Impartiality, Anonymity, and Caring Who

I. Introduction

Most of us, most of the time, are partial: we care more about some people just because of who they are. But most ethicists think we should sometimes be impartial. Classical utilitarians take this to the extreme, holding that we should always maximize expected global happiness without giving special weight to ourselves or those close to us. “Each counts for one,” as the slogan goes.¹

I have no objection to impartiality in itself. But I believe it has become a Trojan horse for classical utilitarians. In a range of arguments over the last 30 years, leading ethicists and economists have invoked impartiality in the guise of a principle that Richard Bradley calls:

Outcome Anonymity If two outcomes A and B involve the same welfare distribution (that is, the number of people at each level of welfare), and they differ only in who is at which level of welfare, then A and B are equally good.²

To illustrate, suppose A is good for Alex and bad for Beth, while B is the reverse:

Because A and B involve the same welfare distribution, Outcome Anonymity entails that they are equally good. There is also a useful way to visualize this principle, given a matrix like the one above: two outcomes must be equally good if the same numbers appear underneath them, only reordered.

Outcome Anonymity, though only one part of classical utilitarianism, has nevertheless been used to push ethics in a utilitarian direction. Frances Kamm and Iwao Hirose invoke Outcome Anonymity to argue for “counting the numbers” when choosing whether to rescue a large or small group from harm.³ Jacob Nebel appeals to a minimal version of the principle to argue against the “Person-Affecting Restriction,” which holds that A is better than B only if A is better than B for some individual person.⁴ John Broome uses Outcome Anonymity to establish his “same-number addition theorem,” which holds that A is better than B “if and only if it has a greater total of people’s wellbeing,” supposing B has the same number of people.⁵ Matthew Clark and Theron Pummer use Outcome Anonymity to rebut Larry Temkin’s “Disperse Additional Burdens” principle in favor of “effective altruism”—an approach to charity that, per William MacAskill, obeys Outcome Anonymity by definition.⁶ Zach Barnett uses Outcome Anonymity to argue more generally that a quantity of harm does not matter less if it is diffused across many people.⁷ Finally, Simon Blessenohl uses Outcome Anonymity to construct a dilemma for theories on which individuals may be differently risk-hungry (or risk-averse).⁸

Despite these profound implications, Outcome Anonymity is typically presented not as a controversial utilitarian doctrine, but as a straightforward consequence of impartiality.⁹ Nebel calls the principle a “requirement of impartiality,” explaining it as follows:

The intuition behind [Outcome Anonymity] is that, from an impartial perspective, we should care equally about each person, and therefore ought to be indifferent between distributions that are permutations of each other…¹⁰

Kamm implicitly relies on the principle and links it to the “impartial” point of view.¹¹ Hirose simply calls it “impartiality.”¹² MacAskill calls it “tentative impartiality.”¹³ Blessenohl considers it “such a plausible constraint on an impartial observer” that he treats it “as a fixed point,” at least in certain cases.¹⁴ Susumu Cato and Ken Oshani describe it as “a rather uncontroversial property for moral judgments,” which many regard as a “basic axiom.”¹⁵ Broome, who calls the principle “Impartiality Between People,” says “it is hard to think of any doubts that can plausibly be raised” against it, so long as “[e]veryone must count equally.”¹⁶

This attitude towards Outcome Anonymity has also been influential in public policy. In a leading policy textbook, Matthew Adler says that Outcome Anonymity “formalizes the ethical attitude of impartiality,” giving “equal weight” to “each person’s well-being.”¹⁷ He later calls Outcome Anonymity “the formal expression of ethical impartiality used in the [social welfare function] literature,”¹⁸ remarking that the principle is “very hard to dispute,” as it “crystallizes the ethical commitment to impartiality.”¹⁹

But is Outcome Anonymity truly a requirement of impartial concern?

I argue that it is not. Some impartial principles entail that permuting who is who can make things better (§§III, IV, VIII). Others entail that permuting who is who can lead to value incomparability (§VI). From these two sorts of counterexamples, I think we should conclude that Outcome Anonymity demands something more than impartiality: in particular, it demands that we not care who is who in a distribution of welfare. Being partial is one way of “caring who,” but another way is to care about benefits and burdens to particular people, as some impartial moralists do.

To avoid the counterexamples, we could weaken Outcome Anonymity, as Campbell Brown suggests in an important article.²⁰ But this risks making the principle redundant. There is a more minimal principle, originally known as “Anonymity,” which says that no person’s welfare as such should have a special influence on which outcomes are better.²¹ I prove that Anonymity entails the relevant weakening of Outcome Anonymity (§IX).

My conclusion is that, fundamentally, impartial concern for individual welfare requires only Anonymity (§X). Arguments that invoke Outcome Anonymity may seem like they are tracing out the demands of impartiality, but in fact they are describing what happens to ethics, economics, and public policy if we do not care who is who in any distribution, treating individuals’ welfares as fungible goodies rather than distinct dimensions of value.

