Among the four axioms used to derive the von Neumann-Morgenstern theorem, one stands out as not being axiomatic when applied to the aggregation of individual utilities into a social utility:

Axiom (Independence): Let A and B be two lotteries with A > B, and let t \in (0, 1] then tA + (1 − t)C > tB + (1 − t)C .

In terms of preferences over social outcomes, this axiom means that if you prefer A to B, then you must prefer A+C to B+C for all C, with A+C meaning adding another group of people with outcome C to outcome A.

It's the social version of this axiom that implies "equity of utility, even among equals, has no utility". To see that considerations of equity violates the social Axiom of Independence, suppose my u(outcome) = difference between the highest and lowest individual utilities in outcome. In other words, I prefer A to B as long as A has a smaller range of individual utilities than B, regardless of their averages. It should be easy to see that adding a person C to both A and B can cause A’s range to increase more than B’s, thereby reversing my preference between them.

Oh! I realized only now that this isn't about average utilitarianism vs. total utilitarianism, but about utilitarianism vs. egalitarianism. As far as I understand the word, utilitarianism means summing people's welfare; if you place any intrinsic value on equality, you aren't any kind of utilitarian. The terminology is sort of confusing: most expected utility maximizers are not utilitarians. (edit: though I guess this would mean only total utilitarianism counts, so there's a case that if average utilitarianism can be called utilitarianism, then egalitarianism can be called utilitarianism... ack)

In this light the question Phil raises is kind of interesting. If in all the axioms of the expected utility theorem you replace lotteries by distributions of individual welfare, then the theorem proves that you have to accept utilitarianism. People who place intrinsic value on inequality would deny that some of the axioms, like maybe transitivity or independence, hold for distributions of individual welfare. And the question now is, if they're not necessarily irrational to do so, is it necessarily irrational to deny the same axioms as applying to merely possible worlds?

(Harsanyi proved a theorem that also has utilitarianism follow from some axioms, but I can't find a good link. It may come down to the same thing.)

I may be misunderstanding here, but I think there's a distinction you're failing to make:

Max expected utility over possible future states (only one of which turns out to be real, so I guess max utility over expected future properties of the amplitude field over configuration space, rather than properties over individual configurations, if one want's to get nitpicky...), while average/total/whatever utilitarianism has to do with how you deal with summing the good experienced/recieved among people that would exist in the various modeled states.

At least that's my understanding.

We should maximize average utility across all living people.

(Actually all people, but dead people are hard to help.)

This post seems incoherent to me :-( It starts out talking about personal utilities, and then draws conclusions about the social utilities used in utilitarianism. Needless to say, the argument is not a logical one.

Proving that average utilitarianism is correct seems like a silly goal to me. What does it even mean to prove an ethical theory correct? It doesn't mean anything. In reality, evolved creatures exhibit a diverse range of ethical theories, that help them to attain their mutually-conflicting goals.

This is back to the original argument, and not on the definition of expected utility functions or the status of utilitiarianism in general.

PhilGoetz's argument appears to contain a contradiction similar to that which Moore discusses in Principia Ethica, where he argues that the principle egoism does not entail utilitarianism.

Egoism: X ought to do what maximizes X's happiness.
Utilitarianism: X ought to do what maximizes EVERYONE's happiness

(or put Xo for X. and X_sub_x for Everyone).

X's happiness is not logically equivalent to Everyone's happiness. The important takeway here is that because happiness is indexed to an individual person (at least as defined in the egoistic principle), each person's happiness is an independent logical term.

We have to broaden the scope of egoism slightly to include whatever concept of the utility function you use, and the discussion of possible selves. However, unless you have a pretty weird concept of self/identity, I don't see why it wouldn't work. In that situation, X's future self in all possible worlds bears a relationship to X at time 0, such that future X's happiness is independent of future Everyone's happiness.

Anyway, using Von-Neumann Morgenstern doesn't work here. There is no logical reason to believe that averaging possible states with regard to an individual's utility has any implications for averaging happiness over many different individuals.

As addendum, neither average nor total utility provides a solution to the fairness, or justice, issue (i.e. how utility is distributed among people, which at least has some common sense gravity to it). Individual utility maximization more or less does not have to deal with that issue at all (their might be some issues with time-ordering of preferences, etc., but that's not close to the same thing). That's another sign Von-Neumann Morgenstern just doesn't give an answer as to which ethical system is more rational.

Average utilitarianism is actually a common position.

"Utility", as Eliezer says, is just the thing that an agent maximizes. As I pointed out before, a utility function need not be defined over persons or timeslices of persons (before aggregation or averaging); its domain could be 4D histories of the entire universe, or other large structures. In fact, since you are not indifferent between any two distributions of what you call "utility" with the same total and the same average, your actual preferences must have this form. This makes questions of "distribution of utility across people" into type errors.

Giving diminished returns on some valuable quantity X, equal distribution of X is preferable anyway.

I think you're quite confused about the constraints imposed by von Neumann-Morgenstern theorem.

In particular, it doesn't in any way imply that if you slice a large region of space into smaller regions of space, the utility of the large region of space has to be equal to the sum of utilities of smaller regions of space considered independently by what ever function gives you the utility within a region of space. Space being the whole universe, smaller regions of space being, say, spheres fitted around people's brains. You get the idea.

[Average utilitarianism] implies that for any population consisting of very good lives there is a better population consisting of just one person leading a life at a slightly higher level of well-being (Parfit 1984 chapter 19). More dramatically, the principle also implies that for a population consisting of just one person leading a life at a very negative level of well-being, e.g., a life of constant torture, there is another population which is better even though it contains millions of lives at just a slightly less negative level of well-being (Parfit 1984). That total well-being should not matter when we are considering lives worth ending is hard to accept. Moreover, average utilitarianism has implications very similar to the Repugnant Conclusion (see Sikora 1975; Anglin 1977).

Average utilitarianism has even more implausible implications. Consider a world A in which people experience nothing but agonizing pain. Consider next a different world B which contains all the people in A, plus arbitrarily many people all experiencing pain only slightly less intense. Since the average pain in B is less than the average pain in A, average utilitarianism implies that B is better than A. This is clearly absurd, since B differs from A only in containing a surplus of arbitrarily many people experiencing nothing but intense pain. How could one possibly improve a world by merely adding lots of pain to it?

The von Neumann-Morgenstern theorem indicates that the only reasonable form for U is to calculate the expected value of u(outcome) over all possible outcomes

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*.

This means that your utility function U is indifferent with regard to whether the distribution of utility is equitable among your future selves. Giving one future self u=10 and another u=0 is equally as good as giving one u=5 and another u=5.

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.