2.1 Weak prioritarianism

Prioritarians believe that, other things being equal, ^{Footnote 2} benefiting a person matters more, from a moral or social point of view, the worse off the person is. As formulated by Parfit (Reference Parfit1991: 105), the thought is that ‘utility [i.e. individual welfare] has diminishing marginal moral importance’.

Prioritarians often favour interventions that increase equality. In fact, given any fixed sum of welfare, the view always favours a more equal distribution of that sum. However, unlike standard versions of egalitarianism, prioritarianism arguably does not care about equality for its own sake. For, according to prioritarianism, the moral value of benefiting a person is completely determined by the person’s own welfare, and depends in no way on how their welfare compares to others’.

To see how this works, we operationalize prioritarianism in the following standard way. Each individual’s well-being has moral value. The moral value of a distribution can be represented as the sum of the moral value of each individual’s well-being in that distribution. ^{Footnote 3} The moral value of a person’s well-being does not depend on the existence or welfare of other people in the distribution. It is entirely a function of her own well-being. But the moral value of a person’s well-being is not equal to her well-being; it is the value of a priority weighting function applied to her well-being. A person’s priority-weighted well-being is a strictly increasing, strictly concave function of that person’s well-being. To say that it is strictly increasing means that increasing a person’s well-being will always increase her priority-weighted well-being. To say that it is strictly concave means that increasing a person’s well-being by a fixed amount from a higher level increases her priority-weighted well-being by less than increasing her well-being by that same amount from a lower level. Together, these features mean that benefits to a person make a diminishing, but always positive, marginal contribution to the person’s priority-weighted well-being.

A simple example of a strictly concave function is the square-root function, which is the blue curve depicted in Figure 1. The horizontal axis represents a person’s well-being. The vertical axis represents her priority-weighted well-being. The slope of the square-root function is positive but decreases. That is what a strictly increasing, strictly concave function looks like.

Figure 1. Strictly and Weakly Concave Priority Weighting Functions.

Our problem applies to a much wider class of views than prioritarianism alone, so described. We will therefore treat the view just described as a special case, which we call strict prioritarianism. Strict prioritarians believe that the priority-weighting function is strictly concave, like the square-root function. Weak prioritarianism holds, more generally, that the priority-weighting function is weakly concave. This means that a fixed increase in well-being from a lower level increases a person’s priority-weighted well-being by at least as much as a same-sized increase from any higher level. On this view, benefits need not have decreasing marginal value, but they cannot have increasing marginal value.

What is the interest of this more general class of views? Some people believe that, among individuals who are sufficiently well off, we should give no priority to those who are worse off (Crisp Reference Crisp2003). On one version of this sufficientarian view, benefits to those who are sufficiently well off still matter, but they should be distributed in a utilitarian manner. This view can be modelled as a version of weak prioritarianism in which the priority weighting function is strictly concave up to some point at which one has a sufficient quantity of well-being, after which it becomes linear. A simple example of such a function is the red curve in Figure 1; the red priority-weighting function follows the square-root graph until welfare level 16, at which point it becomes linear. Utilitarianism is another special case of weak prioritarianism – namely, one where the priority-weighting function is linear throughout. This means that priority is never given to those who are worse off (nor to those who are better off).

To summarize, the weakly prioritarian family of social welfare (i.e. moral value) functions compares distributions according to their sums of priority-weighted well-being, where the priority weighting function is strictly increasing and weakly concave. In the next subsection, we raise a problem for all views within this family except utilitarianism.

In laying out this problem, we appeal freely to numerical representations of welfare. Weak prioritarianism presupposes that such representations are meaningful. We judge that some tradeoffs involving such quantities are reasonable, others unreasonable. In this we follow the methodology suggested by Parfit: using our judgement.

Some may nevertheless find themselves unable to make judgements about tradeoffs involving different quantities of welfare without further information about how we assume welfare to be measured – i.e. what exactly the lives at various welfare levels are like. There are many views about the measure of well-being, and any particular choice of measure would be highly controversial. Our own attitude is that the tradeoffs we find reasonable seem (to us) reasonable on any plausible measure of well-being, and that the tradeoffs we find unreasonable seem (to us) unreasonable on any plausible measure as well. But some may find it helpful, for illustrative purposes, to have a concrete measure on the table.

One common practice in both the philosophical literature and in applications – e.g. in health economics – is to suppose that a unit of well-being corresponds to some duration of life of some quality (see e.g. Parfit Reference Parfit2012; Nord and Johansen Reference Nord and Johansen2014; Otsuka Reference Otsuka2017; Lipman and Attema Reference Lipman and Attema2019). For example, we might take a unit of well-being to correspond to a year of very happy life. Because this sort of measure is both widely used and fairly concrete, we invite the reader to apply it when interpreting our results involving welfare differences. In section 2.3, however, we offer especially troubling results that do not depend on an arbitrary choice of welfare unit (and which, to our knowledge, have no direct precedent in the decision-theoretic literature).

2.2 Calibration results for weak prioritarianism

Suppose that your preferences between distributions are governed by weak prioritarianism. ^{Footnote 4} And suppose that you can benefit either of two people: Ann or Bob. Ann is slightly worse off than Bob: Bob has welfare w, Ann $w-0.9$ . For example, using a longevity-based measure of well-being, this could mean that Ann lives nine-tenths of a year less than Bob, where their lives are otherwise of equal quality. We represent this initial distribution as $(w-0.9,w)$ . You can either benefit Ann so that the two are equally well off, (w, w), or benefit Bob by a slightly greater amount so that Bob becomes even better off, $(w-0.9,w+1)$ . Suppose that, for some value of w, you would provide the smaller benefit to Ann, resulting in the equal distribution. This choice seems to us reasonable because, although benefiting Bob would maximize total welfare, the difference in total welfare is very slight; it is only one-tenth of a unit. For example, suppose that $w=50$ . We might think it reasonable to prefer (50, 50) to $(49.1,51)$ . We will represent this preference in the following way (where ‘ $x \succ y$ ’ means that x is preferred to y):

(1) $$(50,50) \succ (49.1,51)$$

What can we infer from this particular preference about your preferences regarding tradeoffs with larger stakes? By itself, not very much; we need to know more about your preferences. To see what we can infer given slightly more information, let us start with the simplest, but perhaps somewhat unrealistic case: suppose, for now, that you would prefer the equal distribution for any value of w. (This assumption will soon be relaxed; that is, later we will assume that whether you prefer the equal distribution or the unequal one may depend on the size of w.)

