My book Weighing Goods (Broome Reference Broome1991) contains a discussion of egalitarianism and prioritarianism.Footnote 1 At the time the book was written, prioritarianism was well established amongst economists,Footnote 2 but was only just being discovered by philosophers. This note outlines some arguments from the book, very much abridged, in response to Marc Fleurbaey's ‘Equality versus priority’.
1. THE DISTINCTION
Take a fixed population of n people, and imagine various distributions of well-being across those people. Each distribution can be described by a vector of the form (w 1, w 2, . . ., wn), which lists the well-being of each person in turn.Footnote 3 We wish to compare these distributions together, to determine which is better than which. In fact, we wish to put them into an order according to their goodness, with better distributions ranked higher in the order and worse ones lower. Different ethical views will order the distributions differently. I shall consider only views that are consistent with ‘the principle of personal good’, as I call it. This is the principle that, if one distribution gives some person more well-being than another distribution does, and if it gives no person less well-being than the other does, then it is better than the other.
One ethical view is utilitarianism. Utilitarianism says that one distribution is better than another if and only if it has a greater total of well-being, and two distributions are equally good if and only if they have the same total of well-being. This ordering of the distributions can be represented by a particular value function. A value function assigns a value to each distribution. To say it represents the ordering means it assigns a higher value to one distribution than to another if and only if the former is better than the latter, and it assigns two distributions equal value if and only if they are equally good. A function that represents utilitarianism's ordering is the simple total of well-being:
\begin{equation*}
w_{1} { + }w_{2} { + \cdot\cdot\cdot + w}_n {.}
\end{equation*}
Utilitarianism gives no value to equality in the distribution of well-being. It cares only about the total of well-being, not about how well-being is spread amongst the people. In this paper, I shall concentrate on views that, unlike utilitarianism, do give some value to equality. More specifically, I shall concentrate on views that have this implication: that one distribution is better than another if it has the same total of well-being as the other, but has that well-being more equally distributed. I shall take this principle to be more formally expressed by the Pigou–Dalton condition (Fleurbaey Reference Fleurbaey2015).
Any view that assigns value to equality in this sense might be called egalitarian. Daniel Hausman (Reference Hausman2015) urges us to adopt this piece of terminology. But there is an important distinction to be drawn amongst views that give value to equality in this sense, and later in this note I shall argue that some of them cannot count as truly egalitarian. Those ones should be called prioritarian and not egalitarian.
How should we specify the distinction between egalitarianism and prioritarianism? Fleurbaey reminds us that the two views differ in the grounds they offer for valuing equality. Prioritarians believe that a distribution of well-being should be valued on the basis of each person's well-being taken separately, independently of its relationship to other people’s. It is the absolute level of a person's well-being that matters to prioritarians, and not how her well-being compares with other people’s. The lower a person's well-being, the greater the priority prioritarians assign to improving it. This leads them to value equality indirectly. On the other hand, egalitarians are directly concerned with how each person's well-being stands in comparison to other people’s. They value equality directly.
Do these different grounds translate into a concrete difference between the two views’ implications? Specifically, do they imply different orderings of distributions?
I think they do. I am going to suggest a distinction that I think correctly captures and makes precise the views of the two sides, and attributes to them concretely different conclusions. I take it for granted that both views include the Pigou–Dalton condition and the principle of personal good. I suggest that prioritarianism should be understood as the view that, also, the ordering of the distributions is additively separable, and egalitarianism as the view that, also, it is not.
What does this mean, exactly? A complication is that some orderings cannot be represented by a value function. But for brevity in this note, I shall concentrate on the ones that can be. Among those orderings, the additively separable ones are those that can be represented by an additively separable function. Given the Pigou-Dalton condition and the principle of personal good,Footnote 4 an additively separable function is one that has the form:
\begin{equation}
f({w}_{1} ) + f{(w}_{2} {) + } \cdots { + }\,f{(}w_n {)}\end{equation}
where f is some increasing strictly concave function (a function whose graph slopes upwards but bends downwards).
I think this additively separable formula accurately captures the prioritarian idea that each person's well-being should be evaluated independently of other people's well-being. Conversely, I think the requirement that the value function is not additively separable captures the egalitarian idea that comparisons between different people's well-being matter.
