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The problem partly is in utility functions being used as both: a) as a metaphor and b) as an exact mathematical tool with exact properties.

a) can be used to elucidate terminal values in a discussion or to structure and focus a discussion away from vague concepts in ethics. But as a metaphor it cannot be used to derive anything with strength. b) on the other hand can strictly only be used where the preconditions are satisfied. Mixing a) and b) means committing the mathematical fallacy: Believing that to have formulated something in an exact way solves the issue in practice.

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Against utility functions

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Jun 19, 2014

I think we should stop talking about utility functions.

In the context of ethics for humans, anyway. In practice I find utility functions to be, at best, an occasionally useful metaphor for discussions about ethics but, at worst, an idea that some people start taking too seriously and which actively makes them worse at reasoning about ethics. To the extent that we care about causing people to become better at reasoning about ethics, it seems like we ought to be able to do better than this.

The funny part is that the failure mode I worry the most about is already an entrenched part of the Sequences: it's fake utility functions. The soft failure is people who think they know what their utility function is and say bizarre things about what this implies that they, or perhaps all people, ought to do. The hard failure is people who think they know what their utility function is and then do bizarre things. I hope the hard failure is not very common.

It seems worth reflecting on the fact that the point of the foundational LW material discussing utility functions was to make people better at reasoning about AI behavior and not about human behavior.

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It's more than a metaphor; a utility function is the structure any consistent preference ordering that respects probability must have. It may or may not be a useful conceptual tool for practical human ethical reasoning, but "just a metaphor" is too strong a judgment.

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a utility function is the structure any consistent preference ordering that respects probability must have.

This is the sort of thing I mean when I say that people take utility functions too seriously. I think the von Neumann-Morgenstern theorem is much weaker than it initially appears. It's full of hidden assumptions that are constantly violated in practice, e.g. that an agent can know probabilities to arbitrary precision, can know utilities to arbitrary precision, can compute utilities in time to make decisions, makes a single plan at the beginning of time about how they'll behave for eternity (or else you need to take into account factors like how the agent should behave in order to acquire more information in the future and that just isn't modeled by the setup of vNM at all), etc.

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The biggest problematic unstated assumption behind applying VNM-rationality to humans, I think, is the assumption that we're actually trying to maximize something.

To elaborate, the VNM theorem defines preferences by the axiom of completeness, which states that for any two lotteries A and B, one of the following holds: A is preferred to B, B is preferred to A, or one is indifferent between them.

So basically, a “preference” as defined by the axioms is a function that (given the state of the agent and the state of the world in general) outputs an agent’s decision between two or more choices. Now suppose that the agent’s preferences violate the Von Neumann-Morgenstern axioms, so that in one situation it prefers to make a deal that causes it to end up with an apple rather than an orange, and in another situation it prefers to make a deal that causes it to end up with an orange rather than an apple. Is that an argument against having circular preferences?

By itself, it's not. It simply establishes that the function that outputs the agent’s actions behaves differently in different situations. Now the normal way to establish that this is bad is to assume that all choices are between monetary payouts, and that an agent with inconsistent preferences can be Dutch Booked and made to lose money. An alternative way, which doesn't require us to assume that all the choices are between monetary payouts, is to construct a series of trades between resources that leaves us with less resources than when we started.

Stated that way, this sounds kinda bad. But then there are things that kind of fit that description, but which we would intuitively think of as good. For instance, some time back I asked:

Suppose someone has a preference to have sex each evening, and is in a relationship with someone what a similar level of sexual desire. So each evening they get into bed, undress, make love, get dressed again, get out of bed. Repeat the next evening.

How is this different from having exploitable circular preferences? After all, the people involved clearly have cycles in their preferences - first they prefer getting undressed to not having sex, after which they prefer getting dressed to having (more) sex. And they're "clearly" being the victims of a Dutch Book, too - they keep repeating this set of trades every evening, and losing lots of time because of that.

In response, I was told that

The circular preferences that go against the axioms of utility theory, and which are Dutch book exploitable, are not of the kind "I prefer A to B at time t1 and B to A at time t2", like the ones of your example. They are more like "I prefer A to B and B to C and C to A, all at the same time".