We represent this more general preference as follows:

(2) $${\rm{For}}\;{\rm{all}}\,w:{\mkern 1mu} ({\it{w}},\,{\it{w}}) \succ ({\it{w}} - {\rm{0}}.{\rm{9}},{\it{w}} + {\rm{1}})$$

This preference may still seem reasonable because, for any value of w, the difference in total welfare between the equal and unequal distributions remains very small ( $0.1$ ); it seems reasonable to benefit the worse off rather than benefiting the better off by a slightly greater amount.

Knowing only that your preferences between distributions are weakly prioritarian, and that they satisfy (2), what can we infer about your preferences over tradeoffs with larger stakes? In particular, suppose that Ann is 8 units worse off than Bob $(w-8,w)$ , and that you can either benefit Ann by 8 units (w, w) or provide some larger benefit of size G to Bob $(w-8,w+G)$ . What is the largest value of G such that you would prefer (w, w) to $(w-8,w+G)$ that we can infer from our limited knowledge of your distributive preferences?

It might be natural to guess 10, 50, or 100, or to think that we cannot know without knowing more about the shape of your priority weighting function. In fact, however, we can – surprisingly – infer that you would prefer to benefit Ann by 8 units rather than benefiting Bob by G units no matter how large G is.

The above implication holds not only for tradeoffs between two individuals; it holds for tradeoffs between any number of people. And the same is true of the other implications that we shall discuss. The weak prioritarian evaluates a distribution according to the sum of each person’s priority-weighted well-being, which cannot depend on the welfare or existence of other people. (That is the sense in which the prioritarian cares only about each individual’s absolute welfare level.) So, if a weakly prioritarian distributor prefers a distribution in which two people have welfare w to a distribution in which one person has $w-L$ and the other has $w+G$ , then she must also prefer a distribution in which any number of people have welfare w to a distribution in which half that number of people have $w-L$ and the other half have $w+G$ .

It will be helpful, especially later on, to have some symbols to represent these distributions involving populations of arbitrary size. Take any distribution ${\bf{w}}=({{w}}_{{1}},..., {{w}}_n)$ , where $w_i$ represents the welfare of person i. If n people have the same welfare w, we represent this distribution as ${\bf{w}}_n$ : that is, ${\bf{w}}_n$ contains n people at level w. For any distribution $\bf{w}$ and quantity of welfare k, let ${\bf{w}}+k$ represent the distribution $(w_1+k, ..., w_n+k)$ in which k is added to each person’s welfare in $\bf{w}$ . For any distributions ${\bf{w}}=(w_1, ..., w_n$ ) and ${\bf{u}}=({{u}}_1, ..., {{u}}_m$ ), let $(\bf{w},\bf{u})$ represent the distribution $(w_1, ..., w_n, u_1, ..., u_m)$ that concatenates (or ‘stacks’) $\bf{w}$ and $\bf{u}$ together. What we have found is that a weakly prioritarian distributor whose preferences satisfy (2) must also satisfy

(3) $${\rm{For}}\;{\rm{all}}\;w,n,\,{\rm{and}}\,G:{\mkern 1mu} {{\bf{w}}_{{\rm{2n}}}} \succ ({{\bf{w}}_{\rm{n}}} - {\rm{8}},{{\bf{w}}_{\rm{n}}} + {{G}})$$

In words: a distribution in which 2n people are at any level w is preferred to one in which n people are at $w-8$ and n are at $w+G$ , no matter how large G is. This is, we think, an implausibly extreme degree of priority to the worse off. Intuitively, if w and G are very large, it is not reasonable to prefer the equal distribution to the unequal one; after all, a life at level $w-8$ can be as good as we like, and the unequal distribution’s net gain in total welfare, $(G-8)$ times n, can be as large as we like.

The above result is one instance of our prioritarian calibration theorem, which is stated formally and proved in the Appendix; it is based on Rabin’s calibration theorem for expected utility theory, which we mentioned in the Introduction. The general lesson of our prioritarian calibration theorem is this. Suppose that, for any welfare level w, a weakly prioritarian distributor would prefer any distribution in which two people are at level w to a distribution in which one person has welfare $w+g$ (greater) and another has $w-l$ (less), where g is greater than l (and both are positive). Then we can show that, for any welfare level w, such a distributor must prefer any distribution in which everyone would be at level w to a distribution in which half that population would have welfare $w+G$ and the other half would have $w-L$ , where G is greater than L (and both are greater than g and l). If L is large enough, we can show this to hold for any value of G, however large. Put schematically, the theorem says that weak prioritarianism entails conditionals of the following form:

(4) \begin{align*} & {\rm{If}},\;{\rm{for}}\;{\rm{all}}\;w:{\mkern 1mu} (w,w) \succ (w - l,w + g), \\ & {\rm{then}},\;{\rm{for}}\;{\rm{all}}\;w,n:{\mkern 1mu} {{\bf{w}}_{{\rm{2}}n}} \succ ({{\bf{w}}_n} - L,{{\bf{w}}_n}{\rm{ + }}G).\end{align*}

