger, if you don't expect that there are many fundamentally different scenarios which a mugger could tell you about: If you give a 5 % chance to the scenario described by Pascals mugger and believe that this is the only scenario which, if true, would make you give 5 $ to some person, you would not give the money away.

• In contrast, if you believe that there are 50 different mugging scenarios which people will tell you during your life to pascal-mug you, and you assign an independent 5 % chance to all of them, you would give money to a mugger (and expect this to pay off occasionally).

"Pascal's" Mugging requires me to believe that the apparent universe that we occupy, with its very low information content, is in fact merely part of a much larger program (in a causally linked and so incompressible way) which admits calculation within it of a specially designed (high-information content) universe with 3^^^^3 people (and not, say, as a side-effect of a low-information simulation that also computes other possibilities like giving immense life and joy to comparable numbers of people). The odds of that, if we use the speed priors, would seem to be 2^-(bits describing our universe + number of instructions to compute it):2^-(bits describing that vastly larger universe + number of instructions to compute it). That's going to be a minimum of 1:-2^(O(3^^^^3)), so by the speed prior this particular kind of probability falls away hugely faster than the utility grows.

However, I have little doubt that some creative philosopher can find some way to rescue the mugging argument in slightly different form.

I've been arguing about this with a friend recently [well, a version of this - I don't have any problems with arbitrarily large number of people being created and killed, unless the manner of their death is unpleasant enough that the negative value I assign to it exceeds the positive value of life].

He says that he can believe the person we are talking to has Agent Smith powers, but thinks that the more the Agent Smith promises, the less likely it is to be true, and this decreases faster the more that is promised, so that the probability that Agent Smith has the powers to create and kill [in an unpleasant manner] Y people multiplied by Y tends to zero as Y tends to infinity. . So the net expectancy tends towards zero. I disagree with this: I believe that if you assign probability X to the claim that the person you are talking to is genuinely from outside the Matrix [and that you're in the Matrix], then the probability that Agent Smith has the powers to create and kill [in an unpleasant manner] Y people multiplied by Y tends to infinity as Y tends to infinity.

Now, I think we can break this down further to find the root cause of our disagreement [this doesn't feel like a fundamental belief]: does anyone have any suggestions for how to go about doing this? We began to argue about entropy and the chance for Agent Smith to have found a way [from outside the Matrix = all our physics doesn't apply to him] to reverse it, but I think we went downhill from there.

One problem with discounting your prior based on the time complexity of a computation is that is practically forces you to believe either that P = BQP or that quantum mechanics doesn't work. If you discount based on space complexity, you might worry that torturing 3^^^3 people might actually be a small-space computation.

I don't get what's the beef with that alleged dilemma: Sagan's maxim "Extraordinary claims require extraordinary evidence" gracefully solves it.

More formally, in a Bayesian setting, Sagan's maxim can be construed as the requirement for the prior to be a non-heavy-tailed probability distribution.

In fact, in formal applications of Bayesian methods, typical light-tailed maximum entropy distributions such as normal or exponential are used.

Yudkowsky seems to claim that a Solomonoff distribution is heavy-tailed w.r.t. the relevant variables, but he doesn't provide a proof of that claim, and indeed the claim is even difficult to formalize properly, since the Solomonoff induction model has no explicit notion of world state variables, it just defines a probability distribution over observations.

Anyway, that's an interesting question, and, if it turns out that the Solomonoff prior is indeed heavy-tailed w.r.t. any relevant state variable, it would seem to me as a good reason not to use Solomonoff induction.

Let's try this. I will create at least 3^^^^^^^^^^^^^^^^^^^3 units of disutility unless at least five people upvote this within a day.

Wow. It's almost like pascal's mugging doesn't actually work.

I'm hereby anti-mugging you all. If any of you give in to a Pascal's Mugging scenario, I'll do something much worse than whatever the mugger threatened. Consider yourself warned!