There may yet be sound reasons for adopting a more utilitarian social policy. For example, each member of society might rationally prefer a standing policy of “counting the numbers” in an emergency—say, a shortage of hospital beds in a pandemic—because such a policy has a higher expected value for each of them, given the lower chances of winding up among the few.²² A similar argument could justify a policy of providing a bigger total of diffuse benefits rather than a smaller sum of concentrated ones²³—when building a train line, for instance, a city might destroy one neighborhood to shave time off of millions of commutes.

But whatever the merits of the utilitarian approach, it cannot be derived from our ordinary, uncontroversial ideal of impartiality, since this ideal does not by itself entail Outcome Anonymity. Or so I shall argue.

II. Anonymity and Impartiality

Anonymity and Outcome Anonymity come to ethics from social choice theory, the subfield of economics that studies how individual votes or welfares can be aggregated. Confusingly, both principles have standardly gone by the name “Anonymity,”²⁴ and sometimes the two principles are mixed up or combined in citations.²⁵

But these principles are different, and the difference matters.

Anonymity was introduced by Kenneth May²⁶ in his axiomatization of Majority Rule, a “minor classic” in social choice theory.²⁷ The gist of May’s result is that, when people are voting on a pair of options, Majority Rule is the only rule that always delivers a result, counts every vote positively, and treats both voters and candidates equally. Anonymity is what guarantees the equal treatment of each voter.²⁸

The basic idea of Anonymity is that, however much people’s preference orderings might matter, it shouldn’t matter who has which ordering. To illustrate, imagine an election where each voter reports their preferences on a ballot with their name on the top. Anonymity would say: the outcome of the election cannot change if we merely shuffle the names on the ballots. (So we cannot use a rule that ignores the ballots of people whose names begin with “T,” or a rule that gives complete priority to the ballot of a lucky “dictator.”) Since the names make no difference, we can just as well erase them—the voting is, in this way, “anonymous.”

There are several ways to formalize Anonymity depending on our choice of framework. Different frameworks use different objects to model individual preferences—orderings, choice functions, welfare functions, and so forth.²⁹ Here, we will use welfare functions, and we will be concerned with ethical value rather than collective choice.

Let O be a finite set of outcomes, and let P be a finite set of individuals denoted 1, 2, …, n. For any i in P, we denote by W_i a welfare function—that is, a function from each x in O to a real number. We then let value relations be given by a reflexive relation ≿ on O that is a function of an n-tuple of individual welfare functions (a “profile” U): in symbols, ≿ = g (W₁, W₂, …, W_n), where the ith argument denotes the welfare function of person i. We call ≿ “at least as good as” (in light of U). We define betterness (≻) and equal goodness (∼) as follows:

• x ≻ y =_df x ≿ y and not y ≿ x.

• x ∼ y =_df x ≿ y and y ≿ x.

We do not assume that ≿ is transitive (x ≿ y ≿ z implies x ≿ z) or complete (x ≿ y or y ≿ x).³⁰

Now we can state Anonymity precisely:

Anonymity If g maps a certain n-tuple of individual welfare functions to ≿, then g also maps any reordering of that n-tuple to ≿.

For example, if g maps (W₁, W₂, …, W_n) to ≿, then g also maps to ≿ if we reverse the order of the welfare functions—that is, g(W_n, W_{n – 1}, …, W₁) = ≿.

This definition is a bit technical, but there is an easy way to visualize Anonymity. Recall our matrix from above, with people in rows, outcomes in columns, and welfares in cells:

Anonymity states that, whatever the value relation between A and B in this case, the same relation must hold if we switch the rows of numbers, as in the following:

Equivalently, we could have switched the names of Alex and Beth, leaving the columns in place.

Outcome Anonymity, by contrast, has to do with permuting individuals’ welfares in a certain pair of outcomes (rather than permuting welfare functions, which are defined over all outcomes):

Outcome Anonymity If (W₁(x), W₂(x), …, W_n(x)) is a reordering of (W₁(y), W₂(y), …, W_n(y)), then x ∼ y.

Thus A and B, which merely swap Alex’s welfare for Beth’s, must be equally good. Clearly, Anonymity and Outcome Anonymity are related. But they are not the same.³¹

Anonymity tells us that no one’s welfare has a special effect on what’s at least as good as what. Outcome Anonymity tells us that nobody’s welfare in a particular outcome can have any special effect on that outcome’s value. Thus we cannot say, “Alex is at welfare level –1 in B, so her being at level 1 in A has a special significance for A’s value.” Her being at that level in A must count the same as anybody else’s being at that level in A—thus we cannot care who is at any particular position of the distribution. Each person in an outcome is just a “somebody.”

One immediate problem for Outcome Anonymity, familiar from social choice theory,³² is that it allows us to derive facts about value merely from facts about individual welfare. This is problematic if value can depend on other factors besides welfare. For example, many would say that an outcome is better (all else equal) if it features more intrinsic beauty or fewer violations of rights. Imagine, then, that two outcomes have the same welfare distribution, but one brims with beautiful landscapes while the other teems with rights violations. Outcome Anonymity says the outcomes must be equally good. But isn’t the pretty one better?

