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Comment author: Manfred 30 October 2014 02:41:55PM 2 points [-]

In general, the question "how many copies are there" may not be answerable in certain weird situations (or can be answered only arbitrarily).

I agree with this. In probability terms, this is saying that P(there are 9 copies of me) is not necessarily meaningful because the event is not necessarily well defined.

My first response is / was that the event "the internet says it's Monday" seems a lot better-defined than "there are 9 of me," and should therefore still have a meaningful probability, even in anthropic situations. But an example may be necessary here.

I think you'd agree that a good example of "certain weird situations" is the divisible brain. Suppose we ran a mind on transistors and wires of macroscopic size. That is, we could make them half as big and they'd still run the same program. Then one can imagine splitting this mind down the middle into two half-sized copies. If this single amount of material counts as two people when split, does it also count as two people when it's together?

Whether it does or doesn't is, to some extent, mere semantics. If we set up a Sleeping Beauty problem except that there's the same amount of total width on both sides, it then becomes semantics whether there is equal anthropic probability on both sides, or unequal. So the "anthropic probabilities are meaningless" argument is looking pretty good. And if it's okay to define amount of personhood based on thickness, why not define it however you like and make probability pointless?

But I don't think it's quite as bad as all that, because of the restriction that your definition of personhood is part of how you view the world, not a free parameter. You don't try to change your mind about the gravitational constant so that you can jump higher. So agents can have this highly arbitrary factor in what they expect to see, but still behave somewhat reasonably. (Of course, any time an agent has some arbitrary-seeming information, I'd like to ask "how do you know what you think you know?" Exploring the possibilities better in this case would be a bit of a rabbit hole, though.)

Then, if I'm pretending to be Stuart Armstrong, I note that there's an equivalence in the aforementioned equal-total-width sleeping beauty problem between e.g. agents who think that anthropic probability is proportional to total width but have the same payoffs in both worlds ("width-selfish agents"), and agents who ignore anthropic probability, but weight the payoffs to agents by their total widths, per total width ("width-average-utilitarian outside perspective [UDT] predictors").

Sure, these two different agents have different information/probabilities and different internal experience, but to the extent that we only care about the actions in this game, they're the same.

Even if an agent starts in multiple identical copies that then diverge into non-identical versions, a selfish agent will want to self-modify to be an average utilitarian between non-identical versions. But this is a bit different from the typical usage of "average utilitarianism" in population ethics. A population-ethics average utilitarian would feed one of their copies to hungry alligators if it paid of for the other copies. But a reflectively-selfish average utilitarian would expect some chance of being the one fed to the alligators, and wouldn't like that plan at all.

Actually, I think the cause of this departure from average utilitarianism over copies is the starting state. When you start already defined as one of multiple copies, like in the divisible brain case, the UDT agent that naive selfish agents want to self-modify to be no longer looks just like an average utilitarian.

So that's one caveat about this equivalence - that it might not apply to all problems, and to get these other problems right, the proper thing to do is to go back and derive the best strategy in terms of selfish preferences.

Which is sort of the general closing thought I have: your arguments make a lot more sense to me than they did before, but as long as you have some preferences that are indexically selfish, there will be cases where you need to do anthropic reasoning just to go from the selfish preferences to the "outside perspective" payoffs that generate the same behavior. And it doesn't particularly matter if you have some contrived state of information that tells you you're one person on Mondays and ten people on Tuesdays.

Man, I haven't had a journey like this since DWFTTW. I was so sure that thing couldn't be going downwind faster than the wind.

P.S. So I have this written down somewhere, the causal buzzword important for an abstract description of the game with the marbles is "factorizable probability distribution." I may check out a causality textbook and try and figure the application of this out with less handwaving, then write a post on it.

Comment author: Stuart_Armstrong 30 October 2014 03:08:03PM 1 point [-]

then write a post on it.

That would be interesting.

Comment author: lackofcheese 29 October 2014 07:46:39PM *  1 point [-]

As I mentioned earlier, it's not an argument against halfers in general; it's against halfers with a specific kind of utility function, which sounds like this: "In any possible world I value only my own current and future subjective happiness, averaged over all of the subjectively indistinguishable people who could equally be "me" right now."

In the above scenario, there is a 1/2 chance that both Jack and Roger will be created, a 1/4 chance of only Jack, and a 1/4 chance of only Roger.

Before finding out who you are, averaging would lead to a 1:1 odds ratio, and so (as you've agreed) this would lead to a cutoff of 1/2.

