I’m a Postdoctoral Research Fellow at Oxford University’s Global Priorities Institute.
Previously, I was a Philosophy Fellow at the Center for AI Safety.
So far, my work has mostly been about the moral importance of future generations. Going forward, it will mostly be about AI.
You can email me at elliott.thornley@philosophy.ox.ac.uk
So, you would agree that the following is an English description of a theorem:
If an agent has complete, transitive preferences, and it does not pursue dominated strategies, then it must be representable as maximizing expected utility.
Yep, I agree with that.
I feel pretty fine with justifying the transitive part via theorems basically like the one I gave above.
Note that your money-pump justifies acyclicity (The agent does not strictly prefer A to B, B to C, and C to A) rather than the version of transitivity necessary for the VNM and Complete Class theorems (If the agent weakly prefers A to B, and B to C, then the agent weakly prefers A to C). Gustafsson thinks you need Completeness to get a money-pump for this version of transitivity working (see footnote 8 on page 3), and I’m inclined to agree.
when you have intransitive preferences it’s not even clear what a “dominated strategy” would be.
A dominated strategy would be a strategy which leads you to choose an option that is worse in some respect than another available option and not better than that other available option in any respect. For example, making all the trades and getting A- in the decision-situation below would be a dominated strategy, since you could have made no trades and got A:
So I still think that it is basically incorrect to say:
And yet the error seems to have gone uncorrected for more than a decade.
The error is claiming that
There exist theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy.
I haven’t seen anyone point out that that claim is false.
That said, one could reason as follows:
Rohin, John, and others have argued that agents with incomplete preferences can act in accordance with policies that make them immune to pursuing dominated strategies.
Agents with incomplete preferences cannot be represented as maximizing expected utility.
So, if Rohin’s, John’s, and others’ arguments are sound, there cannot exist theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy.
Then one would have corrected the error. But since the availability of this kind of reasoning is easily missed, it seems worth correcting the error directly.
Theorems are typically of the form “Suppose X, then Y”; what is X if not an assumption?
X is an antecedent.
Consider an example. Imagine I claim:
Suppose James is a bachelor. Then James is unmarried.
In making this claim, I am not assuming that James is a bachelor. My claim is true whether or not James is a bachelor.
I might temporarily assume that James is a bachelor, and then use that assumption to prove that James is unmarried. But when I conclude ‘Suppose James is a bachelor. Then James is unmarried’, I discharge that initial assumption. My conclusion no longer depends on it. Any conclusion which can be proved with no undischarged assumptions is a theorem.
Theorem. Suppose there is a set of possible worlds , and an agent that given a current world and a proposed new world specifies how much money it would pay to switch to from . Suppose further than cannot be money pumped, that is, there is no sequence of worlds such that (1) and (2) . Then must be transitive in the following sense: for any , if and , then .
Proof. Suppose is not transitive, so there exists some where , , and . But then then the sequence is a money pump, leading to a contradiction.
I agree that this is a theorem. But it’s not a ‘coherence theorem’ (at least not in the way that I’ve used the term in this post, and not in the way that previous authors seem to have used the term [see the Appendix]): it doesn’t state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. It states only that, unless an agent’s preferences are acyclic, that agent is liable to pursue strategies that are dominated by some other available strategy.
You can call it a ‘coherence theorem’. Then it would be true that coherence theorems exist. But the important point remains: Premise 1 of the coherence argument is false. There are no theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. VNM doesn’t say that, Savage doesn’t say that, Bolker-Jeffrey doesn’t say that, Dutch Books don’t say that, Cox doesn’t say that, Complete Class doesn’t say that.
For the second argument (that the completeness axiom is suspicious): I think this is basically expressing the same sort of objection that I express here, particularly the section “There are no coherence arguments that say you must have preferences”. I didn’t tie it to the Completeness axiom because I think it’s a mistake to get bogged down in the details of the specific assumptions present in theorems when you can make the same point in English, but it is the same conceptual point, as far as I can tell.
I agree. I think the points that you make in that post are good.
For what it’s worth my position here is “you can’t argue for AI risk solely via coherence theorems; you also have to argue for why the AI will be goal-directed in the first place, but there are plausible arguments for that conclusion (which are not based on coherence arguments)”.
I agree with this too.