Table 1 illustrates several representative examples of these results, where $g=1$ . ^{Footnote 5} Consider a weakly prioritarian distributor who, for all w, prefers (w, w) to $(w-l,w+1)$ , for each column’s value of l. Then such a distributor must, for any w and any n, prefer ${\bf{w}}_{2n}$ to $({{w}}_n-L,{{w}}_n+G)$ , where each L is what is subtracted in a cell in the leftmost column in Table 1 while the addends in the other cells are the highest (integer) values of G for each choice of l and L. If a cell contains $\infty$ , this means that G can be arbitrarily large. For example, Table 1 says that a weakly prioritarian distributor who prefers (w, w) to $(w-0.75,w+1)$ for all w must, for all w, n, and G, prefer ${\bf{w}}_{2n}$ to $({\bf{w}}_n-4,{\bf{w}}_n+G)$ . ^{Footnote 6}

Table 1. Weakly prioritarian distributor who prefers (w,w) to $(w-l,w+g)$ for all w when $g=1$ must, for all w, n, prefer ${{\bf{w}}}_{2n}$ to $({\bf{w}}_n-L,{\bf{w}}_n+G)$ for Ls and Gs entered in table

As Table 1 illustrates, the results can be made less extreme by requiring a larger value of l, as in the columns towards the left. Shifting the value of l towards the left can be regarded as shifting our preferences involving small-stakes tradeoffs closer to utilitarianism. For example, suppose we know only that a weakly prioritarian distributor prefers (w, w) to $(w-0.99,w+1)$ for any w – i.e. she is willing to forgo a measly hundredth of a unit of net total welfare to benefit the worse off. Such a distributor can have sensible preferences involving many somewhat larger tradeoffs. For example, she can prefer $(w-25,w+G)$ to (w, w) for any integer G greater than 33. But, as shown in the bottom row, she must, for any w, prefer ${\bf{w}}_{2n}$ to $({\bf{w}}_n-75,{\bf{w}}_n+G)$ , for arbitrarily large G. And even this seems unreasonable for sufficiently high values of w, since then $w-75$ remains an excellent life.

The intuition behind our prioritarian calibration theorem is relatively simple. A weak prioritarian’s aversion to inequality must be determined solely by the shape of her priority weighting function. A robust preference for benefiting the worse off when small quantities of well-being are at stake means that the marginal priority-weighted value of well-being diminishes very quickly, so that arbitrarily large gains above any given well-being level are less valuable than merely modest gains below that level.

To see how this works more concretely, we will walk through an informal sketch of the proof, applied to a particular choice of l and g. Suppose that we prefer (w, w) to $(w-0.5,w+1)$ for any w. And suppose that our preferences are weakly prioritarian. This means that, for some weakly concave priority weighting function $f(\cdot)$ , and for any w, the total priority-weighted well-being of (w, w) is greater than the total priority-weighted well-being of $(w-0.5,w+1$ ):

(5) $$f(w) + f(w) \gt f(w - 0.5) + f(w + 1)$$

We want to know the marginal priority-weighed value of an increment of well-being at various levels. Suppose that some person has welfare level 100. What is the marginal priority-weighted value of an additional unit of welfare from this level? Well, we prefer (100, 100) to $(100-0.5,100+1)$ , so our priority weighting function must be such that the total priority-weighted well-being of (100, 100) is greater than the total priority-weighted well-being of $(99.5,101$ ):

(6) $$f(100) + f(100) \gt f(99.5) + f(101)$$

So the difference in priority-weighted value between 100 and 101 must be less than the difference in priority-weighted value between $99.5$ and 100:

(7) $$f(101) - f(100) \lt f(100) - f(99.5)$$

Next consider the marginal priority-weighted value of an additional unit from level 101. Again, we prefer (101,101) to $(101-0.5,101+1)$ , so the difference in priority-weighted value between 101 and 102 must be less than the difference in priority-weighted value between $100.5$ and 101:

(8) $$f(102) - f(101) \lt f(101) - f(100.5)$$

But what is the difference in priority-weighted value between $100.5$ and 101? Since our priority weighting function is weakly concave, the marginal priority-weighted value of welfare between $100.5$ and 101 cannot exceed the marginal priority-weighted value of welfare between 100 and 101, since the latter interval starts from a lower level:

(9) $${{f(101) - f(100.5)} \over {101 - 100.5}} \le\, {{f(101) - f(100)} \over {101 - 100}}$$

This means that the difference in priority-weighted value between $100.5$ and 101 cannot be more than half the difference in priority-weighted value between 101 and 100:

(10) $$f(101) - f(100.5) \le 1/2\left[ {f(101) - f(100)} \right]$$

So, putting (8) and (10) together, the difference in priority-weighted value between 102 and 101 must be less than half the difference between 101 and 100:

(11) $$f(102) - f(101) \lt 1/2\left[ {f(101) - f(100)} \right]$$

Iterations of this reasoning imply that the difference in priority-weighted well-being between consecutive whole-numbered levels is less than half the difference between the preceding levels. So, for example, the priority-weighted value of a unit of well-being gained from level 149 must be less than $(1/2)^{49}=1/{562949953421312}$ times the difference in priority-weighted value between levels 100 and 101, which must in turn, by (7), be less than the difference between 99.5 and 100. This steep decline in marginal priority-weighted value establishes a firm upper bound on the priority-weighted value of gains in well-being. Specifically, no matter how large G is, the total difference in priority-weighted well-being between $100+G$ and 100 must be less than twice the difference in priority-weighted well-being between 100 and $99.5$ . ^{Footnote 7} And this is true not just for gains from level 100, but from any w whatsoever.

The results in Table 1 seem pretty extreme. We expect that many proponents of weak prioritarianism would agree, and would therefore claim that a distributor should not prefer to benefit the worse off by a slightly lower amount from all levels of well-being. If w is sufficiently high, perhaps the distributor should no longer prefer to benefit the slightly worse off at the expense of total well-being. After all, weak prioritarians give priority to the worse off because of how badly off they are, and therefore need not prioritize the worse off when the worse off are extremely well off.

This move would prevent the weakly prioritarian distributor from having to prioritize the worse-off at the expense of arbitrarily large gains to the moderately better off. But it is not, by itself, enough to avoid implausibly extreme degrees of priority in large-stakes cases.