2. SOME DIFFERENCES BETWEEN THE THEORIES
Marc Fleurbaey adopts the same definition of prioritarianism, but (like Hausman) defines egalitarianism to include prioritarianism. I think he does so because he believes that egalitarianism, as I define it, is not a principled, distinctive view that can be opposed to prioritarianism. But I believe it is a distinctive view, and deserves a distinct name. This note explains why.
True, I define egalitarianism in a negative fashion, as the view that is not prioritarian. But this does not necessarily make it unprincipled or undistinctive. Here is a close parallel. Expected utility theory is the view that the ordering of uncertain prospects is additively separable amongst states of nature. Non-expected-utility theory is the view that it is not. Expected utility theory could formally be treated as a special, extreme case of non-expected-utility theory. But the people who believe in non-expected-utility theory think they have principled reasons for rejecting this special case. Similarly egalitarians think they have principled reasons for rejecting prioritarianism.
A prioritarian ordering of distributions is inevitably different from an egalitarian one, just because one is additively separable and the other is not. However, the difference cannot be effectively displayed in very simple examples that compare together only two distributions. For instance, the examples used in the ‘levelling-down objection’ are ineffective at distinguishing the views. Changing from A to B below is what is called a ‘levelling down’:
\begin{eqnarray*}
A &=& (100,200)\\
B &=& (100,100).
\end{eqnarray*}
Prioritarians think A is better than B; they are opposed to levelling down. So are egalitarians as I have defined them. The principle of personal good implies that A is better than B, and I included the principle of personal good as part of the definition of both egalitarianism and prioritarianism. Levelling down has no tendency to separate prioritarians from egalitarians, as I defined them.
True, some extreme egalitarians reject the principle of personal good. I excluded them from my definition because I see no merit in their view. An egalitarian who accepts the principle of personal good can still be a genuine egalitarian, so levelling down provides no objection to egalitarianism. I agree completely with Fleurbaey about this.
It takes more complicated examples to display the concrete difference between egalitarianism and prioritarianism. Here is one, which is roughly modelled on Maurice Allais's (Reference Allais, Allais and Hagen1979) famous counterexample to expected utility theory. Compare these four distributions:
\begin{eqnarray*}
&& C{\rm = (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2)}\\
&& D{\rm = (4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2)}\\
&& E{\rm = (2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1)}\\
&& F{\rm = (4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)}
\end{eqnarray*}
Prioritarianism implies that C is better than D if and only if E is better than F. This is easily checked from (*). The reason is that the only difference between C and D is in the well-beings of the first two people, and there is exactly the same difference between E and F. Additive separability implies that the well-being of everyone else is irrelevant to the comparison.
However, C in this example has the merit of perfect equality, which E does not share. An egalitarian might find this a reason to think C better than D that is not also a reason to think E better than F. On the other hand, F is plausibly better than E because of its greater total of well-being. So an egalitarian may well think that C is better than D, and F better than E. This view is inconsistent with prioritarianism. It is a principled egalitarian view.
3. A CRUCIAL DIFFERENCE
Nevertheless, it is not incumbent on an egalitarian to think C better than D and F better than E. So this example does not definitively separate egalitarians and prioritarians. In Weighing Goods, I presented an example that does definitively separate them. Fleurbaey discusses this example in his paper. It is this:
\begin{eqnarray*}
&&G{\rm : Either\ (2, 2)\ or\ (1, 1)\ with\ equal\ probability}{\rm .}\\
&&H{\rm : Either\ (2, 1)\ or\ (1, 2)\ with\ equal\ probability}{\rm .}
\end{eqnarray*}
The options G and H are uncertain prospects, so the example introduces a complication that we have not faced till now. I shall assume that expected utility theory tells us the right way to make a valuation under uncertainty.Footnote 5 According to expected utility theory, the way to value a prospect with uncertain results is first to assign something called a ‘utility’ to each of its possible outcomes, and then take the mathematical expectation of this utility. That expectation – ‘expected utility’ – represents the prospect's value; one prospect is better than another if and only if it has a higher expected utility.
