Average utilitarianism must be correct?

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.

u(world) = total_lifetime_veronica_happiness + total_lifetime_betty_happiness - | total_lifetime_veronica_happiness - total_lifetime_betty_happiness | 

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.)

We should maximize average utility across all living people.

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

(A) 2010: 6.5ua; 2011: 6.5ua; 2012: 6.5ua; 2013: 6.5ua
(B) 2010: 6.5ua; 2011: 7.0ua; 2012: 7.0ua; 2013: 7.0ua

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.

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.

[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?

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.

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.

I haven't done the math, so take the following with a grain of salt.

We humans care about what will happen in the future. We care about how things will turn out. Call each possible future an "outcome". We humans prefer some outcomes over another. We ought to steer the future towards the outcomes we prefer. Mathematically, we have a (perhaps partial) order on the set of outcomes, and if we had perfect knowledge of how our actions affected the future, our decision procedure would just be "pick the best outcome".

So far I don't think I've said anything controversial.

But we don't have perfect knowledge of the future. We must reason and act under uncertainty. The best we can hope to do is assign conditional probabilities of to outcomes based on our possible actions. But in order to choose actions based on probabilities rather than certainties, we have to know a little more about what our preferences actually are. It's not enough to know that one outcome is better than another, we have to know how much better. Let me give an example. If you are given the choice between winning a little money with probability .1 and a lot of money with probability .01, which option do you choose? Well, I haven't given you enough information. If "a little" is $1 and "a lot" is $1 million, you should go for "a lot". But if "a lot" is only $2, you're better off going for "a little".

So it turns out that if you want to have a consistent decision theory under these circumstances, there's only one form it can take. Instead of a partial order on outcomes, you have to express your preference for each outcome as a number, called the utility of that outcome. And instead of selecting action that leads to the best outcome, you select the actions that lead to the highest expected utility.

The function that maps outcomes to utilities is called the utility function, and it is unique (given your preferences) up to an positive affine transformation. In other words, you can multiply the whole utility function by a positive scalar, or add any number to it, and it's meaning does not change.

So you see, "maximize expected utility" doesn't mean you have to maximize say, your profit, or your personal happiness, or even the number of human lives you save. All it really means is that your preferences ought to be consistent, because if they are, and if you're trying the best you can to steer the future towards them, then "maximizing expected utility" is what you're already doing.

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.

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.

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 there's still some confusion here, because this isn't right. To take an example, suppose U = ln(u).

Average utilitarianism is actually a common position.

"Utility", as Eliezer says, is the thing that is maximized, not happiness. 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 distributons 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.

Average utilitarianism is actually a common position.

Because if your choices under uncertainty do not maximize the expected value of some utility function, you are behaving inconsistently (in a particular sense, axiomatized by Savage, and others - there's a decent introduction here).

These axioms are contestable, but the reasons for contesting them have little to do with population ethics.

Also, as I've said before, the utility function that consistent agents maximize the expectation of need not be identical with an experienced utility function, though it will usually need to be a positive monotonic transform of one.

Why do we think it's reasonable to say that we should maximize average utility across all our possible future selves

Because that's what we want, even if our future selves don't. If I know I have a 50/50 chance of becoming a werewolf (permanently, to make things simple) and eating a bunch of tasty campers on the next full moon, then I can increase loqi's expected utility by passing out silver bullets at the campsite ahead of time, at the expense of wereloqi's utility.

In other words, one can attempt to improve one's expected utility as defined by their current utility function by anticipating situations where they no longer implement said function.

I figured out what the problem is. Axiom 4 (Independence) implies average utilitarianism is correct.

Suppose you have two apple pies, and two friends, Betty and Veronica. Let B denote the number of pies you give to Betty, and V the number you give to Veronica. Let v(n) denote the outcome that Veronica gets n apple pies, and similarly define b(n). Let u_v(S) denote Veronica's utility in situation S, and u_b(S) denote Betty's utility.

Betty likes apple pies, but Veronica loves them, so much so that u_v(v(2), b(0)) > u_b(b(1), v(1)) + u_v(b(1), v(1)). We want to know whether average utilitarianism is correct to know whether to give Veronica both pies.

Independence, the fourth axiom of the von Neumann-Morgenstern theorem, implies that if the outcome L is preferable to outcome M, then one outcome of L and one outcome of N is preferable to one outcome of M and one outcome of N.

Let L represent giving one pie to Veronica and M represent giving one pastry to Betty. Now let’s be sneaky and let N also represent giving one pastry to Veronica. The fourth axiom says that L + N—giving two pies to Veronica—is preferable to L + M—giving one to Veronica and one to Betty. We have to assume that to use the theorem.

But that’s the question we wanted to ask--whether our utility function U should prefer the solution that gives two pies to Veronica, or one to Betty and one to Veronica! Assuming the fourth axiom builds average utilitarianism into the von Neumann-Morgenstern theorem.

I tend to think that utilitarianism is a pretty naive and outdated ethical philosophy - but I also think that total utilitarianism is a bit less silly than average utilitarianism. Having read this post, my opinion on the topic is unchanged. I don't see why I should update towards Phil's position.

I started reading the Weymark article that conchis linked to. We have 4 possible functions:

• u(world), one person's utility for a world state
• s(world), social utility for a world state
• U(lottery), one person's utility given a probability distribution over future world states
• S(lottery), social utility given a probability distribution over future world states

I was imagining a set of dependencies like this:

• Combine multiple u(world) to get s(world)
• Combine multiple s(world) to get S(lottery)

Weymark describes it like this:

• Combine multiple u(world) to get U(lottery)
• Combine multiple U(lottery) to get S(lottery)

Does anyone have insight into whether there is any important difference between these approaches?

This is the same ethical judgement that an average utilitarian makes when they say that, to calculate social good, we should calculate the average utility of the population

This is the part that I think is wrong. You don't assess your average utility when you evaluate your utility function, you evaluate your aggregate utility. 10 utilons plus 0 utilons is equivalent to 5 utilons plus 5 utilons not because their average is the same but because their total is the same.