After finding out whether you are, in fact, Jack or Roger, you have only one possible self in the TAILS world, and one possible self in the relevant HEADS+Jack/HEADS+Roger world, which leads to a 2:1 odds ratio and a cutoff of 2/3.

Ultimately, I guess the essence here is that this kind of utility function is equivalent to a failure to properly conditionalise, and thus even though you're not using probabilities you're still "Dutch-bookable" with respect to your own utility function.

I guess it could be argued that this result is somewhat trivial, but the utility function mentioned above is at least intuitively reasonable, so I don't think it's meaningless to show that having that kind of utility function is going to put you in trouble.

Comment author: Stuart_Armstrong 30 October 2014 10:01:46AM *  1 point [-]

"In any possible world I value only my own current and future subjective happiness, averaged over all of the subjectively indistinguishable people who could equally be "me" right now."

Oh. I see. The problem is that that utility takes a "halfer" position on combining utility (averaging) and "thirder" position on counterfactual worlds where the agent doesn't exist (removing them from consideration). I'm not even sure it's a valid utility function - it seems to mix utility and probability.

For example, in the heads world, it values "50% Roger vs 50% Jack" at the full utility amount, yet values only one of "Roger" and "Jack" at full utility. The correct way of doing this would be to value "50% Roger vs 50% Jack" at 50% - and then you just have a rescaled version of the thirder utility.

I think I see the idea you're getting at, but I suspect that the real lesson of your example is that that mixed halfer/thirder idea cannot be made coherent in terms of utilities over worlds.

Comment author: lackofcheese 29 October 2014 07:55:08PM *  1 point [-]

OK, the "you cause 1/10 of the policy to happen" argument is intuitively reasonable, but under that kind of argument divided responsibility has nothing to do with how many agents are subjectively indistinguishable and instead has to do with the agents who actually participate in the linked decision.

On those grounds, "divided responsibility" would give the right answer in Psy-Kosh's non-anthropic problem. However, this also means your argument that SIA+divided = SSA+total clearly fails, because of the example I just gave before, and because SSA+total gives the wrong answer in Psy-Kosh's non-anthropic problem but SIA+divided does not.

Ah, subjective anticipation... That's an interesting question. I often wonder whether it's meaningful.

As do I. But, as Manfred has said, I don't think that being confused about it is sufficient reason to believe it's meaningless.

Comment author: Stuart_Armstrong 30 October 2014 09:44:28AM 1 point [-]

The divergence between reference class (of identical people) and reference class (of agents with the same decision) is why I advocate for ADT (which is essentially UDT in an anthropic setting).

Comment author: lackofcheese 29 October 2014 08:00:17PM 1 point [-]

You definitely don't have a 50% chance of dying in the sense of "experiencing dying". In the sense of "ceasing to exist" I guess you could argue for it, but I think that it's much more reasonable to say that both past selves continue to exist as a single future self.

Regardless, this stuff may be confusing, but it's entirely conceivable that with the correct theory of personal identity we would have a single correct answer to each of these questions.

Comment author: Stuart_Armstrong 30 October 2014 09:39:01AM 1 point [-]

Conceivable. But it doesn't seem to me that such a theory is necessary, as it's role seems merely to be able to state probabilities that don't influence actions.

Comment author: lackofcheese 29 October 2014 02:04:41PM *  1 point [-]

Linked decisions is also what makes the halfer paradox go away.

I don't think linked decisions make the halfer paradox I brought up go away. Any counterintuitive decisions you make under UDT are simply ones that lead to you making a gain in a counterfactual possible worlds at the cost of a loss in actual possible worlds. However, in the instance above you're losing both in the real scenario in which you're Jack, and in the counterfactual one in which you turned out to be Roger.

Granted, the "halfer" paradox I raised is an argument against having a specific kind of indexical utility function (selfish utility w/ averaging over subjectively indistinguishable agents) rather than an argument against being a halfer in general. SSA, for example, would tell you to stick to your guns because you would still assign probability 1/2 even after you know whether you're "Jack" or "Roger", and thus doesn't suffer from the same paradox. That said, due to the reference class problem, If you are told whether you're Jack or Roger before being told everything else SSA would give the wrong answer, so it's not like it's any better...

To get a paradox that hits at the "thirder" position specifically, in the same way as yours did, I think you need only replace the ticket with something mutually beneficial - like putting on an enjoyable movie that both can watch. Then the thirder would double count the benefit of this, before finding out who they were.