To see this, consider Table 2. Table 2 is based on the same calibration theorem as Table 1, but assumes only that the weakly prioritarian distributor prioritizes the slightly worse off at all levels of well-being up to 100. The theorem implies, for example, that if the distributor prefers any distribution in which both Ann and Bob have well-being level w to one in which Ann has $w-0.75$ and Bob has $w+1$ for any value of w up to 100, then she must prefer any distribution in which everyone would have well-being level 75 to one in which half that population would have well-being level 55 and the other half would have well-being level $1.67$ million.

Table 2. Weakly prioritarian distributor who prefers (w,w) to $(w-l,w+g)$ for all $w \leq 100$ when $g=1$ must prefer ${\bf{75}}_{2n}$ to $({\bf{75}}_n-L,{\bf{75}}_n+G)$ for Ls and Gs entered in table

Some of the implications shown in Table 2 may seem reasonable – for example, for $l=0.99$ . More extreme results, however, can be delivered by adjusting the other parameters. For example, if a weakly prioritarian distributor would prefer (w, w) to $(w-0.99,w+1)$ for all values of w up to 3000, then she must prefer ${\bf{500}}_{2n}$ to $({\bf{400}}_n,\,{\bf{5.98}} \times {\bf{10^{12}}}_{n})$ . Or, if she would prefer (w, w) to $(w-0.099,w+0.1)$ for all values of w up to 100, then she must prefer ${\bf{75}}_{{2n}}$ to $({\bf{25}}_n, {\bf{{185418}}}_n)$ . And these results, based on seemingly reasonable degrees of priority to the worse off, still seem very extreme. ^{Footnote 8}

In sum, prioritarians face a dilemma: they can give seemingly reasonable priority to the worse off when small differences in well-being are at stake only by giving extreme – and, we think, unreasonable – priority to the worse off when very large differences are at stake.

2.3 Revenge dilemma for ratio-scale invariant prioritarianism

We do not insist on any particular response to this dilemma, but our sympathies lie in the following direction. If we are justified in giving priority to the worse off – that is, if utilitarianism is false – then weak prioritarianism is false. There is, however, room for other responses, corresponding to the different horns of our dilemma. Some weak prioritarians may respond that, though we ought to give some priority to the worse off, it is unreasonable to prioritize the slightly worse off in the specific ways we have considered. Others may respond that it is reasonable to prioritize the moderately worse off even at the expense of the very large gains we have considered. We don’t have anything like a decisive, non-question-begging argument against these responses. We acknowledge that staunch proponents of weak prioritarianism who do not share our intuitions, or who are sufficiently willing to revise their intuitions about cases, may remain unfazed. But we ourselves would prefer a distributive theory, if there is one, that can accommodate the preferences we have reported.

To us, rejecting the intuitions is a desperate last resort. But there is a kind of sceptical attitude towards these intuitions that we would like to acknowledge. We have been freely expressing intuitions about cases involving various quantities of well-being. Some of these quantities seem small, others large. One may reasonably wonder whether we can rely on our intuitions about distributions expressed in these terms. After all, on many views about the measurement of well-being, such numbers are arbitrary. Unlike, say, money, there is no absolute scale of well-being. Numbers below 1 may sound small, and numbers above 100 may sound large. But, the sceptic may argue, we should not trust intuitions that are sensitive to particular numerical representations of welfare.

This response may seem to sap our prioritarian dilemma of its force. But it also inspires a new variation of our dilemma, which we take to be even more forceful than our initial version.

On most views about the measurability of welfare, a quantitative representation of well-being is only unique up to certain kinds of transformations. This is how we understand the idea that welfare has no absolute scale. The different kinds of admissible transformations correspond to different kinds of scales of well-being, and some theories of distribution are compatible with certain kinds of scales and not others. The most common kind of scale assumed by prioritarians is a ratio scale. This is a scale on which ratios of welfare levels are meaningful, but differences are not. For example, if Ann is represented as having welfare 50 and Bob welfare 100, then Bob is twice as well off as Ann, and any accurate representation of their welfare levels must assign Bob a welfare level that is twice as large as Ann’s. But it need not be 50 units greater than Ann’s, because the absolute size of the difference is arbitrary.

In the literature on social welfare functions, this idea is generally implemented in the following way. For any distribution ${\bf{w}}=({{w}}_1, ..., {{w}}_n)$ and number k, let $k\bf{w}$ represent the distribution $(k w_1, ..., k w_n)$ in which each person’s welfare in $\bf{w}$ is multiplied by k. According to the requirement of ratio-scale invariance, for any distributions $\bf{w}$ and $\bf{v}$ , and any positive k, $\bf{w}$ is preferred to $\bf{v}$ iff $k\bf{w}$ is preferred to $k\bf{v}$ . This principle captures the idea that the only significant properties of our well-being measure are captured by well-being ratios: absolute differences and levels are not meaningful or morally significant. This principle is satisfied by the Atkinson social welfare function, which is the most widely endorsed version of prioritarianism. It is, for instance, defended at length by Adler (Reference Adler2012). ^{Footnote 9}

If weak prioritarians deny that absolute differences in welfare are meaningful, then perhaps they can credibly insist that our intuitions about trading off ‘small’ and ‘large’ quantities of welfare should not be trusted. We can, however, derive analogous calibration results stated in terms of ratios. For example, suppose that a weakly prioritarian distributor would prefer a distribution in which Ann and Bob are both at some level w to a distribution in which Bob is 1% better off than w and Ann is $0.5\%$ worse off than w. And suppose that the distributor’s preferences are ratio-scale invariant, in the sense that we have just explained. Then we can show that, for any level w, the distributor must prefer a distribution in which everyone is at w to a distribution in which half that population is 2% worse off than w and the other half has any quantity of welfare whatsoever.