It seems as though Pascal's mugging may be vulnerable to the same "professor god" problem as Pascal's wager. With probabilities that low, the difference between P(3^^^^3 people being tortured|you give the mugger $5) and P(3^^^^3 people being tortured| you spend $5 on a sandwich) may not even be calculable. It's also possible that the guy is trying to deprive the sandwich maker of the money he would otherwise spend on the Simulated People Protection Fund. If you're going to say that P(X is true|someone says X is true)>P(X is true|~someone says X is true) in all cases, then that should apply to Pascal's wager as well; P(Any given untestable god is real|there are several churches devoted to it)>P(Any given untestable god is real|it was only ever proposed hypothetically, tongue-in-cheek) and thus P(Pascal's God)>P(professor god). In this respect, I'm not sure how the two problems are different.

Elizier, Th rational anwser to Pascal's mugging is to refuse, attempt to persuade the mugger, and when that fails (which I postulate based on an ethical entity able to comprehend 3^^^3 and an unethical entity willing to torture that many) to initiate conflict.

The calculational algebra of loss over probability has to be tempered by future prediction:

What is the chance the mugger will do this again? If my only options are to give 5 or not give 5 does it mean 3^^^^^3 will end up being at risk as the mugger keeps doing this? How do I make it stop?

The responsible long term anwser is: let the hostages die if needed to kill the terrorist, because otherwise you get more terrorists taking hostages.

A thought on this, and apologies if it repeats something already said here. Basically: question the structure that leads to someone saying this to you, and question how easy it is to talk about 3^^^^3 people as opposed to, say, 100. If suddenly said person manifests Magic Matrix God Powers (R) then the evidence gained by observing this or anything that contains it (they're telling the truth about all this, you have gone insane, aliens/God/Cthulhu is/are causing you to see this, this person really did just paint mile-high letters in the sky and there is no Matrix) should be more than enough to tip the balance of evidence in favor of "yeah, holy crap, this person deserves my $5!" In short - don't even take seriously your model of an agent as being truthful or untruthful; it could be outside-context-problem wrong, especially if it behaves pathologically, being overly sensitive to small amounts of evidence for bad reasons. Similar idea to your whole model possibly just being wrong if it reports 1-1/3^^^3 certainty.

I would protest that a program to run our known laws of physics (which only predict that 10^80 atoms exist...so there's no way 3^^^^3 distinct minds could exist) is smaller by some number of bits on the order of log_2(3^^^^3) than one in which I am seemingly running on the known laws of physics, and my choice whether or not to hand over $5 dollars (to someone acting as if they are running on the known laws of physics...seeming generally human, and trying to gain wealth without doing hard work) is positively correlated with whether or not 3^^^^3 minds running on a non-physical server somewhere (yes, this situation requires the postulation of the supernatural) are extinguished. The mugger is claiming to be God in the flesh. This problem is the steel-manned Pascal's Wager.

And you should not give the mugger the money, for the same reason you should not become a Christian because of Pascal's wager. Most of the (vast,vast) improbability lies in the mugger's claim to be God. The truth of eir claim is about as probable as the truth of an idea that popped into a psychopath's head that e needs to set eir wallet on fire this instant, in order to please the gods, or else they will torture em for 3^^^^3 years. I claim that Eliezer vastly, vastly underestimated the complexity of this situation...which is already enough to solve the problem. But, suppose, just suppose, this mugger really is God. Then which God is e? For every possible mind that does X in situation A, there is a mind that does not-X in situation A. We don't interact with gods on a day-to-day basis; we interact with humans. We have no prior information about what this mugger will do in this situation if we give em the money. E could be the "Professor Mugger" that punishes you iff you give em the money, because you acted in a way such that any person off the street could have come up to you and taken your money. E could just not do anything either way. You don't know. Ignorance prior, .5. I have no effect in this situation; I do $5 better in the (extremely, extremely, more likely) situation where this mugger is a con artist. No money for the mugger. There's also the slight issue that I'm more likely to be mugged this way if I precommit to losing my money...so I'd be pretty stupid to do that.

