Crossposting this comment from the EA Forum:
Nuno says:
I appreciate the whole post. But I personally really enjoyed the appendix. In particular, I found it informative that Yudkowsk can speak/write with that level of authoritativeness, confidence, and disdain for others who disagree, and still be wrong (if this post is right).
I respond:
(if this post is right)
The post does actually seem wrong though.
I expect someone to write a comment with the details at some point (I am pretty busy right now, so can only give a quick meta-level gleam), but mostly, I feel like in order to argue that something is wrong with these arguments is that you have to argue more compellingly against completeness and possible alternative ways to establish dutch-book arguments.
Also, the title of "there are no coherence arguments" is just straightforwardly wrong. The theorems cited are of course real theorems, they are relevant to agents acting with a certain kind of coherence, and I don't really understand the semantic argument that is happening where it's trying to say that the cited theorems aren't talking about "coherence", when like, they clearly are.
You can argue that the theorems ar...
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.
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.
While I agree that such theorems would count as coherence theorems, I wouldn't consider this to cover most things I think of as coherence theorems, and as such is simply a bad definition.
I think of coherence theorems loosely as things that say if an agent follows such and such principles, then we can prove it will have a certain property. The usefulness comes from both directions: to the extent the principles seem like good things to have, we're justified in assuming a certain property, and to the extent that the property seems too strong or whatever, then one of these principles will have to break.
I don't think the majority of the papers that you cite made the argument that coherence arguments prove that any sufficiently-advanced AI will be representable as maximizing expected utility. Indeed I am very confident almost everyone you cite does not believe this, since it is a very strong claim. Many of the quotes you give even explicitly say this:
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)
The emphasis here is important.
I don't think really any of the other quotes you cite make the strong claim you are arguing against. Indeed it is trivially easy to think of an extremely powerful AI that is VNM rational in all situations except for one tiny thing that does not matter or will never come up. Technically it's preferences can now not be represented by a utility function, but that's not very relevant to the core arguments at hand, and I feel like in your arguments you are trying to tear down some strawman of some extreme position that I don't think anyone holds.
Eliezer has also explicitly written about it being possible to design superintelligences that reflectively coherently believe in logical falsehoods. He thinks this is possible, just very difficult. That alone would also violate VNM rationality.
You misunderstand me (and I apologize for that. I now think I should have made this clear in the post). I’m arguing against the following weak claim:
And my argument is:
‘The relevant situations may not arise’ is a different objection. It’s not the one that I’m making.
(and 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)
Please see this old comment and this one.
Copying my response from the EA forum:
(if this post is right)
The post does actually seem wrong though.
Glad that I added the caveat.
Also, the title of "there are no coherence arguments" is just straightforwardly wrong. The theorems cited are of course real theorems, they are relevant to agents acting with a certain kind of coherence, and I don't really understand the semantic argument that is happening where it's trying to say that the cited theorems aren't talking about "coherence", when like, they clearly are.
Well, part of the semantic nuance is that we don't care as much about the coherence theorems that do exist if they will fail to apply to current and future machines
IMO completeness seems quite reasonable to me and the argument here seems very weak (and 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).
Here are some scenarios:
Well, part of the semantic nuance is that we don't care as much about the coherence theorems that do exist if they will fail to apply to current and future machines
The correct response to learning that some theorems do not apply as much to reality as you thought, surely mustn't be to change language so as to deny those theorems' existence. Insofar as this is what's going on, these are pretty bad norms of language in my opinion.
Also, the title of “there are no coherence arguments” is just straightforwardly wrong. The theorems cited are of course real theorems, they are relevant to agents acting with a certain kind of coherence, and I don’t really understand the semantic argument that is happening where it’s trying to say that the cited theorems aren’t talking about “coherence”, when like, they clearly are.
This seems wrong to me. The post’s argument is that the cited theorems aren’t talking about “coherence”, and it does indeed seem clear that (at least most of, possibly all but I could see disagreeing about maybe one or two) these theorems are not, in fact, talking about “coherence”.
Ngl kinda confused how these points imply the post seems wrong, the bulk of this seems to be (1) a semantic quibble + (2) a disagreement on who has the burden of proof when it comes to arguing about the plausibility of coherence + (3) maybe just misunderstanding the point that's being made?
(1) I agree the title is a bit needlessly provocative and in one sense of course VNM/Savage etc count as coherence theorems. But the point is that there is another sense that people use "coherence theorem/argument" in this field which corresponds to something like "If you're not behaving like an EV-maximiser you're shooting yourself in the foot by your own lights", which is what brings in all the scary normativity and is what the OP is saying doesn't follow from any existing theorem unless you make a bunch of other assumptions
(2) The only real substantive objection to the content here seems to be "IMO completeness seems quite reasonable to me". Why? Having complete preferences seems like a pretty narrow target within the space of all partial orders you could have as your preference relation, so what's the reason why we should expect minds to steer towards this? Do humans have complete preferences...
I think most democratic countries use proportional representation, not FTPT. But talking about "most" is an FTPT error. Enough countries use proportional representation that you can study the effect of voting systems. And the results are shocking to me. The theoretical predictions are completely wrong. Duverger's law is false in every FTPT country except America. On the flip side, while PR does lead to more parties, they still form 1-dimensional spectrum. For example, a Green Party is usually a far-left party with slightly different preferences, instead of a single issue party that is willing to form coalitions with the right.
If politics were two dimensional, why wouldn't you expect Condorcet cycles? Why would population get rid of them? If you have two candidates, a tie between them is on a razor's edge. The larger the population of voters, the less likely. But if you have three candidates and three roughly equally common preferences, the cyclic shifts of A > B > C, then this is a robust tie. You only get a Condorcet winner when one of the factions becomes as big as the other two combined. Of course I have assumed away the other three preferences, but this is robust to them ...