This example follows from our ratio-scale prioritarian calibration theorem, which is also stated and proved in the Appendix. In schematic form, the theorem says that, given ratio-scale invariance, weak prioritarianism entails conditionals of the following form:

(12) \begin{align*} & {\rm{If}},\;{\rm{for}}\;{\rm{some}}\;w:{\mkern 1mu} (w,w) \succ ([1 - l]w,[1 + g]w), \\ & {\rm{then}},\;{\rm{for}}\;{\rm{all}}\;w,n:{\mkern 1mu} {{\bf{w}}_{{\rm{2}}n}} \succ ([{\rm{1}} - L]{\mkern 1mu} {{\bf{w}}_n},[1 + G]{\mkern 1mu} {{\bf{w}}_n}).\end{align*}

Some representative results of this theorem are presented in Table 3. In the examples provided, we suppose that $g=1\%$ – i.e. the better-off person in the slightly unequal distribution is $1 \%$ better off than w. Again, if a cell has $\infty$ , this means that G can be arbitrarily large. For example, the table says that, given ratio-scale invariance, a weakly prioritarian distributor who prefers (w,w) to $(0.9905w,1.01w)$ for some w – e.g. (100, 100) to $(99.05,101)$ – must, for any w, any n, and any G, prefer $({\bf{w}}_{n},{\bf{w}}_{n})$ to $(0.80{\bf{w}}_{n},[1+G]{\bf{w}}_{n})$ .

Table 3. Given ratio-scale invariance, a weakly prioritarian distributor who prefers (w,w) to $([1-l]\,w,\, [1+g]\,w)$ for some w when $g=1\%$ must, for any w and n, prefer ${({\bf{w}}_{n},{\bf{w}}_{n})}$ to $([1-L]\,{\bf{w}}_{n},\,[1+G]\,{\bf{w}}_{n})$ for $1-L$ s and $1+G$ s entered in table

In Table 3, the only pairs of L and G that seem remotely plausible are when ${l}=0.99\%$ – i.e. the worse-off person in the slightly unequal distribution would be only $0.99 \%$ worse off than w. For this value of l (when ${g}=1\%$ ), G approaches infinity only as L grows close to 1: approximately $1-1.12 \times 10^{-30}$ . So perhaps the weak prioritarian should insist that, when ${g}=1\%$ , the small-stakes inequality-introducing tradeoff should only be rejected when l is as high as $0.99\%$ . However, we do not find this to be a very comfortable position for the prioritarian. We think this, first, because it seems no less reasonable to reject the tradeoff for slightly lower values of l, such as $0.95\%$ . And, second, because it seems unreasonable to reject, from extremely low initial levels, the large-stakes tradeoff when L is close to 1 and G is arbitrarily large, since multiplying such extremely low levels by a minuscule 1 – L makes such a small difference, in particular when compared to the difference made by multiplying that same level by an arbitrarily large 1 + G. If Ann’s and Bob’s lives are initially barely positive, then multiplying Ann’s welfare by a factor arbitrarily close to zero will still result in her life being barely positive, but multiplying Bob’s by an arbitrarily large factor could make his life wonderful. ^{Footnote 10}

Some may wonder whether we can rely on our intuitions about percentages of well-being. Even if such percentages are not arbitrary, it might be wondered whether we have a sufficiently good grip on them for us to be confident that certain tradeoffs are or are not reasonable. More generally, the reliability of people’s intuitions about quantities of well-being could be questioned.

Recall, however, the quotations from Parfit at the outset of this paper. Contemporary theories of distribution, such as prioritarianism and egalitarianism, attempt to avoid giving absolute priority to the worst off. But their views do not determine a unique intermediate degree of priority between absolute priority (leximin) and no priority (utilitarianism). To defend their distributive preferences, proponents of such theories must simply use their judgement. But if, as the sceptical attitude maintains, we cannot trust our intuitions about the kinds of tradeoffs we have presented, then how is any distribution-sensitive theorist supposed to form a particular view, or to make distributive choices in practice? If the sceptical response is right, we do not see how any particular degree of priority to the worse off could be justified. And, if we have no reasonable basis for some particular degree of priority, this would seem to make such views fairly uninteresting from a practical perspective, given the need to make tradeoffs in the real world.

3.1 The generalized Gini family

Suppose we wish to avoid both horns of our calibration dilemma – that is, to accommodate moderate aversion to small-stakes tradeoffs without unreasonable aversion to large-stakes tradeoffs – and therefore reject weak prioritarianism. What kind of theory do we need?

The core commitment of prioritarianism that may appear responsible for our calibration results is its additively separable form. Each individual’s well-being makes some contribution to the value of the welfare distribution. This contribution does not depend on the existence or welfare of other people, but is instead determined entirely by the priority-weighting function applied to her well-being. The value of a distribution is then represented as the sum of each individual’s contributive value – that is, her priority-weighted well-being. This feature guarantees that the prioritarian distributor’s responses to various tradeoffs depends entirely on the concavity of the priority weighting function. It therefore allows us to infer, from the rejection of certain small-stakes tradeoffs, severe constraints on the priority weighting function’s degree of concavity at various welfare levels. These severe constraints lead to implausible results regarding large-stakes tradeoffs.

In light of the above observations, one might hope that the solution to our calibration dilemmas is to be found in a nonseparable social welfare function, such as paradigmatically egalitarian theories entail. ^{Footnote 11} Unlike prioritiarians, egalitarians care about how well or badly off people are in comparison to others. In particular, according to egalitarianism, the moral value that a person’s welfare contributes to the total moral value of a welfare distribution partly depends on how well or badly off others are, which means that the moral value (or social welfare) function cannot be additively separable over individual welfare.

Perhaps the most widely endorsed version of egalitarianism is the generalized Gini family (Weymark Reference Weymark1981). We will focus on this particular family of egalitarian views in order to show, first, how egalitarians can avoid the calibration dilemma for prioritarianism, and, second, that egalitarians within this family face their own calibration dilemmas too. The motivation for focusing on this version of egalitarianism will become apparent below. In section 3.3, however, we briefly consider egalitarian views that fall outside this particular family of views.

