Calibration Dilemmas in the Ethics of Distribution∗ Jacob M. Nebel and H. Orri Stefánsson Draft of June 2, 2020 Abstract This paper presents a new kind of problem in the ethics of distribution. The problem takes the form of several "calibration dilemmas," in which intuitively reasonable aversion to small-stakes inequalities requires leading theories of distribution to recommend intuitively unreasonable aversion to large-stakes inequalities-e.g., inequalities in which half the population would gain an arbitrarily large quantity of well-being or resources. We first lay out a series of such dilemmas for a family of broadly prioritarian theories. We then consider a widely endorsed family of egalitarian views and show that, despite avoiding the dilemmas for prioritarianism, they are subject to even more forceful calibration dilemmas. We then show how our results challenge common utilitarian accounts of the badness of inequalities in resources (e.g., wealth inequality). These dilemmas leave us with a few options, all of which we find unpalatable. We conclude by laying out these options and suggesting avenues for further research. ∗Thanks to Gustaf Arrhenius, MatthewAdler, Ralf Bader, Kara Dreher, Marc Fleurbaey, JimmyGoodrich, Ben Holguín, Nils Holtug, AdamKern, Kacper Kowalczyk, Dillon Liu, Mike Otsuka, Dean Spears, and Alex Voorhoeve. The paper has been presented in seminars at University of Copenhagen, Stockholm University, and Institute for Futures Studies, as well as at the Fourth PPE Society Meeting. We are grateful to the participants for the comments and suggestions. 1 1 Introduction Leading contemporary theories in the ethics of distribution can be regarded as attempts to stake out a reasonable middle ground between two extremes. On one extreme is the utilitarian principle of distribution, which evaluates distributions according to their total (or average) well-being. Many philosophers reject utilitarianism because it is insensitive to inequality in the distribution of well-being. Many of us believe, contrary to utilitarianism, that we ought to give some priority to those who are worse off than others. On the other extreme is the leximin (short for "lexicographic maximin") rule, inspired by Rawls's (1971) difference principle. Leximin assigns absolute priority to the very worst off. It says to choose, between any two distributions, the one that is better for the worst-off person whose welfare differs between those distributions. Whereas utilitarianism seems to assign too little (i.e., zero) priority to the worse off, leximin seems to assign too much. For example, consider a large population in which everyone is quite poorly off. Suppose that we can either benefit one member of this population by an arbitrarily small amount, or benefit any number of people better off than her by an arbitrarily large amount. It seems at least permissible to benefit the many. But leximin would require us to benefit the one, even if the many are quite poorly off too. Most contemporary theorists in the ethics of distribution agree that leximin is implausibly extreme. Yet they have said surprisingly little about how much priority we should give, instead of absolute priority, to the worse off.1 Similarly, they have given no precise guidance for how to, say, balance the interests of the very worst off against the interests of the second-worst off. For example, prioritarians believe that benefiting a person matters more the worse off that person is. When introducing this view, Derek Parfit (1991: 20) says that prioritarian1Notable exceptions include Tungodden and Vallentyne (2005) and Fleurbaey, Tungodden, and Vallentyne (2009), whose results are in a similar spirit to ours. 2 ism does not tell us how much priority we should give to those who are worse off. On this view, benefits to the worse off could be morally outweighed by sufficient benefits to the better off. To decidewhat would be sufficient, wemust simply use our judgement. Fair enough. It is hard to see how else a prioritarian could decide when benefits to the better off would outweigh benefits to the worse off without, as Parfit says, simply using her judgment. Similar remarks apply to contemporary versions of egalitarianism. Egalitarians believe that it is bad, because unfair, for some to be worse off than others (some add: through no fault of their own). But, as egalitarians are quick to clarify, they don't care only about inequality. Egalitarians don't judge a situation in which everyone is equally miserable to be better than a situation in which everyone is happy but unequally so. Many egalitarians care both about decreasing inequality and about increasing total well-being (see Barry 1989: 79; Temkin 2003; Persson 2006).But, Parfit (1991: 5) suggests, if we giveweight to both equality and utility [i.e., well-being], we have no principled way to assess their relative importance. To defend a particular decision, we can only claim that it seems right. Again, fair enough. It is hard to see how else an egalitarian could make decisions that involve tradeoffs between equality and total well-being, without simply using her judgment. This does not mean, however, that prioritarians and egalitarians can consistently endorse any combination of distributive judgments about all cases. On theories of both kinds, our judgments about some tradeoffs commit us to judgments about other tradeoffs. These commitments, we argue, pose a problem for contemporary theories of distributive ethics that are supposed to give some priority, but not extreme priority, to those who are worse off than others. 3 Our argument proceeds through a series of "calibration dilemmas," in which seemingly reasonable aversion to inequalities involving small differences in well-being commits prioritarianism and the most commonly defended version of egalitarianism to seemingly unreasonable aversion to inequalities involving larger differences in well-being- e.g., inequalities from which half the population would gain an arbitrarily large quantity of well-being. (We will soon see some examples.) These implications are not as extreme as leximin's. But, we believe, they are nonetheless implausible. We start by characterizing a general class of views that includes prioritarianism, as typically understood, as a special case. We lay out a series of calibration dilemmas for these views. We then consider a widely endorsed family of egalitarian views and show that views in this family avoid the dilemmas for prioritarians. However, they are subject to no less forceful calibration dilemmas of their own, and other versions of egalitarianism fare no better. We then consider the implications of our calibration dilemmas for utilitarianism. Though we take our results to provide some support for the utilitarian's insensitivity to inequality in the distribution of well-being, they also pose a problem for prominent utilitarian explanations of the badness of inequalities in the distribution of resources (e.g., wealth inequality). Those who do not want to interpret our calibration dilemmas as support for utilitarianism have a few options. For instance, prioritarians and egalitarians could respond to these dilemmas by giving up aversion to inequality when small quantities of well-being are at stake. An alternative responsewould be to simply accept the extreme implications of aversion to small-scale inequality. In the concluding section we consider these responses and suggest avenues for further research. 4 2 Prioritarian Calibration Dilemmas 2.1 Weak Prioritarianism Prioritarians believe that, other things being equal,2 benefiting a person matters more, from a moral or social point of view, the worse off the person is. As formulated by Parfit (1991: 105), the thought is that "utility [i.e., individual welfare] has diminishingmarginal moral importance." Prioritarians often favor interventions that increase equality. In fact, given any fixed sum of welfare, the view always favors 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.3 Themoral value of a person'swell-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 themoral 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 wellbeing 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 2 We assume throughout that things other than people's welfare are held fixed across all interventions and situations we consider. So, for instance, questions about desert or responsibility will not arise. We also assume a fixed population size. 3Although this framing is most natural for "telic" prioritarianism, understood as a view about the goodness of distributions, our results apply just as well to deontic versions of prioritarianism (e.g., Nebel 2017; Williams 2012), as long as the choiceworthiness of a distribution, in the kinds of situations we discuss, is evaluated in a prioritarianmanner. Our results are also robust across different views about the "currency" of distribution. Althoughwe explicitly discuss distributions of welfare, the same results apply for distributions of other goods-e.g., primary goods, capability indices, expected welfare, or whatever. 