It seems like you deliberately picked completeness because that's where Dutch book arguments are least compelling, and that you'd agree with the more usual Dutch book arguments.
But I think even the Dutch book for completeness makes some sense. You just have to separate "how the agent internally represents its preferences" from "what it looks like the agent us doing." You describe an agent that dodges the money-pump by simply acting consistently with past choices. Internally this agent has an incomplete representation of preferences, plus a memory. But externally it looks like this agent is acting like it assigns equal value to whatever indifferent things it thought of choosing between first. If humans don't get to control the order this agent considers options, or if we let it run for a long time and it's already experienced the things humans might try to present to it from them on, then it will look like it's acting according to complete preferences.
It feels like this post starts with a definition of "coherence theorem", sees that the so-called coherence theorems don't match this definition, and thus criticizes the use of the term "coherence theorem".
But this claimed definition of "coherence theorem" seems bad to me, and is not how I would use the phrase. Eliezer's definition, OTOH is:
If you are not shooting yourself in the foot in sense X, we can view you as having coherence property Y.
which seems perfectly fine to me. It's significant that this isn't completely formalized, and requires intuitive judgement as to what constitutes "shooting yourself in the foot".
Which makes the criticism feel unwarranted, or at best misdirected.
The point is: 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.
But suppose we instead define 'coherence theorems' as theorems which state that
If you are not shooting yourself in the foot in sense X, we can view you as having coherence property Y.
Then you can fill in X and Y any way you like. Either it will turn out that there are no coherence theorems, or it will turn out that coherence theorems cannot play the role they're supposed to play in coherence arguments.
It is not fine if, whichever way you interpret some premise, either:
(1) the premise comes out false.
Or:
(2) the premise does not support the conclusion.
Reserve the term ‘coherence theorems’ for whatever rough category you like. ‘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’ refers to a precise category of non-existent things.
The title "There are no coherence theorems" seems click-baity to me, when the claim relies on a very particular definition "coherence theorem". My thought upon reading the title (before reading the post) was something like "surely, VNM would count as a coherence theorem". I am also a bit bothered by the confident assertions that there are no coherence theorems in the Conclusion and Bottom-lines for similar reason.
I have now seen this post cited in other spaces, so I am taking the time to go back and write out why I do not think it holds water.
I do not find the argument against the applicability of the Complete Class theorem convincing.
See Charlie Steiner's comment:
You just have to separate "how the agent internally represents its preferences" from "what it looks like the agent us doing." You describe an agent that dodges the money-pump by simply acting consistently with past choices. Internally this agent has an incomplete representation of preferences, plus a memory. But externally it looks like this agent is acting like it assigns equal value to whatever indifferent things it thought of choosing between first.
Decision theory is concerned with external behaviour, not internal representations. All of these theorems are talking about whether the agent's actions can be consistently described as maximising a utility function. They are not concerned whatsoever with how the agent actually mechanically represents and thinks about its preferences and actions on the inside. To decision theory, agents are black boxes. Information goes in, decision comes out. Whatever processes may go on in bet...
Thanks, Lucius. Whether or not decision theory as a whole is concerned only with external behaviour, coherence arguments certainly aren’t. Remember what the conclusion of these arguments is supposed to be: advanced agents who start off not being representable as EUMs will amend their behaviour so that they are representable as EUMs, because otherwise they’re liable to pursue dominated strategies.
Now consider an advanced agent who appears not to be representable as an EUM: it’s paying to trade vanilla for strawberry, strawberry for chocolate, and chocolate for vanilla. Is this agent pursuing a dominated strategy? Will it amend its behaviour? It depends on the objects of preference. If objects of preference are ice-cream flavours, the answer is yes. If the objects of preference are sequences of trades, the answer is no. So we have to say something about the objects of preference in order to predict the agent’s behaviour. And the whole point of coherence arguments is to predict agents’ behaviour.
And once we say something about the objects of preference, then we can observe agents violating Completeness and acting in accordance with policies like ‘if I previously turned down some optio...
Note that you can still get EUM-like properties without completeness: you just can't use a single fully-fleshed-out utility function. You need either several utility functions (that is, your system is made of subagents) or, equivalently, a utility function that is not completely defined (that is, your system has Knightian uncertainty over its utility function).
See Knightian Decision Theory. Part I
Arguably humans ourselves are better modeled as agents with incomplete preferences. See also Why Subagents?
I agree with habryka that the title of this post is a little pedantic and might just be inaccurate, but I nevertheless found the content to be thought-provoking, easy to follow, and well written.
I actually also think the post makes some good points. I think arguing against completeness is a pretty good thing to do, and an approach with a long history of people thinking about the theory of rational agents. I feel like this particular posts's arguments against completeness are not amazing, but they seem like a decent contribution. I just wish it didn't have all the other stuff on how "everyone who ever referenced 'coherence theorems' is making a mistake".
Some different (I think) points against arguments related to the ones that are rebutted in the post:
Money-pump arguments don’t give us much reason to expect that advanced artificial agents will be representable as expected-utility-maximizers.
The space of agents is large; EU maximizers may be a simple, natural subset of all possible agents.
Given any EU maximizer, you can construct a new, more complicated agent which has a preferential gap about something trivial. This new agent will (by VNM) not be an EU maximizer.
Similarly, given an agent with incomplete preferences that satisfies the other axioms, you can (always? trivially??) construct an agent with co...
Great post. I think a lot of the discussion around the role of coherence arguments and what we should expect a super-intelligent agent to behave like is really sloppy and I think this distinction between "coherence theorems as a self-contained mathematical result" and "coherence arguments as a normative claim about what an agent must be like on pain of shooting themselves in the foot" is an important one
The example of how an incomplete agent avoids getting Dutch-booked also seems to look very naturally like how irl agents behave imo. One way of thinking ab...