Are you sure? It doesn't seem to be that this would be paradoxical; since the decisions are linked you could argue that "If I hadn't put on an enjoyable movie for Jack/Roger, Jack/Roger wouldn't have put on an enjoyable movie for me, and thus I would be worse off". If, on the other hand, only one agent gets to make that decision, then the agent-parts would have ceased to be subjectively indistinguishable as soon as one of them was offered the decision.

Comment author: Stuart_Armstrong 29 October 2014 05:10:25PM *  2 points [-]

Did I make a mistake? It's possible - I'm exhausted currently. Let's go through this carefully. Can you spell out exactly why you think that halfers are such that:

  1. They are only willing to pay 1/2 for a ticket.
  2. They know that they must either be Jack or Roger.
  3. They know that upon finding out which one they are, regardless of whether it's Jack or Roger, they would be willing to pay 2/3.

I can see 1) and 2), but, thinking about it, I fail to see 3).

Comment author: Manfred 29 October 2014 01:58:53PM *  2 points [-]

And I'm still not seeing what that either assumption gives you, if your decision is already determined

I'll delay talking about the point of all of this until later.

whether anthropic probabilities are meaningful at all.

Probabilities are a function that represents what we know about events (where "events" is a technical term meaning things we don't control, in the context of Cox's theorem - for different formulations of probability this can take on somewhat different meanings). This is "what they mean."

As I said to lackofcheese:

Probabilities have a foundation independent of decision theory, as encoding beliefs about events. They're what you really do expect to see when you look outside.

This is an important note about the absent-minded driver problem et al, that can get lost if one gets comfortable in the effectiveness of UDT. The agent's probabilities are still accurate, and still correspond to the frequency with which they see things (truly!) - but they're no longer related to decision-making in quite the same way.

"The use" is then to predict, as accurately as ever, what you'll see when you look outside yourself.

If you accept that the events you're trying to predict are meaningful (e.g. "whether it's Monday or Tuesday when you look outside"), and you know Cox's theorem, then P(Monday) is meaningful, because it encodes your information about a meaningful event.

In the Sleeping Beauty problem, the answer still happens to be straightforward in terms of logical probabilities, but step one is definitely agreeing that this is not a meaningless statement.

(side note: If all your information is meaningless, that's no problem - then it's just like not knowing anything and it gets P=0.5)

Comment author: Stuart_Armstrong 29 October 2014 04:49:49PM *  3 points [-]

Probabilities are a function that represents what we know about events

As I said to lackofcheese:

If we create 10 identical copies of me and expose 9 of them one stimuli and 1 to another, what is my subjective anticipation of seeing one stimuli over the other? 10% is one obvious answer, but I might take a view of personal identity that fails to distinguish between identical copies of me, in which case 50% is correct. What if identical copies will be recombined later? Eliezer had a thought experiment where agents were two dimensional, and could get glued or separated from each other, and wondered whether this made any difference. I do to. And I'm also very confused about quantum measure, for similar reasons.

In general, the question "how many copies are there" may not be answerable in certain weird situations (or can be answered only arbitrarily).

EDIT: with copying and merging and similar, you get odd scenarios like "the probability of seeing something is x, the probability of remembering seeing it is y, the probability of remembering remembering it is z, and x y and z are all different." Objectively it's clear what's going on, but in terms of "subjective anticipation", it's not clear at all.

Or put more simply: there are two identical copies of you. They will be merged soon. Do you currently have a 50% chance of dying soon?

Comment author: lackofcheese 29 October 2014 01:44:06PM *  1 point [-]

But SIA also has some issues with order of information, though it's connected with decisions

Can you illustrate how the order of information matters there? As far as I can tell it doesn't, and hence it's just an issue with failing to consider counterfactual utility, which SIA ignores by default. It's definitely a relevant criticism of using anthropic probabilities in your decisions, because failing to consider counterfactual utility results in dynamic inconsistency, but I don't think it's as strong as the associated criticism of SSA.

Anyway, if your reference class consists of people who have seen "this is not room X", then "divided responsibility" is no longer 1/3, and you probably have to go whole UTD.

If divided responsibility is not 1/3, what do those words even mean? How can you claim that only two agents are responsible for the decision when it's quite clear that the decision is a linked decision shared by three agents?

If you're taking "divided responsibility" to mean "divide by the number of agents used as an input to the SIA-probability of the relevant world", then your argument that SSA+total = SIA+divided boils down to this: "If, in making decisions, you (an SIA agent) arbitrarily choose to divide your utility for a world by the number of subjectively indistinguishable agents in that world in the given state of information, then you end up with the same decisions as an SSA agent!"