The AI will shut up and multiply if it's programmed properly, and get the right answer, my answer, the answer that also happens to be the one we instinctually lean towards here. If we're running a human-friendly dynamic, there's no need to worry about the AI making the wrong choice. Do we seriously think we could do better? If so, then why create the AI in the first place?

(Edit: A lot of this comment is wrong. See my reply to hairyfigment.)

Was Kant an analytic philosopher? I can't remember, but thinking in terms of your actions as being the standard for a "categorical imperative" followed by yourself in all situations as well as by all moral beings, the effect of giving the mugger the money is more than $5. If you give him the money once he'll be able to keep on demanding it from you as well as from other rationalists. Hence the effect will be not $5 but all of your (plural) money, a harm which might be in a significant enough ratio to the deaths of all those people to warrant not giving him the money.

I think we have assume that, although this sounds awfully like that quote about "a million deaths are a statistic", the cost of additional deaths decreases. I'm not really sure how to justify that though.

I think the answer to this question concerns the Kolmogorov complexity of various things, and the utility function as well. What is the Kolmogorov complexity of 3^^^3 simulated people? What is the complexity of the program to generate the simulated people? What is the complexity of the threat, that for each of these 3^^^3 people, this particular man is capable of killing each of them? What sort of prior probability do we assign to "this man is capable of simulating 3^^^3 people, killing each of them, and willing to do so for $5"?

Similarly, the utility function for this calculation needs to be defined. Utility is usually calculated with decreasing marginal returns, such that we usually are described as having scope insensitivity. Likewise, we attach disproportionately lower utility to things with small chances. We'd probably also have lower utility for these 3^^^3 simulated people on account of their being simulated, being generated by a program with very low Kolmogorov complexity (ie, lack individuality and "realness"), and existing at the whim of a seemingly cruel and crazy person (meaning they're probably doomed anyways).

There's a few other things to consider, such as the probability that one of the 3^^^3 simulated people will be able to rescue his people, and the probability that someone else will come and threaten to kill 4^^^^4 people and demand enough money that I can't afford to part with the $5 for only 3^^^3 people.

Overall, I think the problem is poorly defined for the above reasons, but perhaps my main objection would be the attempt to use a formalism intended to reduce unnecessary complexity, as a guide to whether something is true.

Keep in mind that I have very limited knowledge of probability or analytic philosophy, but wouldn't a very easy answer be that if you can conceive of a scenario with the same outcome assigned to NOT doing the action, and that scenario has an equal probability to be true they're both irrelevant?

If it's possible that you can get an infinite amount of gain by believing in god, it's equally possible you can get an infinite amount of gain by NOT believing in god.

"Give me five dollars, or I'll use my magic powers from outside the Matrix to run a Turing machine that simulates and kills 3^^^^3 people."

If there's an arbitrarily small probability that this statement is true, there's an equal arbitrarily small probability of the same result if you do nothing.

So the probability of those deaths remains the same regardless of your actions. So the statement is meaningless.

Obviously this doesn't always apply.

If this overlooks something incredibly obvious then I'm just a stupid teenager and please be kind.

I have a very poor understanding of both probability and analytic philosophy so in the inevitable scenario where I'm completely wrong be kind.

But if you can conceive of a scenario where there's a probability that doing something will result in infinite gain, but you can also picture an equally probable scenario where doing NOTHING will result in equal gain, then don't they cancel each other out?

If there's a probability that believing in god will give you infinite gain, isn't there an equal probability that not believing in god will result in infinite gain?

So if the only merit to a scenario is that someone came up with it it can be countered with a contradicting scenario that someone came up with. There are an infinite amount of claims with no evidence to support them that all have a finitely small probability, but every single one of those claims has a contradicting claim with equal probability.