Coming back to this, the policy
if I previously turned down some option X, I will not choose any option that I strictly disprefer to X
seems irrational to me if applied in general. Suppose I offer you and , where both and are random, and is ex ante preferable to , e.g. stochastically dominates , but has some chance of being worse than . You pick . Then you evaluate to get . However, suppose you get unlucky, and is worse than . Suppose f...
EDIT: Looks like a similar point made here.
I wonder if we can "extend" utility maximization representation theorems to drop Completeness. There's already an extension to drop Continuity by using an ordinal-indexed vector (sequence) of real numbers, with entries sorted lexicographically ("lexicographically ordered ordinal sequences of bounded real utilities", Russell and Isaacs, 2020). If we drop Completeness, maybe we can still represent the order with a vector of independent but incomparable dimensions across which it must respect ex ante Pareto eff...
Strongly upvoted.
Humans at least do not satisfy completeness/don't admit a total order over their preferences.
See also:
The post argues a lot against completeness. I have a hard time imagining an advanced AGI (which has the ability to self-reflect a lot) that has a lot of preferences, but no complete preferences.
Your argument seems to be something like "There can be outcomes A and B where neither nor . This property can be preserved if we sweeten A a little bit: then we have but neither nor . If faced with a decision between A and B (or faced with a choice between ), the AGI can do something arbitrary, eg just flip a coin."
I expect advanced AGI syste...
To the extent that humans are general intelligences and have incomplete preferences (for ex. preferential gaps), it seems apparently possible and imaginable to have a generally-intelligent agent with incomplete preferences.
The post argues a lot against completeness. I have a hard time imagining an advanced AGI (which has the ability to self-reflect a lot) that has a lot of preferences, but no complete preferences.
A couple of relevant quotes:
Of all the axioms of utility theory, the completeness axiom is perhaps the most questionable.[8] Like others of the axioms, it is inaccurate as a description of real life; but unlike them, we find it hard to accept even from the normative viewpoint.
...Before stating more carefully our goal and the contribution thereof, let us note that there are several economic reasons why one would like to study incomplete preference relations. First of all, as advanced by several authors in the literature, it is not evident if completeness is a fundamental rationality tenet the way the transitivity property is. Aumann (1962), Bewley (1986) and Mandler (1999), among others, defend this position very strongly from both the normative and positive viewpoints. Indeed, if one takes the psychological preference approach (which derives choices from preferences), and not the revealed preference approach, it seems natural to define a preference relation as a potentiall
I have only skimmed this and am planning a careful read, but my first impression is that Wald's complete class theorem is exactly the desired coherence theorem. Complete preferences seem like the weakest possible assumption one could make and still expect to show non-Bayesian decision rules are dominated. If you want to show that "want something -> should use Bayesian decision theory to get it" there is no need to argue about the premise that an agent wants something, and to me wanting something coherently requires complete preferences.
The LessWrong Review runs every year to select the posts that have most stood the test of time. This post is not yet eligible for review, but will be at the end of 2024. The top fifty or so posts are featured prominently on the site throughout the year.
Hopefully, the review is better than karma at judging enduring value. If we have accurate prediction markets on the review results, maybe we can have better incentives on LessWrong today. Will this post make the top fifty?
More generally, suppose that the agent acts in accordance with the following policy in all decision-situations: ‘if I previously turned down some option X, I will not choose any option that I strictly disprefer to X.’ That policy makes the agent immune to all possible money-pumps for Completeness.
Am I missing something or does this agent satisfy Completeness anytime it faces a decision for the second time?
If the agent doesn’t know in advance what trades they will be offered or is incapable of backward induction, then their pursuit of a dominated strategy need not indicate any defect in their preferences. Their pursuit of a dominated strategy can instead be blamed on their lack of knowledge and/or reasoning ability.
But then wouldn't your proposed policy be dominated by choosing to be indifferent between options with gap, because it works better without knowing trades in advance, and doesn't work worse otherwise?
The arguments typically require agents to make decisions independently of the parts of the decision tree in the past (or that are otherwise no longer accessible, in case they were ruled out). But an agent need not do that. An agent can always avoid getting money pumped by just following the policy of never picking an option that completes a money pump (or the policy of never making any trades, say). They can even do this with preference cycles.
Does this mean money pump arguments don't tell us anything? Such a policy may have other costs that an agent would...
In "A money-pump for Completeness" you say "by the transitivity of strict preference"
This only says that transitive preferences do not need to be complete which is weaker than preferences do not need to be complete.
Looking at Gustafsson, 2022's money pumps for completeness, the precaution principles he uses just seem pretty unintuitive to me. The idea seems to be that if you'll later face a decision situation where you can make a choice that makes you worse off but you can't make yourself better off by getting there, you should avoid the decision situation, even if it's entirely under your control to make a choice in that situation that won't leave you worse off. But, you can just make that choice that won't leave you worse off later instead of avoiding the situation...
The author doesn't seem to realize that there's a difference between representation theorems and coherence theorems.
...The Complete Class Theorem says that an agent’s policy of choosing actions conditional on observations is not strictly dominated by some other policy (such that the other policy does better in some set of circumstances and worse in no set of circumstances) if and only if the agent’s policy maximizes expected utility with respect to a probability distribution that assigns positive probability to each possible set of circumstances.
This theorem
These arguments don't work.
You've mistaken acyclicity for transitivity. The money-pump establishes only acyclicity. Representability-as-an-expected-utility-maximizer requires transitivity.
As I note in the post, agents can make themselves immune to all possible money-pumps for completeness by acting 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.’ Acting in accordance with this policy need never require an agent to act against any of their preferences.
