¹ Otsuka, Michael, ‘Skepticism about Saving the Greater Number’, Philosophy and Public Affairs 32 (2004), p. 415CrossRefGoogle Scholar.

² And likewise, Worse-offedness can be viewed in either individual or aggregate form; see e.g. Temkin, Larry S., ‘Inequality’, Philosophy and Public Affairs 15 (1986), pp. 105, 111Google Scholar; Inequality (Oxford, 1993), pp. 19–27, 50-8.

³ McKerlie, Dennis, ‘Equality and Priority’, Utilitas 6 (1994), p. 28CrossRefGoogle Scholar.

⁴ Kamm, F. M., Morality, Mortality, vol. 1 (Oxford, 1993), p. 280Google Scholar.

⁵ In what follows, the principle of Worse-offedness will stand for both a ‘prioritarian’ and an ‘egalitarian’ view, since the main wedge that divides the two – the ‘levelling-down objection’ – doesn't apply in cases, such as those at issue here, in which a resource must be distributed to one side or the other.

⁶ On the counter-intuitiveness of Taurek's approach, see some of the discussion in Otsuka, ‘Skepticism’, pp. 414 n. 3, 415, 424 n. 23.

⁷ Daniels, Norman, ‘Four Unsolved Problems’, Hastings Center Report 24 (July−August 1994), p. 28Google Scholar. See also Harris, John, ‘Justice and Equal Opportunities in Health Care’, Bioethics 13 (1999), p. 398Google Scholar; and Norcross, Alastair' discussion in ‘Comparing Harms: Headaches and Human Lives’, Philosophy and Public Affairs 26 (1997), pp. 135–67Google Scholar.

⁸ Rakowski, Eric, ‘The Aggregation Problem’, Hastings Center Report 24 (July−August 1994), p. 35Google ScholarPubMed.

⁹ Brink, David, ‘The Separateness of Persons, Distributive Norms, and Moral Theory’, Value, Welfare and Morality, ed. Frey, R. G. and Morris, C. W. (New York, 1994), p. 275Google Scholar.

¹⁰ See, e.g., Crisp, Roger, ‘Equality, Priority and Compassion’, Ethics 113 (2003)Google Scholar; Reibetanz, Sophia, ‘Contractualism and Aggregation’, Ethics 108 (1998), p. 304Google Scholar; Menzel, Paul T., Strong Medicine: The Ethical Rationing of Health Care (New York, 1990), p. 185Google Scholar; and Nord, Erik, ‘The Relevance of Health State after Treatment’, Journal of Medical Ethics 19 (1993), p. 37Google Scholar.

¹¹ Otsuka, ‘Skepticism’, p. 415.

¹² See e.g. Edgar, Andrew et al. , The Ethical QALY (Surrey, 1998)Google Scholar.

¹³ Even non-complete-lives approaches assume intra-personal additivity between measures of well-being at different time periods in a person's life (see, e.g., Lippert-Rasmussen, Kasper, ‘Measuring the Disvalue of Inequality over Time’, Theoria 69 (2003), pp. 32–45CrossRefGoogle Scholar). I will assume the complete-lives approach for what follows, however, partly because it is the general choice of major prevalent-approach philosophers, but also because the alternative I advance can be adapted to non-complete-lives views, though I won't show that in this article.

¹⁴ Kappel, Klemens, ‘Equality, Priority, and Time’, Utilitas 9 (1997), p. 217Google Scholar; Temkin, Larry, ‘Determining the Scope of Egalitarian Concern’, Theoria 69 (2003), p. 47Google Scholar.

¹⁵ See Kamm, Morality, Mortality, ch. 12.

¹⁶ Daniels, ‘Four Unsolved Problems’, p. 28.

¹⁷ They also differ when understood as aggregate claims, but I am leaving such matters till later.

¹⁸ I have not attached cardinal measures to the Easy Cases because, at this stage, I simply want to illustrate them in a general way.

