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conchis comments on Average utilitarianism must be correct? - Less Wrong
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I'm afraid that's not what it says. It says that any consistent set of choices over gambles can be represented as the maximization of some utility function. It does not say that that utility function has to be u. In fact, it can be any positive monotonic transform of u. Call such a transform u*.
I'm afraid this still isn't right either. To take an example, suppose u* = ln(u+1). Assuming 50-50 odds for each outcome, the Eu* for your first gamble is ln(11). The Eu* for your second gamble is ln(36), which is higher. So the second gamble is preferred, contra your claim of indifference. In fact, this sort of "inequality aversion" (which is actually just risk aversion with respect to u) will be present whenever u* is a concave function of u.
The rest of the argument breaks down at this point too, but you do raise an interesting question: are there arguments that we should have a particular type of risk aversion with respect to utility (u), or are all risk preferences equally rational?
EDIT: John Broome's paper "A Mistaken Argument Against the Expected Utility Theory of Rationality" (paywalled, sorry) has a good discussion of some of these issues, responding to an argument made by Maurice Allais along the lines of your original post, and showing that it rested on a mathematical error.
I read summaries on the web yesterday (can't find them now) that concluded that the theorem proves that any utility function U that satisfies the axioms must simply be the expected value of the utility functions u. Wikipedia says U must be a linear combination of the function u and the probabilities of each outcome, which is more vague. But in either case, U = ln(u+1) is not allowed.
If u* is just a function of u, then u* is the single-outcome utility function. What is u, and why do you even bring it up? Are you thinking that I mean pre-diminishing-marginal-utility objects when I say utility?
The second gamble is preferred because it has a higher average utility. The actual utilities involved here are ln(11) and 2ln(5); ln(11) is less than twice ln(5). So this is not a case in which we are compensating for inequity; total utility is higher in the 2ln(5) case.
I think you are misinterpreting me and thinking that I mean u(outcome) is an object measure like dollars. It's not. It's utility.
Wait! I think you're thinking that I'm thinking that the theorem reaches directly down through the level of U, and also says something about u. That's not what I'm thinking.
I'm afraid I have no idea what you're thinking any more (if I ever did). I am, however, fairly confident that whatever you're thinking, it does not imply utilitarianism, average or otherwise (unless you're also using those terms in a non-standard way).
My point is whatever you think of as utility, I can apply a positive monotonic transformation to it, maximize the expectation of that transformation, and this will still be rational (in the sense of complying with the Savage axioms).
Accepting, arguendo, that your analogy to interpersonal utility aggregation holds up (I don't think it does, but that's another matter), this means that, whatever utilities individuals have, I can apply a positive monotonic transformation to them all, maximize the expectation of the transformation, and this will be rational (in some sense that is as yet, not entirely clear). If that transformation is a concave function, then I can take account of inequality. (Or, more precisely, I can be prioritarian.) This is all pretty standard stuff.
u* = ln(u+1), not U=ln(u+1). u* = ln(u+1) is allowed.
It was intended to reflect whatever you conceived of utility as. I brought it up because you did.
No.
It has higher average utility*, but it doesn't have higher average utility.
It's 2ln(6) = ln(36), but that doesn't affect the point.
Nope.
I'm afraid I don't know what you mean by this.
EDIT:
Actually, it's slightly more precise. Maximizing any linear combination of the u and the probabilities will describe the same set of preferences over gambles as maximizing E(u). The same goes for any positive affine transformation of u.
Sure. That has no bearing on what I'm saying. You are still maximizing expectation of your utility. Your utility is not the function pre-transformation. The axioms apply only if the thing you are maximizing the expectation of is your utility function. There's no reason to bring up applying a transformation to u to get a different u. You're really not understanding me if you think that's relevant.
Not at all. You can multiply each probability by a different constant if you do that. Or you can multiply them all by -1, and you would be minimizing E(u).
Did you even read the next paragraph where I tried to explain why it does have a bearing on what you're saying? Do you have a response?
Fair. I assumed a positive constant. I shouldn't have.
I read it. I don't understand why you keep bringing up "u", whatever that is. You use u* to represent the utility function on a possible world. We don't care what is inside that utility function for the purposes of this argument. And you can't get out of taking the expected value of your utility function by transforming it into another utility function. Then you just have to take the expected value of that new utility function.
Read steven0461's comment above. He has it spot on.
But you still haven't engaged with it at all. I'm going to give this one last go before I give up.
Utilitarianism starts with a set of functions describing each individuals' welfare. To purge ourselves of any confusion over what u means, let's call these w(i). It then defines W as the average (or the sum, the distinction isn't relevant for the moment) of the w(i), and ranks certain (i.e. non-risky) states of the world higher if they have higher W. Depending on the type of utilitarianism you adopt, the w(i) could be defined in terms of pleasure, desire-satisfaction, or any number of other things.
The Savage/von Neuman-Morgenstern/Marschak approach starts from a set of axioms that consistent decision-makers are supposed to adhere to when faced with choices over gambles. It says that, for any consistent set of choices you might make, there exists a function f (mapping states of the world into real numbers), such that your choices correspond to maximizing E(f). As I think you realize, it puts no particular constraints on f.
Substituting distributions of individuals for probability distributions over the states of the world (and ignoring for the moment the other problems with this), the axioms now imply that for any consistent set of choices we might make, there exists a function f, such that our choices correspond to maximizing E(f).