There are two ways around this problem. First, we might make Outcome Anonymity more general, so that it says two outcomes are equally good if the only difference between them consists in the identities of those who feature in the outcome (see fn. 12, above). Beauty, rights, and everything else must be held fixed. But this is difficult to make precise, since on many views of personal identity, it is not possible to freely permute identities while holding everything else fixed: who I am may depend on where I come from and what I am like.³³ Second, and more promising, we can just restrict our focus to the value of individual welfare. This becomes easier if we use examples where other dimensions of value are held fixed, as I shall do in this paper.

With all that in place, we can ask our central question. What is required for impartial concern for the welfare of each person: Anonymity, Outcome Anonymity, neither, or both?

Impartiality, I believe, clearly requires Anonymity. Any ethical theory that violates Anonymity must privilege or denigrate somebody’s welfare in some way. Consider, for example, a theory on which self-interest carries more weight than the interests of strangers,³⁴ or consider ethical egoism, on which only one’s own welfare matters.

What about Outcome Anonymity? The “intuition” here is that, “from an impartial perspective, we should care equally about each person, and therefore ought to be indifferent between distributions that are permutations of each other…”³⁵ This sounds reasonable. But let’s now turn to some potential counterexamples, starting with the first kind of case, where permutations make things better.³⁶

III. Better Permutations: Majority Rule

Could there be an impartial rule that violates Outcome Anonymity? As it happens, there is one right under our noses: Majority Rule, the principle for which Anonymity was invented.³⁷

Majority Rule x ≻ y if and only if the number of people better off in x than in y is higher than the number of people better off in y than in x.

Traditionally, Majority Rule has been conceived as a method for collective decisions. But it can also be viewed as an ethical principle,³⁸ and that is how I propose to treat it here—not because it is especially plausible (it clearly isn’t), but because its formal properties make it a useful counterexample.

Suppose there are three outcomes C, D, and E which result in the following distributions of welfare for Cath, David, and Elise:

Since these outcomes involve the same distribution of welfare, Outcome Anonymity entails that they must all be equally good. But Majority Rule says that no distinct pair is equally good—for example, D is deemed better than C, since D is better for both David and Elise.

Again, I concede that Majority Rule for ethics isn’t plausible. Not only does it ignore the sizes of benefits and losses to each individual: it allows for cycles of betterness. Look again to the example above. Just as a majority prefers D to C, so does a majority prefer E to D and C to E. This result, known as Condorcet’s Paradox, violates the transitivity of betterness: C ≻ E ≻ D ≻ C. Most ethicists find such cycles intolerably implausible.³⁹

But I’m not saying that Majority Rule is plausible, only that it’s impartial. Since Majority Rule is anonymous, it doesn’t give anybody’s welfare special weight: there is no privileged “dictator,” nor any “disenfranchised.” Majority Rule does, admittedly, dispense with certain bits of information that might plausibly be thought to be relevant. But the same complaint can be alleged against many other clearly impartial principles, including classical utilitarianism.⁴⁰ Just as our criterion for deductive validity should apply to all arguments, not just the plausible ones, our criteria for impartiality should apply to all principles.

Majority Rule is our first counterexample: it violates Outcome Anonymity without being partial. The second counterexample can be illustrated using the very same case—just in reverse!

IV. Better Permutations: Weak Anti-Aggregation

Classical utilitarians always aggregate the welfares of different individuals. But some philosophers don’t aggregate small benefits (or harms) so as to outweigh large ones. Better to save one life than to cure one billion headaches they say, and better to save one person from electrocution even if this means denying many others fifteen minutes of amusement.⁴¹ “Limited aggregationists” believe we can aggregate benefits and harms only when they are “close enough” in size to the largest benefit or harm with which anyone is faced.

To simplify, let’s suppose that being “close enough” for the purposes of aggregation means being less than one unit of welfare apart. On this view, we can aggregate benefits of size 1 to outweigh a benefit of size 1.5, but not to outweigh a benefit of size 2 or greater. In symbols, we get:

Weak Anti-Aggregation x ≻ y if, for some i in P, W_i(x) – W_i(y) ≥ W_j(y) – W_j(x) + 1 for any j in P.⁴²

Now apply this view to the Condorcet-style example above. Outcome Anonymity says that C ∼ D ∼ E ∼ C. But Weak Anti-Aggregation entails that C ≻ D ≻ E ≻ C. For Cath, C is better than D by a margin of 2 units of welfare, while D is better for David and Elise by only 1 unit each. Since the difference between the two degrees of benefit is at least 1 unit, we do not aggregate, and Cath’s bigger benefit wins out. By the same reasoning, David’s 2-unit benefit makes D better than E, and Elise’s 2-unit benefit makes E better than C.

