¹ A recent review of the literature is Nils Holtug, ‘Prioritarianism’, Oxford Research Encyclopedia of Politics, <http://dx.doi.org/10.1093/acrefore/9780190228637.013.232> (2017).

² Broome, John, ‘General and Personal Good’, The Oxford Handbook of Value Theory, ed. Hirose, Iwao and Olson, Jonas (New York, 2015), pp. 249–66Google Scholar.

³ McCarthy, David, ‘Risk-Free Approaches to the Priority View’, Erkenntnis 78 (2013), pp. 421–49CrossRefGoogle Scholar.

⁴ Greaves, Hilary, ‘Antiprioritarianism’, Utilitas 27 (2015), pp. 1–42CrossRefGoogle Scholar.

⁵ Temkin, Larry, ‘Equality, Priority or What?’, Economics and Philosophy 19 (2003), pp. 61–87CrossRefGoogle Scholar.

⁶ Crisp, Roger, ‘Equality, Priority, and Compassion’, Ethics 113 (2003), pp. 745–63CrossRefGoogle Scholar.

⁷ Voorhoeve, Alex, ‘How Should We Aggregate Competing Claims?’, Ethics 125 (2014), pp. 64–87CrossRefGoogle Scholar.

⁸ Otsuka, Michael and Voorhoeve, Alex, ‘Why it Matters that Some are Worse Off than Others: An Argument against the Priority View’, Philosophy and Public Affairs 37 (2009), pp. 171–99CrossRefGoogle Scholar

⁹ McCarthy, David, ‘Utilitarianism and Prioritarianism II’, Economics and Philosophy 24 (2008), pp. 1–33CrossRefGoogle Scholar.

¹⁰ See e.g. Adler, Matthew D., Well-Being and Fair Distribution: Beyond Cost-Benefit Analysis (New York, 2012)Google Scholar, chs. 5–7; Arneson, Richard J., ‘Desert and Equality’, Egalitarianism: New Essays on the Nature and Value of Equality, ed. Holtug, Nils and Lippert-Rasmussen, Kasper (Oxford, 2007), pp. 262–93Google Scholar; Holtug, Nils, Persons, Interests, and Justice (Oxford, 2010)CrossRefGoogle Scholar; O'Neill, Martin, ‘Priority, Preference and Value’, Utilitas 24 (2012), pp. 332–48CrossRefGoogle Scholar; Parfit, Derek, ‘Another Defence of the Priority View’, Utilitas 24 (2012), pp. 399–440CrossRefGoogle Scholar; Porter, Thomas, ‘In Defence of the Priority View’, Utilitas 24 (2012), pp. 349–64CrossRefGoogle Scholar.

¹¹ Arneson, ‘Desert and Equality’, p. 283.

¹² Arneson, ‘Desert and Equality’, p. 287.

¹³ Adler, Well-Being and Fair Distribution, ch. 5. See Holtug, ‘Prioritarianism’, reviewing the claim approach along with other rationales for prioritarianism.

¹⁴ The conflict between the well-being Pareto principles and Priority for the More Deserving will also have implications for non-prioritarian moral views that take account of desert, e.g. desert-modulated utilitarianism. See Feldman, Fred, ‘Adjusting Utility for Justice: A Consequentialist Reply to the Objection from Justice’, Utilitarianism, Hedonism, and Desert, ed. Feldman, Fred (Cambridge, 1977), pp. 154–74Google Scholar; Rendall, Matthew, ‘Priority and Desert’, Ethical Theory and Moral Practice 16 (2013), pp. 939–51CrossRefGoogle Scholar. Space constraints preclude a discussion of these implications.

¹⁵ Outcomes are either whole possible worlds, or models of possible worlds that are used in moral deliberation by cognitively bounded decision-makers. The analysis in this article doesn't depend on which interpretation of outcome is adopted – except that the pragmatic justification of certain axioms (see below, n. 27) is more straightforward on the ‘models’ view.

¹⁶ A quasi-ordering is a binary relation that is reflexive and transitive but not necessarily complete. I assume, specifically, that the relation of ‘at least as good as’ is (1) reflexive: each outcome is at least as good as itself; and (2) transitive: if x is at least as good as y, and y is at least as good as z, then x is at least as good as z. The relations of ‘better than’ and ‘equally good as’ are in turn derivable from the ‘at least as good as’ relation (in the standard manner for a quasi-ordering). x is better than y iff x is at least as good as y but y is not at least as good as x. x and y are equally good iff x is at least as good as y and y is at least as good as x. It is possible (absent a further Completeness axiom; see below) that two distinct outcomes x and y are incomparable, i.e. it is neither the case that x is at least as good as y, nor that y is at least as good as x.

