# There are no coherence theorems

by 23 min read20th Feb 2023100 comments

# Ω 20

Crossposted from the AI Alignment Forum. May contain more technical jargon than usual.

[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:

1. 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.
2. Sufficiently-advanced artificial agents will not pursue dominated strategies.
3. So, sufficiently-advanced artificial agents will be ‘coherent’: they will be representable as maximizing expected utility.

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:

An agent can be represented as maximizing expected utility if and only if their preferences satisfy the following four axioms:

1. Completeness: For all lotteries X and Y, X is at least as preferred as Y or Y is at least as preferred as X.
2. Transitivity: For all lotteries X, Y, and Z, if X is at least as preferred as Y, and Y is at least as preferred as Z, then X is at least as preferred as Z.
3. Independence: For all lotteries X, Y, and Z, and all probabilities 0<p<1, if X is strictly preferred to Y, then pX+(1-p)Z is strictly preferred to pY+(1-p)Z.
4. Continuity: For all lotteries X, Y, and Z, with X strictly preferred to Y and Y strictly preferred to Z, there are probabilities p and q such that (i) 0<p<1, (ii) 0<q<1, and (iii) pX+(1-p)Z is strictly preferred to Y, and Y is strictly preferred to qX+(1-q)Z.

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:

Agents are rationally required to satisfy Axiom A.

But inside the AI safety community, the claim is:

Sufficiently-advanced artificial agents will satisfy Axiom A.

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:

An agent can be induced to accept a set of bets that guarantee a net loss if and only if that agent’s credences violate one or more of the probability axioms.

The Dutch Book Argument for Conditionalization says:

An agent can be induced to accept a set of bets that guarantee a net loss if and only if that agent updates their credences by some rule other than Conditionalization.

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:

Completeness: For all lotteries X and Y, X is at least as preferred as Y or Y is at least as preferred as X.

Since:

‘X is strictly preferred to Y’ is defined as ‘X is at least as preferred as Y and Y is not at least as preferred as X.’

And:

‘The agent is indifferent between X and Y’ is defined as ‘X is at least as preferred as Y and Y is at least as preferred as X.’

Completeness can be rephrased as:

Completeness (rephrased): For all lotteries X and Y, either X is strictly preferred to Y, or Y is strictly preferred to X, or the agent is indifferent between X and Y.

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:

• That the agent’s strict preferences are transitive.
• That the agent knows in advance what trades they will be offered.
• That the agent is capable of backward induction: predicting what they would choose at later nodes and taking those predictions into account at earlier nodes.

(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

• There are no coherence theorems
• Money-pump arguments don’t give us much reason to expect that advanced artificial agents will be representable as expected-utility-maximizers.

# 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

• Wald’s complete class theorem
• Von-Neumann-Morgenstern utility theorem
• Cox’s Theorem
• Dutch book arguments”

### ‘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:

• You are vulnerable to money pumps only to the extent to which you deviate from the von Neumann-Morgenstern axioms of expected utility.

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.”

1. ^

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.

2. ^

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+.

# Ω 20

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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...

[-]EJT3moΩ133015

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.

2Adele Lopez3mo
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 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.
6EJT3mo
If you use this definition, then VNM (etc.) counts as a coherence theorem. But Premise 1 of the coherence argument (as I've rendered it) remains false, and so you can't use the coherence argument to get the conclusion that sufficiently-advanced artificial agents will be representable as maximizing expected utility.
8habryka3mo
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:  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:

• For any agent who cannot be represented as maximizing expected utility, there is at least some situation in which that agent will pursue a dominated strategy.

And my argument is:

1. There are no theorems which state or imply that claim. VNM doesn’t, Savage doesn’t, Bolker-Jeffrey doesn’t, Dutch Books don't, Cox doesn't, Complete Class doesn't.
2. Money-pump arguments for the claim are not particularly convincing (for the reasons that I give in the post).

‘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.

6habryka3mo
These are both great! I now find that I have strong-upvoted them both at the time. Indeed, I think this kind of concreteness feels like it does actually help the discussion quite a bit.  I also quite liked John's post on this topic: https://www.lesswrong.com/posts/3xF66BNSC5caZuKyC/why-subagents [https://www.lesswrong.com/posts/3xF66BNSC5caZuKyC/why-subagents]

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:

• Our highly intelligent system notices that to have complete preferences over all trades would be too computationally expensive, and thus is willing to accept some, even a large degree of incompleteness.
...
5habryka3mo
I do really want to put emphasis on the parenthetical remark "(at least in some situations, though they may not arise)". Katja is totally aware that the coherence arguments require a bunch of preconditions that are not guaranteed to be the case for all situations, or even any situation ever, and her post is about how there is still a relevant argument here.
4Ben Pace3mo
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.
5NunoSempere3mo
I am not defending the language of the OP's title, I am defending the content of the post.
4NunoSempere3mo
See this comment: <https://www.lesswrong.com/posts/yCuzmCsE86BTu9PfA/there-are-no-coherence-theorems?commentId=v2mgDWqirqibHTmKb>

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”.