5 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 such a 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 0 20 40 60 80 100 0 2 4 6 8 10 12 14 Well-being Pr io rit ywe ig ht ed we llbe in g Strictly Concave Weakly Concave 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 6 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 2003). 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 (which means that welfare above the point in question is not priority weighted). 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. Thismeans that priority is never given to those who are worse off (nor to those who are better off). Since the calibration dilemmas that we raise in the next section assume some priority to the worse off, the arguments in the next two sections are not arguments against (but possibly indirect arguments in favor of) utilitarianism. In section 4, however, we discuss in more detail what our results imply for utilitarianism. To summarize, the weakly prioritarian family of social welfare (i.e., moral value) functions compares distributions according to their sums of priority-weightedwell-being, where the priority weighting function is strictly increasing and weakly concave. In the next section, 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 7 we follow the methodology suggested by Parfit: using our judgment. In subsection 2.3, we say more about the different scales on which welfare might be measured, and explain why our results hold given any scale that is available to a weak prioritarian. Some may nevertheless find themselves unable to make judgments about tradeoffs involving different quantities of welfare without further information about how we assume welfare to bemeasured-i.e., what exactly the lives at variouswelfare 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., Lipman and Attema 2019; Nord and Johansen 2014; Otsuka 2017; Parfit 2012). For example, we might take a unit of well-being to correspond to a year of very happy life. The use of this measure requires that a year of such life is just as good for a person no matter how long or well she has otherwise lived (though, for a prioritarian, the moral weight of such a year diminishes with happy lifespan). This assumption is questionable (though, for defense, see Broome 2004). But, because this sort of measure is both widely used and fairly concrete, we sometimes use it for illustrative purposes; we restrict it to cases in which all lives are of the same quality throughout, so that the lives under comparison differ only in their length. Again, though, we do not take our own judgments to depend on this or any other concrete measure of well-being. It may simply help to make our results easier to understand. It is also worth emphasizing that our results themselves will not depend on any particular measure of well-being; only the interpretation and implications of our results could be taken to depend on such ameasure. We invite those who favor some alternative measure of well-being to apply their fa8 vored measure when interpreting our results. We are open to the possibility that our results may look more or less palatable given different measures of well-being. In that case, our results may affect not only our choices between distributive theories, but also our choices between measures of well-being. And, even for those who are completely untroubled by our results given their favored measure of well-being, our results may be of great practical importance. For, in order to apply any distributive theory such as prioritarianism to real-world tradeoffs, one needs to choose between competing versions of the theory, which support different degrees of priority to the worse off. Our results may, at the very least, help such a theorist to make a more informed choice. 2.2 Calibration Results for Weak Prioritarianism Suppose that your preferences between distributions are governed by weak prioritarianism.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 ofw, youwould 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 0.1. 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 y" means that x is preferred to y): (50, 50) (49.1, 51) (1) 4By this wemean that, for some weakly concave priority weighting function, you prefer x to y just in case x contains at least as much priority-weighted well-being as y. 9 What can we infer from this particular choice 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 that youwould prefer the equal distribution for any value ofw. (This assumption will soon be relaxed; that is, later wewill 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: For all w: (w,w) (w + 1, w − 0.9) (2) 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 areweakly 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 sizeG 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. (This follows from the first theorem we prove in the appendix, which we are about to explain.) 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 10 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 has other w +G, then she must also prefer a distribution in which any number of people havewelfarew to a distribution inwhich half that number of people have w − L and the other half have w + G. In particular, a weakly prioritarian preference that satisfies equation (2) entails that for any level w, and for any population size, a distribution in which everyone is equally well off at level w is preferable to a distribution in which half the population is at level w − 8, no matter how well off the other half would be. It will be helpful, especially later on, to have some symbols to represent these distributions involving populations of arbitrary size. Take any distribution w = (w1, . . . , wn), where wi represents the welfare of person i. If n people have the same welfare w, we represent this distribution aswn: that is,wn contains n people at level w. For any distribution w and quantity of welfare k, let w + k represent the distribution (w1 + k, . . . , wn + k) in which k is added to each person's welfare inw. For any distributionsw = (w1, . . . , wn) and u = (u1, . . . , um), let (w,u) represent the distribution (w1, . . . , wn, u1, . . . , um) that concatenates (or "stacks") w and u together. What we have found is that a weakly prioritarian distributor whose preferences satisfy (2) must also satisfy For all w, n, and G:w2n (wn − 8,wn +G) (3) 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 11 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. It may help to translate this result into more concrete terms, using a longevity-based measure of well-being. Suppose that everymember of some population derives a constant unit of welfare from each year of happy life: every year of happy life is worth just as much to each of them as every other year is to them and everyone else; there is no danger of longevity causing boredom, for example. Let's also imagine that people in this population are equally well off in all respects other than possibly their lifespans. Now, suppose that one of these people will live slightly longer than some other person, (w − 0.9, w). It may seem reasonable to extend the lifespan of the shorter-lived person so that the two enjoy equally many happy years (w,w), rather than extending the lifespan of the longerlived person by slightly longer (w− 0.9, w+1), on grounds of priority to the shorter-lived. Suppose we have this preference, and that our preferences between distributions are governed by weak prioritarianism. Then we must prefer that everyone lives for 110 happy years, (110n, 110n), to half the population living 102 years and the other half living a billion years (102n, 109n). But, on the assumption that each person derives a constant amount of welfare from a year of happy life, no matter how many other years she has lived, it would seem much better for half the population to live for a billion years than for everyone to be mere supercentenarians, when the difference for the other half would be less than a decade. Of course, these assumptions about the relation between welfare and happy life-years can be plausibly denied. We do not wish to commit ourselves to any concrete interpretation of quantities of welfare; we offer the example only for illustrative purposes. (An alternative illustration can be given in terms of relieving pains-e.g., hours of a headache of some severity.) Now, even on the above assumptions about the relationship between welfare and happy life-years, some may prefer the distribution in which everyone lives for 110 years on broadly egalitarian grounds. It may seem unfair for some to live nearly 12 a billion years longer than others. But prioritarians cannot appeal to this kind of egalitarian rationale; they care only