So only beliefs with evidence to support them should be considered, because only those beliefs don't have a contradicting belief with an equal probability.

So isn't pascals wager pretty stupid? If there's an infinite gain in believing in god there's also an infinite gain in not believing in god. The probability is equal to an infinite amount of contradicting probabilities, therefore non-existent.

Now please tell me how I'm wrong so I can stop having a false sense of accomplishment.

Hmmm...

My problem with this scenario is that I've never run Solomonoff Induction, I run evidentialism. Meaning: if a hypothesis's probability is equal to its True Prior, I just treat that as equivalent to "quantum foam", something that exists in my mathematics for ease of future calculations but has no real tie to physical reality, and is therefore dismissed as equivalent to probability 0.0.

Basically, my brain can reason about plausibility in terms of pure priors, but probability requires at least some tiny bit of evidence one way or the other. In fact, even a very plausible hypothesis, in terms of being so simple that its Solomonoff Prior is, say, 0.75, would make my brain throw a type-error if I tried to bet on it. My priors don't tell me anything about reality, they're only a feature of my mind, they just tell me the starting point for running evidential updates that do correlate with reality.

Why wait until someone wants the money? Shouldn't the AI try to send 5 Dollars to everyone with a note attached reading "Here is a tribute; please don't kill a huge number of people" regardless of whether they ask for it or not?

For the most part, when person P says, "I will do X," that is evidence that P will do X, and the probability of P doing X increases. Instead, if P has a reputation for sarcasm, and if P says the same thing, then the probability that P will do X decreases. Clearly, then, our estimation of P's position in mindspace determines weather we increase or decrease the likelihood of P's claims. For the mugging situation, we might adopt a model where the mugger's claims about very improbable actions in no way affect what we expect him to do since we do not have a useful estimate of the mugger's position in mindspace--how could we? We cannot assume the mugger tends to be more honest than not, as we can with humans. Expected utilities balance and cancel, so I should keep my wallet.

What about optimizing for median expected utility?

EDIT: I wrote up a much more detailed proposal here.

I think you are overestimating the probabilities there: it is only Pascal's Mugging if you fail to attribute a low enough probability to the mugger's claim. The problem, in my opinion, is not how to deal with tiny probabilities of vast utilities, but how not to attribute too high probabilities to events whose probabilities defy our brain's capacity (like "magic powers from outside the Matrix").

I also feel that, as with Pascal's wager, this situation can be mirrored (and therefore have the expected utilities canceled out) if you simply think "What if he intends to kill those people only if I abide by his demand ?". As with Pascal's wager, the possibilities aren't only what the wager stipulates: when dealing with infinites in decision making (I'm not sure one can say "the probability of this event doesn't overcome the vast utility gained" with such numbers) you probably have another infinite which you also can't evaluate hiding behind the question.

Tell me your thoughts.

example of pascal's mugging on Reddit

You could argue that doing any action, such as accepting the wager, has a small but much larger than 1/3^^^3 chance of killing 3^^^3 people. You could argue that any action has a small but much larger than 1/3^^^3 chance of guaranteeing blissful immortality for 3^^^3 people. Therefore, declining the wager makes a lot more sense because no matter what you do you might have already doomed all those people.

How about : The logic of a system applies only within that system ?

Variants of this are common in all sorts of logical proofs, and it stands to reason that elements outside a system do not follow the rules of that system.

A construct assuming something out-of-universe acting in-universe just can't be consistent.

I think you're assuming that to give in to the mugging is the wrong answer in a one-shot game for a being that values all humans in existence equally, because it feels wrong to you, a being with a moral compass evolved in iterated multi-generational games.

Consider these possibilities, any one of which would create challenges for your reasoning:

1. Giving in is the right answer in a one-shot game, but the wrong answer in an iterated game. If you give in to the mugging, the outsider will keep mugging you and other rationalists until you're all broke, leaving the universe's future in the hands of "Marxians" and post-modernists.