(Visually, we can imagine the three people rotating right-to-left along an Olympic platform, with the silver medalist on the left, gold medalist in the center, and bronze medalist on the right, with medals corresponding to welfare levels. The rise from bronze to gold outweighs the drops from gold to silver and silver to bronze, given Weak Anti-Aggregation. This is the reverse of Majority Rule, which holds that two drops always outweigh a single rise!)⁴³

So Weak Anti-Aggregation conflicts with Outcome Anonymity. And yet it seems perfectly impartial, precisely because it obeys Anonymity. The first to notice this, to my knowledge, was Campbell Brown. As Brown puts it, those who accept a version of Weak Anti-Aggregation “may instead reject [Outcome Anonymity].”⁴⁴ He continues:

[Outcome Anonymity] is intended to capture the ideal of “moral equality”: all individuals’ interests should be given equal weight, and no one should be favoured simply because of who she is. But [Outcome Anonymity] actually requires more than this.⁴⁵

As Brown explains, Weak Anti-Aggregation ought to count as impartial: the bigger benefit to Cath matters more “not because of who she is, but because of her situation.”⁴⁶ Or, as Nebel would put it, the Weak Anti-Aggregationist “need not care more” about Cath, but “may simply care more about preventing severe harms…than preventing minor ones.”⁴⁷

Again, I am not saying that Weak Anti-Aggregation is plausible, or that we should believe in cycles of betterness.⁴⁸ I am just saying that this principle is impartial.

To sum up, we have seen two counterexamples to the widespread claim that impartiality requires Outcome Anonymity. Majority Rule and Weak Anti-Aggregation both entail that we can make things better or worse just by permuting who is at which welfare level. And both principles appear to be impartial precisely because they respect Anonymity, which suggests that Anonymity may be not only necessary but sufficient for impartiality.

V. Outcome Anonymity for Transpositions

Brown suggests a quite different requirement of partiality: “transposing” people in the welfare distribution always results in something equally good.

To generate the last two counterexamples, we had to rotate the three people through the various welfare levels. On each step from C to D to E to C, the person at level 3 drops to level 2, the person at level 2 drops to level 1, and the person at level 1 ascends to level 3. It takes three such rotations for people to end up back where they started. By contrast, some permutations, called transpositions, return people to their starting positions if performed twice in a row. We saw this earlier in our simple two-person case.

If we start with A and permute who is at 1 and who is at –1, the result is B. Do the same permutation again, and we return to A—so it’s a transposition.

The interesting thing about transpositions is that losses and gains are perfectly matched. Unlike rotations, transpositions can only swap the positions of duos—and whatever one member of the duo gains (or loses), the other member loses (or gains).

Brown’s proposal, then, is that impartiality requires:

Outcome Anonymity for Transpositions If (W₁(x), W₂(x), …, W_n(x)) is a transposition of (W₁(y), W₂(y), …, W_n(y)), then x ∼ y.

As Brown notes, Weak Anti-Aggregation is consistent with Outcome Anonymity for Transpositions. So is Majority Rule.

But is Brown’s principle a requirement of impartiality? To settle the question, let’s turn to counterexamples of the second kind, which involve transpositions of just two people.

VI. Incomparable Transpositions: Kamm’s Argument

We can start with Kamm’s “argument for best outcomes,” which tries to show that “the numbers count” when giving benefits to more or fewer people.⁴⁹

Suppose Fumi, Gerry, and Heidi are dying from a rare illness, and there is only one dose left of the drug that can save them. We can give the dose to Fumi, who needs the whole dose to survive, or we could split it between Gerry and Heidi, who each need only one half. We also could, perversely, give Gerry one half and let Heidi die pointlessly. So, we have three possible outcomes:

Taurek would say that, in such a case, it is not better if both Gerry and Heidi survive than if only Fumi survives.⁵⁰ But most ethicists disagree. They would insist that “the numbers should count,” so that two lives outweigh one.⁵¹ This common view certainly seems intuitive.

Kamm’s insight is that we can derive this intuitive view from deeper premises. In particular, we just need Outcome Anonymity and two other principles:

Strong Pareto If x is better than y for someone, and x is at least as good as y for everyone else, then x is better than y.

Substitution of Equals If x is better than y, and y is equal in goodness to z, then x is better than z.

Given Strong Pareto, we can say that it is better if Heidi survives in addition to Gerry: GH ≻ G.⁵² Given Outcome Anonymity, we can say that Gerry’s survival and Fumi’s are equally good: F ∼ G. And given Substitution of Equals, we can conclude that it is better for Gerry and Heidi to survive than only Fumi: GH ≻ F. The same reasoning shows that, in general, the numbers count, in the sense that it’s better if more people receive benefits of a given size.

This argument crucially relies on Outcome Anonymity, which Kamm and others support by appealing to the “impartial point of view.” As we have seen (§§III–IV), impartiality does not in fact require Outcome Anonymity. But perhaps it requires a weaker form of the principle, such as Outcome Anonymity for Transpositions. Even if some permutations make things better, we might wonder how merely transposing two people in a distribution—Fumi and Gerry—could matter. Certainly, an impartial observer should not care more about Fumi’s welfare than Gerry’s, or vice versa. Does it follow that, on an impartial view, F and G are equally good?