8keith_wynroe3mo
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? (3) In some other comments you're saying that this post is straw-manning some extreme position because people who use coherence arguments already accept you could have e.g. >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 This seems to be entirely missing the point/confused - OP isn't saying that agents can realistically get away with not being VNM-rational because its inconsistencies/incompletenesses aren't efficiently exploitable, they're saying that you can have an agent that aren't VNM-rational and aren't exploitable in principle - i.e., your example is an agent that could in theory be money-pumped by another sufficiently powerful agent that was able to steer the world to where their corner-c
4NunoSempere3mo
Copying my second response from the EA forum: I think that there is some sense in which the character in your example would be right, since: * Arrow's theorem doesn't bind approval voting. * Generalizations of Arrow's theorem don't bind probabilistic results, e.g., each candidate is chosen with some probability corresponding to the amount of votes he gets. Like, if you had someone saying there was "a deep core of electoral process" which means that as they scale to important decisions means that you will necessarily get "highly defective electoral processes", as illustrated in the classic example of the "dangers of the first pass the post system". Well in that case it would be reasonable to wonder whether the assumptions of the theorem bind, or whether there is some system like approval voting which is much less shitty than the theorem provers were expecting, because the assumptions don't hold. The analogy is imperfect, though, since approval voting is a known decent system, whereas for AI systems we don't have an example friendly AI.
5habryka3mo
Sorry, this might have not been obvious, but I indeed think the voting impossibility theorems have holes in them because of the lotteries case and that's specifically why I chose that example.  I think that intellectual point matters, but I also think writing a post with the title "There are no voting impossibility theorems", defining "voting impossibility theorems" as "theorems that imply that all voting systems must make these known tradeoffs", and then citing everyone who ever talked about "voting impossibility theorems" as having made "an error" would just be pretty unproductive. I would make a post like the ones that Scott Garrabrant made being like "I think voting impossibility theorems don't account for these cases", and that seems great, and I have been glad about contributions of this type.
1Noosphere893mo
Unfortunately, most democratic countries do use first past the post. The 2 things that are inevitable is condorcet cycles and strategic voting (Though condorcet cycles are less of a problem as you scale up the population, and I have a sneaking suspicion that condorcet cycles go away if we allow a real numbered infinite amount of people.)
9Douglas_Knight3mo
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 being small, not merely nonexistent. I don't know what happens in the following model: there are three issues A,B,C. Everyone, both voter and candidate, is for all of them, but in a zero-sum way, represented a vector a,b,c, with a+b+c = 11, a,b,c>=0. Start with the voters as above, at (10,1,0), (0,10,1), (1,0,10). Then the candidates (11,0,0), (0,11,0), (0,0,11) form a Condorcet cycle. By symmetry there is no Condorcet winner over all possible candidates. Randomly shift the proportion of voters. Is there a candidate that beats the three given candidates? One that beats all possible candidates? I doubt it. Add noise to make the individual voters unique. Now, I don't know.
2NunoSempere3mo
You don't have strategic voting with probabilistic results. And the degree of strategic voting can also be mitigated.
1Noosphere893mo
Hm, I remember Wikipedia talked about Hylland's theroem that generalizes the Gibbard-Sattherwaite theorem to the probabilistic case, though Wikipedia might be wrong on that.
1[comment deleted]3mo

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.

4EJT3mo
Great points. Thinking about these kinds of worries is my next project, and I’m still trying to figure out my view on them.
3JenniferRM22d
I don't know if you're still working on this, but if don't already know of the literature on choice supportive bias [https://en.wikipedia.org/wiki/Choice-supportive_bias] and similar processes that occur in humans, they look to me a lot like heuristics that probably harden a human agent into being "more coherent" over time (especially in proximity to other ways of updating value estimation processes), and likely have an adaptive role in improving (regularizing [https://en.wikipedia.org/wiki/Regularization_(mathematics)]?) instrumental value estimates. Your essay seemed consistent with the claim that "in the past, as verifiable by substantial scholarship, no one ever proved exactly X" but your essay never actually showed "X is provably false" that I noticed? And, indeed, maybe you can prove it one way or the other for some X, where X might be (as you seem to claim) "naive coherence is impossible" or maybe where some X' or X'' are "sophisticated coherence is approached by algorithm L as t goes to infinity" (or whatever)? For my money, the thing to do here might be to focus on Value-of-Information [https://en.wikipedia.org/wiki/Value_of_information], since VoI seems to me like a super super super important concept, and potentially a way to bridge questions of choice and knowledge and costly information gathering actions.
3EJT10d
Thanks! I'll have a think about choice-supportive bias and how it applies. I think it is provably false that any agent not representable as an expected-utility-maximizer is liable to pursue dominated strategies. Agents with incomplete preferences aren't representable as expected-utility-maximizers, and they can make themselves immune from pursuing dominated strategies 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.’
2JenniferRM5d
I don't know about you, but I'm actually OK dithering a bit, and going in circles, and doing things that mere entropy can "make me notice regret based on syntactically detectable behavioral signs" (like not even active adversarial optimization pressure like that which is somewhat inevitably generated in predator prey contexts). For example, in my twenties I formed an intent, and managed to adhere to the habit somewhat often, where I'd flip a coin any time I noticed decisions where the cost to think about it in an explicit way was probably larger than the difference in value between the likely outcomes. (Sometimes I flipped coins and then ignored the coin if I noticed I was sad with that result, as a way to cheaply generate that mental state of having an intuitive internally accessible preference without having to put things into words or do math. When I noticed that that stopped working very well, I switched to flipping a coin, then "if regret, flip again, and follow my head on head, and follow the first coin on tails". The double flipping protocol seemed to help make ALL the first coins have "enough weight" for me to care about them sometimes, even when I always then stopped for a second to see if I was happy or sad or bored by the first coin flip. And of course I do such things much much much less now, and lately have begun to consider taking a personal vow to refuse to randomize, except towards enemies, for an experimental period of time.) The plans and the hopes here sort of naturally rely on "getting better at preferring things wisely over time"! And the strategy relies pretty critically on having enough MEMORY to hope to build up data on various examples of different ways that similar situations went in the past, such as to learn from mistakes and thereby rarely "lack a velleity [https://en.wikipedia.org/wiki/Velleity]" and to reduce the rate at which I justifiably regret past velleities or choices.  And a core reason I think that sentience and sapience r
3Lucius Bushnaq2mo
That separation between internal preferences and external behaviour is already implicit in Dutch books. Decision theory is about external behaviour, not internal representations. It talks about what agents do, not how agents work inside. As parts of decision theory, a preference, to them, is about something the system does or does not do in a given situation. When they talk about someone preferring pizza without pineapple, it's about that person paying money to not have pineapple on their pizza in some range of situations, not some definition related to computations about pineapples and pizzas in that person's brain.