about each person's absolute level of welfare, not about how people fare relative to each other. The above result is one instance of our prioritarian calibration theorem, which is stated more formally and proved in an appendix. Our theorem is inspired by Matthew Rabin's (2000) celebrated calibration theorem for expected utility theory. Rabin's theorem establishes that an expected utility maximizer can only be risk averse when small sums of money are at stake if she is absurdly risk averse when more is at stake.5 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 two people are at level w to a distribution in which one has welfare w +G and the other has 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 ofG, however large. To take another example: if a weakly prioritarian distributor prefers (w,w) to (w − 0.5, w + 1) for any w, then she must, for any w and n, prefer w2n to (wn − 2,wn +G), no matter how large G is. Put schematically, the theorem says that weak prioritarianism entails conditionals of the following form: If, for all w: (w,w) (w − l, w + g), then, for all w, n:w2n (wn − L,wn +G). (4) 5There are at least three differences between our theorem and Rabin's. First, our theorem is slightly stronger, and our proof is slightly simpler. Second, Rabin's discussion is formulated in terms of gains and losses, whereas we use no such framing; this is important because Rabin's preferred explanation of the observation that people do not behave as his theoremwould predict appeals to (non-expected utilitymaximizing) loss aversion (Kahneman and Tversky 1979), which cannot explain our non-loss-involving judgments. Third, Rabin's theorem involves quantities of money, whereas our focus is on welfare; this opens up variations of the kind introduced in the next subsection, which exploit common assumptions about the numerical representation of welfare that do not apply to money. 13 Table 1 illustrates several representative examples of these results, where g = 1. 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, preferw2n to (wn−L,wn+G), where eachL 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 ofG for each choice of l and L. If a cell contains ∞, 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 w2n to (wn − 4,wn +G). Table 1: Weakly prioritarian distributor who prefers (w,w) to (w− l, w+ g) for all w when g = 1must, for all w, n, prefer w2n to (wn − L,wn +G) for Ls and Gs entered in table. If, for all w, (w,w) (w − l, w + 1) Then, for all w, n, w2n l = 0.99 l = 0.95 l = 0.9 l = 0.75 l = 0.5 (wn − 2,wn+ 2) 2) 2) 5) ∞) (wn − 4,wn+ 4) 5) 7) ∞) ∞) (wn − 6,wn+ 6) 8) 20) ∞) ∞) (wn − 8,wn+ 8) 13) ∞) ∞) ∞) (wn − 10,wn+ 11) 21) ∞) ∞) ∞) (wn − 15,wn+ 17) ∞) ∞) ∞) ∞) (wn − 20,wn+ 25) ∞) ∞) ∞) ∞) (wn − 25,wn+ 33) ∞) ∞) ∞) ∞) (wn − 50,wn+ 105) ∞) ∞) ∞) ∞) (wn − 75,wn+ ∞) ∞) ∞) ∞) ∞) The examples in table 1 all assume g = 1. But the results can be easily generalized. What matters is the ratio between g and l. If the values of g and l in table 1 are multiplied by a common constant k, the corresponding values of G and L should be multiplied by k. For example, a weakly prioritarian distributor who, for any w, would prefer (w,w) to (w−9, w+10)must, for anyw, n, andG, preferw2n to (wn−80,wn+G). Our theorem also allows us to derive similar implications involving groups of different sizes. For example, a weakly prioritarian distributor who prefers (w,w) (w− 0.5, w+ 1) for any w must, for any w, n, and G, prefer w100n to (wn − 8,w99n +G). 14 As table 1 illustrates, the results can bemade 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 w2n to (wn − 75,wn + G), for arbitrarily large G. And even this seems unreasonable for sufficiently high values of w, since then w − 75 remains an excellent life. To use our earlier example involving life-years, if we assume that each year of happy life makes a constant contribution to lifetimewell-being, it would seem unreasonable to prefer a distribution in which everyone lives for 1075 years to one in which half the population lives for billions of years and the other half lives for "only" a millennium. It seems too extreme to sacrifice so much welfare on grounds of priority to the worse off, who are, in any case, extremely well off. 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 theworse offwhen 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(*), and for any w, the total priority-weightedwell-being of (w,w) is greater than the total priority-weightedwell15 being of (w − 0.5, w + 1): f(w) + f(w) > f(w − 0.5) + f(w + 1) (5) 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): f(100) + f(100) > f(99.5) + f(101) (6) So the difference in priority-weighted value between 100 and 101-i.e., one unit-must be less than the difference in priority-weighted value between 99.5 and 100-i.e., half a unit: f(101)− f(100) < f(100)− f(99.5) (7) 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: f(102)− f(101) < f(101)− f(100.5) (8) Butwhat 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 16 between 100 and 101, since the latter interval starts from a lower level: f(101)− f(100.5) 101− 100.5 ≤ f(101)− f(100) 101− 100 (9) 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: f(101)− f(100.5) ≤ 1/2 [f(101)− f(100)] (10) So, putting (8) and (10) together, the difference in priority-weighted value between 102 and 101must be less than half the difference between 101 and 100: f(102)− f(101) < 1/2 [f(101)− f(100)] (11) 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 = 1/562 949 953 421 312 times the difference in priority-weighted value between 99.5 and 100. This steep decline in marginal priorityweighted value establishes a firm upper bound on the priority-weighted value of gains in well-being. Specifically, no matter how largeG is, the total difference in priority-weighted well-being between 100 + G and 100 must be less than twice the difference in priorityweighted well-being between 100 and 99.5.6 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 6This is because the series∑∞i=1(1/2)i−1 converges to 2. 17 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 theworse off because of how badly off they are, and therefore need not prioritize theworse 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 ofwell-beingup to 100. The theorem implies, for example, that if the distributor prefers anydistribution inwhich bothAnn andBobhavewell-being levelw to one inwhich 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 1.67million. Table 2: Weakly prioritarian distributor who prefers (w,w) to (w− l, w+ g) for all w ≤ 100 when g = 1must prefer 752n to (75n − L,75n +G) for Ls and Gs entered in table. If, for all w up to 100, (w,w) (w − l, w + 1) Then, when w = 75, for any n, w2n l = 0.99 l = 0.95 l = 0.9 l = 0.75 l = 0.5 (wn − 2,wn+ 2) 2) 2) 5) 1.34× 108) (wn − 4,wn+ 4) 5) 7) 6197) 9.40× 108) (wn − 6,wn+ 6) 8) 20) 19 263) 4.16× 109) (wn − 8,wn+ 8) 13) 80) 42 491) 1.70× 1010) (wn − 10,wn+ 11) 21) 155) 83 786) 6.86× 1010) (wn − 15,wn+ 17) 56) 432) 387 147) 2.20× 1012) (wn − 20,wn+ 25) 101) 902) 1.67× 106) 7.04× 1013) (wn − 25,wn+ 33) 160) 1696) 7.05× 106) 2.25× 1015) (wn − 50,wn+ 80) 837) 26 784) 9.39× 109) 7.56× 1022) (wn − 75,wn+ 141) 3728) 376 242) 1.25× 1013) 2.54× 1030) 18 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 5002n to (400n, 5.98× 1012n). 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 752n to (25n,185418n). And these results, based on seemingly reasonable degrees of priority to the worse off, still seem very extreme. Let us clarify the problemwe take these results to pose for weak prioritarianism. Many of us want to give some priority to those who are worse off. If such priority is justified, it should be reasonable to prioritize slightly smaller benefits to the slightly worse off from a wide range of welfare levels, as in the values of g, l, and w we have considered. But our priority to the worse off has limits. We do not prefer small benefits the slightly worse off at the expense of extremely large gains to half the population. At least, we (the authors) have those distributive preferences, as do (in our experience)many other theoristswith broadly prioritarian inclinations. But these preferences are inconsistentwithweak prioritarianism. Something has to go. Thus, the dilemma: weak prioritarians 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 Responses to the Prioritarian Dilemma 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. Views in the weakly prioritarian family cannot fully capture our reasonable priority to the worse off. This does not mean that weak prioritarianism is false, because utilitarianism is a version of weak prioritarianism (where the weighting function is linear) and we are not sure-largely for the reasons 19 explored in this paper-that we should give priority to the worse off. But if we should give priority to the worse off, then we should reject weak prioritarianism, and should instead accept some other theory