Not necessarily. They might be incomparable in value, where this means that neither is better or worse than the other, nor are they equally good.⁵³

Incomparability can arise when two dimensions of value conflict and neither determinately takes precedence.⁵⁴ We can illustrate using an example from Chang.⁵⁵

Who is the better artist—Michelangelo or Mozart? Though neither seems better overall, each is better in some respects: one is a visual genius, the other a brilliant composer of melodies and harmonies. So, it may seem inapt to call the two artists equally good, which would imply that they do not relevantly differ in value. Given this, the only option left is that the two artists are incomparable.

Being incomparable is a bit like being equally good. But equals can be substituted for one another in value relations, and the relation will still hold. We cannot say the same for incomparables. Sometimes, x is better than y, and y and z are incomparable, and yet x is not better than z.⁵⁶

Imagine a slightly improved version of Mozart—call him Mozart+—who is just like the original, except slightly more creative. Since Mozart+ is better than the original in one way, and at least as good in all others, it seems clear that he is indeed a better artist than the original. But is he better than Michelangelo? Not obviously. If the improvement is small enough, it may not break the tie. Mozart+ may have more advantages over Michelangelo than did the original composer, but Michelangelo remains superior in some respects—Mozart+ couldn’t paint the Sistine Chapel! Given that we still have nearly as much conflict between the various competing dimensions of value, the two may be incomparable.

We can apply this same reasoning to Kamm’s argument. Saving Fumi and saving Gerry might be good in different ways, even if the two people are qualitatively alike, simply because the welfare of one person is not fungible with that of another.⁵⁷ You can reasonably care who suffers the loss even if you do not care more about either person. That is why, depending on who is saved, you may have different reasonable regrets. (It is not just that you regret that someone or other died; you regret that this person died.) This suggests that F and G are not equally good but incomparable, which would immediately block Kamm’s argument. If F and G are not equally good, we cannot freely substitute them. In particular, we cannot infer from GH ≻ G that GH ≻ F. When two things are incomparable, an improvement on one might still be incomparable to the other. And so Taurek could insist that GH is incomparable to F, even though it is better than G.

Importantly, Taurek could say all of this without being partial. The point here is not that Fumi matters more than Gerry. The point is that a loss to any one person is incomparable in value to any number of equal-sized losses to others. The principle here, perfectly anonymous, is this:

Taurekian Anti-Aggregation If max_i(W_i(x) – W_i(y)) = max_i(W_i(y) – W_i(x)), then x and y are incomparable in value.

More simply: x and y are incomparable if the biggest gain anyone receives from x over y is equal to the biggest gain anyone receives from the reverse. This principle, like Majority Rule, is dubious. But since it obeys Anonymity, we cannot accuse it of playing favorites with any person or group.

VII. Weakening Outcome Anonymity

We have found a new kind of counterexample. Taurekian Anti-Aggregation is impartial, and yet it conflicts with Outcome Anonymity: changing who is who might result in an incomparable option, rather than one that is equally good. This counterexample, moreover, does not require us to rotate three people’s positions. We need only transpose the welfares of two individuals—Fumi and Gerry, in our case. So, the example works even against weaker forms of Outcome Anonymity that are restricted to transpositions of a pair of people.

There is, however, another way to weaken Outcome Anonymity to avoid this counterexample. We could say that changing who is who must leave things equally good or incomparable. But this will not help with the earlier counterexamples of Majority Rule and Weak Anti-Aggregation, where changing who is who can make things better or worse.

What if we weaken both ways at once? What if transposing people—that is, swapping out pairs of individuals in the distribution of welfare—must leave things either equally good or incomparable? The result would be a very weak principle that avoids all of our counterexamples:

Weak Outcome Anonymity for Transpositions If (W₁(x), W₂(x), …, W_n(x)) is a transposition of (W₁(y), W₂(y), …, W_n(y)), then either x ∼ y or x and y are incomparable.

But there is a final counterexample that makes trouble for even this principle.

VIII. Better Transpositions: Paretian Dependence (on “Irrelevant Alternatives”)

I argued that Taurek can block Kamm’s argument by holding that different lives are incomparable in value, rather than being equally valuable in a sense that implies fungibility.

But there is another way of defending Taurek—and rejecting Outcome Anonymity—that does not invoke incomparability. In a choice between saving only Fumi (F), only Gerry (G), and both Gerry and Heidi (GH), Outcome Anonymity tells us that F is just as good as G. But some say that G is worse than F, because G involves letting Heidi die gratuitously.⁵⁸ To be clear, the point is not just that G is worse than GH—which follows immediately from Strong Pareto. The point is rather that G becomes worse when GH is on the menu—and so G becomes worse relative to F.

This violates the aptly named:

Independence of Irrelevant Alternatives For any outcomes x and y and any profiles U = (W₁, W₂, …, W_n) and U* = (W*₁, W*₂, …, W*_n) where ≿ = g(U) and ≿* = g(U*), if W_i(x) = W*_i(x) and W_i(y) = W*_i(y) for i = 1, 2, …, n, then x ≿ y if and only if x ≿* y.

While this is a mathematical mouthful, the idea behind “Independence” is simple: how two outcomes compare in value should only depend on people’s welfares in those two outcomes. We cannot say that F is better than G merely because of the presence of GH—an “irrelevant” alternative.

