Newcomb and Dominance

Hey All,

We never really talk about the Newcomb problem around here (never really == less than 75% of the time), so I thought I’d throw (what I think is) a new substantive line in the ring.

I assume a standard formulation of the Newcomb problem with an infallible predictor, which can be found here.  In the PDF below, I put forth an intuitively appealing argument for two-boxing in the Newcomb problem which employs dominance reasoning. I then suggest a potential issue with this argument as formulated. Much of the thinking that went into this was inspired by, among others, David Wiens, Stephen Campbell, Shen-yi Liao, and Jason Konek.

I’d appreciate any feedback here, especially reformulations of the two-box argument that stick to the intention of the original but employ different notions of dominance.

Some thoughts: Against a Newcomb Dominance Argument

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14 Responses to Newcomb and Dominance

  1. Steve C. says:

    Hi Dan, here’s a half-baked thought.

    On the face of it, your Pref Possibility principle seems implausible. A person can have preferences about all sorts of things (that go beyond the action under consideration) and I don’t see why one cannot, say, prefer that he dine with Captain Kangaroo this evening, rather than his mother-in-law (though the former is epistemically impossible for him while the latter is not). It’s not clear to me that that is an irrational preference, much less a preference one cannot have.

    But I’m assuming you can address this worry by specifying the sort of action-related preferences that you (presumably) have in mind. (I take it that your mention of “ideal actor” is gesturing in this direction. But I don’t think it’s enough since any agent is going to have all sorts of preferences that are irrelevant to the action at hand–and that doesn’t seem to conflict with being an ideal agent.)

    That said, I have some sympathy for where you seem to be going. It does seem odd that, for instance, an epistemically impossible state of affairs should play this sort of role in deliberation.

  2. Thanks for the thoughts, Steve!

    I am not convinced that the example you’ve given is a counterexample to the principle though. The principle commands that no preference that is BOTH epistemically and metaphysically impossible be ordered before a preference that is either epistemically OR metaphysically possible. So, in your example, the actor prefers (A) dine with Cpt. Kangaroo to (B) dine with m-i-l. We agree that (B) is e and m possible and that (A) is epistemically impossible, but it is not obvious to me that (A) is metaphysically impossible. You might say that it is necessarily the case that Cpt. Kangaroo is not something you can dine with in which case, this would in fact be a counter example. But, to me, it seems like I have a pretty good and coherent idea of what it would be like to dine with Cpt. Kangaroo. At least, it does not represent a clear case of a logical contradiction.

    The kind of cases that the principle is meant to rule out are cases like the following:
    (PPP) I prefer that p&~p, rather than dine with my mother-in-law, and I know that p&~p is a contradiction.

    I argue that the consequent of line 4 is a preference like this. Essentially, this is because there is no world in which the laws of the infallible case hold, there is a million in the box, and you two-box. Let’s call the conjunction of these conditions (TB). Let the analogous set of conditions were you one-box be (OB). Then Line 4 says “I prefer (TB) to (OB).” But (TB) is metaphysically and epistemically impossible! So, line 4 is a statement like the ones above.

    Yea, the prima facie, the principle put forth does come off as implausible, but I have yet to come up with an obvious counterexample. I am open to some though …

  3. dtlocke says:

    “Let S be a possible world (to the Newcomb-problem world) in which there is a million dollars in the opaque box… in S, if I were to two-box, then the predictor would have predicted that I would two-box, so there would not be a million dollars in the opaque box. =><= So (!!!), it is not metaphysically possible that I two-box [in S].”

    “(!!!)” added. Compare:

    Let S be a possible world in which I stand-up. In S, if I were to stay sitting, then I would not stand-up. So (!!!), it is not metaphysically possible that I stay sitting in S.

    In general:

    Let S be a possible world in which P. In S, if it were that not-P, then it would not be that P. So (!!!), it is not metaphysically possible that not-P in S.

  4. Seriously Dustin? Yes, it does literally say that “it is not metaphysically possible that I two-box,” but clearly what this means is that it is metaphysically impossible that (1) the conditions of Newcomb obtain, (2) there is a million dollars in the opaque box, and (3) I two-box, at the same time.