The couple, if they had to pay a third party a cent to get undressed and then a cent to get dressed, would probably do it and consider it worth it---they end up two cents short but having had an enjoyable experience. Nothing irrational about that. To someone with the other "bad" kind of circular preferences, we can offer a sequence of trades (first A for B and a cent, then C for A and a cent, then B for C and a cent) after which they end up three cents short but otherwise exactly as they started (they didn't actually obtain enjoyable experiences, they made all the trades before anything happened). It is difficult to consider this rational.

But then I asked that, if we accept this, then what real-life situation does count as an actual circular preference in the VNM sense, given that just about every potential circularity that I can think of is the kind "I prefer A to B at time t1 and B to A at time t2"? And I didn't get very satisfactory replies.

Intuitively, there are a lot of real-life situations that feel kind of like losing out due to inconsistent preferences, like someone who wants to get into a relationship when he's single and then wants to be single when he gets into a relationship, but there our actual problem is that the person spends a lot of time being unhappy, rather than with the fact that he makes different choices in different situations. Whereas with the couple, we think that's fine because they get enjoyment from the "trades".

The general problem that I'm trying to get at is that in order to hold up VNM rationality as a normative standard, we would need to have a meta-preference: a preference over preferences, stating that it would be better to have preferences that lead to some particular outcomes. The standard Dutch Book example kind of smuggles in that assumption by the way that it talks about money, and thus makes us think that we are in a situation where we are only trying to maximize money and care about nothing else. And if you really are trying to only maximize a single concrete variable or resource and care about nothing else, then you really should try to make sure that your choices follow the VNM axioms. If you run a betting office, then do make sure that nobody can Dutch Book you.

But we don't have such a clear normative standard for life in general. It would be reasonable to try to construct an argument for why the couple having sex were rational but the person who kept vacillating about being in a relationship was irrational by suggesting that the couple got happiness whereas the other person was unhappy... but we also care about other things than just happiness (or pleasure) and thus aren't optimizing just for pleasure either. And unless you're a hedonistic utilitarian, you're unlikely to say that we should optimize only for pleasure either.

So basically, if you want to say that people should be VNM-rational, then you need to have some specific set of values or goals that you think people should strive towards. If you don't have that, then VNM-rationality is basically irrelevant aside for the small set of special cases where people really do have a clear explicit goal that's valued above other things.

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Now suppose that the agent’s preferences violate the Von Neumann-Morgenstern axioms, so that in one situation it prefers to make a deal that causes it to end up with an apple rather than an orange, and in another situation it prefers to make a deal that causes it to end up with an orange rather than an apple. Is that an argument against having circular preferences?

I'm not sure I follow in what sense this is a violation of the vNM axioms. A vNM agent has preferences over world-histories; in general one can't isolate the effect of having an apple vs. having an orange without looking at how that affects the entire future history of the world.

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Right, I was trying to say "it prefers an apple to an orange and an orange to an apple in such a way that does violate the axioms". But I was unsure of what example to actually give of that, since I'm unsure of what real-life situations really would violate the axioms.

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The example that comes to mind to show the how the sex thing isn't a problem is that of a robot car with a goal to drive as many miles as possible. Every day it will burn through all its fuel and fuel up. Right after it fuels up, it will have no desire for further fuel - more fuel simply does not help it go further at this point, and forcing it can be detrimental. Clearly not contradictory

You could have a similar situation with a couple wanting sex iff they haven't had sex in a day, or wanting an orange if you've just eaten an apple but wanting an apple if you've just eaten an orange.

To strictly show that something violates vNM axioms, you'd have to show that this behavior (in context) can't be fulfilling any preferences better than other options that the agent is aware of - or at least be able to argue that the revealed utility function is contrived and unlikely to hold up in other situations (not what the agent "really wants").

Constantly wanting what one doesn't have can have this defect. If I keep paying you to switch my apple for your orange and back (without actually eating either), then you have a decent case, if you're pretty confident I'm not actually fulfilling my desire to troll you ;)

The "want's a relationship when single" and "wants to be single when not" thing does look like such a violation to me. If you let him flip flop as often as he desires, he's not going to end up happily endorsing his past actions. If you offered him a pill that would prevent him from flip flopping, he very well may take it. So there's a contradiction there.