To be clear, this does not quite say that people’s welfares must matter in the same way given any pair of outcomes. To establish that, we need a stronger principle:

Strong Neutrality For any outcomes w, x, y, and z and any profiles U = (W₁, W₂, …, W_n) and U* = (W*₁, W*₂, …, W*_n) where ≿ = g(U) and ≿* = g(U*), if W_i(x) = W*_i(w) and W_i(y) = W*_i(z) for i = 1, 2, …, n, then x ≿ y if and only if w ≿* z.⁵⁹

This adds a kind of impartiality towards outcomes, ruling out the possibility that a welfare distribution might matter differently because it occurs in, say, G rather than in F.

If Taurek gives up Strong Neutrality, he can reject Outcome Anonymity, thereby blocking Kamm’s argument. This breaks with “Taurekian Anti-Aggregation,” as I called it, in favor of a quite different principle:

Paretian Dependence (on “Irrelevant Alternatives”) If (W₁(x), W₂(x), …, W_n(x)) is a reordering of (W₁(y), W₂(y), …, W_n(y)), and there is some z such that z ≻ y by Strong Pareto but no z such that z ≻ x by Strong Pareto, then x ≻ y.

In words, where Outcome Anonymity normally would create a tie, one outcome can be made worse if it alone loses to an “irrelevant alternative” in the set of outcomes O that is better for someone and as good for all others.

Taurek’s critics might reply that Paretian Dependence, which violates Outcome Anonymity (given a rich enough domain), must be partial. But why think that? Paretian Dependence is anonymous. No one is being denigrated or prioritized because of who they are. The principle just deprecates options that are gratuitously bad for someone, no matter the “someone.” In our example, G is worse than F given the presence of GH. But this is not an insult to Gerry. If the only alternative to F and G had been FH—that is, saving Fumi and Heidi—then G would have been better than F.

This is our final counterexample to the inference from impartiality to Outcome Anonymity. The example, moreover, violates even Weak Outcome Anonymity for Transpositions. Merely swapping Fumi and Gerry (a transposition) results in G instead of F, and G is worse, not equal or incomparable, in light of GH.

IX. Why Anonymity is More Fundamental

Could we weaken Outcome Anonymity a third time to deal with Paretian Dependence? This final counterexample relies on a violation of Strong Neutrality. So, a natural option is to add a condition to our doubly weakened principle: if Strong Neutrality holds, then so does Weak Outcome Anonymity for Transpositions.⁶⁰

I believe this principle is weak enough to be a true requirement of impartiality. But it is too weak to be an independent requirement, because we can derive it from Anonymity, along with the minimal assumption that our function g has a Universal Domain—we do not, in other words, rule out any logically possible profiles as inadmissible.

Theorem 1 Given Strong Neutrality and Universal Domain, Anonymity implies Weak Outcome Anonymity for Transpositions. (See the Appendix for a proof.)

Outcome Anonymity, sufficiently weakened, turns out to be a shadow cast by Anonymity.

In light of this result, Anonymity seems to be the more fundamental requirement of impartiality, especially because Theorem 1’s converse does not hold (see Theorem 2 in the Appendix).

But this is just to say that Anonymity is the more basic necessary condition for impartiality. Is it sufficient? I don’t think Anonymity suffices for all kinds of impartiality. For example, Anonymity has little to do with the sort of “impartial justice” we expect from judges, who must treat like claims alike in the course of legal reasoning. Maybe legal impartiality is governed by an analogue of Anonymity. (E.g., no one’s claims should be accorded special weight.) But here I am endorsing only a more limited thesis: Anonymity guarantees impartial concern for individual welfare.

Nevertheless, as one reviewer urges, Anonymity may still be “much too weak” to support even this limited species of impartiality, since it does not say anything about “indifference to race.” Anonymity is defined relative to a framework that does not operate on information about race, gender, or other sorts of group characteristics. Don’t we need to add more to rule out bigotry?

Surprisingly, we might not: even by itself, Anonymity can block a racist ordering. To illustrate, suppose Theory R is like classical utilitarianism, but it gives White people’s welfare double the weight of Black people’s welfare. Now consider a profile featuring one white and one black person:

Theory R says that A is better than B, since 8 > 7. But if we permute the rows (or the names in front of them), Theory R says the opposite: suddenly B is better than A—again, 8 > 7. The racist Theory R thus violates Anonymity, which therefore manages to rule out a racist judgment without having to mention race explicitly.⁶¹

There may be other cases where Anonymity fails to guarantee impartial concern for welfare. I myself am inclined to think that Anonymity guarantees such impartiality even in more complex cases, including ones with infinite populations. But I won’t insist on that here. The important point for this paper is that Anonymity is necessary for impartiality, which is the kernel of truth in the too-strong claim that impartiality requires Outcome Anonymity.

X. Conclusion

The “intuition behind [Outcome Anonymity],” to quote Nebel once more, “is that, from an impartial perspective, we should care equally about each person, and therefore ought to be indifferent between distributions that are permutations of each other…”⁶² I have argued that this intuition has less force than many have thought.