that can accommodate, without extreme consequences, the seemingly reasonable distributive preferences we have considered. In section 3, we ask whether some version of egalitarianism could be such a theory. That is our favored response, but again, we do not insist on it. There is 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 themoderatelyworse off even at the expense of the very large gains we have considered. We don't have anything like a decisive, nonquestion-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 judgments about cases, may remain unfazed. But we ourselves- including one authorwho has defended a version of prioritarianism in otherwork-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. It just does not seem unreasonable to us to benefit worse-off Ann by 0.75 units, resulting in an equal distribution, rather than better-off Bob by 1 unit, resulting in a more unequal distribution, from initial welfare levels up to 100. And it just does not seem reasonable to us, when half the population is at level 55 and the other is at 1.67× 106, to equalize their welfare to level 75. The falsity of one of these judgments may, in the end, turn out to be a highly nonobvious consequence of our best distributive theory. But, to us, it would be an unattractive consequence. There is a kind of skeptical 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 20 reasonably wonder whether we can rely on our intuitions about distributions expressed in these terms. After all, onmany views about themeasurement ofwell-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 skeptic may argue, we should not trust intuitions that are sensitive to particular numerical representations of welfare. This responsemay 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. This is all somewhat abstract, so let us consider a concrete example. 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. This kind of scale is assumed by the Atkinson social welfare function, which is the most widely endorsed version of prioritarianism. It is, for instance, defended at length by Adler (2011).7 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" 7Atkinson prioritarians hold that the priority weighting function f(*) has the form f(w) = (1− δ)−1w1−δ for δ > 0 (except when δ = 1, in which case f(w) = logw), where δ represents the degree of priority to the worse off. 21 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 inwhich Ann and Bob are both at some levelw to a distribution in which Bob is 1% better off than w and Ann is 0.5%worse off than w. And suppose that welfare is measurable on a ratio scale, 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 ratioscale measurability of welfare, weak prioritarianism entails conditionals of the following form: If, for some w: (w,w) ([1− l]w, [1 + g]w), then, for all w, n:w2n ([1− L]wn, [1 +G]wn), (12) where, for anydistributionw andnumber k, kw represents the distribution (k w1, . . . , k wn) in which each person's welfare inw is multiplied by k. 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∞, this means that G can be arbitrarily large. For example, the table says that if welfare is measurable on a ratio scale, then 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 and any G, prefer (w,w) to (0.80w, [1 +G]w). In table 3, the only pairs of L andG 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× 10−30. So perhaps the weak prioritarian should insist that, when g = 1%, the small-stakes tradeoff should only be rejected when l is as low as 22 Table 3: Given ratio-scale measurability of welfare, a weakly prioritarian distributor who prefers (w,w) to ([1− l]w, [1+g]w) for some wwhen g = 1%must, for any w and n, prefer (w,w) to ([1− L]w, [1 +G]w) for 1− Ls and 1 +Gs entered in table. If, for some w, (w,w) ([1− l] w, 1.01w) Then, for all w, n, w2n l = 0.99% l = 0.95% l = 0.9% l = 0.75% l = 0.5% (0.98wn, 1.02wn) 1.02wn) 1.02wn) 1.03wn) ∞wn) (0.96wn, 1.04wn) 1.04wn) 1.05wn) ∞wn) ∞wn) (0.94wn, 1.06wn) 1.07wn) 1.15wn) ∞wn) ∞wn) (0.92wn, 1.08wn) 1.11wn) ∞wn) ∞wn) ∞wn) (0.90wn, 1.11wn) 1.17wn) ∞wn) ∞wn) ∞wn) (0.85wn, 1.16wn) 1.93wn) ∞wn) ∞wn) ∞wn) (0.80wn, 1.23wn) ∞wn) ∞wn) ∞wn) ∞wn) (0.75wn, 1.30wn) ∞wn) ∞wn) ∞wn) ∞wn) (0.50wn, 1.97wn) ∞wn) ∞wn) ∞wn) ∞wn) (0.25wn, 4.02wn) ∞wn) ∞wn) ∞wn) ∞wn) 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 greater values of l, such as 0.995%. And, second, because it seems unreasonable to reject, from extremely low initial levels, the large-stakes tradeoff when L is minuscule and G is arbitrarily large, sincemultiplying such extremely low levels by aminusculeLmakes such a small difference, in particularwhen compared to the differencemade bymultiplying that same level by an arbitrarily large 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. We regard this result as our most troubling dilemma for weak prioritarians. It just seems reasonable to us to prefer some distribution in which two people are equally well off at some level w to another in which one person is 1% better off than w and the other is 0.9% worse off than w. On a longevity-based measure of well-being, for example, this seems reasonable to us when a life at levelw contains, say, 50 very worthwhile years. And, for some values of w (e.g., 100 years of equally worthwhile life), it just does not seem 23 reasonable to us to prefer any distribution in which any number of people are at level w to any distribution in which half that population is 8% worse off than w while the other half has any arbitrarily high level of welfare. Priority to the worse off must sometimes be outweighed, we think, by arbitrarily large gains in well-being.8 In response to these judgments, some may continue to 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 theworst 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 theoriesmust simply use their judgment. But if, as the skeptical 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 tomake distributive choices in practice? If the skeptical 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 prior8An Atkinson prioritarian might try to rationalize these extreme implications by postulating that the moral value of welfare is bounded. After all, the particular members of the Atkinson family that have the most extreme results discussed above imply that the moral value of welfare is bounded. But even if one has a principled explanation for why the moral value of welfare should be bounded (which we do not find very plausible), this does not make it seem reasonable to reject the particular large-stakes tradeoffs that we have identified (e.g., tradeoffs in which one person is 2% worse off than w and another is better off by an arbitrarily large factor). Similarly, even though many people think that the utility of money is bounded (e.g., as a possible lesson of the St. Petersburg paradox), that does not make the particular results of Rabin's calibration theorem seem any more reasonable (e.g., that a risk-averse decision-maker would turn down a 50-50 gamble between losing $1 000 and gaining an infinite amount of money). In both cases, what is extreme is not really the willingness to give up an infinite gain, but rather the small (potential) loss that is needed to outweigh any arbitrarily large (potential) gain. Moreover, even if there were independent reasons for thinking that the moral value of welfare is bounded, we take it to be instructive to learn that, assuming an Atkinson function, moderate inequality aversion when stakes are small implies that the moral value of welfare is bounded. This would still be relevant for a committed Aktinson-prioritarian because not all Atkinson functions imply that the moral value of welfare is bounded. 24 ity, this would seem to make such views fairly uninteresting from a practical perspective, given the need to make tradeoffs in the real world. We do not insist that the skeptical attitude is wrong. But we do not think it is helpful for those who wish to give priority to the worse off. It only makes it harder to navigate between the extremes of utilitarianism and leximin. Again, our preferred response to the dilemma is not necessarily to reject weak prioritarianism (since, again, utilitarianism is weakly prioritarian). But, if we can reasonably give priority to the worse off (which perhaps we can't), then we would be inclined to reject weak prioritarianism. At the very least, we would prefer an alternative theory, if there is one, that can accommodate our seemingly reasonable aversion to small-stakes tradeoffs without leading to seemingly unreasonable aversion to large-stakes tradeoffs. In the next section, we askwhether a version of egalitarianism can serve as that alternative. 3 Egalitarian Calibration Dilemmas 3.1 The Generalized Gini Family Supposewewish 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 priorityweighting 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 25 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.9 Unlike prioritiarians, egalitarians care about howwell 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 1981). 