Your coherence conjecture sounds good! It sounds like it roughly matches this theorem:
Screenshot is from this paper.
And this avoids the Complete Class Theorem conclusion of dominated strategies, how?
The Complete Class Theorem assumes that the agent’s preferences are complete. If the agent’s preferences are incomplete, the theorem doesn’t apply. So, you have to try to get Completeness some other way.
You might try to get Completeness via some money-pump argument, but these arguments aren’t particularly convincing. Agents can make themselves immune to all possible money-pumps for Completeness by acting 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.’
Again, we care about domination, not representability at all.
Can you expand on this a little more? Agents cannot be (or appear to be) expected utility maximizers unless they are representable as expected utility maximizers, so if we care about whether agents will be (or will appear to be) expected utility maximizers, we have to care about whether they will be representable as expected utility maximizers.
In the limit
You’re pushing towards the wrong limit. A rock can be represented as indifferent between all options and hence as having complete preferences.
As I explain in the post, an agent’s preferences are incomplete if and only if they have a preferential gap between some pair of options, and an agent has a preferential gap between two options A and B if and only if they lack any strict preference between A and B and this lack of strict preference is insensitive to some sweetening or souring (such that, e.g., they strictly prefer A to A- and yet have no strict preferences either way between A and B, and between A- and B).
Spell it out more concretely
Sure. Imagine an agent as powerful and future-steering as you like. Among its options are A, A-, and B: the agent strictly prefers A to A-, and has a preferential gap between A and B, and between A- and B. Its preferences are incomplete, so the Complete Class Theorem doesn’t apply.
[Suppose that you tried to use the proof of the Complete Class Theorem to prove that this agent would pursue a dominated strategy. Here’s why that won’t work:
[Written by EJT as part of the CAIS Philosophy Fellowship. Thanks to Dan for help posting to the Alignment Forum]
Introduction
For about fifteen years, the AI safety community has been discussing coherence arguments. In papers and posts on the subject, it’s often written that there exist 'coherence 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. Despite the prominence of these arguments, authors are often a little hazy about exactly which theorems qualify as coherence theorems. This is no accident. If the authors had tried to be precise, they would have discovered that there are no such theorems.
I’m concerned about this. Coherence arguments seem to be a moderately important part of the basic case for existential risk from AI. To spot the error in these arguments, we only have to look up what cited ‘coherence theorems’ actually say. And yet the error seems to have gone uncorrected for more than a decade.
More detail below.^{[1]}
Coherence arguments
Some authors frame coherence arguments in terms of ‘dominated strategies’. Others frame them in terms of ‘exploitation’, ‘money-pumping’, ‘Dutch Books’, ‘shooting oneself in the foot’, ‘Pareto-suboptimal behavior’, and ‘losing things that one values’ (see the Appendix for examples).
In the context of coherence arguments, each of these terms means roughly the same thing: a strategy A is dominated by a strategy B if and only if A is worse than B in some respect that the agent cares about and A is not better than B in any respect that the agent cares about. If the agent chooses A over B, they have behaved Pareto-suboptimally, shot themselves in the foot, and lost something that they value. If the agent’s loss is someone else’s gain, then the agent has been exploited, money-pumped, or Dutch-booked. Since all these phrases point to the same sort of phenomenon, I’ll save words by talking mainly in terms of ‘dominated strategies’.
With that background, here’s a quick rendition of coherence arguments:
Typically, authors go on to suggest that these expected-utility-maximizing agents are likely to behave in certain, potentially-dangerous ways. For example, such agents are likely to appear ‘goal-directed’ in some intuitive sense. They are likely to have certain instrumental goals, like acquiring power and resources. And they are likely to fight back against attempts to shut them down or modify their goals.
There are many ways to challenge the argument stated above, and many of those challenges have been made. There are also many ways to respond to those challenges, and many of those responses have been made too. The challenge that seems to remain yet unmade is that Premise 1 is false: there are no coherence theorems.
Cited ‘coherence theorems’ and what they actually say
Here’s a list of theorems that have been called ‘coherence theorems’. None of these theorems state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue dominated strategies. Here’s what the theorems say:
The Von Neumann-Morgenstern Expected Utility Theorem:
The Von Neumann-Morgenstern Expected Utility Theorem is as follows:
Note that this theorem makes no reference to dominated strategies, vulnerabilities, exploitation, or anything of that sort.
Some authors (both inside and outside the AI safety community) have tried to defend some or all of the axioms above using money-pump arguments. These are arguments with conclusions of the following form: ‘agents who fail to satisfy Axiom A can be induced to make a set of trades or bets that leave them worse-off in some respect that they care about and better-off in no respect, even when they know in advance all the trades and bets that they will be offered.’ Authors then use that conclusion to support a further claim. Outside the AI safety community, the claim is often:
But inside the AI safety community, the claim is:
This difference will be important below. For now, the important thing to note is that the conclusions of money-pump arguments are not theorems. Theorems (like the VNM Theorem) can be proved without making any substantive assumptions. Money-pump arguments establish their conclusion only by making substantive assumptions: assumptions that might well be false. In the section titled ‘A money-pump for Completeness’, I will discuss an assumption that is both crucial to money-pump arguments and likely false.
Savage’s Theorem
Savage’s Theorem is also a Von-Neumann-Morgenstern-style representation theorem. It also says that an agent can be represented as maximizing expected utility if and only if their preferences satisfy a certain set of axioms. The key difference between Savage’s Theorem and the VNM Theorem is that the VNM Theorem takes the agent’s probability function as given, whereas Savage constructs the agent’s probability function from their preferences over lotteries.
As with the VNM Theorem, Savage’s Theorem says nothing about dominated strategies or vulnerability to exploitation.
The Bolker-Jeffrey Theorem
This theorem is also a representation theorem, in the mould of the VNM Theorem and Savage’s Theorem above. It makes no reference to dominated strategies or anything of that sort.