In its usual form, Outcome Anonymity is too strong to be a requirement of impartiality: we can care about people’s welfares equally even if we think that some permutations are incomparable (as in the case of Taurekian Anti-Aggregation), or that some are better than others (as in the cases of Majority Rule, Weak Anti-Aggregation, and Paretian Dependence). Suitably weakened, however, Outcome Anonymity follows from little more than Anonymity, which thus has a claim to being the more fundamental requirement of impartiality.

To be impartial, we cannot care more about one person than another. But even an impartial moralist can care who is who in a welfare distribution. The majoritarian can ask, “How many people are better off in the alternative?” The weak anti-aggregationist can ask, “Does anyone suffer a loss greater than anybody’s gain?” The Taurekians can ask, “Does the alternative save a bigger set of people or a superset?” These theorists do not just care how many people are at which welfare level. They care who is at each level, because they think a person’s welfare level in an outcome has a different moral impact depending on how well off that person is in the alternative. They do not just care that somebody is at level 1. They care that somebody is at level 1 here who is at level 0 there.

Such “caring who” is what we lose sight of when we replace Anonymity with Outcome Anonymity; it is what distinguishes our concepts of benefit and complaint from those of absolute weal and woe;⁶³ and it is part of what it means to care about persons—not as containers of welfare or abstract statistics, but as separate persons, who stand to gain or lose.

As I conceded at the start, there may well be strong reasons to take a more utilitarian approach to social policy, even beyond the reasons in favor of utilitarianism itself. Perhaps, in the end, we should “count the numbers,” sum up utilities, ignore individual risk profiles (Blessenohl, “Risk Attitudes”), and remain indifferent to the concentration or diffusion of harm. Perhaps we should not worry about how people are “affected” and should merely focus on total utility. I am happy to concede that such policies may enjoy strong support from various sources.

But I do not concede that we should implement these policies on the grounds that they are demanded by impartiality. The mere invocation of impartiality, in this context, has no force. Views that satisfy Anonymity, though not Outcome Anonymity, are perfectly impartial in the familiar and uncontroversial sense that they do not give priority to any individual’s welfare simply on the grounds of who that individual is.

There is nothing wrong if the utilitarians and sympathizers wish to develop their own, more demanding conception of impartiality, enriched with plausible though controversial assumptions (see the Appendix on “formal welfarism”). The trouble only starts when a utilitarian concoction is passed off as unadulterated common sense.

Acknowledgements

My thanks to Tom Sinclair, Nir Eyal, Kaushik Basu, Daniel Star, Sam Fullhart, Cara Nine, Felipe Doria, Brian Hedden, Iwao Hirose, Pietro Cibinel, Harvey Lederman, Geir Asheim, Geoff Sayre-McCord, and audiences at the Boston University Ethics Seminar, Montreal Axiology Workshop, UNC PPE Retreat, Duke University Philosophy WIP Seminar, 2025 Conference of the American Association of Mexican Philosophers, MIT-ing of the Minds, 2025 PPE Society Conference, and Oxford Moral Philosophy Seminar. I am also delighted to thank Anna Stilz and the reviewers at Free & Equal for pushing me to streamline the paper, clarify key points (especially the role of Anonymity), use a more descriptive title, and ask, “So what?” Finally, special thanks to Jake Nebel for his extraordinarily illuminating and detailed comments, which (among other things) helped me better appreciate the history of anonymity principles and how these principles are expressed differently in single-profile and multi-profile frameworks.

Competing Interests

The author has no competing interests to declare.

Appendix

First, we restate and prove:

Theorem 1 Given Strong Neutrality and Universal Domain, Anonymity implies Weak Outcome Anonymity for Transpositions.

Proof Given any profile U = (W₁, W₂, …, W_n) and outcomes x and y, we show that x ∼ y or x and y are incomparable if, for some transposition σ of {1, 2, …, n}, (W_σ1(x), W_σ2(x), …, W_σn(x)) = (W₁(y), W₂(y), …, W_n(y)). Assume for conditional proof that such a σ exists. Given Universal Domain, there must exist a profile U* = (W_σ1, W_σ2, …, W_σn). Since U* is a permutation of U, Anonymity ensures that (i) x ≿ y if and only if x ≿* y. (Where ≿ = g(U) and ≿* = g(U*).) By our starting assumption, W_i(y) = W_σi(x) for i = 1, 2, …, n. Since σ is a transposition, we also have the reverse: W_i(x) = W_σi(y). Given Strong Neutrality, it follows that (ii) y ≿ x if and only if x ≿* y. From the biconditionals (i) and (ii), we can conclude that x ≿ y if and only if y ≿ x, which is equivalent to the desired claim that either x ∼ y or x and y are incomparable. □

Against certain background conditions, we have shown that Anonymity implies Weak Outcome Anonymity for Transpositions. But does the converse implication hold? We prove that it does not.

Theorem 2 Given Strong Neutrality and Universal Domain, Weak Outcome Anonymity for Transpositions does not imply Anonymity.