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. Strict egalitarianism is the version of this viewwhere the sequence of weights 9When it comes to the distinction between egalitarianism and prioritarianism, we follow a terminological convention that has become common in philosophy since Parfit (1991); see also Adler and Holtug (2019), Broome (2015), and Otsuka and Voorhoeve (2018). 26 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. Finally, utilitarianism is the version of weak egalitarianism where each person (irrespective of her rank) gets the same weight. Wedefineweak egalitarianism in thisway for convenience, because it is the only paradigmatically egalitarian view that we extensively discuss.10 We do not mean to imply that the generalized Gini family is the only kind of view that deserves the name "egalitarianism," andwe 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.11 It is compatible with a cardinal scale of well-being; it plays nicely with both positive and negativewelfare levels; and it has been claimed to have plausible implications in variable-population and risky cases (Asheim and Zuber 2014; 2016). Finally, and most importantly, the generalized Gini family seems especially relevant to the present discussion in light of recent work by Lara Buchak. Buchak (2017) defends the generalized Gini family by appealing to her structurally analogous theory of decision under uncertainty. Buchak (2013: 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. 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 10Buchak (2017), however, defends the Gini family on what she calls prioritarian grounds. Since the issue of nomenclature is irrelevant to our argument, we have no interest in defending our characterization of prioritarianism as committed to additive separability (thereby excluding nonutilitarian Gini functions). 11See, e.g., d'Aspremont and Gevers (2002) and Fleurbaey (2010). For more specific members of the Gini family, see Donaldson and Weymark (1980) and Blackorby, Bossert, and Donaldson (2005: ch. 4). 27 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+2L. 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,12 which seems to us no less 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 npeople 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 peoplewould be equallywell off, at some levelw. 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 relativelyworse 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, 12This dilemma is loosely based on the decision-theoretic argument of Sadiraj (2014); see also Cox and Sadiraj (2006) and Cox, Sadiraj, Vogt, and Dasgupta (2013). 28 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 secondand 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 fourthworst-off person's welfare is no lower than w + 1, and that the worst-off person's 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 ofwelfare levels- again, for some (possibly different for each pair) value ofw 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 = 10 000-that is, we are assuming a population a third of the size of tiny San Marino's (wewould get evenmore 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× 10229. 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 29 distribution inwhich 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 = 10 000. 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 10 000 must, for any w, prefer w10 000 to (w5000 − L,w5000 +G), for Ls and Gs entered in table. If, for each pair of adjacent positions, there is some w such that (w,w) (w − l, w + 1) Then, for any w, w10 000 l = 0.99 l = 0.95 l = 0.9 l = 0.75 l = 0.5 (w5000 − 2, w5000+ 1.33× 1022) 4.82× 10111) 1.23× 10229) 9.88× 10624) 2.82× 101505) (w5000 − 4, w5000+ 2.67× 1022) 9.64× 10111) 2.45× 10229) 1.98× 10625) 5.65× 101505) (w5000 − 6, w5000+ 4.00× 1022) 1.45× 10112) 3.68× 10229) 2.96× 10625) 8.47× 101505) (w5000 − 8, w5000+ 5.33× 1022) 1.93× 10112) 4.90× 10229) 3.95× 10625) 1.13× 101506) (w5000 − 10, w5000+ 6.67× 1022) 2.41× 10112) 6.13× 10229) 4.94× 10625) 1.41× 101506) (w5000 − 15, w5000+ 1.00× 1023) 3.61× 10112) 9.19× 10229) 7.41× 10625) 2.12× 101506) (w5000 − 20, w5000+ 1.33× 1023) 4.82× 10112) 1.23× 10230) 9.88× 10625) 2.82× 101506) (w5000 − 25, w5000+ 1.67× 1023) 6.02× 10112) 1.53× 10230) 1.23× 10626) 3.53× 101506) (w5000 − 50, w5000+ 3.33× 1023) 1.20× 10113) 3.06× 10230) 2.47× 10626) 7.06× 101506) (w5000 − 75, w5000+ 5.00× 1023) 1.81× 10113) 4.60× 10230) 3.70× 10626) 1.06× 101507) 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 30 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 welfarew−L and everyone else would havew+G, for some extremely large values ofG. 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 10 000 must, for any w, prefer w10 000 to (w1000 − L,w9000 +G), for Ls and Gs entered in table. If, for each pair of adjacent positions in the lowest decile, there is some w such that (w,w) (w − l, w + 1) Then, for any w, w10 000 l = 0.99 l = 0.95 l = 0.9 l = 0.75 l = 0.5 (w1000 − 2, w9000+ 509) 7.98× 1019) 1.14× 1043) 5.79× 10121) 2.38× 10297) (w1000 − 4, w9000+ 1019) 1.60× 1020) 2.29× 1043) 1.16× 10122) 4.76× 10297) (w1000 − 6, w9000+ 1528) 2.39× 1020) 3.43× 1043) 1.74× 10122) 7.14× 10297) (w1000 − 8, w9000+ 2038) 3.19× 1020) 4.58× 1043) 2.32× 10122) 9.52× 10297) (w1000 − 10, w9000+ 2547) 3.99× 1020) 5.72× 1043) 2.89× 10122) 1.19× 10298) (w1000 − 15, w9000+ 3821) 5.98× 1020) 8.58× 1043) 4.34× 10122) 1.79× 10298) (w1000 − 20, w9000+ 5095) 7.98× 1020) 1.14× 1044) 5.79× 10122) 2.38× 10298) (w1000 − 25, w9000+ 6369) 9.97× 1020) 1.43× 1044) 7.24× 10122) 2.98× 10298) (w1000 − 50, w9000+ 12 739) 1.99× 1021) 2.86× 1044) 1.45× 10123) 5.95× 10298) (w1000 − 75, w9000+ 19 109) 2.99× 1021) 4.29× 1044) 2.17× 10123) 8.93× 10298) 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 evenmore 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× 10449. That is extreme. Some egalitarians may defend this extreme implication on the grounds that, when G 31 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 responsewould putmost contemporary egalitarians in distributive ethics in a difficult position-particularly those who favor the generalized Gini family. This is because such theorists are opposed to leveling 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 thewell-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 thewell-being of some (in the case above, 999 990 000 people)while making others very slightly worse off (e.g., by 2 units) makes things worse, owing to the resulting inequality. The calibration dilemma for weak egalitarianism is that intuitively reasonable aversion to small-stakes inequalities between small numbers of people (e.g., individual pairs, in our cases), evenwhen limited to a small worse-off subset of the population, leads tomanifestly 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. Our dilemma for weak egalitarianism seems to us even more forceful than our dilem32 mas 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 smallstakes inequalities is not best explained by the diminishing marginal importance of wellbeing (as the prioritarian thinks) or the diminishing weight given to the well-being of the better-off (as the Gini egalitarian thinks). 3.3 Radically Nonseparable Egalitarianism Wementioned, on page 25, that the feature ofweak prioritarianism that seemsmost 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, that 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 33 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 2010: 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,13 which of those distributions is better cannot depend on thewelfare of unaffected individuals, that is, those individuals who have the samewelfare 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 equalitywhile avoiding absurd consequences, wemust reject separability in amore 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 1930: 252). Rejecting comonotonic separability, however, is not enough to avoid our calibration dilemmas. To see this, consider a simple egalitarian view, due to Rescher (1966: 35), that violates comonotonic separability. 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.14 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 13More precisely, if one person is better off than another in one distribution, then she isn't worse off than that person in the other distribution. 14That is, Average − x × Standard Deviation. Rescher sets x = 1/2. The problem we raise also applies if the weighted standard deviation is subtracted from total rather than average well-being. 