What does prioritarianism say about the relative goodness of G and H? Each of these prospects has two possible outcomes, and each outcome is a distribution of well-being. Prioritarians value a distribution of well-being according to the additively separable formula (*) – in this case f(w 1) + f(w 2). For example, the first outcome in option G has the value f(2) + f(2), and the first outcome in H the value f(2) + f(1). So far as expected utility theory is concerned, the utility of an outcome need not be exactly equal to its value found this way. But in fact prioritarians must identify utility with value, as I shall soon explain. For that reason, a prioritarian must value G and H by the expectation of their value, calculated according to the additively separable formula. For G this gives us
\begin{equation*}
\frac{1}{2}{\rm \{ }f{\rm (2) \,{+}\, }f{\rm (2)\} \,{+}\, }\frac{1}{2}{\rm \{ }f{\rm (1) \,{+}\, }f{\rm (1)\} ,}
\end{equation*}
and for H
\begin{equation*}
\frac{1}{2}{\rm \{ }f{\rm (2) \,{+}\, }f{\rm (1)\} \,{+}\, }\frac{1}{2}{\rm \{ }f{\rm (1) \,{+}\, }f{\rm (2)\} ,}
\end{equation*}
These two expected values are the same. The conclusion is that prioritarians must take G and H to be equally good.
Fleurbaey points out that prioritarians might have a way to escape this conclusion. A prioritarian need not value G and H by taking the expectation of their value according to the additively separable formula. Instead, she might take the expectation of some transform of this value: g(f(w 1) + f(w 2)), where g is some increasing function. Then she would calculate the value of G as
\begin{equation*}
\frac{1}{2}g(f(2) + f(2)) + \frac{1}{2}g(f(1) + f(1)),
\end{equation*}
and the value of H as
\begin{equation*}
\frac{1}{2}g(f(2) + f(1)) + \frac{1}{2}g(f(1) + f(2)).
\end{equation*}
A suitable transformation g will make G come out better than H.
A prioritarian has potentially two ways to justify transforming the value function. First, the additively separable function (*) only represents the ordering of the distributions by their goodness; it does not pretend to measure their goodness more precisely than that. It only gets distributions in the right order: it assigns one distribution a higher value than another if and only if that distribution is better. Any increasing transform of the value function will represent the ordering equally well.
Second, even if the value function measures goodness more strictly than this, expected utility theory allows utility to be any increasing transform of value. So expected utility theory would itself permit the transformation. All this Fleurbaey points out.
However, this transformation of value is not really available to a prioritarian. Prioritarianism requires a distribution to be evaluated on the basis of each person's well-being, taken separately from other people’s. Look at the two individual's separate well-beings under the options G and H. Each option gives each individual either two units of well-being or one unit, with equal probability. So each person has exactly the same prospect of well-being under G as she has under H. A prioritarian must therefore take G and H to be equally good.
This argument extends prioritarianism a little. Prioritarianism was defined for distributions of well-being, and I am applying it to an uncertain prospect (‘ex ante’, in Fleurbaey's terms). But the case is special, because each individual has the very same prospect of well-being under G as under H. I do not think a prioritarian could justifiably deny the extension for this special case. So prioritarianism directly implies G and H are equally good. We do not need to consider the formulae to tell that.
Any manoeuvre on the part of a prioritarian to give H and G unequal value must therefore be mistaken. This conclusion blocks the transformation of the value function by g. Prioritarians must evaluate the prospects G and H by taking the expectation of the additively separable value function (*), untransformed.Footnote 6
On the other hand, a true egalitarian must think G better than H. When valuing a distribution, an egalitarian thinks we should compare the positions of the different people, to see how they stand relative to one another. In G they stand equal for sure, and in H they stand unequal for sure. So G must be better than H from the point of view of an egalitarian. No one deserves the name of ‘egalitarian’ unless she thinks G better than H.Footnote 7
I conclude that this example strictly separates prioritarians from egalitarians. Prioritarians must think G and H equally good, whereas egalitarians must think G better than H.
4. THE MERITS OF THE THEORIES
Who are right, prioritarians or egalitarians? Weighing Goods offers in Chapter 10 an argument against prioritarianism, though not a conclusive one.Footnote 8 Once more, I can only sketch it here. Its core is this. To give their theory meaning, prioritarians need a measure of a person's well-being that is distinct from the value of her well-being. They may not be able to find one.
Take some given quantity of improvement in a person's well-being. According to prioritarians, the value of this improvement depends on whom it comes to. By its ‘value’, I mean the amount it contributes to the overall goodness of a distribution. An improvement has more value if it comes to someone who is initially worse off than if it comes to someone who is initially better off. So prioritarians make a distinction between the quantity of a change in well-being, and the value of the change. They distinguish the value of well-being from the quantity of well-being.
