And this was my reply:
This is an unfinished part of the theory that I've also thought about, though your example puts it very crisply (you might consider posting it to LW?)
My current thoughts on resolution tend to see two main avenues:
1) Construct a full-blown DAG of math and Platonic facts, an account of which mathematical facts make other mathematical facts true, so that we can compute mathematical counterfactuals.
2) Treat differently mathematical knowledge that we learn by genuinely mathematical reasoning and by physical observation. In this case we...
Logical uncertainty has always been more difficult to deal with than physical uncertainty; the problem with logical uncertainty is that if you analyze it enough, it goes away. I've never seen any really good treatment of logical uncertainty.
But if we depart from TDT for a moment, then it does seem clear that we need to have causelike nodes corresponding to logical uncertainty in a DAG which describes our probability distribution. There is no other way you can completely observe the state of a calculator sent to Mars and a calculator sent to Venus, and yet remain uncertain of their outcomes yet believe the outcomes are correlated. And if you talk about error-prone calculators, two of which say 17 and one of which says 18, and you deduce that the "Platonic answer" was probably in fact 17, you can see that logical uncertainty behaves in an even more causelike way than this.
So, going back to TDT, my hope is that there's a neat set of rules for factoring our logical uncertainty in our causal beliefs, and that these same rules also resolve the sort of situation that you describe.
If you consider the notion of the correlated error-prone calculators, two returning 17 and one re...
I think this problem is based (at least in part) on an incoherence in the basic transparent box variant of Newcomb's problem.
If the subject of the problem will two-box if he sees the big box has the million dollars, but will one-box if he sees the big box is empty. Then there is no action Omega could take to satisfy the conditions of the problem.
In this variant that introduces the digit of pi, there is an unknown bit such that whatever strategy the subject takes, there is a value of that bit that allows Omega an action consistant with the conditions. Howev...
I'm not clear at all what the problem is, but it seems to be symantic. It's disturbing that this post can get 17 upvotes with almost no (2?) comments actually referring to what you're saying- indicating that no one else here really gets the point either.
It seems you have an issue with the word 'dependent' and the definition that Eliezer provided. Under that definition, E (the ith digit of pi) would be dependent on C (our decision to one or two box) if we two-boxed and got a million dollars, because then we would know that E = 0, and we would not have kno...
In UDT1, I would model this problem using the following world program. (For those not familiar with programming convention, 0=False, and 1=True.)
def P(i):
E = (Pi(i) == 0)
D = Omega_Predict(S, i, "box contains $1M")
if D ^ E:
C = S(i, "box contains $1M")
payout = 1001000 - C * 1000 + E * 1e9
else:
C = S(i, "box is empty")
payout = 1000 - C * 1000 + E * 1e9
We then ask, what function S maximizes the expected payout at the end of P? When S sees "box is empty" clearly it ...
TDT is Timeless Decision Theory. It wouldn't be bad to say that in the first paragraph somewhere.
EDIT: Excellent. Thanks.
So let's say I'm confronted with this scenario, and I see $1M in the large box.
So lets get the facts:
1) There is $1M in the large box and thus (D xor E)=true
2) I know that I am an one boxing agent
3) Thus D="one boxing"
4) Thus I know D/=E since the xor is true
5) I one-box and live happily with $1,000,000
When Omega simulates me with the same scenario and without lying there is no problem.
Seems like much of the mindgames are hindered by simply precommitting to choices.
For the red-and-green just toss a coin (or whatever choice of randomness you have).
We could make an ad-hoc repair to TDT by saying that you're not allowed to infer from a logical fact to another logical fact going via a physical (empirical) fact.
In this case, the mistake happened because we went from "My decision algorithm's output" (Logical) to "Money in box" (Physical) to "Digits of Pi" (Logical), where the last step involved following an arrow on a causal graph backwards: The digits of Pi has a causal arrow going into the "money in box" node.
The TDT dependency inference could be implemented by...
Consider the following version of the transparent-boxes scenario.
I'm trying to get a grip on what this post is about, but I don't know enough of the literature about Newcomb's Problem to be sure what is referred to here by "the transparent-boxes scenario". Can someone who knows briefly recap the baseline scenario of which this is a version?
So let's say I'm confronted with this scenario, and I see $1M in the large box.
So lets get the facts:
When Omega simulates me with the same scenario and without lying there is no problem.