Dutch Books
The Dutch Book Argument for Probabilism says:
The Dutch Book Argument for Conditionalization says:
These arguments do refer to dominated strategies and vulnerability to exploitation. But they suggest only that an agent’s credences (that is, their degrees of belief) must meet certain conditions. Dutch Book Arguments place no constraints whatsoever on an agent’s preferences. And if an agent’s preferences fail to satisfy any of Completeness, Transitivity, Independence, and Continuity, that agent cannot be represented as maximizing expected utility (the VNM Theorem is an ‘if and only if’, not just an ‘if’).
Cox’s Theorem
Cox’s Theorem says that, if an agent’s degrees of belief satisfy a certain set of axioms, then their beliefs are isomorphic to probabilities.
This theorem makes no reference to dominated strategies, and it says nothing about an agent’s preferences.
The Complete Class Theorem
The Complete Class Theorem says that an agent’s policy of choosing actions conditional on observations is not strictly dominated by some other policy (such that the other policy does better in some set of circumstances and worse in no set of circumstances) if and only if the agent’s policy maximizes expected utility with respect to a probability distribution that assigns positive probability to each possible set of circumstances.
This theorem does refer to dominated strategies. However, the Complete Class Theorem starts off by assuming that the agent’s preferences over actions in sets of circumstances satisfy Completeness and Transitivity. If the agent’s preferences are not complete and transitive, the Complete Class Theorem does not apply. So, the Complete Class Theorem does not imply that agents must be representable as maximizing expected utility if they are to avoid pursuing dominated strategies.
Omohundro (2007), ‘The Nature of Self-Improving Artificial Intelligence’
This paper seems to be the original source of the claim that agents are vulnerable to exploitation unless they can be represented as expected-utility-maximizers. Omohundro purports to give us “the celebrated expected utility theorem of von Neumann and Morgenstern… derived from a lack of vulnerabilities rather than from given axioms.”
Omohundro’s first error is to ignore Completeness. That leads him to mistake acyclicity for transitivity, and to think that any transitive relation is a total order. Note that this error already sinks any hope of getting an expected-utility-maximizer out of Omohundro’s argument. Completeness (recall) is a necessary condition for being representable as an expected-utility-maximizer. If there’s no money-pump that compels Completeness, there’s no money-pump that compels expected-utility-maximization.
Omohundro’s second error is to ignore Continuity. His ‘Argument for choice with objective uncertainty’ is too quick to make much sense of. Omohundro says it’s a simpler variant of Green (1987). The problem is that Green assumes every axiom of the VNM Theorem except Independence. He says so at the bottom of page 789. And, even then, Green notes that his paper provides “only a qualified bolstering” of the argument for Independence.
Money-Pump Arguments by Johan Gustafsson
It’s worth noting that there has recently appeared a book which gives money-pump arguments for each of the axioms of the VNM Theorem. It’s by the philosopher Johan Gustafsson and you can read it here.
This does not mean that the posts and papers claiming the existence of coherence theorems are correct after all. Gustafsson’s book was published in 2022, long after most of the posts on coherence theorems. Gustafsson argues that the VNM axioms are requirements of rationality, whereas coherence arguments aim to establish that sufficiently-advanced artificial agents will satisfy the VNM axioms. More importantly (and as noted above) the conclusions of money-pump arguments are not theorems. Theorems (like the VNM Theorem) can be proved without making any substantive assumptions. Money-pump arguments establish their conclusion only by making substantive assumptions: assumptions that might well be false.
I will now explain how denying one such assumption allows us to resist Gustafsson’s money-pump arguments. I will then argue that there can be no compelling money-pump arguments for the conclusion that sufficiently-advanced artificial agents will satisfy the VNM axioms.
Before that, though, let’s get the lay of the land. Recall that Completeness is necessary for representability as an expected-utility-maximizer. If an agent’s preferences are incomplete, that agent cannot be represented as maximizing expected utility. Note also that Gustafsson’s money-pump arguments for the other axioms of the VNM Theorem depend on Completeness. As he writes in a footnote on page 3, his money-pump arguments for Transitivity, Independence, and Continuity all assume that the agent’s preferences are complete. That makes Completeness doubly important to the ‘money-pump arguments for expected-utility-maximization’ project. If an agent’s preferences are incomplete, then they can’t be represented as an expected-utility-maximizer, and they can’t be compelled by Gustafsson’s money-pump arguments to conform their preferences to the other axioms of the VNM Theorem. (Perhaps some earlier, less careful money-pump argument can compel conformity to the other VNM axioms without assuming Completeness, but I think it unlikely.)
So, Completeness is crucial. But one might well think that we don’t need a money-pump argument to establish it. I’ll now explain why this thought is incorrect, and then we’ll look at a money-pump.
Completeness doesn’t come for free
Here’s Completeness again:
Since:
And:
Completeness can be rephrased as:
And then you might think that Completeness comes for free. After all, what other comparative, preference-style attitude can an agent have to X and Y?
This thought might seem especially appealing if you think of preferences as nothing more than dispositions to choose. Suppose that our agent is offered repeated choices between X and Y. Then (the thought goes), in each of these situations, they have to choose something. If they reliably choose X over Y, then they strictly prefer X to Y. If they reliably choose Y over X, then they strictly prefer Y to X. If they flip a coin, or if they sometimes choose X and sometimes choose Y, then they are indifferent between X and Y.
Here’s the important point missing from this thought: there are two ways of failing to have a strict preference between X and Y. Being indifferent between X and Y is one way: preferring X at least as much as Y and preferring Y at least as much as X. Having a preferential gap between X and Y is another way: not preferring X at least as much as Y and not preferring Y at least as much as X. If an agent has a preferential gap between any two lotteries, then their preferences violate Completeness.