34 would prefer the first distribution. Then we must prefer any distribution in which all four people would be equally well off, at any levelw, to a distribution in which twowould have welfare w + 1000 000 and the other two would have w − 1.15 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.16 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. Butwe 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 15This is because rejection of the small-stakes tradeoff requires x to exceed about 0.88, and acceptance of the large-stakes tradeoff requires x to be less than about 0.87. 16Consider, for example, the "total pairwise difference" model considered by Arrhenius (2013), based on Temkin (1993)'s "additive principle of equality" and "relative to all those better off view of complaints" (see also Rabinowicz 2008). On this view, inequality is measured by the sum of (absolute) differences for each distinct pair of individuals. Suppose that the value of a distribution is given by aggregate welfare minus x-weighted inequality, so measured, for some real number x. Suppose, using the example raised above, that we prefer the first distribution (w,w,w + 100, w + 100) to the second (w − 0.99, w + 1, w + 100, w + 100) for some w. Then, for any w, we must prefer two thousand people having welfare w to one thousand having welfare w + 1000 000 and the other thousand having welfare w − 1. 35 larger. We leave the search for such a view as a project for further research. 4 Utilitarianism and Inequality We take our calibration dilemmas for prioritarianism and egalitarianism to weaken the appeal that these views have over the extreme opposites of utilitarianism on the one hand and leximin on the other hand. But since leximin is, in our view, clearly too extreme, we think that our discussion so far should be seen as providing some indirect evidence in favor of utilitarianism. More on this in the next, concluding section. But first, we explain, in this penultimate section, why our calibration dilemmas also have some troubling implications for utilitarianism. Utilitarianism, of course, is consistent with some aversion to certain kinds of inequality. For instance, utilitarians typically claim to be averse to inequality in the distribution of goods, such as money, that are assumed to have decreasing marginal utility. According to Bentham (1843: 206), "The loss of a portion of wealth will produce a loss of happiness to each individual, more or less great, according to the proportion between the portion he loses and the portion he retains." And Pigou (2013) finds it "evident that any transference of income from a relatively rich man to a relatively poor man of similar temperament, since it enables more intense wants, to be satisfied at the expense of less intense wants, must increase the aggregate sum of satisfaction." Some utilitarians (e.g., Greene and Baron 2001) appear to believe that the diminishing marginal utility of resources best explains and justifies our aversion to inequality in the distribution of resources, and indeed explains away our nonutilitarian intuitions about the badness of inequality in welfare. We take our calibration results to count strongly against this utilitarian account of inequality aversion when relatively little is at stake. After all, one can simply re-interpret the numbers in our calibration results for prioritarianism, from section 2.2, as, say, dollar amounts rather than magnitudes of welfare. Therefore, if a utilitarian wants to accom36 modate general inequality aversion with respect to money by postulating that people's welfare functions are strictly concave over money, then the resulting view will only be moderately inequality averse when relatively little money is at stake if it is absurdly inequality averse when more money is at stake. To take an example, suppose that a utilitarian distributor would, on account of the decreasing marginal utility of money, prefer a situation where each of Ann and Bob have an equal total wealthw, up to $100 000, to a situationwhere Ann's total wealth isw+$1 000 while Bob's is w − $500. Then our calibration result for weak prioritarianism shows that the distributor would have to prefer a situation where each of Ann and Bob have a total wealth of $75 000 to a situation where Bob has a total wealth of $67 000 and Ann has a total wealth of $17 000 000 000 000 (i.e., about 227 million times her initial wealth of $75 000). Now it is possible that inequality as extreme as that in the above paragraph has negative effects on total welfare that cannot be explained in terms of the decreasing marginal utility of money. For instance, compare a situation where everyone has an equal wealth of $75 000 to one where Ann has a wealth of $17 000 000 000 000 while nobody else has a wealth greater than $75 000. It is not hard to imagine that the extreme difference in wealth in the second scenario will bring with it difference in political power, say, that could negatively affect the welfare of everyone except Ann. In that case, we might find that total welfare decreases when we go from the equal situation to the unequal one. The observation in the last paragraph does not, however, undermine calibration arguments against the claim that inequality aversion with respect to money can be completely explained by the decreasing marginal utility of money. For note that a utilitarian who justifies aversion to small-stakes wealth inequality (such as that in the example two paragraphs back) by pointing to the decreasing marginal utility of money, would have to accept extreme consequences for large stakes even if the inequality in question would have no effects on total welfare that are not due to the decreasing marginal utility of money. For instance, if a utilitarian wants to justify the preference for ($100 000, $100 000) over 37 ($101 000, $99 500) in terms of the decreasing marginal utility of money, then she has to be able to justify the preference for ($75 000, $75 000) over ($67 000, $17 000 000 000 000) in terms of only the decreasing marginal utility of money. But surely that is implausible.17 Although the social, structural, or political effects of great differences in wealth might justify the latter preference, it really is hard to see how an $8 000 dollar wealth loss to one person, from an initial wealth of $75 000, could by itself so drastically affect their standard of living that their welfare loss would not be smaller than the welfare gain to someone who, from the same initial wealth, increases their wealth by a factor of more than two hundred millions. There are other ways in which utilitarians can try to accommodate aversion to resource inequality. For example, some utilitarians claim that inequality is bad for people (see, e.g., Broome 1991). On this kind of view, there might be some good-e.g., some instrumentally good resource, or perhaps individual well-being before considering the badness of inequality for each person-that ought to be distributed in an egalitarian manner, at least when other ingredients of well-being are held fixed. The idea would be that doing so would be required for total all-things-considered well-being to be maximized. We take our calibration results to count strongly against that view, as well, at least if an egalitarian distributionmeans that the distribution satisfies some version of the egalitarian formulas we have considered (e.g., the Gini family). One can re-interpret the numbers in our egalitarian calibration results, from section 3, as quantities of some good rather than amounts of all-things-considered welfare. Therefore, a utilitarian like the one under consideration could only be moderately averse to inequality with respect to resources (or welfare without the inequality factor) when little is at stake and few people are involved if 17This implication seems implausible even if the welfare effect of wealth has some limit-that is, even if there is some level of wealth beyond which no gain in wealth will increase a person's welfare. For such limits to justify the preference for ($75 000, $75 000) over ($67 000, $17 000 000 000 000), one would have to assume that the welfare increase from extra wealth is satiated somewhere not too far from a total wealth of $75 000 dollars. But Kahneman andDeaton (2010: 16491) find that "the effects of income on individuals' life evaluations showno satiation, at least to an amountwell over $120 000" per year. Assuming that 'individuals' life evaluations' correlate with or contribute to their welfare it seems therefore unlikely that the welfare effects of total wealth on welfare could be satiated not far from the $75 000 dollar level. 