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.
Because he said so, and people tend to be true to their word more often than dictated by chance.
They claim to not be a human. They're still a person, in the sense of a sapient being. As a larger class, you'd expect lower correlation, but it would still be above zero.
Let's say you're a sociopath, that is, the only factors in your utility function are your own personal security and happiness.
Can we use the less controversial term 'economist'?
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 havi...
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 e...
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 sug...
create 3^^^^3 units of disutility according to your utility function
For all X:
If your utility function assigns values to outcomes that differ by a factor of X, then you are vulnerable to becoming a fanatic who banks on scenarios that only occur with probability 1/X. As simple as that.
If you think that banking on scenarios that only occur with probability 1/X is silly, then you have implicitly revealed that your utility function only assigns values in the range [1,Y], where Y<X, and where 1 is the lowest utility you assign.
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.
I don't think the QWERTYUIOP thing is literally zero Bayesian evidence either. Suppose the thought of that particular possibility was manually inserted into your mind by the simulation operator.
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:
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 ...
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 o...
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 thi...
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...
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...
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 t...
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 ...
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 Cop...
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 sho...
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.
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:
If ...
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 muc...
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...
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 ...
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 ...
"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 o...
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 comp...
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."
If you don't donate $5 to SIAI, some random guy in China will die of a heart attack because we couldn't build FAI fast enough. Please donate today.
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 periodi...
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 justifi...
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 mach...
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 aro...
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.....
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 propos...
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 ...
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 ...
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 a...
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 t...
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.
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 p...
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 the...
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,...
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...
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...
The most common formalizations of Occam's Razor, Solomonoff induction and Minimum Description Length, measure the program size of a computation used in a hypothesis, but don't measure the running time or space requirements of the computation. What if this makes a mind vulnerable to finite forms of Pascal's Wager? A compactly specified wager can grow in size much faster than it grows in complexity. The utility of a Turing machine can grow much faster than its prior probability shrinks.
Consider Knuth's up-arrow notation:
In other words: 3^^^3 describes an exponential tower of threes 7625597484987 layers tall. Since this number can be computed by a simple Turing machine, it contains very little information and requires a very short message to describe. This, even though writing out 3^^^3 in base 10 would require enormously more writing material than there are atoms in the known universe (a paltry 10^80).
Now suppose someone comes to me and says, "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."
Call this Pascal's Mugging.
"Magic powers from outside the Matrix" are easier said than done - we have to suppose that our world is a computing simulation run from within an environment that can afford simulation of arbitrarily large finite Turing machines, and that the would-be wizard has been spliced into our own Turing tape and is in continuing communication with an outside operator, etc.
Thus the Kolmogorov complexity of "magic powers from outside the Matrix" is larger than the mere English words would indicate. Therefore the Solomonoff-inducted probability, two to the negative Kolmogorov complexity, is exponentially tinier than one might naively think.
But, small as this probability is, it isn't anywhere near as small as 3^^^^3 is large. If you take a decimal point, followed by a number of zeros equal to the length of the Bible, followed by a 1, and multiply this unimaginably tiny fraction by 3^^^^3, the result is pretty much 3^^^^3.
Most people, I think, envision an "infinite" God that is nowhere near as large as 3^^^^3. "Infinity" is reassuringly featureless and blank. "Eternal life in Heaven" is nowhere near as intimidating as the thought of spending 3^^^^3 years on one of those fluffy clouds. The notion that the diversity of life on Earth springs from God's infinite creativity, sounds more plausible than the notion that life on Earth was created by a superintelligence 3^^^^3 bits large. Similarly for envisioning an "infinite" God interested in whether women wear men's clothing, versus a superintelligence of 3^^^^3 bits, etc.
The original version of Pascal's Wager is easily dealt with by the gigantic multiplicity of possible gods, an Allah for every Christ and a Zeus for every Allah, including the "Professor God" who places only atheists in Heaven. And since all the expected utilities here are allegedly "infinite", it's easy enough to argue that they cancel out. Infinities, being featureless and blank, are all the same size.
But suppose I built an AI which worked by some bounded analogue of Solomonoff induction - an AI sufficiently Bayesian to insist on calculating complexities and assessing probabilities, rather than just waving them off as "large" or "small".
If the probabilities of various scenarios considered did not exactly cancel out, the AI's action in the case of Pascal's Mugging would be overwhelmingly dominated by whatever tiny differentials existed in the various tiny probabilities under which 3^^^^3 units of expected utility were actually at stake.
You or I would probably wave off the whole matter with a laugh, planning according to the dominant mainline probability: Pascal's Mugger is just a philosopher out for a fast buck.
But a silicon chip does not look over the code fed to it, assess it for reasonableness, and correct it if not. An AI is not given its code like a human servant given instructions. An AI is its code. 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? What is the mere Earth at stake, compared to a tiny probability of 3^^^^3 lives?
How do I know to be worried by this line of reasoning? How do I know to rationalize reasons a Bayesian shouldn't work that way? A mind that worked strictly by Solomonoff induction would not know to rationalize reasons that Pascal's Mugging mattered less than Earth's existence. It would simply go by whatever answer Solomonoff induction obtained.
It would seem, then, that I've implicitly declared my existence as a mind that does not work by the logic of Solomonoff, at least not the way I've described it. What am I comparing Solomonoff's answer to, to determine whether Solomonoff induction got it "right" or "wrong"?
Why do I think it's unreasonable to focus my entire attention on the magic-bearing possible worlds, faced with a Pascal's Mugging? Do I have an instinct to resist exploitation by arguments "anyone could make"? Am I unsatisfied by any visualization in which the dominant mainline probability leads to a loss? Do I drop sufficiently small probabilities from consideration entirely? Would an AI that lacks these instincts be exploitable by Pascal's Mugging?
Is it me who's wrong? Should I worry more about the possibility of some Unseen Magical Prankster of very tiny probability taking this post literally, than about the fate of the human species in the "mainline" probabilities?
It doesn't feel to me like 3^^^^3 lives are really at stake, even at very tiny probability. I'd sooner question my grasp of "rationality" than give five dollars to a Pascal's Mugger because I thought it was "rational".
Should we penalize computations with large space and time requirements? This is a hack that solves the problem, but is it true? Are computationally costly explanations less likely? 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 exponentially 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?
Given that, in general, a Turing machine can increase in utility vastly faster than it increases in complexity, how should an Occam-abiding mind avoid being dominated by tiny probabilities of vast utilities?
If I could formalize whichever internal criterion was telling me I didn't want this to happen, I might have an answer.
I talked over a variant of this problem with Nick Hay, Peter de Blanc, and Marcello Herreshoff in summer of 2006. I don't feel I have a satisfactory resolution as yet, so I'm throwing it open to any analytic philosophers who might happen to read Overcoming Bias.