The key contrast between indifference and preferential gaps is that indifference is sensitive to all sweetenings and sourings. Consider an example. C is a lottery that gives the agent a pot of ten dollar-bills for sure. D is a lottery that gives the agent a different pot of ten dollar-bills for sure. The agent does not strictly prefer C to D and does not strictly prefer D to C. How do we determine whether the agent is indifferent between C and D or whether the agent has a preferential gap between C and D? We sweeten one of the lotteries: we make that lottery just a little but more attractive. In the example, we add an extra dollar-bill to pot C, so that it contains $11 total. Call the resulting lottery C+. The agent will strictly prefer C+ to D. We get the converse effect if we sour lottery C, by removing a dollar-bill from the pot so that it contains $9 total. Call the resulting lottery C-. The agent will strictly prefer D to C-. And we also get strict preferences by sweetening and souring D, to get D+ and D- respectively. The agent will strictly prefer D+ to C and strictly prefer C to D-. Since the agent’s preference-relation between C and D is sensitive to all such sweetenings and sourings, the agent is indifferent between C and D.
Preferential gaps, by contrast, are insensitive to some sweetenings and sourings. Consider another example. A is a lottery that gives the agent a Fabergé egg for sure. B is a lottery that returns to the agent their long-lost wedding album. The agent does not strictly prefer A to B and does not strictly prefer B to A. How do we determine whether the agent is indifferent or whether they have a preferential gap? Again, we sweeten one of the lotteries. A+ is a lottery that gives the agent a Fabergé egg plus a dollar-bill for sure. In this case, the agent might not strictly prefer A+ to B. That extra dollar-bill might not suffice to break the tie. If that is so, the agent has a preferential gap between A and B. If the agent has a preferential gap, then slightly souring A to get A- might also fail to break the tie, as might slightly sweetening and souring B to get B+ and B- respectively.
The axiom of Completeness rules out preferential gaps, and so rules out insensitivity to some sweetenings and sourings. That is why Completeness does not come for free. We need some argument for thinking that agents will not have preferential gaps. ‘The agent has to choose something’ is a bad argument. Faced with a choice between two lotteries, the agent might choose arbitrarily, but that does not imply that the agent is indifferent between the two lotteries. The agent might instead have a preferential gap. It depends on whether the agent’s preference-relation is sensitive to all sweetenings and sourings.
A money-pump for Completeness
So, we need some other argument for thinking that sufficiently-advanced artificial agents’ preferences over lotteries will be complete (and hence will be sensitive to all sweetenings and sourings). Let’s look at a money-pump. I will later explain how my responses to this money-pump also tell against other money-pump arguments for Completeness.
Here's the money-pump, suggested by Ruth Chang (1997, p.11) and later discussed by Gustafsson (2022, p.26):
‘≻’ denotes strict preference and ‘||’ denotes a preferential gap, so the symbols underneath the decision tree say that the agent strictly prefers A to A- and has a preferential gap between A- and B, and between B and A.
Now suppose that the agent finds themselves at the beginning of this decision tree. Since the agent doesn’t strictly prefer A to B, they might choose to go up at node 1. And since the agent doesn’t strictly prefer B to A-, they might choose to go up at node 2. But if the agent goes up at both nodes, they have pursued a dominated strategy: they have made a set of trades that left them with A- when they could have had A (an outcome that they strictly prefer), even though they knew in advance all the trades that they would be offered.
Note, however, that this money-pump is non-forcing: at some step in the decision tree, the agent is not compelled by their preferences to pursue a dominated strategy. The agent would not be acting against their preferences if they chose to go down at node 1 or at node 2. And if they went down at either node, they would not pursue a dominated strategy.
To avoid even a chance of pursuing a dominated strategy, we need only suppose that the agent acts in accordance with the following policy: ‘if I go up at node 1, I will go down at node 2.’ Since the agent does not strictly prefer A- to B, acting in accordance with this policy does not require the agent to change or act against any of their preferences.
More generally, suppose that the agent acts in accordance with the following policy in all decision-situations: ‘if I previously turned down some option X, I will not choose any option that I strictly disprefer to X.’^{ }That policy makes the agent immune to all possible money-pumps for Completeness.^{[2]} And (granted some assumptions), the policy never requires the agent to change or act against any of their preferences.
Here’s why. Assume:
(If the agent doesn’t know in advance what trades they will be offered or is incapable of backward induction, then their pursuit of a dominated strategy need not indicate any defect in their preferences. Their pursuit of a dominated strategy can instead be blamed on their lack of knowledge and/or reasoning ability.)
Given the agent’s knowledge of the decision tree and their grasp of backward induction, we can infer that, if the agent proceeds to node 2, then at least one of the possible outcomes of going to node 2 is not strictly dispreferred to any option available at node 1. Then, if the agent proceeds to node 2, they can act on a policy of not choosing any outcome that is strictly dispreferred to some option available at node 1. The agent’s acting on this policy will not require them to act against any of their preferences. For suppose that it did require them to act against some strict preference. Suppose that B is strictly dispreferred to A, so that the agent’s policy requires them to choose C, and yet C is strictly dispreferred to B. Then, by the transitivity of strict preference, C is strictly dispreferred to A. That means that both B and C are strictly dispreferred to A, contrary to our original assumption that at least one of the possible outcomes of going to node 2 is not strictly dispreferred to any option available at node 1. We have reached a contradiction, and so we can reject the assumption that the agent’s policy will require them to act against their preferences. This proof is easy to generalize so that it applies to decision trees with more than three terminal outcomes.