Seems like much of the mindgames are hindered by simply precommitting to choices.
For the red-and-green just toss a coin (or whatever choice of randomness you have).
I have a question that is probably stupid and/or already discussed in the comments. But I don't have time to read all the comments, so, if someone nonetheless would kindly explain why I'm confused, I would be grateful.
The OP writes
...So E does indeed "depend on" C, in the particular sense you've specified. Thus, if I happen to have a strong enough preference that E output True, then TDT (as currently formulated) will tell me to two-box for the sake of that goal. But that's the wrong decision, of course. In reality, I have no choice about the spec
So let's say I'm confronted with this scenario, and I see $1M in the large box.
So lets get the facts:
When Omega simulates me with the same scenario and without lying there is no problem.
Seems like much of the mindgames are hindered by simply precommitting to choices.
For the red-and-green just toss a coin (or whatever choice of randomness you have).
So let's say I'm confronted with this scenario, and I see $1M in the large box.
So lets get the facts:
When Omega simulates me with the same scenario and without lying there is no problem.
Seems like much of the mindgames are hindered by simply precommitting to choices.
For the red-and-green just toss a coin (or whatever choice of randomness you have).
Let:
When:
Omega fails.
Omega chooses M or !M. I get $1M or 0.
Omega chooses M=false. I get $0.1.
Omega chooses M=true. I get $1M.
M chooses either M or !M. I get either $1.1 or $0.1 depending on Omega's whims
Omega has no option. I make Omega look like a fool.
So, depending on how 'Omega ...
First thought: We can get out of this dilemma by noting that the output of C also causes the predictor to choose a suitable i, so that saying we cause the ith digit of pi to have a certain value is glossing over the fact that we actually caused the i[C]th digit of pi to have a certain value.
In the setup in question, D goes into an infinite loop (since in the general case it must call a copy of C, but because the box is transparent, C takes as input the output of D).
In Eliezer's similar red/green problem, if the simulation is fully deterministic and the initial conditions are the same, then the simulator must be lying, because he must've told the same thing to the first instance, at a time when there had been no previous copy. (If those conditions do not hold, then the solution is to just flip a coin and take your 50-50 chance.)
Are these still problems when you change them to fix the inconsistencies?
Logical uncertainty has always been more difficult to deal with than physical uncertainty; the problem with logical uncertainty is that if you analyze it enough, it goes away. I've never seen any really good treatment of logical uncertainty.
But if we depart from TDT for a moment, then it does seem clear that we need to have causelike nodes corresponding to logical uncertainty in a DAG which describes our probability distribution. There is no other way you can completely observe the state of a calculator sent to Mars and a calculator sent to Venus, and yet remain uncertain of their outcomes yet believe the outcomes are correlated. And if you talk about error-prone calculators, two of which say 17 and one of which says 18, and you deduce that the "Platonic answer" was probably in fact 17, you can see that logical uncertainty behaves in an even more causelike way than this.
So, going back to TDT, my hope is that there's a neat set of rules for factoring our logical uncertainty in our causal beliefs, and that these same rules also resolve the sort of situation that you describe.
If you consider the notion of the correlated error-prone calculators, two returning 17 and one returning 18, then the most intuitive way to handle this would be to see a "Platonic answer" as its own causal node, and the calculators as error-prone descendants. I'm pretty sure this is how my brain is drawing the graph, but I'm not sure it's the correct answer; it seems to me that a more principled answer would involve uncertainty about which mathematical fact affects each calculator - physically uncertain gates which determine which calculation affects each calculator.
For the (D xor E) problem, we know the behavior we want the TDT calculation to exhibit; we want (D xor E) to be a descendant node of D and E. If we view the physical observation of $1m as telling us the raw mathematical fact (D xor E), and then perform mathematical inference on D, we'll find that we can affect E, which is not what we want. Conversely if we view D as having a physical effect, and E as having a physical effect, and the node D xor E as a physical descendant of D and E, we'll get the behavior we want. So the question is whether there's any principled way of setting this up which will yield the second behavior rather than the first, and also, presumably, yield epistemically correct behavior when reasoning about calculators and so on.
That's if we go down avenue (2). If we go down avenue (1), then we give primacy to our intuition that if-counterfactually you make a different decision, this logically controls the mathematical fact (D xor E) with E held constant, but does not logically control E with (D xor E) held constant. While this does sound intuitive in a sense, it isn't quite nailed down - after all, D is ultimately just as constant as E and (D xor E), and to change any of them makes the model equally inconsistent.