    Similarly, to show that it is metaphysically impossible to be both standing and sitting, one may argue as follows: take any possible world in which I am standing, it is not the case in this world that I am sitting. Hence, there is no possible world in which I am standing and sitting; therefore, it is metaphysically impossible that I both stand and sit (in the same respect).

    Thanks for the “(!!!)” though. That really made my day a little more dramatically entertaining.

  5. dtlocke says:

    (Indeed, I was taking it as assumed that there is no metaphysically possible world at which you both stand and sit.)

    What you say is that is that at S it is not metaphysically possible that you two-box. But now you’re telling me that what you meant is that at S it is not metaphysically possible that you two-box, the conditions of Newcomb obtain, and there is a million in the opaque box. First, if you mean Q&R&P I suggest not writing just P. Second, let’s just put Q&R&P into your argument explicitly, and see if your conclusion follows.

    “Let S be a possible world (to the Newcomb-problem world) in which there is a million dollars in the opaque box… in S, if I were to two-box, then the predictor would have predicted that I would two-box, so there would not be a million dollars in the opaque box. =><= So, it is *it is metaphysically impossible that the conditions of Newcomb obtain, there is a million dollars in the opaque box, and I two-box*. Also, because I am aware of the Newcomb constraints, I know that *it is not the case that the conditions of Newcomb obtain, there is a million dollars in the opaque box, and I two-box*. So, in S, it is neither metaphysically nor epistemically possible that *the conditions of Newcomb obtain, there is a million dollars in the opaque box, and I two-box*. Therefore, by the preference possibility principle [!!!], I do not prefer two-boxing to one-boxing in S.”

    Nope, we’ve still got a “[!!!]” in there! Now your final conclusion doesn’t follow. But I think it would follow if it were changed to:

    “I do not prefer that *the conditions of Newcomb obtain, there is a million dollars in the opaque box, and I two-box* to one-boxing.

    But that of course is not equivalent to what you are trying to prove:

    “Claim. It is not the case that if there is a million dollars in the opaque box, I strictly prefer to take two boxes.”

    The relevant preferences are between one-boxing and two-boxing, not between one-boxing (and such and such other conditions holding) and two-boxing and such and such other conditions holding.

  6. dtlocke says:

    I’m sorry if I’m still not understanding. Perhaps you would just like to reconstruct your “proof”, saying explicitly what you mean?

  7. Steve C. says:

    Hey Dan,
    Hmmm…how about the following?

    Joe prefers the SOA of him having a pleasant dinner on a sphere-cube table with Captain Kangaroo this evening to the SOA of him being eaten alive by rats this evening. (Joe believes the former SOA to be e and m impossible, and the latter to be e and m possible.)

    On what grounds do you reject this alleged counterexample to your PP? Do you deny that one can have such a preference? Or are you granting that but denying that it is rational?

  8. Re: Steve’s Comment
    Hey there Steve,

    You’re right to suggest that this is another instance that seems to violate the principle I’ve put forth. I do not think it actually does violate this principle though. You’re right in that it wouldn’t seem so odd to assert such a preference, that is say “I prefer having a pleasant dinner on a sphere-cube table with Captain Kangaroo this evening to being eaten alive by rats this evening.” I’d like to stick by my guns though and claim that if I know that there is no possible world in which I eat a pleasant dinner on a sphere-cube table with Captain Kangaroo, I cannot actually have this preference. My intuition here tells me that if when I assert “I prefer having a pleasant dinner on a sphere-cube table with Captain Kangaroo this evening …” I do not really strictly mean anything. What could I strictly mean here?