Summarizing this section
Money-pump arguments for Completeness (understood as the claim that sufficiently-advanced artificial agents will have complete preferences) assume that such agents will not act in accordance with policies like ‘if I previously turned down some option X, I will not choose any option that I strictly disprefer to X.’ But that assumption is doubtful. Agents with incomplete preferences have good reasons to act in accordance with this kind of policy: (1) it never requires them to change or act against their preferences, and (2) it makes them immune to all possible money-pumps for Completeness.
So, the money-pump arguments for Completeness are unsuccessful: they don’t give us much reason to expect that sufficiently-advanced artificial agents will have complete preferences. Any agent with incomplete preferences cannot be represented as an expected-utility-maximizer. So, money-pump arguments don’t give us much reason to expect that sufficiently-advanced artificial agents will be representable as expected-utility-maximizers.
Conclusion
There are no coherence theorems. Authors in the AI safety community should stop suggesting that there are.
There are money-pump arguments, but the conclusions of these arguments are not theorems. The arguments depend on substantive and doubtful assumptions.
Here is one doubtful assumption: advanced artificial agents with incomplete preferences will not act 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.’ Any agent who acts in accordance with that policy is immune to all possible money-pumps for Completeness. And agents with incomplete preferences cannot be represented as expected-utility-maximizers.
In fact, the situation is worse than this. As Gustafsson notes, his money-pump arguments for the other three axioms of the VNM Theorem depend on Completeness. If Gustafsson’s money-pump arguments fail without Completeness, I suspect that earlier, less-careful money-pump arguments for the other axioms of the VNM Theorem fail too. If that’s right, and if Completeness is false, then none of Transitivity, Independence, and Continuity has been established by money-pump arguments either.
Bottom-lines
Appendix: Papers and posts in which the error occurs
Here’s a selection of papers and posts which claim that there are coherence theorems.
‘The nature of self-improving artificial intelligence’
“The appendix shows how the rational economic structure arises in each of these situations. Most presentations of this theory follow an axiomatic approach and are complex and lengthy. The version presented in the appendix is based solely on avoiding vulnerabilities and tries to make clear the intuitive essence of the argument.”
“In each case we show that if an agent is to avoid vulnerabilities, its preferences must be representable by a utility function and its choices obtained by maximizing the expected utility.”
‘The basic AI drives’
“The remarkable “expected utility” theorem of microeconomics says that it is always possible for a system to represent its preferences by the expectation of a utility function unless the system has “vulnerabilities” which cause it to lose resources without benefit.”
‘Coherent decisions imply consistent utilities’
“It turns out that this is just one instance of a large family of coherence theorems which all end up pointing at the same set of core properties. All roads lead to Rome, and all the roads say, "If you are not shooting yourself in the foot in sense X, we can view you as having coherence property Y."”
“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—derive some way in which this behavior corresponds to shooting off your own foot.”
“And that's at least a glimpse of why, if you're not using dominated strategies, the thing you do with relative utilities is multiply them by probabilities in a consistent way, and prefer the choice that leads to a greater expectation of the variable representing utility.”
“The demonstrations we've walked through here aren't the professional-grade coherence theorems as they appear in real math. Those have names like "Cox's Theorem" or "the complete class theorem"; their proofs are difficult; and they say things like "If seeing piece of information A followed by piece of information B leads you into the same epistemic state as seeing piece of information B followed by piece of information A, plus some other assumptions, I can show an isomorphism between those epistemic states and classical probabilities" or "Any decision rule for taking different actions depending on your observations either corresponds to Bayesian updating given some prior, or else is strictly dominated by some Bayesian strategy".”
“But hopefully you've seen enough concrete demonstrations to get a general idea of what's going on with the actual coherence theorems. We have multiple spotlights all shining on the same core mathematical structure, saying dozens of different variants on, "If you aren't running around in circles or stepping on your own feet or wantonly giving up things you say you want, we can see your behavior as corresponding to this shape. Conversely, if we can't see your behavior as corresponding to this shape, you must be visibly shooting yourself in the foot." Expected utility is the only structure that has this great big family of discovered theorems all saying that. It has a scattering of academic competitors, because academia is academia, but the competitors don't have anything like that mass of spotlights all pointing in the same direction.”
‘Things To Take Away From The Essay’
“So what are the primary coherence theorems, and how do they differ from VNM? Yudkowsky mentions the complete class theorem in the post, Savage's theorem comes up in the comments, and there are variations on these two and probably others as well. 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. One thing to emphasize: these theorems generally do not assume any pre-existing probabilities (as VNM does); an agent's implied probabilities are instead derived. Yudkowsky's essay does a good job communicating these concepts, but doesn't emphasize that this is different from VNM.”
‘Sufficiently optimized agents appear coherent’
“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, like paying someone a cent to throw a switch and then paying them another cent to throw it back.”
“There is a large literature on different sets of coherence constraints that all yield expected utility, starting with the Von Neumann-Morgenstern Theorem. No other decision formalism has comparable support from so many families of differently phrased coherence constraints.”
‘What do coherence arguments imply about the behavior of advanced AI?’
“Coherence arguments say that if an entity’s preferences do not adhere to the axioms of expected utility theory, then that entity is susceptible to losing things that it values.”
Disclaimer: “This is an initial page, in the process of review, which may not be comprehensive or represent the best available understanding.”
‘Coherence theorems’
“In the context of decision theory, "coherence theorems" are theorems saying that an agent's beliefs or behavior must be viewable as consistent in way X, or else penalty Y happens.”
Disclaimer: “This page's quality has not been assessed.”
“Extremely incomplete list of some coherence theorems in decision theory
‘Coherence arguments do not entail goal-directed behavior’
“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. There must be some other strategy that never does any worse than your strategy, but does strictly better than your strategy with certainty in at least one situation. There’s a good explanation of these arguments here.”