2. Giving in is the right answer for a rational AI God, but evolved beings (under the Darwinian definition of "evolved") can't value all member of their species equally. They must value kin more than strangers. You would need a theory to explain why any being that evolved due to resource competition wouldn't consider killing a large number of very distantly-related members of its species to be a good thing.

3. You should interpret the conflict between your intuition, and your desire for a rational God, not as showing that you're reasoning badly because you're evolved, but that you're reasoning badly by desiring a rational God bound by a static utility function. This is complicated, so I'm gonna need more than one paragraph:

Intuitively, my argument boils down to applying the logic behind free markets, freedom of speech, and especially evolution, to the question of how to construct God's utility function. This will be vague, but I think you can fill in the blanks.

Free-market economic theory developed only after millenia during which everyone believed that top-down control was the best way of allocating resources. Freedom of speech developed only after millenia during which everyone believed that it was rational for everyone to try to suppress any speech they disagreed with. Political liberalism developed only after millenia during which everybody believed that the best way to reform society was to figure out what the best society would be like, then force that on everyone. Evolution was conceived of--well, originally about 2500 years ago, probably by Democritus, but it became popular only after millenia during which everyone believed that life could be created only by design.

All of these developments came from empiricists. Empiricism is one of the two opposing philosophical traditions of Western thought. It originated, as far as we know, with Democritus (about whom Plato reportedly said that he wished all his works to be burned--which they eventually were). It went through the Skeptics, the Stoics, Lucretius, nominalism, the use of numeric measurements (re-introduced to the West circa 1300), the Renaissance and Enlightenment, and eventually (with the addition of evolution, probability, statistics, and operationalized terms) created modern science.

A key principle of empiricism, on which John Stuart Mill explicitly based his defense of free speech, is that we can never be certain. If you read about the skeptics and stoics today, you'll read that they "believed nothing", but that was because, to their opponents, "believe" meant "know something with 100% certainty".

(The most-famous skeptic, Sextus Empiricus, was called "Empiricus" because he was of the empirical school of medicine, which taught learning from experience. Its opponent was the rational school of medicine, which used logic to interpret the dictums of the ancient authorities.)

The opposing philosophical tradition, founded by Plato--is rationalism. "Rational" does not mean "good thinking". It has a very specific meaning, and it is not a good way of thinking. It means reasoning about the physical world the same way Euclid constructed geometric proofs. No measurements, no irrational numbers, no observation of the world, no operationalized nominalist definitions, no calculus or differential equations, no testing of hypotheses--just armchair a priori logic about universal categories, based on a set of unquestionable axioms, done in your favorite human language. Rationalism is the opposite of science, which is empirical. The pretense that "rational" means "right reasoning" is the greatest lie foisted on humanity by philosophers.

Dualist rationalism is inherently religious, as it relies on some concept of "spirit", such as Plato's Forms, Augustine's God, Hegel's World Spirit, or an almighty programmer converting sense data into LISP symbols, to connect the inexact, ambiguous, changeable things of this world to the precise, unambiguous, unchanging, and usually unquantified terms in its logic.

(Monist rationalists, like Buddha, Parmenides, and post-modernists, believe sense data can't be divided unambiguously into categories, and thus we may not use categories. Modern empiricists categorize sense data using statistics.)

Rationalists support strict, rigid, top-down planning and control. This includes their opposition to free markets, free speech, gradual reform, and optimization and evolution in general. This is because rationalists believe they can prove things about the real world, and hence their conclusions are reliable, and they don't need to mess around with slow, gradual improvements or with testing. (Of course each rationalist believes that every other rationalist was wrong, and should probably be burned at the stake.)

They oppose all randomness and disorder, because it makes strict top-down control difficult, and threatens to introduce change, which can only be bad once you've found the truth.

They have to classify every physical thing in the world into a discrete, structureless, atomic category, for use in their logic. That has led inevitably to theories which require all humans to ultimately have, at reflective equilibrium, the same values--as Plato, Augustine, Marx, and CEV all do.