38 she is extremely averse to inequality when more is at stake and more people are involved. Of course, there may be other ways for utilitarians to explain our intuitive aversion to small-stakes inequalities in the distribution of certain goods. For instance, it may well be the case that some small-scale inequalities are bad for aggregatewelfare, not (only) because of the decreasing marginal utility of money, but because of some other psychological phenomena such as envy or aversion to perceived distributive injustice. We ourselves however suspect that if utilitarianism is the correct theory of distribution, then at least most smallstakes tradeoffs like those that we have considered (both in welfare and in other goods) should probably be accepted, even though it seems reasonable to reject them. Moreover, since we think that our calibration results do provide some support for utilitarianism, we are inclined to take these results to cast doubt on egalitarian judgments condemning such small-scale tradeoffs. It might be worth emphasizing, however, that we are not claiming that a utilitarian ought to, say, maximize total wealth, thus giving no weight to wealth equality. Our suggestion, rather, is that small-stakes inequalities are not as bad as our intuitive aversion to themwould suggest; this is compatible with there being strong utilitarian reason to worry about much larger inequalities in, say, wealth. At the very least, our arguments suggest that a utilitarian's aversion to such small inequalities cannot be explained by such simple mechanisms as the diminishing marginal utility of money. Again, we do not insist on any particular response to our dilemmas. But, however we respond to them, it appears that some of our considered judgments about the ethics of distribution will have to go. 5 Conclusion Let us summarize our results. We have considered a broad generalization of prioritarianism and the most widely endorsed family of egalitarian views. We have shown that these 39 theories can bemoderately averse to small-stakes tradeoffs, in seemingly reasonable ways, only if they are extremely averse to large-stakes tradeoffs, in seemingly unreasonableways. Moreover, we have seen that the most common utilitarian justification of inequality aversion (with respect to resources) can only explain such aversion when relatively little is at stake by implying extreme inequality aversion when more is at stake. This leaves us with a few options. One option is to conclude that the theorieswe have considered are false, on the grounds that our moderate aversion to small-stakes tradeoffs is reasonable but that the theories' resulting aversion to large-stakes tradeoffs is unreasonable. The research program suggested by this optionwould be to find some other theory that can successfully navigate both horns of our dilemmas. Our hopes lie with this option, but we are somewhat pessimistic about it. Though there may be various theories that avoid the particular calibration dilemmas we have raised, such theories may face other kinds of calibration dilemmas (as we saw with radically nonseparable egalitarianism) or be implausible in other ways. So, while we hope that others will discover a superior alternative to the distributive theories we have considered, we cannot take the existence of such an alternative for granted. Another option is to embrace the extreme implications of the views we have considered. Those who are attracted to this option may take our results to demonstrate that considerations of justice and priority to the worse off are more demanding than we may have thought. We ourselves are not attracted to this option. If one is comfortable with the demands of justice or priority being so extreme, why not go all the way-that is, to leximin? Of course, our extreme implications for large-stakes tradeoffs are not quite as extreme as leximin's. But the position of rejecting leximin as implausibly extreme while welcoming the extreme attitude towards large-stakes tradeoffs in our calibration dilemmas seems to us somewhat unstable. At the very least, this option would seem to significantly weaken the attraction of the theories we have considered, which are often pitched asmoderate and reasonable in contrast to leximin. 40 A third option is to maintain that our aversion to small-stakes tradeoffs is simply unreasonable. This option is compatible with many possible members of the weakly prioritarian and weakly egalitarian families, including versions that assign strict priority to those who are worse off. But it pushes these families closer to their common member: the utilitarian principle of distribution. If the small-stakes tradeoffs we have considered are unreasonable, then we ourselves would be inclined to place our bets on utilitarianism about distribution. If it were reasonable to give (strict) priority to the worse off, then we would have thought it reasonable to prefer a distribution in which two people are equally well off to an unequal distribution that has one-tenth of a unit less of aggregate welfare. At the very least, if we cannot reasonably be averse to the kinds of small-stakes tradeoffs we have considered, this would significantly weaken the attraction (for us) of nonutilitarian versions of weak prioritarianism and egalitarianism. Onemight defend either of the second or third options by offering some debunking account of the intuitions to which we have appealed. For instance, proponents of the second option might argue that the large-stakes implications we have drawn out involve quantities that are greater than those that we can intuitively grasp. Proponents of the third option could argue that our aversion to small-stakes inequalities results from an unreliable overgeneralization of our attitudes towards real-world inequalities involving large quantities of welfare. Although there might be some truth to either or both of the above debunking claims, they do not-even if they are true-undermine the main message of this paper. Our main aim has been to show that some common and seemingly plausible intuitions about the ethics of distribution are not mutually satisfiable. And that is true, of course, regardless of the possibility of debunking some of these intuitions. Indeed, our calibration results are, in some ways, even more practically important for the theorist who is skeptical of the moral intuitions that make our results seem unpalatable. As competing versions of prioritarianism and egalitarianism support different degrees of priority to the worse off, our results may, at the very least, help such a skeptic to choose between different 41 distributive theories. Appendix Throughout the appendix, "bxc" denotes the greatest integer less than or equal to x, and "dxe" denotes the smallest integer greater than or equal to x. Prioritarian Calibration Theorem. Suppose that f(*) is strictly increasing and weakly concave over all welfare levels. Suppose that there are some welfare levels w > w and g > l > 0 (where g ≤ 2l) such that, for all w ∈ [w,w], 2f(w) > f(w + g) + f(w − l). Then, for all w ∈ [w,w], L > 0, and G > 0, (i) f(w)−f(w−L) >  bL/gc∑ i=1 (g/l)i [f(w)− f(w − l)], if L ≤ w − w(L− [w − w]) (g/l)b(w−w)/gc + b(w−w)/gc∑ i=1 (g/l)i  [f(w)− f(w − l)], if L > w − w (ii) f(w+G)−f(w) <  dG/ge∑ i=1 (l/g)i−1 [f(w)− f(w − l)], if G ≥ w − w(G− [w − w]) (l/g)d(w−w)/ge−1 + d(w−w)/ge∑ i=1 (l/g)i−1  [f(w)− f(w − l)], if G < w − w Proof. Our proof follows the same strategy as Rabin's, except we use increments of g in the proof of part (i). This is slightly simpler and obtains a stronger result. We walk through part (i) in special detail to make the steps of subsequent proofs more readily apparent to the unfamiliar reader. We first prove part (i). f(*) is weakly concave, and w− g < w− l. So the marginal priorityweighted value of each unit of well-being over the interval [w − g, w]-graphically, the slope of the priority weighting function over that interval-must be at least as great as it is over the 42 interval [w − l, w]: f(w)− f(w − g) g ≥ f(w)− f(w − l) l . Thus, f(w)− f(w − g) ≥ g/l [f(w)− f(w − l)]. More generally, for any natural number k, f(w − [k − 1]g)− f(w − kg) ≥ g/l [f (w − [k − 1] g)− f (w − [k − 1] g − l)] . If w − g ≥ w, then the tradeoff would be rejected from initial level w − g. So, f(w − g) − f(w − g − l) > f(w) − f(w − g). And we have, from the paragraph above (letting k = 2), f(w−g)−f(w−2g) ≥ g/l [f(w−g)−f(w−g−l)]. So, f(w−g)−f(w−2g) > g/l [f(w)−f(w−g)]. More generally, for any k such that w − (k − 1)g ≥ w, f(w − [k − 1]g)− f(w − [k − 1]g − l) > f(w − [k − 2]g)− f(w − [k − 1]g). Therefore, for any such k, f(w − [k − 1]g)− f(w − kg) > g/l [f(w − [k − 2]g)− f(w − [k − 1]g)]. So f(w) − f(w − kg) > ∑ki=1 (g/l)i [f(w) − f(w − l)]. This yields the lower bound on f(w) − f(w − L) stated in the theorem. We next prove part (ii). The tradeoff would be rejected from initial level w, so f(w + g) + f(w) < f(w) + f(w − l). And, for any natural numberm such that w + [m− 1] g ≤ w, f(w +mg)− f(w + [m− 1]g) < f(w + [m− 1]g)− f(w + [m− 1]g − l). And, by the weak concavity of f(*), f(w + [m− 1]g)− f(w + [m− 1]g − l) ≤ l/g [f(w + [m− 1]g)− f(w + [m− 2]g)]. 