    It appears to me that the initial plausibility of this example could be explained away in one of two ways. (1) Maybe by asserting “I prefer having a pleasant dinner on a sphere-cube table with Captain Kangaroo this evening to being eaten alive by rats this evening.”, what I’m actually trying to convey is that I would dislike being eaten alive by ants this evening so much that I’d prefer anything before that (anything, such as something so absurd as being at a sphere-cube table). Or maybe what is really being asserted here is (2) that I have some preference related to what I have asserted, but which is not strictly picked out by what I have asserted. For instance, maybe “I prefer having a pleasant dinner on a sphere-cube table with Captain Kangaroo this evening …” is saying something like “I prefer to have dinner at a WEIRD table with …”. This approach to explaining the initial plausibility of your suggestion essentially points out that MOST of the assertion is completely understandable, that is that it has something to do with eating dinner and with the Cpt. The plausibility on this account comes the parts that DO make sense.

    Anyway, those are my thoughts on that example. Recently, since we spoke about the principle that I suggested, I have been inclined to except a stronger principle than the one I have provided. That one is:

    (PP’) If an ideal actor prefers it to be the case that A, then the actor must believe A to be metaphysically possible.

    This principle seems much more defensible, but I am not prepared to provide a defense of it here.

  9. Steve C. says:

    I like this response. Off the top of my head, I can’t think of a good reason to resist (PP’). It seems like a promising way to motivate the weaker (PP).

  10. Shen-yi Liao says:

    does “believe A to be metaphysically possible” just mean that A is epistemically possible?

    i am not as convinced by (PP’). it seems to me that even in a world where everything that is green is also red (or your favorite harmless impossibility), i can still prefer two-boxing in the standard newcomb case, and i am being epistemically rational in doing so. an ideal actor can prefer one epistemic world over another even if both are impossible. in this respect, i think the original (PP) does better. at any rate, it shows that (PP’) cannot be used to motivate (PP), which is not weaker but different.

  11. Dustin,

    First, sorry for taking so long to respond.

    Second, I just noticed that in your first response to me that you added in a “[in S]” which was not in my original argument. It is important to my original argument that I am not claiming that (i.e. the version with the “[in S]”). To recover the original, you should take “So, it is not metaphysically possible that I two-box.” to mean the same as “So, it is not metaphysically possible that I two-box [in the relevant Newcomb world].”

    Second’, as you suggested in your communication to me, there is a reformulation of this argument that may be successful, but here I will try to suggest that the given formulation still holds water as it is sufficiently close to the version suggested.

    The general form of the dominance argument is to claim that for every epistemically possible world in which the conditions of the Newcomb obtain, the ideal actor prefers two-boxing to one-boxing (line 6). Line 6 in the argument is supposed to be a consequent of line 3-5 (and the dominance principle). Line 3 partitions the class of epistemically possible worlds into two partitions: those in which there is a million dollars in the box and those in which there is not. Then, lines 4 and 5 are supposed to show that regardless of which of those partitions the actual world is in, the dominant strategy is two-boxing. So, if that is the role of lines 4 and 5, surely interpreting them as you have (that, for instance, line 4 tells us that if there is a million dollars in the opaque box, I strictly prefer to take two boxes [in all cases]), then the dominance argument is almost trivially unsound. Consider for instance the case where there is a hand grenade in the second box, which under your interpretation, is relevant since we are not requiring that the realm of discussion is limited to worlds in which the conditions of the Newcomb problem obtain. So, I take it that this way of interpreting lines 4 and 5 is wholly uncharitable.

    Rather, the best way to interpret the dominance argument at line 4 (“If there is a million dollars in the opaque box, I strictly prefer to take two boxes.”) is as saying that if we restrict the class of possible worlds that we’re talking about to those in which there is a million the box and the conditions of Newcomb obtain (one of the partitions established by line 3), then I prefer to be in a world in which I two-box over one in which I one-box. That is, I prefer to be in a world in which (Newcomb & Million & Two-box) over one in which (Newcomb & Million & One-box). To avoid the possibility of the dominance argument being trivially unsound, we should interpret it this way. But, if we take this to be the claim of line 4, then the argument I presented does serve to show that line 4 is false (if you treat my argument like line 4, in that we assume that only worlds in which the million dollars is present and Newcomb conditions obtain are in play). Here is how that would work: (with minor edits for clarity)

    “Let S be a possible world (to the Newcomb-problem world) in which there is a million dollars in the opaque box and Newcomb conditions obtain. … in S, if I were to two-box, then the predictor would have predicted that I would two-box, so there would not be a million dollars in the opaque box. So, it is not metaphysically possible that I two-box and the million is present and Newcomb conditions. Also, because I am aware of the Newcomb constraints, I know that I cannot two-box in this world. So, in S, it is neither metaphysically nor epistemically possible that I two-box and Newcomb conditions obtain. Therefore, by the preference possibility principle, I do not prefer two-boxing to one-boxing in in a world in which the million dollar is present and the Newcomb conditions obtain.”