To bring human-specific psychology into it, its not that his inherent desires are contradictory, but that he wants something like "freedom", which he doesn't know how to get in a relationship and something like "intimacy", which he doesn't know how to get while single. It's not that he want's intimacy when single and freedom when not, it's that he wants both always, but the unfulfilled need is the salient one.

Picture me standing on your left foot. "Oww! Get off my left foot!". Then I switch to the right "Ahh! Get off my right foot!". If you're not very quick and/or the pain is overwhelming, it might take you a few iterations to realize the situation you're in and to put the pain aside while you think of a way to get me off both feet (intimacy when single/freedom in a relationship). Or if you can't have that, it's another challenge to figure out what you want to do about it.

I wouldn't model you as "just VNM-irrational", even if your external behaviors are ineffective for everything you might want. I'd model you as "not knowing how to be VNM-rational in presence of strong pain(s)", and would expect you to start behaving more effectively when shown how.

(and that is what I find, although showing someone how to be more rational is not trivial and "here's a proof of the inconsistency of your actions now pick a side and stop feeling the desire for the other side" is almost never sufficient. You have to be able to model the specific way that they're stuck and meet them there)

tl;dr: We're not VNM-rational because we don't know how to be, not because it's not something we're trying to do.

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How do you distinguish his preferences being irrationally inconsistent (he is worse off from entering and leaving relationships repeatedly) from him truly wanting to be in relationships periodically (like how it's rational to alternate between sleeping and waking rather than always doing one or the other)?

If there's a pill that can make him stop switching (but doesn't change his preferences), one of two things will happen: either he'll never be in a relationship (prevented from entering), or he'll stay in his current relationship forever (prevented from leaving). I wouldn't be surprised if he dislikes both of the outcomes and decides not to take the pill.

The pill could instead change his preferences so that he no longer wants to flip-flop, but this argument seems too general - why not just give him a pill that makes him like everything much more than he does now? If my behavior is irrational, I should be able to make myself better off simply by changing my behavior, without having to modify my preferences.

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Relevant question: what does the cognitive science literature on choice-making, preference, and valuation have to say about all this? What mathematical structure actually does model human preferences?

Given that we run on top of neural networks and seem to use some Bayesian algorithms for certain forms of learning (citations available), I currently expect that our choice-making mechanisms might involve conditioning on features or states of our environment at some fundamental level.

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I think your original post would have been better if it included any arguments against utility functions, such as those you mention under "e.g." here.

Besides being a more meaningful post, we would also be able to discuss your comments. For example, without more detail, I can't tell whether your last comment is addressed sufficiently by the standard equivalence of normal-form and extensive-form games.

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Essentially every post would have been better if it had included some additional thing. Based on various recent comments I was under the impression that people want more posts in Discussion so I've been experimenting with that, and I'm keeping the burden of quality deliberately low so that I'll post at all.

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I appreciate you writing this way -- speaking for myself, I'm perfectly happy with a short opening claim and then the subtleties and evidence emerges in the following comments. A dialogue can be a better way to illuminate a topic than a long comprehensive essay.

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Let me rephrase: would you like to describe your arguments against utility functions in more detail?

For example, as I mentioned, there's an obvious mathematical equivalence between making a plan at the beginning of time and planning as you go, which is directly analogous to how one converts games from extensive-form to normal-form. As such, all aspects of acquiring information is handled just fine (from a mathematical standpoint) in the setup of vNM.

The standard response to the discussion of knowing probabilities exactly and to concerns about computational complexity (in essence) is that we may want to throw aside epistemic concerns and simply learn what we can from a theory that is not troubled by them (a la air resistance in physics..)? Is your objection essentially that those factors are more dominant in human morality than LW acknowledges? And if so, is the objection to the normal-form assumption essentially the same?