43 So, for any such m, f(w +mg)− f(w + [m− 1]g) < l/g [f(w + [m− 1]g)− f(w + [m− 2]g)]. This means that f(w +mg)− f(w) <∑mi=1 (l/g)i−1 [f(w)− f(w − l)]. Ratio-Scale Prioritarian Calibration Theorem. Suppose that f(*) is strictly increasing and weakly concave over all welfare levels, and satisfies ratio-scale invariance (i.e., is unique only up to similarity transformations). Suppose there is some welfare level w and positive real numbers l < 1 and ĝ > 1/l such that 2f(w) > f(ĝw) + f(lw). Then, for all w, (i) f(w)− f(Lw) > b−log(L)/log(ĝ)c∑ i=1  1− ĝ ĝ ( l − 1 ) i [f(w)− f(lw)] . (ii) f(Ĝw)− f(w) < dlog Ĝ/log ĝe∑ i=1 ( ĝ − ĝl ĝ − 1 )i−1 f(w)− f(lw). Proof. We first prove part (i). By hypothesis, f(w)− f(lw) > f(ĝw)− f(w). So, by ratio-scale invariance, for any natural number k, f(w/ĝk)− f(lw/ĝk) > f(w/ĝk−1)− f(w/ĝk). And, by weak concavity, f(w/ĝk−1)−f(w/ĝk) ≥ w/ĝ k−1 − w/ĝk w/ĝk−1 − lw/ĝk−1 [ f(w/ĝk−1)− f(l/wĝk−1) ] = 1− ĝ ĝ ( l − 1 ) [f(w/ĝk−1)− f(lw/ĝk−1)] . 44 So, for any k, f(w/ĝk−1)− f(w/ĝk) > 1− ĝ ĝ ( l − 1 ) [f(w/ĝk−2)− f(w/ĝk−1)] . Therefore, f(w)− f(w/gk) > k∑ i=1  1− ĝ ĝ ( l − 1 ) i [f(w)− f(lw)] . We next prove part (ii). By hypothesis, f(ĝw) − f(w) < f(w) − f(lw). So, by ratio-scale invariance, f(ĝkw)− f(ĝk−1w) < f(ĝk−1w)− f(lĝk−1w). And, by concavity, f(ĝkw)− f(lĝkw) ≤ ĝ kw − lĝkw ĝkw − ĝk−1w [ f(ĝkw)− f(ĝk−1w) ] = ĝ − ĝl ĝ − 1 [ f(ĝkw)− f(ĝk−1w) ] . So, for any k, f(ĝkw)− f(ĝk−1w) < ĝ − ĝl ĝ − 1 [ f(ĝk−1w)− f(ĝk−2w) ]. Therefore, f(ĝkw)− f(w) < k∑ i=1 ( ĝ − ĝl ĝ − 1 )i−1 f(w)− f(lw). Egalitarian Calibration Theorem. Let I = {1, . . . , n} and Im = {m, . . . , n− 1}. Suppose that for all adjacent pairs j, j + 1 ∈ Im, there is some initial level w = w(j) = w(j+1), where w(j−1) ≥ w + g and w(j+2) ≥ w− l, such that ∑ i∈I aiw > ∑ i∈I\{j,j+1} aiw + aj(w + g) + aj+1(w− l). Then, for any w, (i) n∑ i=k aix ≥  n−k∑ i=1 (g/l)i−1akx, if k ≥ m k∑ i=1 akx+ n−m∑ i=1 (g/l)i−1 akx if k < m. 45 (ii) k∑ i=1 aix ≤  k∑ i=1 (l/g)i−1akx, if m = 1 m∑ i=1 akx+ k−m∑ i=1 (l/g)i−1akx, if 1 < m < k k∑ i=1 akx, if m ≥ k Proof. From∑i∈I aiw >∑i∈I\{j,j+1} aiw+aj(w+g)+aj+1(w− l), subtract∑i∈I\{j,j+1} aiw from both sides. This yields ajw + aj+1w > aj(w + g) + aj+1(w − l). Thus aj+1l > ajg, so for every j ∈ Im, aj+1 > (g/l) aj . Alongwith the nondecreasingness of the weights, this yields the upper and lower bounds on total rank-weighted utilities stated in the theorem. References Adler, Matthew (2011), Well-Being and Fair Distribution: Beyond Cost-Benefit Analysis (Oxford University Press). Adler, Matthew and Holtug, Nils (2019), "Prioritarianism: A Response to Critics", Politics, Philosophy & Economics (): 1470594X19828022. doi: 10.1177/1470594X19828022. Arrhenius, Gustaf (2013), "Egalitarian Concerns and Population Change", in Nir Eyal, Samia A. Hurst, Ole F. Norheim, and Dan Wikler (eds.), Inequalities in Health (Oxford University Press), 74–92. doi: 10.1093/acprof:oso/9780199931392.003.0007. Asheim, Geir B. and Zuber, Stéphane (2014), "Escaping the Repugnant Conclusion: RankDiscountedUtilitarianismwithVariable Population: Rank-DiscountedUtilitarianism",Theoretical Economics, 9/3 (): 629–50. doi: 10.3982/TE1338. Asheim, Geir B. and Zuber, Stéphane (2016), "Evaluating Intergenerational Risks", Journal of Mathematical Economics, 65 (): 104–17. doi: 10.1016/j.jmateco.2016.05.005. 46 Barry, Brian (1989),Theories of Justice: A Treatise on Social Justice (University of California Press), 452 pp., Google Books: Uhu60F7sXi0C. Bentham, Jeremy (1843), TheWorks of Jeremy Bentham, Vol. 1 (Principles of Morals and Legislation, Fragment on Government, Civil Code, Penal Law), ed. John Browning (William Tait), 712 pp., Google Books: GGpVAAAAcAAJ. Blackorby, Charles, Bossert, Walter, and Donaldson, David (2005), Population Issues in Social Choice Theory, Welfare Economics, and Ethics (Cambridge University Press). Broad, C. D. (1930), Five Types of Ethical Theory (K. Paul, Trench, Trubner), 314 pp. Broome, John (1991),Weighing Goods: Equality, Uncertainty and Time (Wiley-Blackwell). Broome, John (2004),Weighing Lives (Oxford; New York: Oxford University Press), 278 pp. Broome, John (2015), "Equality versus Priority: A Useful Distinction", Economics and Philosophy, 31/02: 219–28. Buchak, Lara (2013),Risk and Rationality (OUPOxford), 273 pp., Google Books: kBgbAgAAQBAJ. Buchak, Lara (2017), "Taking Risks behind the Veil of Ignorance", Ethics, 127/3 (): 610–44. doi: 10.1086/690070. Cox, James C. and Sadiraj, Vjollca (2006), "Smalland Large-Stakes Risk Aversion: Implications of Concavity Calibration for Decision Theory", Games and Economic Behavior, 56/1 (): 45–60. doi: 10.1016/j.geb.2005.08.001. Cox, James C., Sadiraj, Vjollca, Vogt, Bodo, andDasgupta, Utteeyo (2013), "Is There a Plausible Theory for Decision under Risk? A Dual Calibration Critique", Economic Theory, 54/2 (): 305–33. doi: 10.1007/s00199-012-0712-4. Crisp, Roger (2003), "Equality, Priority, andCompassion", Ethics, 113/4: 745–63. doi: 10.1086/ 373954. 47 D'Aspremont, Claude and Gevers, Louis (2002), "Chapter 10 Social Welfare Functionals and Interpersonal Comparability", in Handbook of Social Choice and Welfare, i (Handbook of Social Choice and Welfare; Elsevier), 459–541. doi: 10.1016/S1574-0110(02)80014-5. Donaldson, David and Weymark, John A (1980), "A Single-Parameter Generalization of the Gini Indices of Inequality", Journal of Economic Theory, 22/1 (): 67–86. doi: 10.1016/0022 -0531(80)90065-4. Fleurbaey,Marc (2010), "Assessing Risky Social Situations", Journal of Political Economy, 118/4: 649–80. doi: 10.1086/656513, JSTOR: 10.1086/656513. Fleurbaey,Marc, Tungodden, Bertil, andVallentyne, Peter (2009), "On the Possibility ofNonaggregative Priority for the Worst Off", Social Philosophy and Policy, 26/01 (): 258–85. doi: 10.1017/S0265052509090116. Greene, Joshua and Baron, Jonathan (2001), "Intuitions about Declining Marginal Utility", Journal of Behavioral Decision Making, 14/3 (): 243–55. doi: 10.1002/bdm.375. Kahneman, Daniel and Deaton, Angus (2010), "High Income Improves Evaluation of Life but Not EmotionalWell-Being",Proceedings of the National Academy of Sciences, 107/38 (): 16489– 93. doi: 10.1073/pnas.1011492107, pmid: 20823223. Kahneman, Daniel and Tversky, Amos (1979), "Prospect Theory: An Analysis of Decision under Risk", Econometrica, 47/2: 263–91. doi: 10.2307/1914185, JSTOR: 1914185. Lipman, Stefan A. and Attema, Arthur E. (2019), "Rabin's Paradox for Health Outcomes", Health Economics, 28/8 (): 1064–71. doi: 10.1002/hec.3918. Nebel, Jacob M. (2017), "Priority, Not Equality, for Possible People", Ethics, 127/4 (): 896–911. doi: 10.1086/691568. Nord, Erik and Johansen, Rune (2014), "Concerns for Severity in Priority Setting in Health Care: A Review of Trade-off Data in Preference Studies and Implications for Societal Willingness to Pay for a QALY",Health Policy, 116/2 (): 281–8. doi: 10.1016/j.healthpol.2014 .02.009. 48 Otsuka, Michael (2017), "How It Makes a Moral Difference That One Is Worse off than One Could Have Been", Politics, Philosophy & Economics (): 1470594X17731394. doi: 10.1177/ 1470594X17731394. Otsuka, Michael and Voorhoeve, Alex (2018), "Equality Versus Priority", in Serena Olsaretti (ed.), Oxford Handbook of Distributive Justice (Oxford: Oxford University Press), 65–85. Parfit, Derek (1991), Equality or Priority (The Lindley Lecture; University of Kansas, Department of Philosophy). Parfit, Derek (2012), "Another Defence of the Priority View", Utilitas, 24/3 (): 399–440. doi: 10.1017/S095382081200009X. Persson, Ingmar (2006), "A Defence of Extreme Egalitarianism", in Nils Holtug and Kasper Lippert-Rasmussen (eds.), Egalitarianism: New Essays on the Nature and Value of Equality (Clarendon Press), 83–98. Pigou, A. C. (2013), The Economics of Welfare (Palgrave Macmillan), 897 pp., Google Books: 3qoAAwAAQBAJ. Rabin,Matthew (2000), "RiskAversion andExpected-Utility Theory: ACalibration Theorem", Econometrica, 68/5 (): 1281–92. doi: 10.1111/1468-0262.00158. Rabinowicz, Wlodek (2008), "The Size of Inequality and Its Badness Some Reflections around Temkin's Inequality", Theoria, 69/1-2 (): 60–84. doi: 10.1111/j.1755-2567.2003.tb00754 .x. Rawls, John (1971), A Theory of Justice (Harvard University Press). Rescher, Nicholas (1966), Distributive Justice: A Constructive Critique of the Utilitarian Theory of Distribution (Bobbs-Merrill). Sadiraj, Vjollca (2014), "Probabilistic Risk Attitudes and Local Risk Aversion: A Paradox", Theory and Decision, 77/4 (): 443–54. doi: 10.1007/s11238-013-9410-3. Temkin, Larry S. (1993), Inequality (New York: Oxford University Press). 49 Temkin, Larry S. (2003), "Egalitarianism Defended", Ethics, 113/4: 764–82. doi: 10.1086/ 373955. Tungodden, Bertil and Vallentyne, Peter (2005), "On the Possibility of Paretian Egalitarianism:" Journal of Philosophy, 102/3: 126–54. doi: 10.5840/jphil200510234. Wakker, Peter P. (2010), Prospect Theory: For Risk and Ambiguity (Cambridge University Press), 517 pp. Weymark, John A. (1981), "Generalized Gini Inequality Indices",Mathematical Social Sciences, 1/4 (): 409–30. doi: 10.1016/0165-4896(81)90018-4. Williams, Andrew (2012), "The Priority View Bites the Dust?", Utilitas, 24/3 (): 315–31. doi: 10.1017/S0953820812000106.