That argument is, of course, trivially true because the the number of agents you're dividing by will be the ratio between the SIA odds and the SSA odds of that world. If you allow me to choose arbitrary constants to scale the utility of each possible world, then of course your decisions will not be fully specified by the probabilities, no matter what decision theory you happen to use. Besides, you haven't even given me any reason why it makes any sense at all to measure my decisions in terms of "responsibility" rather than simply using my utility function in the first place.

On the other hand, if, for example, you could justify why it would make sense to include a notion of "divided responsibility" in my decision theory, then that argument would tell me that SSA+total responsibility must clearly be conceptually the wrong way to do things because it uses total responsibility instead.

All in all, I do think anthropic probabilities are suspect for use in a decision theory because
1. They result in reflective inconsistency by failing to consider counterfactuals.
2. It doesn't make sense to use them for decisions when the probabilities could depend upon the decisions (as in the Absent-Minded Driver)

That said, even if you can't use those probabilities in your decision theory there is still a remaining question of "to what degree should I anticipate X, given my state of information". I don't think your argument on "divided responsibility" holds up, but even if it did the question on subjective anticipation remains unanswered.

Comment author: Stuart_Armstrong 29 October 2014 04:35:56PM 2 points [-]

"If, in making decisions, you (an SIA agent) arbitrarily choose to divide your utility for a world by the number of subjectively indistinguishable agents in that world in the given state of information, then you end up with the same decisions as an SSA agent!"

Yes, that's essentially it. However, the idea of divided responsibility has been proposed before (though not in those terms) - it's not just a hack I made up. Basic idea is, if ten people need to vote unanimously "yes" for a policy that benefits them all, do they each consider that their vote made the difference between the policy and no policy, or that it contributed a tenth of that difference? Divided responsibility actually makes more intuitive sense in many ways, because we could replace the unanimity requirement with "you cause 1/10 of the policy to happen" and it's hard to see what the difference is (assuming that everyone votes identically).

But all these approaches (SIA and SSA and whatever concept of responsibility) fall apart when you consider that UDT allows you to reason about agents that will make the same decision as you, even if they're not subjectively indistinguishable from you. Anthropic probability can't deal with these - worse, it can't even consider counterfactual universes where "you" don't exist, and doesn't distinguish well between identical copies of you that have access to distinct, non-decision relevant information.

the question on subjective anticipation remains unanswered.

Ah, subjective anticipation... That's an interesting question. I often wonder whether it's meaningful. If we create 10 identical copies of me and expose 9 of them one stimuli and 1 to another, what is my subjective anticipation of seeing one stimuli over the other? 10% is one obvious answer, but I might take a view of personal identity that fails to distinguish between identical copies of me, in which case 50% is correct. What if identical copies will be recombined later? Eliezer had a thought experiment where agents were two dimensional, and could get glued or separated from each other, and wondered whether this made any difference. I do to. And I'm also very confused about quantum measure, for similar reasons.

Comment author: lackofcheese 29 October 2014 09:02:30AM *  1 point [-]

That's not true. The SSA agents are only told about the conditions of the experiment after they're created and have already opened their eyes.

Consequently, isn't it equally valid for me to begin the SSA probability calculation with those two agents already excluded from my reference class?

Doesn't this mean that SSA probabilities are not uniquely defined given the same information, because they depend upon the order in which that information is incorporated?

Comment author: Stuart_Armstrong 29 October 2014 09:27:10AM 2 points [-]

Doesn't this mean that SSA probabilities are not uniquely defined given the same information, because they depend upon the order in which that information is incorporated?

Yep. The old reference class problem. Which is why, back when I thought anthropic probabilities were meaningful, I was an SIAer.

But SIA also has some issues with order of information, though it's connected with decisions ( http://lesswrong.com/lw/4fl/dead_men_tell_tales_falling_out_of_love_with_sia/ ).

Anyway, if your reference class consists of people who have seen "this is not room X", then "divided responsibility" is no longer 1/3, and you probably have to go whole UTD.

Comment author: lackofcheese 29 October 2014 01:19:50AM *  1 point [-]

I think that argument is highly suspect, primarily because I see no reason why a notion of "responsibility" should have any bearing on your decision theory. Decision theory is about achieving your goals, not avoiding blame for failing.