    Surely, I should have been more explicit about this in the original formulation, but it still appears that the argument presented works as claimed under the assumption that we take the consequent of line 4 and the argument to be referring only to worlds in which the million is present and the conditions of the Newcomb obtain. So, in essence I reject your claim that “The relevant preferences are between one-boxing and two-boxing, not between one-boxing (and such and such other conditions holding) and two-boxing and such and such other conditions holding.” If this were the case, then we need not assume that the Newcomb conditions obtain, in which case the world in which there is a flesh-eating bacteria in the second box will make the dominance argument unsound. Actually, when we discussing the Newcomb case, the relevant alternatives are not just one-boxing and two-boxing in this strict sense. Rather the alternatives are (one-boxing in a world in which Newcomb conditions and a plethora of other background assumptions obtain) and (two-boxing a world in which Newcomb conditions and a plethora of other background assumptions obtain).

  12. Hey there Sam! Thanks for the thoughts.

    Here I am definitely not taking “believe A to be metaphysically possible” to mean that A is epistemically possible. I am not very well versed on the possibility stuff (though I am hoping to take the seminar this semester on it), but here’s what I’m thinking:

    MP: P is metaphysically possible iff there is a possible world in which P.

    EP: P is epistemically possible for S iff it is compatible with S’s beliefs (or knowledge?) about the actual world that, in the actual world, P.

    So then, we see that believing P to be metaphysically possible and P being epistemically possible come apart in case in which S believes that in the actual world ~P, S’s belief is veridical, but S also believes that there is a possible world in which P. If those conditions obtain, then S could believe P is metaphysically possible (because there is a possible world in which P), but P is not epistemically possible since S has true belief that ~P in the actual world.

    Then you make two claims:
    (1) “an ideal actor can prefer one epistemic world over another even if both are impossible.”
    and
    (2) “that (PP’) cannot be used to motivate (PP), which is not weaker but different.”

    Re (1):
    I am not 100% sure I know what you mean by “epistemic worlds” here, so there are two interpretations that I’ll consider:

    (1a) It is possible for an ideal actor to prefer one metaphysically possible world over another metaphysically possible world when he knows that both are metaphysically impossible.
    and
    (1b) It is possible for an ideal actor to prefer that P over Q even if both P and Q are epistemically impossible.

    Re (1a):
    Frankly, my intuitions tell me that this is false. It seems really odd to me to say that I’d prefer a world in which there are round squares over a world in which there everything is both red and green even though I know both of them are worlds that are metaphysically impossible. I don’t think we can prefer anything that we can’t at least think we have a clear conceptual picture of. When I try to imagine a world with square-circles, I get nothing at all. I am inclined to say that there is not even an intension in these cases. Strictly, my sentences have no meaning at all when I assert that I have a preference for something impossible.

    The only way I see to motive something like this is as follows: It certainly appears that I can have preferential schemes such as `I’d prefer to be in any world in which P over any world in which ~P.’ Then it seems like I could substitute in ‘the world in which everything is both green and red’ into the schema to get “I prefer the world in which everything is both green and red and P to the world in which everything is green and red and ~P.” Then let’s say that I know that “the world in which everything is red and green” attempts to pick out a metaphysically impossible world. Then because “the world in which everything is both green and red” doesn’t refer and I know that it doesn’t refer, I certainly cannot use it as an object of substitution into such a preferential scheme! So, I don’t see how the claim (1a) could be right.

    Re (1b):
    (1b) says “It is possible for an ideal actor to prefer that P over Q even if both P and Q are epistemically impossible.” There are two cases: either (1ba) P or Q is metaphysically impossible or (1bb) neither P nor Q are metaphysically impossible.