Two points, made in order of importance:
(1) How we define the term ‘coherence theorems’ doesn’t matter. What matters is that Premise 1 (striking out the word ‘coherence’, if you like) is false.
(2) The way I define the term ‘coherence theorems’ seems standard.
Now making point (1) in more detail:
Reserve the term ‘coherence theorems’ for whatever you like. Premise 1 is false: there are no theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. The VNM Theorem doesn’t say that, nor does Savage’s Theorem, nor does Bolker-Jeffrey, nor do Dutch Books, nor does Cox’s Theorem, nor does the Complete Class Theorem. That is the error in coherence arguments. Premise 1 is false.
Now for point (2):
I take the Appendix to make plausible enough that my use of the term ‘coherence theorems’ is standard, at least in online discussions. Here are some quotations.
1.
Now, by the general idea behind coherence theorems, since we can’t view this behavior as corresponding to expected utilities, we ought to be able to show that it corresponds to a dominated strategy somehow
2.
Roughly, the general claim these theorems make is that any system either (a) acts like an expected utility maximizer under some probabilistic model, or (b) throws away resources in a pareto-suboptimal manner.
3.
Summary: Violations of coherence constraints in probability theory and decision theory correspond to qualitatively destructive or dominated behaviors.
Again, we see a manifestation of a powerful family of theorems showing that agents which cannot be seen as corresponding to any coherent probabilities and consistent utility function will exhibit qualitatively destructive behavior
4.
One of the most pleasing things about probability and expected utility theory is that there are many coherence arguments that suggest that these are the “correct” ways to reason. If you deviate from what the theory prescribes, then you must be executing a dominated strategy.
5.
‘Coherence arguments’ mean that if you don’t maximize ‘expected utility’ (EU)—that is, if you don’t make every choice in accordance with what gets the highest average score, given consistent preferability scores that you assign to all outcomes—then you will make strictly worse choices by your own lights than if you followed some alternate EU-maximizing strategy (at least in some situations, though they may not arise). For instance, you’ll be vulnerable to ‘money-pumping’—being predictably parted from your money for nothing.
6.
The overall message here is that there is a set of qualitative behaviors and as long you do not engage in these qualitatively destructive behaviors, you will be behaving as if you have a utility function.
7.
I think that to contain the concept of Utility as it exists in me, you would have to do homework exercises I don’t know how to prescribe. Maybe one set of homework exercises like that would be showing you an agent, including a human, making some set of choices that allegedly couldn’t obey expected utility, and having you figure out how to pump money from that agent (or present it with money that it would pass up).
8.
The view that utility maximizers are inevitable is supported by a number of coherence theories developed early on in game theory which show that any agent without a consistent utility function is exploitable in some sense.
Maybe the term ‘coherence theorems’ gets used differently elsewhere. That is okay. See point (1).
Using ‘coherence theorems’ with a meaning that is as standard as any, and explaining that meaning within two sentences, seems fine to me.
I would have hoped you reached the second sentence before skimming! I define what I mean (and what I take previous authors to mean) by ‘coherence theorems’ there.
I cite John in the post!
I’m following previous authors in defining ‘coherence theorems’ as
theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy.
On that definition, there are no coherence theorems. VNM is not a coherence theorem, nor is Savage’s Theorem, nor is Bolker-Jeffrey, nor are Dutch Book Arguments, nor is Cox’s Theorem, nor is the Complete Class Theorem.
there are theorems that are relevant to the question of agent coherence
I’d have no problem with authors making that claim.
I would urge the author to create an actual concrete situation that doesn’t seem very dumb in which a highly intelligence, powerful and economically useful system has non-complete preferences
Working on it.
Sounds good!
I haven’t read this post yet, but it sounds like you might be interested in this paper on existential risks from a Thomist Christian perspective if you haven’t seen it already.
Nice post! Consider this a vote for more summaries.
All good points, but Tarsney’s argument doesn’t depend on the assumption that longtermist interventions cannot accidentally increase x-risk. It just depends on the assumption that there’s some way that we could spend $1 million that would increase the epistemic probability that humanity survives the next thousand years by at least 2x10^-14.
Thanks! This is valuable feedback.
By ‘persistent difference’, Tarsney doesn’t mean a difference that persists forever. He just means a difference that persists for a long time in expectation: long enough to make the expected value of the longtermist intervention greater than the expected value of the neartermist benchmark intervention.