Again, there are no particular constraints on f. As a result, (and this is the crucial part) nothing in the axioms say that f has to have anything to do with the w(i). Because the f does not have to have anything in particular to do with the w(i), E(f) does not have to have anything to do with W, and so the fact that we are maximizing E(f) says nothing about we are or should be good little utilitarians.
I think you may be missing PhilGoetz's point.
E(f) needn't have anything to do with W. But it has to do with the f-values of all the different versions of you that there might be in the future (or in different Everett branches, or whatever). And it treats all of them symmetrically, looking only at their average and not at their distribution.
That is: the way in which you decide what to do can be expressed in terms of some preference you have about the state of the world, viewed from the perspective of one possible-future-you, but all those possible-future-yous have to be treated symmetrically, just averaging their preferences. (By "preference" I just mean anything that plays the role of f.)
So, Phil says, if that's the only consistent way to treat all the different versions of you, then surely it's also the only consistent way to treat all the different people in the world.
(This is of course the controversial bit. It's far from obvious that you should see the possible-future-yous in the same way as you see the actual-other-people. For instance, because it's more credible to think of those possible-future-yous as having the same utility function as one another. And because we all tend to care more about the welfare of our future selves than we do about other people's. And so on.)
If so, then the following would be true: To act consistently, your actions must be such as to maximize the average of something (which, yes, need have nothing to do with the functions w, but it had better in some sense be the same something) over all actual and possible people.
I think Phil is wrong, but your criticisms don't seem fair to me.
I think this is part of the problem. Expected utility is based on epistemic uncertainty, which has nothing to do with objective frequency, including objective frequency across Everett branches or the like.
Nothing to do with objective frequency? Surely that's wrong: e.g., as you gain information, your subjective probabilities ought to converge on the objective frequencies.
But I agree: the relevant sense of expectation here has to be the subjective one (since all an agent can use in deciding what to prefer is its own subjective probabilities, not whatever objective ones there may be), and this does seem like it's a problem with what Phil's saying.
What? So if you had accurate information about the probability distribution of outcomes, then you couldn't use expected utility? I don't think that's right. <EDIT> In fact, it's exactly the reverse. Expected utility doesn't really work when you have epistemic uncertainty. </EDIT>
Yes, but maximizing the average of something implies neither utilitarianism nor indifference to equity, as Phil has claimed it does. I don't see how pointing this out is unfair.
You understand what I'm saying, so I'd very much like to know why you think it's wrong.
Note that I'm not claiming that average utilitarianism must be correct. The axioms could be unreasonable, or a strict proof could fail for some as-yet unknown reason. But I think the axioms are either reasonable in both cases, or unreasonable in both cases; and so expected-value maximization and average utilitarianism go together.
See (1) the paragraph in my comment above beginning "This is of course the controversial bit", (2) Wei_Dai's comment further down and my reply to it, and (3) Nick Tarleton's (basically correct) objection to my description of E(f) as being derived from "the f-values of all the different versions of you".
Here we disagree. f is the utility function for a world state. If it were an arbitrary function, we'd have no reason to think that the axioms should hold for it. Positing the axioms is based on our commonsense notion of what utility is like.
I'm not assuming that there are a bunch of individual w(i) functions. Think instead of a situation where one person is calculating only their private utility. f is simply their utility function. You may be thinking that I have some definition of "utilitarianism" that places restrictions on f. "Average utilitarianism" does, but I don't think "utilitarianism" does; and if it did, then I wouldn't apply it here. The phrase "average utilitarianism" has not yet come into play in my argument by this point. All I ask at this point in the argument, is what the theorem asks - that there be a utility function for the outcome.
I thikn you're thinking that I'm saying that the theorem says that f has to be a sum or average of the w(i), and therefore we have to be average utilitarians. That's not what I'm saying at all. I tried to explain that already before. Read steven0461's comment above, and my response to it.
The claim I am taking exception to is the claim that the vNM axioms provide support to (average) utilitarianism, or suggest that we need not be concerned with inequality. This is what I took your bullet points 6 and 8 (in the main post) to be suggesting (not to mention the title of the post!)
If you are not claiming either of these things, then I apologize for misunderstanding you. If you are claiming either of these things, then my criticisms stand.
As far as I can tell, most of your first two paragraphs are inaccurate descriptions of the theory. In particular, f is not just an individual's private utility function. To the extent that the vNM argument generalizes in the way you want it to, f can be any monotonic transform of a private utility function, which means, amongst other things, that we are allowed to care about inequality, and (average) utilitarianism is not implied.
But I've repeated myself enough. I doubt this conversation is productive any more, if it ever was, so I'm going to forego adding any more noise from now on.
I read both of them when they were originally posted, and have looked over them again at your exhortation, but have sadly not discovered whatever enlightenment you want me to find there.
As steven0461 said,
Not "proven", really, but he's got the idea.
I am pretty confident that you're mistaken. f is a utility function. Furthermore, it doesn't matter that the vNM argument can apply to things that satisfy the axioms but aren't utility functions, as long as it applies to the utility functions that we maximize when we are maximizing expected utility.
Either my first two bullet points are correct, or most of the highest-page-ranked explanations of the theory on the Web are wrong. So perhaps you could be specific about how they are wrong.