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.

[-]EJT3moΩ8127

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.

5habryka3mo
That seems totally fine. A term like "coherence theorems" clearly is just like a rough category of things. The definition of the term should not itself bake in the validity of arguments built on top of the elements that the term is trying to draw a definition around.
8EJT3mo
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.

4EJT3mo
Fair enough. I don’t think it’s click-baity: 1. My use of the term matches common usage. See the Appendix [https://www.lesswrong.com/posts/yCuzmCsE86BTu9PfA/there-are-no-coherence-theorems#Appendix__Papers_and_posts_in_which_the_error_occurs]. 2. ‘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’ would have been too long for a title. 3. I (reasonably, in my view) didn’t expect anyone to interpret me as denying the existence of the VNM Theorem, Savage’s Theorem, Bolker-Jeffrey, etc. In any case, I explain how I’m using the term ‘coherence theorems’ in the second sentence of the post.

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).

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".

5EJT3mo
Thanks. I appreciate that. But I do want to insist on the first thing too. Reserve the term ‘coherence theorems’ for whatever you like. The fact remains. Anyone who claims 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 is making a mistake. And anyone who claims that: * VNM/Savage/Bolker-Jeffrey/Dutch Books/Cox's Theorem/the Complete Class Theorem is such a theorem is making a mistake that could have been avoided by looking up what those theorems actually say.

Some different (I think) points against arguments related to the ones that are rebutted in the post:

• requiring a strategy be implementable with a utility function restricts the strategy to a portion of the total strategy space. But, it doesn't follow that any strategy implementable with a utility function actually has to be implemented that way.
• even if a strategy lives in the "utility function" portion of the strategy space, it might be implemented using  additional restrictions such that arguments that would apply to a "typical" utility fu
...
1harfe3mo
What is the function evaluateAction supposed to do when human values contain non-consequentialist components? I assume ExpectedValue is a real number. Maybe there could be a way to build a utility function that corresponds to the code, but that is hard to judge since you have left the details out.
4simon3mo
(edited the code after this comment, corresponding edits below, to avoid noisiness the original is not shown; the original code did not make explicit what I discuss in the "main reason" paragraph:) evaluateWithCorrelations uses both the ProbDistributionOfWorldPath(unknowns) and the Action to generate the ExpectedValue (not explicit, but implicitly the WorldPath can take into account the past and present as well). So, yes, ExpectedValue is a real number, but it doesn't necessarily depend only on the consequences of the action. However, my main reason for thinking that this would be hard to express as a utility function is that the calculation of the ExpectedValue is supposed to take into account the future actions of the AI (not just the Action being chosen now), and is supposed to take into account correlations between ProbDistributionOfHumanValues(t,unknowns) and ProbDistributionOfWorldPath(unknowns). Note, I don't mean taking into account changes in actual human values - it should only be using current ones in the evaluation, though it should take into account possible changes for the prediction. But, the future actions of humans depend on current human values. So, ideally it should be able to predict that asking humans what they want will lead to an update of the model at t' that is correlated to the unknowns in ProbDistributionOfHumanValues(t,unknowns) that will then lead to different actions by the AI depending on what the humans respond with so that it can then assess a better ExpectedValue to this course of action than not asking, whereas if it was a straight utility function maximizer I would expect it would assign the same value in the short run and reduced value in the long run to such asking.   Obviously yes a real AI would be much more complicated.

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...

Strongly upvoted.

Humans at least do not satisfy completeness/don't admit a total order over their preferences.