A generalized Gini function evaluates distributions according to their weighted sums of well-being. Suppose that there are n people. We rank the n people from best-off to worst-off. Each person’s welfare is multiplied by a weight, which depends on her rank. These weights can be represented as positive real numbers. The value of a distribution is the sum of the individuals’ rank-weighted well-being. According to what we will call weak egalitarianism, the sequence of weights is nondecreasing, in the sense that the weight assigned to a person’s well-being cannot be less than the weight assigned to anyone better-off than her. Utilitarianism is the version of weak egalitarianism where each person (irrespective of her rank) gets the same weight. Strict egalitarianism is the version of this view where the sequence of weights is increasing, so that the weight assigned to a person’s well-being must be greater than the weight assigned to anyone better off than her. This is a generalization of the standard Gini social welfare function, whose weights are $1, 3, 5, 7, ...$ – i.e. the sequence of odd numbers. For our purposes, there is nothing special about this eponymous member of the generalized Gini family, whose weights seem to reflect ‘arbitrary distributional judgements’ (Weymark Reference Weymark1981: 414). We only require, with Weymark, that the weights be nondecreasing.

We define weak egalitarianism in this way for convenience, because it is the only paradigmatically egalitarian view that we extensively discuss. ^{Footnote 12} We do not mean to imply that the generalized Gini family is the only kind of view that deserves the name ‘egalitarianism’, and we do not claim that our conclusions generalize to all other views that go by the name. There is, however, good reason to focus on the generalized Gini family. It can be derived from some axioms that should seem attractive to many theorists with broadly egalitarian intuitions. ^{Footnote 13} It is compatible with a merely cardinal (as opposed to ratio) scale of well-being; it plays nicely with both positive and negative welfare levels; and it has been claimed to have plausible implications in variable-population and risky cases (Asheim and Zuber Reference Asheim and Zuber2014, Reference Asheim and Zuber2016).

Finally, and most importantly, the generalized Gini family seems especially relevant to the present discussion in light of recent work by Lara Buchak. Buchak (Reference Buchak2017) defends the generalized Gini family by appealing to her structurally analogous theory of decision under uncertainty. Buchak (Reference Buchak2013: sec 2.3) claims that her decision theory avoids the problem posed by Rabin’s calibration theorem – i.e. that it can accommodate reasonable aversion to small-stakes risks without implying absurd aversion to larger-stakes risks. We might therefore suspect that the generalized Gini family – what we are calling weak egalitarianism – can avoid the problem posed by our prioritarian calibration theorem in a structurally analogous way. ^{Footnote 14}

And indeed it does. To see this, suppose that a weakly egalitarian distributor would prefer a distribution in which both Ann and Bob are equally well off, at any level w, to one in which Bob has welfare $w+1$ and Ann has welfare $w-0.5$ . This implies that the weight assigned to the worst-off, in a two-person society, is more than twice as great as the weight assigned to the better off person. This commits the distributor to preferring, for any level w, a distribution in which two people have welfare w to one in which one person has welfare $w-L$ and the other has $w+2\,L$ . That seems reasonable, not extreme. In sum, our prioritarian calibration theorem has no unwelcome implications for weak egalitarianism.

3.2 Calibration results for the generalized Gini family

We can, however, introduce a different calibration dilemma for weak egalitarianism, which seems to us even more forceful than our prioritarian dilemma. This result has a somewhat different setup, which will require us to consider several distributions over a larger population. Suppose that there are n people in some population, where n is even. And suppose that our preferences between distributions are governed by weak egalitarianism. Suppose, at first, that we can only affect the welfare of the two worst-off people in this population. In one distribution, both of these people would be equally well off, at some level w. In another distribution, one would be slightly better off than the other – one would have welfare $w+1$ , the other $w-0.9$ . Suppose further that, in the latter distribution, these two people would still be at the very bottom of the welfare distribution. This just requires that the third-worst-off person’s welfare is no lower than $w+1$ ; otherwise, their welfare can be whatever we like. It seems to us reasonable to prefer the former distribution – that is, to prefer equality between the two worst-off members of the population to small-stakes inequality between them – at least, for some values of w and of the welfare of other people in the population. Surely our priority to the relatively worse off should not get outweighed by a measly net gain of one-tenth of a unit of total welfare.

What could we conclude from this preference for equality when the stakes are small, given weak egalitarianism? By itself, not very much: just that the weight assigned to the second-worst-off person must be no more than nine-tenths of the weight assigned to the worst-off person. But suppose that this preference is not just confined to tradeoffs between the two worst-off individuals. Consider an analogous choice involving the second- and third-worst-off members of the population. In one distribution, both would be equally well off, at some level w (it needn’t be the same value as in the previous paragraph). In another, one would have welfare $w+1$ , the other $w-0.9$ , where (again) this would not affect the rank ordering of members of the population. This just requires that the fourth-worst-off person’s welfare is no lower than $w+1$ , and that the worst-off person’s welfare is no greater than $w-0.9$ ; otherwise, their welfare can be whatever we like. Again, it seems to us reasonable to prefer equality to small-stakes inequality between these two people – at least, for some values of w and of the welfare of other people in the population.

Suppose that we would have this pattern of preference for equality to small-stakes inequality between every pair of adjacent individuals in the rank ordering of welfare levels – again, for some (possibly different for each pair) value of w and some distribution over the rest of the population. (Perhaps this preference for equality between pairs of individuals of any rank is unreasonable; we will address that issue in a moment.) And suppose that $n={10000}$ – that is, we are assuming a population a third of the size of tiny San Marino’s (we would get even more extreme results by assuming a larger population). Then, for any welfare level w, we must prefer a distribution in which ten thousand people have welfare w to one in which five thousand people have welfare $w-2$ and the other five thousand have $w+1.23 \times 10^{229}$ . That seems to us implausibly extreme, particularly for high values of w.

