Raz has criticized egalitarianism as follows: equality is not in and of itself the sort of thing that can be good for anybody, but anything that is of intrinsic value "can benefit people."

I'm going to argue that either (a) Raz's critique also applies to act utilitarianism, or (b) there are views that are appropriately characterized as egalitarian that escape Raz's critique. Both options are interesting. Which one we take depends on how we read the phrase "benefit people".

Consider the following family CONSEQ of consequentialist views: We can describe the utility-state of the world as a vector *U*=(*u*_{1},...,*u*_{n}) of utilities, where *n* is the number of persons in history and *u*_{i} is a real number. There is a combination function *C* that assigns a real number *C*(*U*) to each such vector *U* (it's a multigrade function) and that satisfies two formal conditions:

- If
*U*' is any permutation of*U*, then*C*(*U*')=*C*(*U*). - If
*U*and*U*' are*n*-dimensional vectors, and every component of*U*is less than or equal to the corresponding component of*U*', and for at least one of the components equality does not hold, then*C*(*U*)<*C*(*U*').

*C*

_{1}(

*U*) where

*U*is the utility-state of the world.

Standard act utilitarianism is a member of CONSEQ—just let *C*(*U*)=*u*_{1}+...+*u*_{n}. But there are other views taht are members of CONSEQ. For instance, suppose that we believe that utilities cannot be negative (life is always worth living). Then we can define *p*-utilitarianism, for any positive real number *p*, by means of the combination function *C*_{p}(*U*)=(*u*_{1}^{p}+...+*u*_{n}^{p})^{1/p} (this is just the *L*^{p} norm on *R*^{n}, assuming the *u*_{i} are non-negative).

Now, let's ask whether Raz's critique applies to act utilitarianism (i.e., to 1-utilitarianism). Can it "benefit people" to maximize *C*_{1}(*U*)? I think there is at least a good *prima facie* case for a negative, and if so, we have a powerful critique of utilitarianism. Of course, if I am the *i*th person, it benefits me to maximize *u*_{i}. I am not benefited by maximizing *C*_{1}(*U*) in and of itself, unless this increase happens to be accompanied by an increase of *u*_{i}. And the same is true for every individual. Hence, maximizing *C*_{1}(*U*) is of no benefit to anybody.

On one interpretation of the "can benefit people" criterion, this has shown that utilitarianism fails the criterion, and hence (a) is true. I don't think this is the best reading of the "can benefit people". For instead, to release a virus that kills exactly one person "harm people" even if the concept of releasing such a virus does not entail of any specific person that *she* is killed.

Here seems to be the way to argue that utilitarianism escapes Raz's condition. That *U*' and *U* are such that *C*_{1}(*U*')>*C*_{1}(*U*) entails that at least one of the *u*'_{i} is greater than the corresponding *u*_{i}. Hence, it is better for somebody if we maximize *C*_{1}(*U*).

But now note that what I just said holds not just for 1-utilitarianism but for any theory in CONSEQ (the crucial condition is (2)). In particular, it holds for *p*-utilitarianism for all positive *p*. But *p*-utilitarianism is, in a precise sense, an egalitarian theory if 0<*p*<1. Why? Because the combination function *C*_{p}(*U*) favors more equal distributions when 0<*p*<1. Here is an easy exercise for the reader (just use the concavity of *f*(*x*)=*x*^{p}): if every component of *U*' is equal to the arithmetic mean of all the components of *U*, and equality does not reign among the components of *U*, then *C*_{p}(*U*')>*C*_{p}(*U*), assuming 0<*p*<1. Moreover, *C*_{p}(*U*) increases if we equalize any subset of the components of *U* (by equalizing a set of components, I mean replacing it by its arithmetic mean).

I suppose Raz could ask: What reason would we have for picking *p*-utilitarianism, other than our valuing equality in and of itself? And if we do value equality in and of itself, don't we fall prey to the "cannot benefit people" argument? But I think the response can simply be: We do not on this view separate out equality as a value to be added on—rather, we have a combination function which fits our intuitions better than *C*_{1}.

Do I think *p*-utilitarianism is plausible in the final analysis? Not at all. We would still have to answer the question of which value of *p* is the right one to choose, and the answer to that would seem to be *ad hoc*. One might think that *clearly* one should choose *p*=1 as that gives the simplest and most natural theory. But to a mathematician, *p*=2 (which is *anti*-egalitarian, in that it disfavors more equal distributions) is at least as natural, on the assumption that utilities can't be negative, since *C*_{2} is just the Euclidean norm on a quadrant of *R*^{n}. This, I think, is a fine argument against 1-utilitarianism (I blogged a version of it before).

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