See also:

1Noosphere893mo
Yes, I generally view human values as partially ordered, not totally ordered. However, the third post answers your second question well. Humans don't have complete preferences, but they still are expected utility maximizers. It's a partial order, not a total order, but it still disagrees with shard theory on relevant details.
7cfoster03mo
Where are you seeing that conclusion in the 3rd post [https://www.lesswrong.com/posts/3xF66BNSC5caZuKyC/why-subagents]? AFAICT the message is that for an agent made up of parts [https://www.lesswrong.com/s/Rm6oQRJJmhGCcLvxh/p/ChierESmenTtCQqZy] that want different things / agent with incomplete preferences, there is no corresponding utility function that would uniquely correspond to its preferences, so humans (having incomplete preferences) are not EUMs. At best, such an agent is more like a market / committee of internal EUMs whose utility functions differ, which accords very well with the mainline "shard"-based picture.
1Noosphere893mo
Sorry for misrepresenting the third post. Though does shard theory agree with the implication of the third post that the shards/sub-agents are utility maximizers themselves?
1cfoster03mo
Sorta? I mean, if you construct an agent via learning, then for a long time the shards within the agent will be much more like reflexes than like full sub-agents/utility maximizers. But in the limit of sophistication, yes there will be some pressure pushing those shards towards individual coherence (EUM-ness), though it's hard to say how the balance shakes out compared to coalitional & other pressures.

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.

9green_leaf3mo
Indeed. What would it even mean for an agent not to prefer A over B, and also not to prefer B over A, and also not be indifferent between A and B?
3Said Achmiz3mo
See my comments on this post for links to several answers to this question.
6green_leaf3mo
I read it, but I'm not at all sure it answers the question. It makes three points: 1. "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 potentially incomplete preorder, thereby allowing for the occasional “indecisiveness” of the agents" I don't see how an agent being indecisive is relevant to preference ordering. Not picking A or B is itself a choice - namely, the agent chooses not to pick either option. 2. "Secondly, there are economic instances in which a decision maker is in fact composed of several agents each with a possibly distinct objective function. For instance, in coalitional bargaining games, it is in the nature of things to specify the preferences of each coalition by means of a vector of utility functions (one for each member of the coalition), and this requires one to view the preference relation of each coalition as an incomplete preference relation." So, if the AI is made of multiple agents, each with its own utility function and we use a vector utility function to describe the AI... the AI still makes a particular choice between A and B (or it refuses to choose, which itself is a choice). Isn't this a flaw of the vector-utility-function description, rather than a real property of the AI? 3. "The same reasoning applies to social choice problems; after all, the most commonly used social welfare ordering in economics, the Pareto dominance" I'm not sure how this is related to AI. Do you have any ideas?
9Said Achmiz3mo
A couple of relevant quotes: (Aumann 1962 [http://share.obormot.net/papers/Utility_Theory_without_the_Completeness_Axiom_Robert_J_Aumann_plus_Correction.pdf]) (Dubra et. al. 2001 [https://cowles.yale.edu/sites/default/files/files/pub/d12/d1294.pdf])

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.

1. You've mistaken acyclicity for transitivity. The money-pump establishes only acyclicity. Representability-as-an-expected-utility-maximizer requires transitivity.

2. 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.

3Eliezer Yudkowsky3mo
And this avoids the Complete Class Theorem conclusion of dominated strategies, how? Spell it out with a concrete example, maybe? Again, we care about domination, not representability at all.

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.

9Eliezer Yudkowsky3mo
In the limit, you take a rock, and say, "See, the complete class theorem doesn't apply to it, because it doesn't have any preferences ordered about anything!"  What about your argument is any different from this - where is there a powerful, future-steering thing that isn't viewable as Bayesian and also isn't dominated?  Spell it out more concretely:  It has preferences ABC, two things aren't ordered, it chooses X and then Y, etc.  I can give concrete examples for my views; what exactly is a case in point of anything you're claiming about the Complete Class Theorem's supposed nonapplicability and hence nonexistence of any coherence theorems?

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:

• Without Completeness, we can’t get a real-valued utility function.
• Without a real-valued utility funct
...
3Eliezer Yudkowsky3mo
I want you to give me an example of something the agent actually does, under a couple of different sense inputs, given what you say are its preferences, and then I want you to gesture at that and say, "Lo, see how it is incoherent yet not dominated!"
2eapi3mo
Say more about what counts as incoherent yet not dominated? I assume "incoherent" is not being used here as an alias for "non-EU-maximizing" because then this whole discussion is circular.
5Eliezer Yudkowsky3mo
Suppose I describe your attempt to refute the existence of any coherence theorems:  You point to a rock, and say that although it's not coherent, it also can't be dominated, because it has no preferences.  Is there any sense in which you think you've disproved the existence of coherence theorems, which doesn't consist of pointing to rocks, and various things that are intermediate between agents and rocks in the sense that they lack preferences about various things where you then refuse to say that they're being dominated?
3eapi3mo
This is pretty unsatisfying as an expansion of "incoherent yet not dominated" given that it just uses the phrase "not coherent" instead. I find money-pump arguments to be the most compelling ones since they're essentially tiny selection theorems for agents in adversarial environments, and we've got an example in the post of (the skeleton of) a proof that a lack-of-total-preferences doesn't immediately lead to you being pumped. Perhaps there's a more sophisticated argument that Actually No, You Still Get Pumped but I don't think I've seen one in the comments here yet. If there are things which cannot-be-money-pumped, and yet which are not utility-maximizers, and problems like corrigibility are almost certainly unsolvable for utility-maximizers, perhaps it's somewhat worth looking at coherent non-pumpable non-EU agents?
3Eliezer Yudkowsky3mo
Things are dominated when they forego free money and not just when money gets pumped out of them.
6keith_wynroe3mo
How is the toy example agent sketched in the post dominated?
4keith_wynroe2mo
Want to bump this because it seems important? How do you see the agent in the post as being dominated?
4eapi3mo
...wait, you were just asking for an example of an agent being "incoherent but not dominated" in those two senses of being money-pumped? And this is an exercise meant to hint that such "incoherent" agents are always dominatable? I continue to not see the problem, because the obvious examples don't work. If I have (1 apple,$0) as incomparable to (1 banana,$0) that doesn't mean I turn down the trade of −1 apple,+1 banana,+$10000 (which I assume is what you're hinting at re. foregoing free money). If one then says "ah but if I offer$9999 and you turn that down, then we have identified your secret equivalent utili-" no, this is just a bid/ask spread, and I'm pretty sure plenty of ink has been spilled justifying EUM agents using uncertainty to price inaction like this. What's an example of a non-EUM agent turning down free money which doesn't just reduce to comparing against an EUM with reckless preferences/a low price of uncertainty?
2keith_wynroe3mo
This seems totally different to the point OP is making which is that you can in theory have things that definitely are agents, definitely do have preferences, and are incoherent (hence not EV-maximisers) whilst not "predictably shooting themselves in the foot" as you claim must follow from this I agree the framing of "there are no coherence theorems" is a bit needlessly strong/overly provocative in a sense, but I'm unclear what your actual objection is here - are you claiming these hypothetical agents are in fact still vulnerable to money-pumping? That they are in fact not possible?
2eapi3mo
The rock doesn't seem like a useful example here. The rock is "incoherent and not dominated" if you view it as having no preferences and hence never acting out of indifference, it's "coherent and not dominated" if you view it as having a constant utility function and hence never acting out of indifference, OK, I guess the rock is just a fancy Rorschach test. IIUC a prototypical Slightly Complicated utility-maximizing agent is one with, say, u(apples,bananas)=min(apples,bananas), and a prototypical Slightly Complicated not-obviously-pumpable non-utility-maximizing agent is one with, say, the partial order (a1,b1)≼(a2,b2)=a1≼a2∧b1≼b2 plus the path-dependent rule that EJT talks about in the post (Ah yes, non-pumpable non-EU agents might have higher complexity! Is that relevant to the point you're making?). What's the competitive advantage of the EU agent? If I put them both in a sandbox universe and crank up their intelligence, how does the EU agent eat the non-EU agent? How confident are you that that is what must occur?
1Eve Grey3mo
Hey, I'm really sorry if I sound stupid, because I'm very new to all this, but I have a few questions (also, I don't know which one of all of you is right, I genuinely have no idea). Aren't rocks inherently coherent, or rather, their parts are inherently coherent, for they align with the laws of the universe, whereas the "rock" is just some composite abstract form we came up with, as observers? Can't we think of the universe in itself as an "agent" not in the sense of it being "god", but in the sense of it having preferences and acting on them? Examples would be hot things liking to be apart and dispersion leading to coldness, or put more abstractly - one of the "preferences" of the universe is entropy. I'm sorry if I'm missing something super obvious, I failed out of university, haha! If we let the "universe" be an agent in itself, so essentially it's a composite of all simples there are (even the ones we're not aware of), then all smaller composites by definition will adhere to the "preferences" of the "universe", because from our current understanding of science, it seems like the "preferences" (laws) of the "universe" do not change when you cut the universe in half, unless you reach quantum scales, but even then, it is my unfounded suspicion that our previous models are simply laughably wrong, instead of the universe losing homogeneity at some arbitrary scale. Of course, the "law" of the "universe" is very simple and uncomplex - it is akin to the most powerful "intelligence" or "agent" there is, but with the most "primitive" and "basic" "preferences". Also apologies for using so many words in quotations, I do so, because I am unsure if I understand their intended meaning. It seems to me that you could say that we're all ultimately "dominated" by the "universe" itself, but in a way that's not really escapeable, but in opposite, the "universe" is also "dominated" by more complex "agents", as individuals can make sandwiches, while it'd take the "universe" muc
3the gears to ascension3mo
The question is how to identify particular bubbles of seekingness in the universe. How can you tell which part of the universe will respond to changes in other parts' shape by reshaping them, and how? How do you know when a cell wants something, in the sense that if the process of getting the thing is interfered with, it will generate physical motions that end up compensating for the interference. How do you know if it wants the thing, if it responds differently to different sizes of interference? Can we identify conflict between two bubbles of seekingness? etc. The key question is how to identify when a physical has a preference for one thing over another. The hope is that, if we find a sufficiently coherent causal mechanism description that specifies what physical systems qualify as For what it's worth, I think you're on a really good track here, and I'm very excited about views that have the one you're starting with. I'd invite browsing my account and links, as this is something I talk about often, from various perspectives, though mostly I defer to others for getting the math right. Speaking of getting the math right: read Discovering Agents [https://www.semanticscholar.org/reader/0c1f1fe5497971479b33e31b236c11d44fbf7a98] (or browse [https://arxivxplorer.com/?query=https%3A%2F%2Farxiv.org%2Fabs%2F2208.08345] related papers [https://www.semanticscholar.org/paper/Discovering-Agents-Kenton-Kumar/0c1f1fe5497971479b33e31b236c11d44fbf7a98]), it's a really great paper. it's not an easy first paper to read, but I'm a big believer in out-of-order learning and jumping way ahead of your current level to get a sense of what's out there. Also check out the related paper Interpreting systems as solving POMDPs [https://ar5iv.labs.arxiv.org/html/2209.01619] (or browse [https://arxivxplorer.com/?query=https%3A%2F%2Farxiv.org%2Fpdf%2F2209.01619.pdf] ) related papers [https://www.semanticscholar.org/paper/Interpreting-systems-as-solving-POMDPs%3A-a-step-a-of-Biehl-Virgo/a7c5bdc