Compare these two distributions of well-being:
\begin{eqnarray*}
&&I{\rm = (3,3)}\\
&&J{\rm = (2,4)}
\end{eqnarray*}
A prioritarian will think I better than J. Imagine a change from J to I. In this change, the first person gains one unit of well-being and the second person loses one. But the first person has priority, because she is worse off than the second. A unit change in her well-being is more valuable than a unit change in the second person's well-being. So in a change from J to I, the two people's well-beings change by the same quantity – positive for one and negative for the other – but the changes differ in value.
Prioritarianism presupposes a quantitative scale for the quantity of well-being: a cardinal scale, to be exact. Philosophers often take it for granted that we have a rough quantitative notion of well-being. No doubt that is so, but we need to ask what its source is. To have a cardinal scale, we have to be able to make sense of statements that compare the quantity of changes in people's well-being. The example contains one change from 2 to 3 and another from 4 to 3, and these changes are supposed to be the same in quantity: one unit. What exactly does it mean to say they are the same in quantity?
One possible meaning is that they have the same value. But that cannot be a prioritarian's meaning, because a prioritarian thinks these changes differ in value. She has to make sense of the comparison some other way.
She may call on the valuation of uncertain prospects, because those also implicitly compare differences in well-being. Compare these two prospects from the point of view of a single person:
\begin{eqnarray*}
&&I\hspace*{-1pt}^\prime = b\ {\rm for}\ {\rm sure,}\\
&&J\hspace*{-2pt}^\prime = {\rm Either}\ a\ {\rm or}\ c{\rm ,}\ {\rm with}\ {\rm equal}\ {\rm probability}{\rm .}
\end{eqnarray*}
a, b and c are outcomes of some sort, for example states of the person's health. Assume a gives our subject more well-being than b, and b more than c. Compared with I’, the risky prospect J’ offers a possible gain in well-being from c to b, but also a possible loss from a to b. If we ask which prospect is better for the person, we are in effect comparing the possible gain with the possible loss. Suppose we conclude that I’ and J’ are equally good. We could take that to mean that the possible gain and the possible loss are the same in quantity: the difference in well-being between c and b is the same as the difference between b and a. We can assign meaning to quantities of well-being by generalizing this idea.
A prioritarian might think we give a quantitative meaning to well-being this way, through evaluations of uncertain prospects. Provided these evaluations are independent of the value of distributions of well-being, she is then free to distinguish the value of well-being from the quantity of well-being, as she needs to.
However, actually the evaluation of uncertain prospects is not independent of the value of distributions of well-being. The two correspond exactly, so that if the quantity of well-being gets its meaning through the value of uncertain prospects, the quantity of well-being will turn out to be exactly the same as the value of well-being. This conclusion can be drawn from a theorem that originates with John Harsanyi (Reference Harsanyi1955).Footnote 9 I am sorry to say I cannot explain it here. It means the prioritarian cannot get her quantities of well-being this way.
A prioritarian still needs to separate the value of well-being from the quantity of well-being. Perhaps she can find a way of doing so. But until she does, her theory is shaky.
Finally, on the other side, a good case can be made for the egalitarian view. It accords well with our natural, intuitive understand of fairness. Fairness comes into play when some good is to be distributed amongst people. It requires people to receive a share of the good that is in proportion to their claim to it. It is not concerned with how much of the good they receive absolutely. Even if each person receives very little, or indeed none at all, each is fairly treated so long as each receives a share in proportion to her claim.
I have been assuming implicitly that people have an equal claim to well-being. If they do, then fairness requires that they receive an equal share of well-being. In the example of G and H, G for sure leads to a fair result, whereas H for sure leads to an unfair result. So G is undoubtedly better from the point of view of fairness. This is the egalitarian conclusion.
ACKNOWLEDGEMENTS
This paper is a response to Fleurbaey (Reference Fleurbaey2015). It was commissioned by the World Health Organization, and I am grateful to the WHO for permission to publish it in this symposium. It was presented at a conference organized by the WHO in Thiruvananthapuram. I am grateful to the participants for their helpful comments. Apart from some minor copy-editing, this paper is the original version written in 2000.