This result is an instance of our egalitarian calibration theorem, which is stated and proved in the Appendix. The general lesson is this. Suppose that, for every pair of adjacent positions in a population’s rank ordering of welfare levels, we prefer some distribution in which the two people in those positions have some equal welfare level, (w, w), to a distribution in which one is slightly better off than the other, $(w-l,w+g)$ , where $g>l>0$ , and everyone else’s welfare and the rank-ordering of individuals are unaffected by the choice (i.e. no one else’s welfare level is between $w-l$ and $w+g$ ). And suppose that our preferences between distributions obey weak egalitarianism. Then, if the population is large enough, we must prefer a distribution in which everyone is equally well off at any w to any distribution in which half the population is moderately worse off ( $w-L$ ) and the other half is much better off ( $w+G$ ). This is because the weights assigned to better-off people must diminish so quickly that large benefits to such people are given very little weight. Table 4 shows some representative values of this theorem when $g=1$ and $n={10000}$ .

Table 4. Weakly egalitarian distributor who prefers, for some w, (w,w) to $(w-l,w+g)$ at each pair of adjacent positions in a population of 10000 must, for any w, prefer ${\bf{w}}_{{10000}}$ to $({\bf{w}}_{5000}-L,{\bf{w}}_{5000}+G)$ , for Ls and Gs entered in table

The results shown in Table 4 are extremely implausible, even when $l=0.99$ . Egalitarians might reply that the initial pattern of aversion to small-stakes inequality is unreasonable. What might seem unreasonable about this pattern is that it prefers equality in small-stakes tradeoffs even for the very best off. Egalitarians might insist that the aversion to small-stakes inequality is reasonable only when that inequality would be between members of some worse-off subset of the population. Suppose, for example, that it is only reasonable to prefer equality to small-stakes inequality between members of the worst-off decile of the population. The distributor would then still have to prefer any distribution in which everyone is equally well off at any level w to a distribution in which the one decile of the population would have welfare $w-L$ and everyone else would have $w+G$ , for some extremely large values of G. Table 5 shows some representative values when, again, $g=1$ and the population contains ten thousand people.

Table 5. Weakly egalitarian distributor who prefers, for some w, (w,w) to $(w-l,w+g)$ at each pair of adjacent positions in the worst-off decile of a population of 10000 must, for any w, prefer ${\bf{w}}_{{10000}}$ to $({\bf{w}}_{1000}-L,{\bf{w}}_{9000}+G)$ , for Ls and Gs entered in table

Since many egalitarians care about the distribution of welfare over all people (on some views, all animals) who ever live, they care about a population that is much larger than ten thousand people. And our calibration results are even more extreme for larger populations. For example, if there are one billion people, then even if the preference for equality over small-stakes inequality (where $l/g=0.9$ ) is confined to the worst-off ten thousand people, one must prefer a distribution in which everyone has equal welfare at any level w to one in which ten thousand people have welfare $w-2$ and everyone else has welfare $w+6.76 \times 10^{449}$ . That is extreme.

Some egalitarians may defend this extreme implication on the grounds that, when G is very large, the unequal distributions are vastly unequal, and at some point such vast inequalities should outweigh the gains in total well-being. But this response would put most contemporary egalitarians in distributive ethics in a difficult position – particularly those who favour the generalized Gini family. This is because such theorists are opposed to levelling down – i.e. decreasing the welfare of the better-off, leaving others unaffected, for the sake of equality. Most contemporary egalitarians believe, and the generalized Gini family implies, that increasing the welfare of some people, leaving others unaffected, always makes things better overall, even if it makes things worse with respect to equality. (Such egalitarians are, in Parfit’s taxonomy, ‘moderate.’) This is so even when the resulting inequality is vast. But it seems strange to insist that greatly increasing the well-being of some while leaving others unaffected always makes things better, no matter how much doing so increases inequality, while accepting the extreme implications we have identified – i.e. that greatly increasing the well-being of some (in the case above, 999990000 people) while making others very slightly worse off (e.g. by 2 units) makes things worse, owing to the resulting inequality.

Other egalitarians may respond to our result by choosing weights on welfare levels that avoid the extreme horn of our dilemma – that is, weights that don’t imply absurd inequality aversion for high-stakes – by giving up seemingly reasonable aversion to inequality for smaller stakes. For example, as mentioned in the previous subsection, the weights of the standard Gini social welfare function are the odd numbers in increasing order. It is easy to see that using these weights would avoid the extreme consequences in Table 4. But this is because, in our view, the standard Gini is not even moderately inequality averse when stakes are relatively small and the population is sufficiently large. For example, the ratio of the weights between the 10th and the 11th best-off is $19/21$ , which is greater than $9/10$ . So the eponymous member of the generalized Gini family would not prefer that the 10th and 11th best-off people (or any adjacent pairs of individuals worse off than them) both be at welfare level w rather than one of them being at $w-0.9$ and the other at $w+1$ . Nor would it prefer that the 20th and the 21st best-off people (or adjacent pairs worse off than them) both be at w rather than one being at $w-0.95$ and the other at $w+1$ , or that the 100th and the 101st best-off people (or adjacent pairs worse off than them) both be at w rather than one of them being at $w-0.99$ and the other at $w+1$ . So, given the standard Gini weights, none of the antecedents of the conditionals we have considered in this section would be satisfied, when applied to a population of more than 100 people. This, we suspect, would seem problematic to many who have egalitarian intuitions. The standard Gini function prefers equality to small-stakes inequality between adjacent individuals only among the very best off. But it seems to us strange to care so much more about inequality between, say, the best- and second-best-off individuals than about inequality between worse-off individuals in adjacent ranks. ^{Footnote 15}