1Eve Grey3mo
I'll read the papers once I get on the computer - don't worry, I may have not finished uni, but I always loved reading papers over a cup of tea. I'm kind of writing about this subject right now, so maybe there you can find something that interests you. How do I know what parts of the universe will respond to what changes? To me, at least, this seems like a mostly false question, for you to have true knowledge of that, you'd need to become the Universe itself. If you don't care about true knowledge just good % chances, then you do it with heuristic. First you come up with composites that are somewhat self similar, but nothing is exactly alike in the Universe, except the Universe itself. Then you create a heuristic for predicting those composites and you use it, as long as the composite is similar enough to the original composite that the heuristic was based on. Of course, heuristics work differently in different environments, but often there are only a few environments even relevant for each composite, for if you take a fish out of water, it will die - now you may want a heuristic for an alive fish in the air, but I see it as much more useful to recompile the fish into catch at that point. This of course applies on any level of composition, from specific specimens of fish, to ones from a specific family, to a single species, then to all fish, then to all living organisms, with as many steps in between these listed as you want. How do we discriminate between which composite level we ought to work with? Pure intuition and experiment, once you do it with logic, it all becomes useless, because logic will attempt to compression everything, even those things which have more utility being uncompressed. I'll get to the rest of your comment on PC, my fingers hurt. Typing on this new big phone is so hard lol.
2Max H3mo
Plus some other assumptions (capable of backwards induction, knowing trades in advance), right? I'm curious whether these assumptions are actually stronger than, or related to, completeness. Both sets (representable and not) are non-empty. The question remains about which set the interesting agents are in. I think that CCT + VNM, money pump arguments, etc. strongly hint, but do not prove, that the EU maximizers are the interesting ones. Also, I personally don't find the question itself particularly interesting, because it seems like one can move between these sets in a relatively shallow way [https://www.lesswrong.com/posts/yCuzmCsE86BTu9PfA/there-are-no-coherence-theorems?commentId=NEavrWsteQ4ZhEa72] (I'd be interested in seeing counterexamples, though). Perhaps that's what Yudkowsky means by not caring about representability?
1EJT3mo
Yep, that’s right! Since the Completeness assumption is about preferences while the backward-induction and knowing-trades-in-advance assumptions are not, they don’t seem very closely related to me. The assumption that the agent’s strict preferences are transitive is more closely related, but it’s not stronger than Completeness in the sense of implying Completeness. Can you say a bit more about what you mean by ‘interesting agents’? From your other comment: I think this could well be right. The main thought I want to argue against is more like:  * Even if you initially succeed in creating a powerful agent that doesn’t maximize expected utility, VNM/CCT/money-pump arguments make it likely that this powerful agent will later become an expected utility maximizer.
2Max H3mo
I meant stronger in a loose sense:  you argued that "completeness doesn't come for free",  but it seems more like actually what you've shown is that not-pursuing-dominated-strategies is the thing that doesn't come for free.  You either need a bunch of assumptions about preferences, or you need one less of those assumptions, plus a few other assumptions about knowing trades, induction, and adherence to a specific policy.  And even given all these other assumptions, the proposed agent with a preferential gap seems like it's still only epsilon-different from an actual EU maximizer. To me this looks like a strong hint that these assumptions actually do point at a core of something simple which one might call "coherence", which I expect to show up in (all minus epsilon) advanced agents, even if there are pathological points in advanced-agent-space which don't have these properties (and even if expected utility theory as a whole isn't quite correct [https://www.lesswrong.com/posts/XYDsYSbBjqgPAgcoQ/why-the-focus-on-expected-utility-maximisers?commentId=a5tn6B8iKdta6zGFu]).
9EJT3mo
I see. I think this is right. I agree with this too, but note that the agent with a single preferential gap is just an example. Agents can have arbitrarily many preferential gaps and still avoid pursuing dominated strategies. And agents with many preferential gaps may behave quite differently to expected utility maximizers.
2quetzal_rainbow3mo
You need only non-transitivity for money pump. Let's suppose that you prefer A to B, B to C and you are indifferent between A and C (not cyclic, not transitive preference). You start with C, you pay me 1 dollar to switch to B, then you pay 1 dollar to switch to A, then I pay you 1 dollar to switch to C (which you do, because A = C implies C + 1 > A) and I have 1 free dollar. Note that your proposed policy doesn't work here, because you do not strictly disprefer C + 1.
2EJT3mo
Nice point but this money-pump only rules out one kind of transitivity-violation (the agent strictly prefers A to B, strictly prefers B to C, and is indifferent between A and C). It doesn't rule out this other kind of transitivity-violation: the agent strictly prefers A to B, strictly prefers B to C, and has a preferential gap between A and C.
3DaemonicSigil3mo
Wait, I can construct a money pump for that situation. First let the agent choose between A and C. If there's a preferential gap, the agent should sometimes choose C. Then let the agent pay a penny to upgrade from C to B. Then let the agent pay a penny to upgrade from B to A. The agent is now where it could have been to begin with by choosing A in the first place, but 2 cents poorer. Even if we ditch the completeness axiom, it sure seems like money pump arguments require us to assume a partial order [https://en.wikipedia.org/wiki/Partially_ordered_set#Partial_order]. What am I missing?
2EJT3mo
So this won't work if the agent knows in advance what trades they'll be offered and is capable of reasoning by backward induction. In that case, the agent will reason that they'd choose A-2p over B-1p if they reached that node, and would choose B-1p over C if they reached that node. So (they will reason), the choice between A and C is actually a choice between A and A-2p, and so they will reliably choose A. And plausibly we should make assumptions like 'the agent knows in advance what trades they will be offered' and 'the agent is capable of backward induction' if we're arguing about whether agents are rationally required to conform their preferences to the VNM axioms.  (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.) That said, I've recently become less convinced that 'knowing trades in advance' is a reasonable assumption in the context of predicting the behaviour of advanced artificial agents. And your money-pump seems to work if we assume that the agent doesn't know what trades they will be offered in advance. So maybe we do in fact have reason to expect that advanced artificial agents will have transitive preferences. (I say 'maybe' because there are some other relevant considerations pushing the other way, discussed in a paper-in-progress by Adam Bales.)
2DaemonicSigil3mo