¹⁷ On the structure and measurement of individual well-being levels and differences, see Adler, Matthew D., ‘Extended Preferences’, The Oxford Handbook of Well-Being and Public Policy, ed. Adler, Matthew D. and Fleurbaey, Marc (New York, 2016), pp. 476–517CrossRefGoogle Scholar.

¹⁸ In the special cases covered by the well-being Pareto principles (see below), with all claims of null valence, or at least one claim in favour of one of the two outcomes and the other claims with the same valence or null, the strength of claims is irrelevant to the ranking of the outcomes. In general, however, the pattern of claims in terms of strength as well as valence will be relevant.

¹⁹ Nagel, Thomas, ‘Equality’, Mortal Questions, ed. Nagel, Thomas (Cambridge, 1979), pp. 106-27Google Scholar; Nagel, Thomas, Equality and Partiality (New York, 1991)Google Scholar. For a discussion of how the claims view builds upon Nagel's insights, see Adler, Well-Being and Fair Distribution, pp. 329–37.

²⁰ More precisely: let π(.) be a one-to-one and onto mapping from the set of individuals {1, 2, . . ., N} onto itself, a so-called permutation mapping. Anonymity then says: if x and y are such that each i in x has the same level of well-being as π(i) in y, x and y are equally good.

²¹ Any given outcome x is equally good as itself. But the pattern of claims between x and itself is such that each individual has a null claim. Since the pattern of claims between x and y is exactly the same, there's no warrant for x to be incomparable with y rather than equally good.

²² See Hall, Marshall Jr, The Theory of Groups (New York, 1959), p. 60Google Scholar.

²³ For a defence of this characterization of prioritarianism, with reference to the literature, see Adler, Well-Being and Fair Distribution, pp. 360–7. A different approach is to equate prioritarianism with what I below term ‘continuous prioritarianism’. See Holtug, ‘Prioritarianism’. On either definition, the prioritarian goodness ranking satisfies the well-being Pareto principles, Pigou–Dalton, Anonymity, and Separability.

²⁴ This is sometimes referred to as ‘strong separability’. A more precise definition is as follows. An individual is ‘unaffected’ by a pair of outcomes if she is equally well off in the two, and she is ‘affected’ by a pair of outcomes if she is not unaffected. For a given pair of outcomes, x and y, let M denote the individuals who are unaffected by the x/y pair, and M⁺ the affected individuals. Let x* and y* be any two outcomes such that (1) all individuals in M are unaffected by the x*/y* pair; and (2) all individuals in M⁺ are unaffected by the x/x* pair and by the y/y* pair. Then Separability requires that: x at least as good as y iff x* at least as good as y*.

²⁵ Adler, Well-Being and Fair Distribution, pp. 351–6.

²⁶ See Adler, ‘Extended Preferences’. As discussed in ‘Extended Preferences’, Measurability requires the quasi-orderings of well-being levels and differences to be complete.

²⁷ Measurability means that each individual's welfare-relevant attributes can be summarized as a single well-being number. A decision-maker can then think about the ranking of outcomes as a ranking of well-being vectors, rather than – in a much more complex way – as a comparison of allocations of attribute bundles to all the individuals in the population of concern. Consistency allows the decision-maker to develop a single ranking of well-being vectors that will guide her ranking of each set of outcomes, independent of the specific membership of that set – rather than needing to have a plurality of rankings of vectors. Completeness and Continuity, together, imply that the goodness ranking of vectors can be represented via a continuous real-valued function G(.).Vector v at least as good as vector v* iff G(v) ≥ G(v*). A wide range of mathematical tools become available for determining what moral goodness recommends.

²⁸ See below, section IV.A.

²⁹ The Appendix is available as an online supplement, on the Utilitas web site, at https://doi.org/10.1017/S0953820817000164. See McCarthy, ‘Utilitarianism and Prioritarianism II’, for a similar proof.

³⁰ The phrase ‘concavely transformed’ is a shorthand for the (yet more awkward) ‘strictly increasingly and strictly concavely transformed’.

³¹ On this possibility, see Vallentyne, Peter, ‘Brute Luck Equality and Desert’, Desert and Justice, ed. Olsaretti, Serena (Oxford, 2003), pp. 169–85Google Scholar.

³² Arneson, ‘Desert and Equality’.