Perhaps you want to know why we should think that we can make this kind of persistent difference. I can talk a little about that in another comment if so.
A point that seems worth noting, from Puzzles for Everyone:
In an especially striking example of conflating utilitarianism with anything remotely approaching systematic thinking, popular substacker Erik Hoel recently characterized the Beckstead & Thomas paper on decision-theoretic paradoxes as addressing “how poorly utilitarianism does in extreme scenarios of low probability but high impact payoffs.” Compare this with the very first sentence of the paper’s abstract: “We show that every theory of the value of uncertain prospects must have one of three unpalatable properties.” Not utilitarianism. Every theory.
(Alas, when I tried to point this out in the comments section, after a brief back-and-forth in which Erik initially doubled down on the conflation, he abruptly decided to instead delete my comments explaining his mistake.)
Yes, nice point. We could depart from the total view and go for a neutral band. But it’s worth noting that this move comes with problems of its own.
Great point. But note that if lives of monk-like tranquility are neutral, that makes the Mirrored Repugnant Conclusion harder to accept:
For any population of hellish lives, there is a population of barely bad lives that is worse.
The total view in population ethics implies this Mirrored Repugnant Conclusion.
If lives of monk-like tranquility are neutral, then lives of monk-like tranquility plus a mosquito bite are barely bad, and so the total view implies:
For any population of hellish lives, there is a population of lives of monk-like tranquility plus a mosquito bite that is worse.
I think in ordinary cases, necessitarianism ends up looking a lot like presentism. If someone presently exists, then they exist regardless of my choices. If someone doesn’t yet exist, their existence likely depends on my choices (there’s probably something I could do to prevent their existence).
Necessitarianism and presentism do differ in some contrived cases, though. For example, suppose I’m the last living creature on Earth, and I’m about to die. I can either leave the Earth pristine or wreck the environment. Some alien will soon be born far away and then travel to Earth. This alien’s life on Earth will be much better if I leave the Earth pristine. Presentism implies that it doesn’t matter whether I wreck the Earth, because the alien doesn’t exist yet. Necessitarianism implies that it would be bad to wreck the Earth, because the alien will exist regardless of what I do.
I’m just trying to fix that discrepancy.
I see. That seems like a good thing to do.
Here’s another good argument against person-affecting views that can be explained pretty simply, due to Tomi Francis.
Person-affecting views imply that it’s not good to add happy people. But Q is better than P, because Q is better for the hundred already-existing people, and the ten billion extra people in Q all live happy lives. And R is better than Q, because moving to R makes one hundred people’s lives slightly worse and ten billion people’s lives much better. Since betterness is transitive, R is better than P. R and P are identical except for the extra ten billion people living happy lives in R. Therefore, it’s good to add happy people, and person-affecting views are false.
Okay, that seems fair. And I agree that the Dutch book is a good argument against the person-affecting intuitions you lay out. But the argument only shows that people initially attracted to those person-affecting intuitions should move to a non-Dutch-bookable person-affecting view. If we want to move people away from person-affecting views entirely, we need other arguments.
The person-affecting views endorsed by philosophers these days are more complex than the families I listed. They’re not so intuitively ridiculous (though I think they still have problems. I have a couple of draft papers on this.).
Also a minor terminological note, you’ve called your argument a Dutch book and so have I. But I think it would be more standard to call it a money pump. Dutch books are a set of gambles all taken at once that are guaranteed to leave a person worse off. Money pumps are a set of trades taken one after the other that are guaranteed to leave a person worse off.
Yup!
Thanks for the comment! In this context, where we’re arguing about whether sufficiently-advanced artificial agents will satisfy the VNM axioms, I only have to give up Decision-Tree Separability*:
And Decision-Tree Separability* isn’t particularly plausible. It’s false if any sufficiently-advanced artificial agent acts in accordance with the following policy: ‘if I previously turned down some option X, I will not choose any option that I strictly disprefer to X.’ And it’s easy to see why agents might act in accordance with that policy: it makes them immune to money-pumps for Completeness.
Also, it seems as if one of the major downsides of resolute choice is that agents sometimes have to act against their preferences. But, as I argue in the post, artificial agents with incomplete preferences who act in accordance with the policy above will never have to act against their preferences.