These sorts of issues are something I'm still thinking through, as I think I've mentioned, so let me think out loud for a bit.
In order to observe anything that you think has already been controlled by your decision - any physical thing in which a copy of D has already played a role - then (leaving aside the question of Omega's strategy that simulated alternate versions of you to select a self-consistent problem, and whether this introduces conditional-strategy-dependence rather than just decision-dependence into the problem) there have to be other physical facts which combine with D to yield our observation.
Some of these physical facts may themselves be affected by mathematical facts, like an implemented computation of E; but the point is that in order to have observed anything controlled by D, we already had to draw a physical, causal diagram in which other nodes descended from D.
So suppose we introduce the rule that in every case like this, we will have some physical node that is affected by D, and if we can observe info that depends on D in any way, we'll view the other mathematical facts as combining with D's physical node. This is a rule that tells us not to draw the diagram with a physical node being determined by the mathematical fact D xor E, but rather to have a physical node determined by D, and then a physical descendent D xor E. (Which in this particular problem should descend from a physical node E that descends from the mathematical fact E, because the mathematical fact E is correlated with our uncertainty about other things, and a factored causal graph should have no remaining correlated sources of background uncertainty; but if E didn't correlate to anything else in particular, we could just have D descending to (D xor E) via the (xor with E) rule.)
When I evaluate this proposed solution for ad-hoc-ness, it does admittedly look a bit ad-hoc, but it solves at least one other problem than the one I started with, and which I didn't think of until now. Suppose Omega tells me that I make the same decision in the Prisoner's Dilemma as Agent X. This does not necessarily imply that I should cooperate with Agent X. X and I could have made the same decision for different (uncorrelated) reasons, and Omega could have simply found out by simulating the two of us that X and I gave the same response. In this case, presumably defecting; but if I cooperated, X wouldn't do anything differently. X is just a piece of paper with "Defect" written on it.
If I draw a causal diagram of how I came to learn this correlation from Omega, and I follow the rule of always drawing a causal boundary around the mathematical fact D as soon as it physically affects something, then, given the way Omega simulated both of us to observe the correlation, I see that D and X separately physically affected the correlation-checker node.
On the other hand, if I can analyze the two pieces of code D and X and see that they return the same output, without yet knowing the output, then this knowledge was obtained in a way that doesn't involve D producing an output, so I don't have to draw a hard causal boundary around that output.
If this works, the underlying principle that makes it work is something along the lines of "for D to control X, the correlation between our uncertainty about D and X has to emerge in a way that doesn't involve anyone already computing D". Otherwise D has no free will (said firmly tongue-in-cheek). I am not sure that this principle has any more elegant expression than the rule, "whenever, in your physical model of the universe, D finishes computing, draw a physical/causal boundary around that finished computation and have other things physically/causally descend from it".
If this principle is violated then D ends up "correlated" to all sorts of other things we observe, like the price of fish and whether it's raining outside, via the magic of xor.
I'm rereading past discussions to find insights. This jumped out at me:
Suppose Omega tells me that I make the same decision in the Prisoner's Dilemma as Agent X. This does not necessarily imply that I should cooperate with Agent X.
Do you still believe this?
According to Ingredients of Timeless Decision Theory, when you set up a factored causal graph for TDT, "You treat your choice as determining the result of the logical computation, and hence all instantiations of that computation, and all instantiations of other computations dependent on that logical computation", where "the logical computation" refers to the TDT-prescribed argmax computation (call it C) that takes all your observations of the world (from which you can construct the factored causal graph) as input, and outputs an action in the present situation.
I asked Eliezer to clarify what it means for another logical computation D to be either the same as C, or "dependent on" C, for purposes of the TDT algorithm. Eliezer answered:
I replied as follows (which Eliezer suggested I post here).
If that's what TDT means by the logical dependency between Platonic computations, then TDT may have a serious flaw.
Consider the following version of the transparent-boxes scenario. The predictor has an infallible simulator D that predicts whether I one-box here [EDIT: if I see $1M]. The predictor also has a module E that computes whether the ith digit of pi is zero, for some ridiculously large value of i that the predictor randomly selects. I'll be told the value of i, but the best I can do is assign an a priori probability of .1 that the specified digit is zero.