And yet, as our results illustrate, alternative members of the generalized Gini family that do seem sufficiently inequality averse when stakes are small are unreasonably extreme in their inequality aversion when stakes are large – hence the dilemma. More generally, our calibration dilemma for weak egalitarians is that intuitively reasonable aversion to small-stakes inequalities between small numbers of people (e.g. individual pairs, in our cases), even when limited to a small worse-off subset of the population, leads to manifestly unreasonable aversion to large-stakes inequalities between large numbers of people. The analogy between this dilemma and the calibration dilemma for weak prioritarianism is, very roughly, this: when the prioritarian prefers equality to small-stakes inequality at various welfare levels, this requires the marginal value of incremental benefits to diminish so quickly that very high welfare levels are given very little weight; when the egalitarian prefers equality to small-stakes inequality between individuals at various rank-positions, this requires the weights assigned to better-off people to diminish so quickly that large numbers of better-off people are given very little weight. ^{Footnote 16}

Our dilemma for weak egalitarianism seems to us even more forceful than our dilemmas for weak prioritarianism, because the aversion to large-stakes inequalities is so extreme even when $l/g$ is so close to 1 and L is relatively small. Although the egalitarian dilemma requires small-stakes inequality aversion to take a slightly more complicated form than the small-stakes inequality aversion assumed in the prioritarian dilemma, the assumed inequality aversion in the egalitarian dilemma is still quite weak. For each pair of ranks, we need only assume that the egalitarian prefers to benefit the worse off for some distribution meeting minimal constraints – i.e. for some value of w and some assignment of welfare levels to others that preserves the rank-ordering.

Of course, we have not considered other theories, besides the generalized Gini family, that deserve the name ‘egalitarian’. So we cannot claim to have identified a problem that afflicts all such views. However, our calibration results suggest that aversion to small-stakes inequalities is not best explained by the diminishing marginal importance of well-being (as the prioritarian thinks) or the diminishing weight given to the well-being of the better-off (as the generalized Gini egalitarian thinks).

3.3 Radically nonseparable egalitarianism

We mentioned above that the feature of weak prioritarianism that seems most clearly responsible for its calibration dilemma is its additive separability. We then showed that a similar dilemma afflicts the generalized Gini family, which is the most widely endorsed family of nonseparable views. One lesson we might draw from this result is that separability is not to blame for the prioritarian calibration dilemma. Another possible response, which should be more appealing to some egalitarians, is to claim that the generalized Gini family is still too separable, in a sense that we now explain.

In our egalitarian calibration dilemma, we considered small-stakes tradeoffs that each affect only two people. As long as those tradeoffs do not affect the rank ordering of anyone else in the population, the generalized Gini family allows us to ignore everyone unaffected by each tradeoff. And that is what allows us to infer constraints on the relative weight of each rank from the rejection of each small-stakes tradeoff. This is a kind of separability, although not the full-fledged separability assumed by prioritarianism. It has been called comonotonic separability (see e.g. Wakker Reference Wakker2010: Ch. 10). Roughly, the moral value of distributions is comonotically separable just in case, between two distributions that share the same rank ordering of persons, ^{Footnote 17} which of those distributions is better cannot depend on the welfare of unaffected individuals, that is, those individuals who have the same welfare in both distributions.

For theorists with broadly egalitarian inclinations, the lesson of our calibration results might be that comonotonic separability is still too strong. Such theorists might claim that, in order to value equality while avoiding absurd consequences, we must reject separability in a more radical way than the generalized Gini family does. We should, according to such theorists, accept a radically nonseparable version of egalitarianism. Such a theory may seem independently attractive to those who believe that distributions are organic unities (e.g. Broad Reference Broad1930: 252).

Rejecting comonotonic separability, however, is not enough to avoid our calibration dilemmas. To see this, consider a simple egalitarian view, due to Rescher (Reference Rescher1966: 35), that violates comonotonic separability. ^{Footnote 18} This view cares about two things: average well-being and inequality, measured (somewhat crudely) by a distribution’s standard deviation. The relative importance of the standard deviation is represented by a positive weight x. The view values a distribution according to the difference between average well-being and this x-weighted standard deviation. ^{Footnote 19}

Suppose that our preferences between distributions are governed by the egalitarian view just described. Now compare two distributions. In one, Ann and Bob have welfare w, and Cat and Dan have welfare $w+100$ . In another, Ann has welfare $w+1$ , Bob has $w-0.99$ , and Cat and Dan are still at $w+100$ . Suppose that, for some value of w, we would prefer the first distribution. Then we must prefer any distribution in which all four people would be equally well off, at any level w, to a distribution in which two would have welfare $w+{1000000}$ and the other two would have $w-1$ . ^{Footnote 20}

Of course, this example is quite limited. It does not show that any radically nonseparable egalitarian view will be subject to calibration dilemmas. But it shows that not all such views avoid them. Radical nonseparability is not, by itself, sufficient to avoid calibration dilemmas. The example also suggests a more general challenge, if only in broad strokes. Any plausible egalitarian view would value at least two things: increasing aggregate welfare and decreasing inequality. In the radically nonseparable case, inequality is measured in such a way that the impact of a change in someone’s well-being is sensitive to everyone else’s well-being even when rank-ordering is held fixed. But, on any plausible measure of inequality, the effect of a small-stakes tradeoff between two members of a larger, already unequal population may be minuscule. So, to reject such a tradeoff, the distributor must assign extreme value to equality. This leads her to reject large-stakes transfers that greatly increase both aggregate welfare and inequality. That is what happens in our simple example, in which inequality is measured crudely by standard deviation. But similar dilemmas can be designed for other, more plausible measures of inequality. ^{Footnote 21}

We therefore suspect that aversion to small-stakes inequality is not best explained by a concern for inequality understood as a holistic feature of distributions. But we do not insist on this lesson. We have not demonstrated the impossibility of a plausible, radically nonseparable version of egalitarianism that can capture reasonable aversion to small-stakes inequality without licensing extreme degrees of inequality aversion when the stakes are larger. We leave the search for such a view as a project for further research.