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It's full of hidden assumptions that are constantly violated in practice, e.g. that an agent can know probabilities to arbitrary precision, can know utilities to arbitrary precision, can compute utilities in time to make decisions, makes a single plan at the beginning of time about how they'll behave for eternity (or else you need to take into account factors like how the agent should behave in order to acquire more information in the future and that just isn't modeled by the setup of vNM at all), etc.

Those are not assumptions of the von Neumann-Morgenstern theorem, nor of the concept of utility functions itself. Those are assumptions of an intelligent agent implemented by measuring its potential actions against an explicitly constructed representation of its utility function.

I get the impression that you're conflating the mathematical structure that is a utility function on the one hand, and representations thereof as a technique for ethical reasoning on the other hand. The former can be valid even if the latter is misleading.

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the mathematical structure that is a utility function

Can you describe this "mathematical structure" in terms of mathematics? In particular, the argument(s) to this function, what do they look like mathematically?

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Certainly, though I should note that there is no original work in the following; I'm just rephrasing standard stuff. I particularly like Eliezer's explanation about it.

Assume that there is a set of things-that-could-happen, "outcomes", say "you win $10" and "you win $100". Assume that you have a preference over those outcomes; say, you prefer winning $100 over winning $10. What's more, assume that you have a preference over probability distributions over outcomes: say, you prefer a 90% chance of winning $100 and a 10% chance of winning $10 over a 80% chance of winning $100 and a 20% change of winning $10, which in turn you prefer over 70%/30% chances, etc.

A utility function is a function f from outcomes to the real numbers; for an outcome O, f(O) is called the utility of O. A utility function induces a preference ordering in which probability-distribution-over-outcomes A is preferred over B if and only if the sum of the utilities of the outcomes in A, scaled by their respective probabilities, is larger than the same for B.

Now assume that you have a preference ordering over probability distributions over outcomes that is "consistent", that is, such that it satisfies a collection of axioms that we generally like reasonable such orderings to have, such as transitivity (details here). Then the von Neumann-Morgenstern theorem says that there exists a utility function f such that the induced preference ordering of f equals your preference ordering.

Thus, if some agent has a set of preferences that is consistent -- which, basically, means the preferences scale with probability in the way one would expect -- we know that those preferences must be induced by some utility function. And that is a strong claim, because a priori, preference orderings over probability distributions over outcomes have a great many more degrees of freedom than utility functions do. The fact that a given preference ordering is induced by a utility function disallows a great many possible forms that ordering might have, allowing you to infer particular preferences from other preferences in a way that would not be possible with preference orderings in general. (Compare this LW article for another example of the degrees-of-freedom thing.) This is the mathematical structure I referred to above.

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Right.

So, keeping in mind that the issue is separating the pure mathematical structure from the messy world of humans, tell me what outcomes are, mathematically. What properties do they have? Where can we find them outside of the argument list to the utility function?

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"a utility function is the structure any consistent preference ordering that respects probability must have."

Yes, but humans still don't have one. It's not even clear they can make themselves have one.

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I don't think I have much to add to this discussion that you guys aren't already going to have covered, except to note that Qiaochu definitely understands what a utility function is and all of the standard arguments for why they "should" exist, so his beliefs are not a function of not having heard these arguments (just noting this because this thread and some of the siblings seem to be trying to explain basic concepts to Qiaochu that I'm confident he already knows, and I'm hoping that pointing this out will speed up the discussion).

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Like the word "rational" is sometimes used instead of "optimal" or "good", words "utility function" are probably used to mean "good" or "our values" or something like that.

Therefore, analogically to the suggestion of only using the word 'rational' when talking about cognitive algorithms and thinking techniques, we should only use the words 'utility function' when talking about computer programs. When speaking about humans, "good / better / the best" probably expresses what we need well enough.

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It seems worth reflecting on the fact that the point of the foundational LW material discussing utility functions was to make people better at reasoning about AI behavior and not about human behavior.

I think part of Eliezer's point was also to introduce decision theory as an ideal for human rationality. (See http://lesswrong.com/lw/my/the_allais_paradox/ for example.) Without talking about utility functions, we can't talk about expected utility maximization, so we can't define what it means to be ideally rational in the instrumental sense (and we also can't justify Bayesian epistemology based on decision theory).