I don't know, this still seems kind of sketchy to me. Say we change the experiment so that it costs the agent a penny to choose A in the initial choice: it will still take that choice, since A-1p is still preferable to A-2p. Compare this to a game where the agent can freely choose between A and C, and there's no cost in pennies to either choice. Since there's a preferential gap between A and C, the agent will sometimes pick A and sometimes pick C. In the first game, on the other hand the agent always picks A. Yet in the first game, not only is picking A more costly, but we've only added options for the agent if it picks C. In other words, an agent that has A>B, B>C, and A~C sure looks like it's paying to take options away from itself, since adding options makes it less likely to pick C, even when it costs a penny to avoid it.
1EJT3mo
Nice! This is a cool case. The behaviour does indeed seem weird. I'm inclined to call it irrational. But the agent isn't pursuing a dominated strategy: in neither game does the agent settle on an option that they strictly disprefer to some other available option. This discussion is interesting and I'm happy to keep having it, but perhaps it's worth saying (if not for your sake then for other readers) that this is a side-thread. The main point of the post is that there are no money-pumps for Completeness. I think that there are probably no money-pumps for Transitivity either, but it's the claim about Completeness that I really want to defend.
3DaemonicSigil3mo
Cool. For me personally, I think that paying to avoid being given more options looks enough like being dominated that I'd want to keep the axiom of transitivity around, even if it's not technically a money pump. So in the case where we have transitivity but no completeness, it seems kind of like there might be a weaker coherence theorem, where the agent's behaviour can be described by rolling a dice to pick a utility function before beginning a game, and then subsequently playing according to that utility function. Under this interpretation, if A > B then that means that A is preferred to B under all utility functions the agent could pick, while a preferential gap between A and B means that sometimes A will be ranked higher and sometimes B will be ranked higher, depending on which utility function the die roll happens to land on. Does this match your intuition? Is there an obvious counterexample to this "coherence conjecture"?
9EJT2mo
Your coherence conjecture sounds good! It sounds like it roughly matches this theorem:  Screenshot is from this paper [https://drive.google.com/file/d/11AxoKTcXFoYNxXc9Gyu_OVYOxwh00I2A/view?usp=sharing].
1quetzal_rainbow3mo
It's not a money pump, because money pump implies infinite cycle of profit. If your loses are bounded, you are fine.
1quetzal_rainbow3mo
Does I understand correctly that preferential gaps have size, like, i do not prefer A to B, I do not prefer A to B+1, but some large N exists that I prefer B + N to A?
1EJT3mo
That can be true (and will often be true when it comes to - e.g. - a human agent with a preferential gap between a Fabergé egg and a long-lost wedding album), but it's not a necessary feature of preferential gaps.
1keith_wynroe3mo
Kind of tangential but I'd be interested in your take on how strongly money-pumping etc is actually an argument against full-on cyclical preferences? One way to think about why getting money-pumped is bad is because you have an additional preference to not pay money to go nowhere. But it feels like all this tells us is that "something has to go", and if an agent is rationally permitted to modify its own preferences to avoid these situations then it seems a priori acceptable for it to instead just say something like "well actually I weight my cyclical preferences more highly so I'll modify the preference against arbitrarily paying" In other words, it feels like the money-pumping arguments presume this other preference that in a sense takes "precedence" over the cyclical ones and I'm not sure how to think about that still
1eapi3mo
(I'm not EJT, but for what it's worth:) I find the money-pumping arguments compelling not as normative arguments about what preferences are "allowed", but as engineering/security/survival arguments about what properties of preferences are necessary for them to be stable against an adversarial environment (which is distinct from what properties are sufficient for them to be stable, and possibly distinct from questions of self-modification).
1keith_wynroe3mo
Yeah I agree that even if they fall short of normative constraints there’s some empirical content around what happens in adversarial environments. I think I have doubts that this stuff translates to thinking about AGIs too much though, in the sense that there’s an obvious story of how an adversarial environment selected for (partial) coherence in us, but I don’t see the same kinds of selection pressures being a force on AGIs. Unless you assume that they’ll want to modify themselves in anticipation of adversarial environments which kinda begs the question
3eapi3mo
Hmm, I was going to reply with something like "money-pumps don't just say something about adversarial environments, they also say something about avoiding leaking resources" (e.g. if you have circular preferences between proximity to apples, bananas, and carrots, then if you encounter all three of them in a single room you might get trapped walking between them forever) but that's also begging your original question - we can always just update to enjoy leaking resources, transmuting a "leak" into an "expenditure". Another frame here is that if you make/encounter an agent, and that agent self-modifies into/starts off as something which is happy to leak pretty fundamental resources like time and energy and material-under-control, then you're not as worried about it? It's certainly not competing as strongly for the same resources as you whenever it's "under the influence" of its circular preferences.
5cfoster03mo
If I'm merely indifferent between A and B, then I will not object to trades exchanging A for B. But if A and B are incomparable for me, then I definitely may object!
4Eliezer Yudkowsky3mo
Say more about behaviors associated with "incomparability"?
7cfoster03mo
Depending on the implementation details of the agent design, it may do some combination of: * Turning down your offer, path-dependent [https://www.lesswrong.com/posts/3xF66BNSC5caZuKyC/why-subagents#Path_Dependence]ly preferring whichever option is already in hand [https://elischolar.library.yale.edu/cgi/viewcontent.cgi?article=2049&context=cowles-discussion-paper-series] / whichever option is consistent with its history of past trades. * Noticing unresolved conflicts within its preference framework, possibly unresolveable without self-modifying into an agent that has different preferences from itself. * Halting and catching fire, folding under the weight of an impossible choice. EDIT: The post also suggests an alternative (better) policy [https://www.lesswrong.com/posts/yCuzmCsE86BTu9PfA/there-are-no-coherence-theorems#Summarizing_this_section] that agents with incomplete preferences may follow.
3Seth Herd3mo
I don't think this goes through. If I have no preference between two things, but I do prefer to not be money-pumped, it doesn't seem like I'm going to trade those things so as to be money-pumped. I am commenting because I think this might be a crucial crux: do smart/rational enough agents always act like maximizers? If not, adequate alignment might be much more feasible than if we need to find exactly the right goal and how to get it into our AGI exactly right. Human preferences are actually a lot more complex. We value food very highly when hungry and water when we're thirsty. That can come out of power-seeking, but that's not actually how it's implemented. Perhaps more importantly, we might value stamp collecting really highly until we get bored with stamp collecting. I don't think these can be modeled as a maximizer of any sort. If humans would pursue multiple goals [https://www.lesswrong.com/posts/Sf99QEqGD76Z7NBiq/are-you-stably-aligned] even if we could edit them (and were smart enough to be consistent), then a similar AGI might only need to be minimally aligned for success. That is, it might stably value human flourishing as a small part of its complex utility function. I'm not sure whether that's the case, but I think it's important.