    If (1ba), then my argument against the possibility of (1a) from above applies here.

    If (1bb), then the instance in question is compatible my principle.
    Proof: Let P and Q be metaphysically possible yet epistemically impossible. Then P is true at some merely possible world and Q is true at some merely possible world. So, it’s reasonable to think that an ideal actor could have the belief that P is true is some possible and Q is true at some possible world. Then my principle allows for the actor to have preferences that order P and Q. No problem here …

    In order for an instance of (1bb) to serve as a counterexample to (PP’), it must be strengthened to an instance of

    (1bb’) It is possible for an ideal actor to prefer that P over Q even if both P and Q are epistemically impossible, metaphysically possible, and either the actor does not believe that P is metaphysically possible or (non-exclusively) the actor does not believe that Q is metaphysically possible.

    (1bb’) entials (1bb) and is incompatible with (PP’).

    I do not believe that this is possible since any instance of a claim that is incompatible with (PP’) would have to say that an actor prefers a SOA which he does not believe to obtain in any possible worlds. Then, this instance would be susceptible to the the arguments against (1a).

    Re (2):
    I am frankly not sure whether strictly speaking (PP’) entails (PP). With some other assumptions, it seems like one may successfully argue from (PP’) to (PP), but I’m not willing to bet on it. I’d be willing to hear counterexamples, but as I suggested above, I do not believe (1) is one.

  13. Shen-yi Liao says:

    I think I do mean (1a). “It seems really odd to me to say that I’d prefer a world in which there are round squares over a world in which there everything is both red and green even though I know both of them are worlds that are metaphysically impossible” is a weird way of putting it though.

    Let me spell out what I have in mind. Consider two worlds, w and w*. In both worlds, there is a tiny ball that is both red and green in some nether regions of Siberia that has no causal influences on anyone’s actions, so both worlds are metaphysically impossible. In w, you have a trillion dollars, cancer and AIDS are cured, and everyone is happy forever. In w*, you are a trillion dollars in debt, the human race is infected by flesh-eating nanobots, and everyone is sad forever. Does it makes sense for you to prefer w to w*?

    From what you write earlier, you seem to think that this question is simply nonsense. I am pretty convinced that it is not, by people like Nolan who argue that it’s useful to reason about situations we take to be impossible. (Though the semantics would obviously not be what you described.) In which case, I think it makes sense to say that you rationally ought to prefer w to w* in the above case. I am not sure why it would be different for an ideal agent.

  14. Yea, you’re right that I’m inclined to say that strictly speaking I don’t think we can reason about objects that we take to instantiate contradictions without first realizing that they don’t instantiate contradictions. My bare intuition here tells me that if I’m going to think about something, I must have an idea of it, and it’s not clear to me that I could have an idea of a world in which there is this ball that is both red and green.

    As I mentioned above, I certainly admit that we can reason to preferential schemes, such as “I prefer any world in which P over any world in which Q.” and then we can think that we’re producing a proper instance of that when we instantiate it with something like “the red/green ball world.” So then, we think we have established preferences about this world. But, if we do realize that there is no object referred to by “the red/green ball world,” then it’s tough for me to say what we’re reasoning about there.

    Maybe, just as an even sketchier thought, we can order worlds that instantiate contradictions by the closeness relation in virtue of considering descriptions of them (since we cannot actually fully consider the world on my account). Then when we reason about that world, we actually reason about the closest non-contradictory world (i.e. the world that best satisfies the description given).

    These seems to be what’s going on in the case you’ve presented anyway. Take w*. On my account we do not have any preferences that order w* compared to any other “world.” But, certainly, I share your intuition that w* is somehow worse than w. Maybe this is because the closest world that best satisfies the description of w* is a world in which you are a trillion dollars in debt, the human race is infected by flesh-eating nanobots, and everyone is sad forever. I can certainly prefer the closet world to the description of w over this world.

    I hope to be in the possible worlds seminar this semester, so hopefully, I’ll have more to say about this kind of thing in the future.

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