However, even if we assume that we do include some notion of responsibility, I think that your argument is still incorrect. Consider this version of the incubator Sleeping Beauty problem, where two coins are flipped.
HH => Sleeping Beauties created in Room 1, 2, and 3
HT => Sleeping Beauty created in Room 1
TH => Sleeping Beauty created in Room 2
TT => Sleeping Beauty created in Room 3
Moreover, in each room there is a sign. In Room 1 it is equally likely to say either "This is not Room 2" or "This is not Room 3", and so on for each of the three rooms.

Now, each Sleeping Beauty is offered a choice between two coupons; each coupon gives the specified amount to their preferred charity (by assumption, utility is proportional to $ given to charity), but only if each of them chose the same coupon. The payoff looks like this:
A => $12 if HH, $0 otherwise.
B => $6 if HH, $2.40 otherwise.

I'm sure you see where this is going, but I'll do the math anyway.

With SIA+divided responsibility, we have
p(HH) = p(not HH) = 1/2
The responsibility is divided among 3 people in HH-world, and among 1 person otherwise, therefore
EU(A) = (1/2)(1/3)$12 = $2.00
EU(B) = (1/2)(1/3)$6 + (1/2)$2.40 = $2.20

With SSA+total responsibility, we have
p(HH) = 1/3
p(not HH) = 2/3
EU(A) = (1/3)$12 = $4.00
EU(B) = (1/3)$6 + (2/3)$2.40 = $3.60

So SIA+divided responsibility suggests choosing B, but SSA+total responsibility suggests choosing A.

Comment author: Stuart_Armstrong 29 October 2014 08:37:03AM *  2 points [-]

The SSA probability of HH is 1/4, not 1/3.

Proof: before opening their eyes, the SSA agents divide probability as: 1/12 HH1 (HH and they are in room 1), 1/12 HH2, 1/12 HH3, 1/4 HT, 1/4 TH, 1/4 TT.

Upon seeing a sign saying "this is not room X", they remove one possible agent from the HH world, and one possible world from the remaining three. So this gives odds of HH:¬HH of (1/12+1/12):(1/4+1/4) = 1/6:1/2, or 1:3, which is a probability of 1/4.

This means that SSA+divided responsibility says EU(A) is $3, and EU(B) is $3.3. - exactly the same ratios as the first setup, with B as the best choice.

Comment author: lackofcheese 28 October 2014 11:17:44PM *  1 point [-]

There's no "should" - this is a value set.

The "should" comes in giving an argument for why a human rather than just a hypothetically constructed agent might actually reason in that way. The "closest continuer" approach makes at least some intuitive sense, though, so I guess that's a fair justification.

The halfer is only being strange because they seem to be using naive CDT. You could construct a similar paradox for a thirder if you assume the ticket pays out only for the other copy, not themselves.

I think there's more to it than that. Yes, UDT-like reasoning gives a general answer, but under UDT the halfer is still definitely acting strange in a way that the thirder would not be.

If the ticket pays out for the other copy, then UDT-like reasoning would lead you to buy the ticket regardless of whether you know which one you are or not, simply on the basis of having a linked decision. Here's Jack's reasoning:

"Now that I know I'm Jack, I'm still only going to pay at most $0.50, because that's what I precommited to do when I didn't know who I was. However, I can't help but think that I was somehow stupid when I made that precommitment, because now it really seems I ought to be willing to pay 2/3. Under UDT sometimes this kind of thing makes sense, because sometimes I have to give up utility so that my counterfactual self can make greater gains, but it seems to me that that isn't the case here. In a counterfactual scenario where I turned out to be Roger and not Jack, I would still desire the same linked decision (x=2/3). Why, then, am I stuck refusing tickets at 55 cents?"

It appears to me that something has clearly gone wrong with the self-averaging approach here, and I think it is indicative of a deeper problem with SSA-like reasoning. I'm not saying you can't reasonably come to the halfer conclusion for different reasons (e.g. the "closest continuer" argument), but some or many of the possible reasons can still be wrong. That being said, I think I tend to disagree with pretty much all of the reasons one could be a halfer, including average utilitarianism, the "closest continuer", and selfish averaging.

Comment author: Stuart_Armstrong 29 October 2014 08:10:06AM *  2 points [-]

simply on the basis of having a linked decision.

Linked decisions is also what makes the halfer paradox go away.

To get a paradox that hits at the "thirder" position specifically, in the same way as yours did, I think you need only replace the ticket with something mutually beneficial - like putting on an enjoyable movie that both can watch. Then the thirder would double count the benefit of this, before finding out who they were.

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