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?

1EJT3mo
Nice point. But making your preferences complete won’t protect you from pursuing dominated strategies if you don’t know what’s coming. For example, suppose at node 1 you face a choice between taking A and proceeding to node 2. You think that at node 2 you’ll face a choice between A- and A+. So, you proceed to node 2, with the intention of taking A+. But you were mistaken. At node 2, you face a choice between A- and A--. You take A-. In that case, you’ve pursued a dominated strategy: you’ve ended up with A- when you could have had A. But your preferences are not to blame. Instead, it was your mistaken beliefs about what options you would have.
1Signer3mo
My intuition was something like "you would get better satisfaction of preference in expectation even if you are uncertain about the future", but I guess it doesn't exactly work without first defining utility function. But what about first choosing between A- and B-, and then between A or B- in A- branch, and B or A- in B- branch - this way you get (A|B-|B|A-) with gaps vs. (A|B) in indifference case - wouldn't the mixture with worse variants intuitively be worse than one with only good ones even if we can't strictly say that incomplete preferences are contradicted?

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?

1EJT1d
I don't think so, Suppose the agent first chooses A when we offer it a choice between A and B. After that, the agent must act as if it prefers A to B-. But it can still lack a preference between A and B, and this lack of preference can still be insensitive to some sweetening or souring: the agent could also lack a preference between A and B+, or lack a preference between A+ and B, or lack a preference between B and A-. What is true is that, given a sufficiently wide variety of past decisions, the agent must act as if its preferences are complete. But depending on the details, that might never happen or only happen after a very long time. If you're interested, these kinds of points got discussed in a bit more detail over in this comment thread [https://forum.effectivealtruism.org/posts/FoRyordtA7LDoEhd7/there-are-no-coherence-theorems?commentId=NXzbgBLoF9BpSSnBh].