³³ See Lippert-Rasmussen, Kasper, ‘Luck-Egalitarianism: Faults and Collective Choice’, Economics and Philosophy 27 (2011), pp. 151–73Google Scholar.

³⁴ More precisely: let π(.) be a permutation mapping on the set of individuals (see n. 20). If x and y are such that, for each i, the well-being level of i in x is equal to the well-being level of π(i) in y and the desert level of i in x is equal to the desert level of π(i) in y, then: x and y are equally good.

³⁵ Why has proviso (5) been added to this axiom? After all, proviso (4) suffices to establish that everyone other than Desi and Lesi has null claims between x and y. The answer is that Priority for the More Deserving without proviso (5) may be internally inconsistent. See Appendix.

³⁶ See immediately below, section II.B. If DM Measurability holds true, then clearly the ranking of any set with intrapersonally fixed desert using the formula ∑f(w_i, d_i) satisfies the well-being Pareto principles, DM Pigou–Dalton, DM Anonymity, and Priority for the More Deserving. I have not established that there is always such a ranking absent DM Measurability.

³⁷ If each person is equally well off in x as she is in y, then – with intrapersonally fixed desert – each person's f value does not change. And if some person is better off in y than x, and her desert does not change, her f value goes up, since f(.) is strictly increasing in well-being.

³⁸ Desert is ‘intrapersonally variable’ in some set of outcomes if it is not intrapersonally fixed. There is at least one person, and at least one pair of outcomes, such that the person's desert level in the first outcome is not the same as her desert level in the second.

³⁹ Consider, for example, a two-step approach that is continuous prioritarian except in using the desert-modulated continuous prioritarian formula as a tiebreaker. This says: (1) Outcome x is better than y if ranked higher by the ∑g(w_i) formula; (2) if the two outcomes are ranked equal by the ∑g(w_i) formula, then x is better than y if ranked higher by the ∑f(w_i, d_i) formula; (3) otherwise x and y are equally good. This two-step approach (which can violate DM Continuity) always satisfies Priority for the More Deserving and Well-Being Strong Pareto.

⁴⁰ Namely, x and z are equally good, while y is better than both.

⁴¹ Namely, y is better than x if the further outcomes z, zz,. . . are such that the set comprised of these outcomes, together with x and y, can be ranked consistently with both Priority for the More Deserving and the well-being Pareto principles.

⁴² The ∑f(w_i, d_i) formula doesn't necessarily satisfy DM Pigou–Dalton if S is characterized by intrapersonally variable desert, but this can be rectified by weakening that axiom to apply only if each person is at the same desert level in x as in y.

⁴³ Arneson, ‘Desert and Equality’, p. 286.

⁴⁴ Parfit, Derek, ‘Equality or Priority’, The Ideal of Equality, ed. Clayton, Matthew and Williams, Andrew (Houndmills, 2000), pp. 81–125Google Scholar, at 103–5.

⁴⁵ Holtug, Persons, Interests, and Justice, p. 204.

⁴⁶ David McCarthy, ‘The Priority View’, Economics and Philosophy, <https://doi.org/10.1017/S0266267116000225> (2016).

⁴⁷ Arneson, ‘Desert and Equality’, p. 281.

⁴⁸ This seems clear from his discussion at Arneson, ‘Desert and Equality’, pp. 278–84.

⁴⁹ See e.g. Sen, Amartya, ‘The Impossibility of a Paretian Liberal’, Journal of Political Economy 78 (1970), pp. 152–7CrossRefGoogle Scholar; Kaplow, Louis and Shavell, Steven, ‘Any Non-welfarist Method of Policy Assessment Violates the Pareto Principle’, Journal of Political Economy 109 (2001), pp. 281–6CrossRefGoogle Scholar; Fleurbaey, Marc and Trannoy, Alain, ‘The Impossibility of a Paretian Egalitarian’, Social Choice and Welfare 21 (2003), pp. 243–63Google Scholar.

⁵⁰ Many thanks for comments to Richard Arneson, Luc Bovens, Vincent Conitzer, Richard Fallon, Jimmy Goodrich, Jerry Green, Till Grüne-Yanoff, Frances Kamm, David McCarthy, Marcus Pivato, Wlodek Rabinowicz, Caleb South, Larry Temkin, Alex Voorhoeve, an anonymous referee, and workshop participants at Duke, Harvard, LSE, Lund, Oxford, Rutgers, Stockholm, UNC, and the University of Virginia. All errors are my own.