¹⁹ One might, in looking at (say) an Easy Case 2(a), conclude that the two sides are so close in WA, B and WP that we should flip a coin. That, of course, depends on the actual cardinal values involved. But in any event, recall – and I will say more about this presently – that we are deciding how to allocate the resource in an original Hard Case by determining which Easy Case is nearest to it. And that in turn means that if the WAs, Bs and WPs in that original Hard Case are roughly equivalent – even if one side dominates in all – Easy Case (c) will be closest (more on this below), and we will flip a coin. In addition, although I cannot show this here, 1(a), 1(b)(i), 1(b)(ii), 2(a), 2(b)(i), 2(b)(ii) and (c) are the only possible combinations of differences and ties in B, WA and WP that issue forth in Easy Cases. One might argue that cases in which one criterion points clearly in one direction while the other two are tied are also Easy Cases: they uncontroversially suggest we go with the one. As the diagrams suggest, however, if any two of WA, B and WP are tied, the third will be as well.

²⁰ In what follows, for purposes of simplicity, I will measure the distances between the Hard Case at hand and the various Easy Cases by differences in B and WA, although of course B, WA and WP all count independently in designating a case Easy or Hard. Having changed each person's B and WA in the Hard Case to get to the closest version of (say) Easy Case 1(b)(ii), the WPs will then automatically conform to 1(b)(ii)'s. But as criteria, WA and B do not additively recapitulate WP, since the Bs on one Side are counter-factual to those on the other. Thus the three criteria count independently.

²¹ We can add these amounts together because of the assumption, shared with the prevalent approach, that one individual's B can be placed on the same cardinal scale as another's, i.e. cardinal intra-criterion inter-personal QALY measurements can be made.

²² In some Hard Cases, we could actually make an even closer version of Easy Case 1(b)(ii) than the one I offer above, were we (for example) to bring each Side to its own Side's average WA, rather than using the average WA for both Sides as a whole. However, this procedure wouldn't work for finding the nearest version of the reverse Easy Case, 2(b)(ii), and so wouldn't preserve a symmetrical approach.

²³ Specifically, the Hard Case is 74 QALYs away from the closest versions of each of 1(a), 1(b)(ii), 2(b)(ii), and (c); 77.74 QALYs away from the closest version of 2(b)(i); 76.26 QALYs away from the closest version of 2(a); and 73.7 QALYs away from the closest version of 1(b)(i).

²⁴ Let me address one possible concern. Take, as an example, the closest version of Easy Case 1(b)(ii). It in effect assumes (see Figure D) that, among many other things, p6's WA, which in the original Hard Case is a meager 6 QALYs, is not as bad as it actually is by 7.37 QALYs – it assumes, in other words, that p6's WA claim is 7.37 QALYs less serious – and it also assumes that p2's WA claim is actually 1.63 QALYs stronger than it really is, i.e. than it is in the Hard Case itself. More exactly, 1(b)(ii) would (if it were the closest Easy Case) impose the solution that would be appropriate if inter alia that were true. In proposing distributions that treat persons inequitably in this way, can Easy Cases work? Yes, because what matters is not whether the distance between p6's WA in the Hard and in a given Easy Case is greater than the distance between p2's, but whether those distances added together are, inter alia, necessarily and systematically greater in the Easy Cases in which p6 wins than those in which p2 wins. Only then would it be more difficult for p6 to reach a favorable outcome than p2; only then, in other words, would p6 find herself treated inequitably in the approach I am advancing. And as it happens, this isn't so. I'll use the closest version of 1(b)(i) to illustrate this, since although it isn't the Easy Case depicted in Figure D, it is the closest to the Hard Case of all Easy Cases, thus, on my approach, delivering the resource to p2's side, Side 1. The difference between p2's WA in the closest version of 1(b)(i) and p2's WA in the Hard Case itself is 1.63; likewise, the difference between p6's WA in the closest version of 1(b)(i) and p6's WA in the Hard Case is 7.37 (the same figures as for 1(b)(ii)). Summed together they make 9. But when it comes to the closest version of the mirror case in which p6's side, Side 2, would win – 2(b)(i) – that distance is even less. It is .63 for p2 plus 6.37 for p6 equals 7, meaning that any ‘inequitable’ treatment within an Easy Case vanishes across all Easy Cases, and so makes it no more difficult for p6 than for p2 to win. If Easy Case 1(b)(i) is nevertheless closest to this particular Hard Case, as it is, it is because of differences involving the other p's, and not because the approach I am advancing treats those p's who least resemble themselves in the Easy Cases unequally.