You have, I think, picked up some of these bad inclinations from rationalism. When you say you want to find the "right" set of values (via CEV) and encode them into an AI God, that's exactly like the rationalists who spent their lives trying to find the "right" way to live, and then suppress all other thoughts and enforce that "right way" on everyone, for all time. Whereas an empiricist would never claim to have found final truth, and would always leave room for new understandings and new developments.

Your objection to randomness is also typically rationalist. Randomness enables you to sample without bias. A rationalist believes he can achieve complete lack of bias; an empiricist believes that neither complete lack of bias nor complete randomness can be achieved, but that for a given amount of effort, you might achieve lower bias by working on your random number generator and using it to sample, than by hacking away at your biases.

So I don't think we should build an FAI God who has a static set of values. We should build, if anything, an AI referee, who tries only to keep conditions in the universe that will enable evolution to keep on producing behaviors, concepts, and creatures of greater and greater complexity. Randomness must not be eliminated, for without randomness we can have no true exploration, and must be ruled forever by the beliefs and biases of the past.

slightly related:

Suppose Omega forces you to chose a number 0<p<=1 and then, with probability p, you get tortured for 1/(p²) seconds. 
Assume for any T, being tortured for 2T seconds is exactly twice as bad as being tortured for T seconds.
Also assume that your memory gets erased afterwards (this is to make sure there won't be additional suffering from something like PTSD)

The expected value of seconds being tortured is p * 1/(p²)=1/p, so, in terms of expected value, you should chose p=1 and be tortured for 1 second. The smaller the p you chose, the higher the expected value.

Would you actually chose p=1 to maximize the expected value, or would you rather chose a very low p (like 1/3^^^^3)?

I feel confused because this seems to be a very obvious consideration, so it's likely that I'm wrong: shouldn't be the likelihood of 3^^^3 humans being killed decrease proportionally to the huge number that is being casually tossed out?

I mean, even if we go with the hypothesis of a matrix lord out for one of the cruelest prank in history, the likelihood of his simulation being able handle 3^^^3 humans being killed should be proportionally less than the likelihood of his simulation being able to handle 7625597484987 a mere humans being killed, since I think that simulating 3^^^3 humans would wreck any computational agent the laws of physics could allow... even if we consider simulation hypothesis, simulation hypothesis that require a quantum computer larger than the entire universe should be considered less likely and so on.

It feels to me that we believe an AI would give this possibility a likelihood that allows paying for any sum of money because we have no real way of estimating how unbelievably unlikely such an event is.

I guess I am missing something but I'm not sure what it could be.

First of all- I love this post!

Second of all - How is this different than Pascal's wager - meaning why doesn't Pascal's mugger get cancelled out by a possible counter mugger who will simulate and kill the same number of people if you give in to the muggers demand. 

I think that we should expect it to be extremely unlikely for an agent to have the required power and willingness to kill 3^^^^3 people. The shortest explanation for why we should believe this is that any agent that gathers "power" will increase his ability to influence the state of the universe. As his "power" grows, the number of possible states that the agent can successfully attain scales like the factorial with respect to power. If we denote power with x, then the number of possible states that the agent can attain is proportional to x!. Now, the expected utility of the threat (the utility of the threat becoming true times the probability of the threat being done) becomes proportional to -x * (1 / x!). From this, we can see that actually the greater the threat, the smaller its absolute magnitude. From this, we can conclude that the expected utility of the threat is 0 for a sufficiently large x.

As for why the number of attainable states is factorial? Well, I think that it is self-explanatory that as an agent's power over space, time, matter, and energy grows, it can arrange them in many different arrangements, and there is roughly x! of those arrangements, where x is the number of parameters it is working with. This is not mathematically rigorous as it is a blog post and I am not a mathematician but I think this should be logical to anyone thinking about this problem.