So I agree with the problem stated here, but "let's stop talking about utility functions" can't be the right solution. Instead we need to emphasize more that having the wrong values is often worse than being irrational, so until we know how to obtain or derive utility functions that aren't wrong, we shouldn't try to act as if we have utility functions.

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The trouble is the people who read the Sequences and went "EY said it, it's probably right, I'll internalise it." This is an actual hazard around here. (Even Eliezer can't make people think, rather than just believe in thinking.)

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Without talking about utility functions, we can't talk about expected utility maximization, so we can't define what it means to be ideally rational in the instrumental sense

I like this explanation of why utility-maximization matters for Eliezer's overarching argument. I hadn't noticed that before.

But it seems like utility functions are an unnecessarily strong assumption here. If I understand right, expected utility maximization and related theorems imply that if you have a complete preference over outcomes, and have probabilities that tell you how decisions influence outcomes, you have implicit preferences over decisions.

But even if you have only partial information about outcomes and partial preferences, you still have some induced ordering of the possible actions. We lose the ability to show that there is always an optimal 'rational' decision, but we can still talk about instances of irrational decision-making.

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Related: We Don't Have A Utility Function, van Gelder on the usefulness of utility functions, Applying utility functions to humans considered harmful.

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Thanks for the links!

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There's a problem with discussing ethics in terms of UFs, which is that no attempt is made to separate morally relevant preferences from others. Which is a wider issue than UFs. There may be some further issue with UFs.

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The problem partly is in utility functions being used as both: a) as a metaphor and b) as an exact mathematical tool with exact properties.

a) can be used to elucidate terminal values in a discussion or to structure and focus a discussion away from vague concepts in ethics. But as a metaphor it cannot be used to derive anything with strength. b) on the other hand can strictly only be used where the preconditions are satisfied. Mixing a) and b) means committing the mathematical fallacy: Believing that to have formulated something in an exact way solves the issue in practice.

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It seems worth reflecting on the fact that the point of the foundational LW material discussing utility functions was to make people better at reasoning about AI behavior and not about human behavior.

For value extrapolation problem, you need to consider both what an AI could do with a goal (how to use it, what kind of thing it is), and which goal represents humane values (how to define it).

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I still think there's too much confusion between ethics-for-AI and ethics-for-humans discussions here. There's no particular reason that a conceptual apparatus suited for the former discussion should also be suited for the latter discussion.

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Yes! Thank you for saying this clearly and distinctly.

Real-world objects are never perfect spheres or other mathematical entities. However, math is quite useful for modeling them. But the way we decide which math is the right math to use to model a particular sort of object is through repeated experiment. And sometimes the trajectory through spacetime of a given object (say, a gold coin of a certain mass) is best modeled by certain math (e.g. ballistics) and sometimes by other very different math (e.g. economics).

Utility functions belong to the math, not the territory.

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Can you give some specific examples of people misusing utility functions? Or if you don't want to point fingers, can you construct examples similar to those you've seen people use?

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This thread was prompted by this comment in the Open Thread.

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That comment is about utilitarianism and doesn't mention "utility functions" at all.

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I can't help but suspect, though, that LW people are drawn to utilitarianism because of what they see as the inevitability of using utility functions to model preferences. Maybe this impression is mistaken.

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For practical purposes I agree that it does not help a lot to talk about utility functions. As the We Don't Have a Utility Function article points out, we simply do not know our utility functions but only vague terminal values. However, as you pointed out yourself that does not mean that we do not "have" a utility function at all.

The soft (and hard) failure seems to be a tempting but unnecessary case of pseudo-rationalization. Still, the concept of an agent "having" (maybe in the sense of "acting in a complex way towards optimizing") a utility funktion seems to be very important for defining utilitarian (hence the name, I guess...) ethical systems. In contrast, the notion of terminal values seems to be a lot more vague and not sufficient for defining utilitarianism. Similar things (practical uselessness but theoretical importance) apply to the evaluation of the intelligence of an agent. Therefore, I think that the term 'utility function' is essential for theoretical debate, even though I agree that it is sometimes used in the wrong place.

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