“The VNM axioms are often justified on the basis that if you don't follow them, you can be Dutch-booked: you can be presented with a series of situations where you are guaranteed to lose utility relative to what you could have done. So on this view, we have "no Dutch booking" implies "VNM axioms" implies "AI risk".”
‘Coherence arguments imply a force for goal-directed behavior.’
“‘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.^{3}”
‘AI Alignment: Why It’s Hard, and Where to Start’
“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.”
‘Money-pumping: the axiomatic approach’
“This post gets somewhat technical and mathematical, but the point can be summarised as:
In other words, using alternate decision theories is bad for your wealth.”
‘Ngo and Yudkowsky on alignment difficulty’
“Except that to do the exercises at all, you need them to work within an expected utility framework. And then they just go, "Oh, well, I'll just build an agent that's good at optimizing things but doesn't use these explicit expected utilities that are the source of the problem!"
And then if I want them to believe the same things I do, for the same reasons I do, I would have to teach them why certain structures of cognition are the parts of the agent that are good at stuff and do the work, rather than them being this particular formal thing that they learned for manipulating meaningless numbers as opposed to real-world apples.
And I have tried to write that page once or twice (eg "coherent decisions imply consistent utilities") but it has not sufficed to teach them, because they did not even do as many homework problems as I did, let alone the greater number they'd have to do because this is in fact a place where I have a particular talent.”
“In this case the higher structure I'm talking about is Utility, and doing homework with coherence theorems leads you to appreciate that we only know about one higher structure for this class of problems that has a dozen mathematical spotlights pointing at it saying "look here", even though people have occasionally looked for alternatives.
And when I try to say this, people are like, "Well, I looked up a theorem, and it talked about being able to identify a unique utility function from an infinite number of choices, but if we don't have an infinite number of choices, we can't identify the utility function, so what relevance does this have" and this is a kind of mistake I don't remember even coming close to making so I do not know how to make people stop doing that and maybe I can't.”
“Rephrasing again: we have a wide variety of mathematical theorems all spotlighting, from different angles, the fact that a plan lacking in clumsiness, is possessing of coherence.”
‘Ngo and Yudkowsky on AI capability gains’
“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).
Like, just actually doing that a few dozen times.
Maybe it's not helpful for me to say this? If you say it to Eliezer, he immediately goes, "Ah, yes, I could see how I would update that way after doing the homework, so I will save myself some time and effort and just make that update now without the homework", but this kind of jumping-ahead-to-the-destination is something that seems to me to be... dramatically missing from many non-Eliezers. They insist on learning things the hard way and then act all surprised when they do. Oh my gosh, who would have thought that an AI breakthrough would suddenly make AI seem less than 100 years away the way it seemed yesterday? Oh my gosh, who would have thought that alignment would be difficult?
Utility can be seen as the origin of Probability within minds, even though Probability obeys its own, simpler coherence constraints.”
‘AGI will have learnt utility functions’
“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.”
Thanks to Adam Bales, Dan Hendrycks, and members of the CAIS Philosophy Fellowship for comments on a draft of this post. When I emailed Adam to ask for comments, he replied with his own draft paper on coherence arguments. Adam’s paper takes a somewhat different view on money-pump arguments, and should be available soon.
Gustafsson later offers a forcing money-pump argument for Completeness: a money-pump in which, at each step, the agent is compelled by their preferences to pursue a dominated strategy. But agents who act in accordance with the policy above are immune to this money-pump as well. Here’s why.
Gustafsson claims that, in the original non-forcing money-pump, going up at node 2 cannot be irrational. That’s because the agent does not strictly disprefer A- to B: the only other option available at node 2. The fact that A was previously available cannot make choosing A- irrational, because (Gustafsson claims) Decision-Tree Separability is true: “The rational status of the options at a choice node does not depend on other parts of the decision tree than those that can be reached from that node.” But (Gustafsson claims) the sequence of choices consisting of going up at nodes 1 and 2 is irrational, because it leaves the agent worse-off than they could have been. That implies that going up at node 1 must be irrational, given what Gustafsson calls ‘The Principle of Rational Decomposition’: any irrational sequence of choices must contain at least one irrational choice. Generalizing this argument, Gustafsson gets a general rational requirement to choose option A whenever your other option is to proceed to a choice node where your options are A- and B. And it’s this general rational requirement (‘Minimal Unidimensional Precaution’) that allows Gustafsson to construct his forcing money-pump. In this forcing money-pump, an agent’s incomplete preferences compel them to violate the Principle of Unexploitability: that principle which says getting money-pumped is irrational. The Principle of Preferential Invulnerability then implies that incomplete preferences are irrational, since it’s been shown that there exists a situation in which incomplete preferences force an agent to violate the Principle of Unexploitability.
Note that Gustafsson aims to establish that agents are rationally required to have complete preferences, whereas coherence arguments aim to establish that sufficiently-advanced artificial agents will have complete preferences. These different conclusions require different premises. In place of Gustafsson’s Decision-Tree Separability, coherence arguments need an amended version that we can call ‘Decision-Tree Separability*’: sufficiently-advanced artificial agents’ dispositions to choose options at a choice node will not depend on other parts of the decision tree than those that can be reached from that node. But this premise is easy to doubt. 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 all possible money-pumps for Completeness, and (as I am about to prove back in the main text) it never requires them to change or act against any of their preferences.
John Wentworth’s ‘Why subagents?’ suggests another policy for agents with incomplete preferences: trade only when offered an option that you strictly prefer to your current option. That policy makes agents immune to the single-souring money-pump. The downside of Wentworth’s proposal is that an agent following his policy will pursue a dominated strategy in single-sweetening money-pumps, in which the agent first has the opportunity to trade in A for B and then (conditional on making that trade) has the opportunity to trade in B for A+. Wentworth’s policy will leave the agent with A when they could have had A+.