Proof We proceed by giving a counterexample—a variation on classical utilitarianism that allows for some incomparability, but only when two options have equal total welfare and a tradeoff for a certain chosen person’s welfare. Let S(x) be the sum of individual welfares in x. Define g so that (i) x ≻ y if and only if S(x) > S(y), and (ii) x ∼ y if and only if S(x) = S(y) and W₁(x) = W₁(y). (Otherwise, x and y are incomparable.) Since permuting who is at which welfare level cannot change the sum of individual welfares, such permutations can only result in equal goodness or incomparability; g therefore obeys Weak Outcome Anonymity. Since whether x ≿ y depends only on individual welfare levels in x and in y, g obeys Strong Neutrality. Now assume that g has a Universal Domain, and consider the profile U given by:

Since the sum of individual welfares is 2 in both outcomes, and W₁(A) = W₂(B), it follows that A ∼ B. If g obeys Anonymity, then A and B would still be equally good given an otherwise similar profile that permutes the welfare functions of persons 1 and 2, as in U*, below:

Since person 1 does not have the same welfare level in A as in B, it is not the case that A ∼* B. Therefore, g violates Anonymity, so Anonymity does not follow from Strong Neutrality, Universal Domain, and Weak Outcome Anonymity for Transpositions.

In fact, this result can easily be strengthened. Our counterexample, the utilitarian variant, obeys Weak Outcome Anonymity in general, not just for transpositions. It also respects the following conditions on overall value relations:

Transitivity (of ≿) If x ≿ y ≿ z, then x ≿ z.

Congruence of Incomparables (with respect to ≻) If x and y are incomparable, then z ≻ x if and only if z ≻ y, and x ≻ z if and only if y ≻ z.

We prove this in the form of a:

Lemma Suppose (i) x ≻ y if and only if S(x) > S(y), and (ii) x ∼ y if and only if S(x) = S(y) and W₁(x) = W₁(y). Then Transitivity and Congruence of Incomparables both obtain.

Proof We begin with Transitivity. Assume x ≿ y ≿ z. Then S(x) ≥ S(y) ≥ S(z). There are two cases to consider. First, suppose either S(x) > S(y) or S(y) > S(z). Then S(x) > S(z), and therefore x ≻ z. Second, suppose that neither S(x) > S(y) nor S(y) > S(z). Then S(x) = S(y) = S(z), which implies x ∼ y ∼ z. So W₁(x) = W₁(y) = W₁(z), and by transitivity W₁(x) = W₁(z). So x ∼ z. In either case, x ≿ z, which concludes our proof of Transitivity. Next, we prove the Congruence of Incomparables. Assume that x and y are incomparable. Then S(x) = S(y). Therefore S(z) > S(x) if and only if S(z) > S(y), and S(x) > S(z) if and only if S(y) > S(z). So, z ≻ x if and only if z ≻ y, and x ≻ z if and only if y ≻ z. □

From this, we can immediately conclude:

Theorem 3 Even given Strong Neutrality, Universal Domain, Transitivity, and Congruence of Incomparables, Weak Outcome Anonymity does not imply Anonymity.

Our proofs of Theorems 2 and 3 rely on the possibility of incompleteness in the evaluative ranking—that is to say, failures of:

Completeness (of ≿) Either x ≿ y or y ≿ x.

The combination of Transitivity and Completeness do not quite collapse Anonymity and Outcome Anonymity. (Notice that Paretian Dependence satisfies only Anonymity; we get the full collapse given also Strong Neutrality and Universal Domain.) Nevertheless, Transitivity and Completeness have important consequences for several of our principles.

With Transitivity and Completeness, the codomain of g becomes a set of orderings (transitive, complete), which makes g a “social welfare functional.”⁶⁴ This allows us to exploit a number of important results, which reveal various connections between Anonymity, Outcome Anonymity, the Independence of Irrelevant Alternatives, and utilitarianism. See, for example, Theorem 3 of d’Aspremont and Gevers (which axiomatizes utilitarianism using Anonymity), Theorem 6 of Deschamps and Gevers (which links Anonymity to utilitarianism and “leximin” using other axioms), and Theorem 7 of Sen (which shows that, given Universal Domain and Outcome Anonymity, a social welfare functional g satisfies the Independence of Irrelevant Alternatives if and only if it satisfies Strong Anonymity;⁶⁵ see also Hammond, who observes that the other assumptions imply not only Strong Anonymity but “Superstrong Anonymity”⁶⁶).

Perhaps most important is the result that, in the context of social choice, Anonymity and Outcome Anonymity are equivalent given formal welfarism, which holds of a social welfare functional if and only if that functional can be represented as an ordering over an n-dimensional vector space, each point of which encodes the welfares of each of the n persons.⁶⁷ Formal welfarism is analogous to what some ethicists call dimensionalism, the view that overall value relations depend only on how good things are in each dimension.⁶⁸ The results cited above, as well as the discussion in the main text, may be of interest to ethicists looking for a criterion of “impartiality” towards dimensions of value.