²⁵ To say that p6's WA is not low enough – in the context of all the other WAs, WPs and Bs in the case – to justify our giving the resource to Side 2 is to point out, among other things, that while p6's WA is 6, two individuals on Side 1, p1 and p3, have WAs almost as low, at 7 and 8. But suppose p6's WA was lower – say it was 2, not 6 – and p1's and p3's were higher, say 13 and 14 respectively. On the approach I am advancing, the Hard Case would now be closest to Easy Case 2(a), which would give the resource to Side 2. (Specifically, it would be 74.68 QALYs from 1(a), 73.02 QALYs from 1(b)(i), 73.02 QALYs from 1(b)(ii), 70.42 QALYs from 2(a), 74.76 QALYs from 2(b)(i), 71.02 QALYs from 2(b)(ii), and 72.02 QALYs from (c).) In a similar vein, suppose that we internally redistribute p6's QALYs within Side 2. Intuitively, for example, we would want it to make a difference if we were to change the Hard Case so that p6 didn't bear so much of Side 2's WA QALYs, and at least some of them were instead redistributed among Side 2's other members. In other words, we'd like to think that the way in which the same amount of Worse-offedness, in QALYs, is distributed over a side, individual to individual, should have some impact on a Hard Case's resolution. And, on the approach I am advancing, it does. Suppose that we revised the Hard Case in Figure D by ‘redistributing’ the WA's on Side 2, improving p6's from 6 to 12 QALYs, and compensating by lowering p5's, p7's and p8's each by 2 QALYs. Side 1 would still win, but what's critical is that it would do so more easily – by a greater amount – and that conforms to our intuitions.

²⁶ A point of clarification: ‘Numbers’, as I am using the term, means that we are prepared to aggregate individual WA, B and WP claims on each side (see, e.g., Hirose, Iwao, ‘Aggregation and Numbers’, Utilitas 16 (2004), pp. 62–79CrossRefGoogle Scholar), and not necessarily that the number of people on each side is unequal. As will become apparent, however, making the numbers of people on each side unequal is required to drive a wedge between individual and aggregate understandings of each of the three criteria – not necessarily in Hard but in some of the Easy Cases – and so raises the toughest kind of situations for the approach I am advancing.

²⁷ See some of Larry Temkin's discussion in Inequality, pp. 105–6.

²⁸ The closest versions of 1(b)(i) and 1(b)(ii) are 81.74 QALYs away, of 2(a) 86.26 QALYs away, of 2(b)(i) 89.5 QALYs away, of 2(b)(ii) 83.5 QALYs away, and of (c) 82.24 QALYs away.

²⁹ This Hard Case is 102.84 QALYs away from 2(a), 109.26 away from 2(b)(i), 106.86 away from 2(b)(ii), 104.95 away from 1(a), 107.86 away from 1(b)(i) and 1(b)(ii), and 106.86 away from (c).

³⁰ I have applied the approach I am advancing to a wide variety of cases, and have found that it consistently generates answers that are intuitively plausible when it comes to any Hard Case, such as those in Figures A/D and E, in which at least some individuals on each side are WA than some on the other side, and in which some individuals on each side can Benefit more than some on the other. It needs modification in Hard Cases in which one side, specifically the side with the smaller aggregate Benefit claim, features individuals who can Benefit more than every individual on the other side, and are WA than every individual on the other side. But (although I can't show this here) the modification can be generated by relying on constructs internal to my approach; in particular, on the differences not between the kinds of Easy Cases that support different sides but between those that support the same side – the difference between an Easy Case 1(b)(i) and an Easy Case 1(b)(ii), for example – that I have not here explored or exploited.

³¹ Anthony Culyer, Don Herzog and Michael Krashinsky offered helpful suggestions on earlier versions of this article.