A. Fundamental principles: Well-Being Pareto, Pigou–Dalton, Anonymity

With undifferentiated desert, a very strong case can be made for each of the following three clusters of principles. I list the principles, and then summarize the case for each.

The well-being Pareto principles flow directly from the claims set-up. If each person is equally well off in x as she is in y, then each person has a null claim between the two. It follows that x is neither better than y, nor worse than y. If the moral comparison of x and y is meant to depend upon the collection of rankings from the N individual perspectives, and all of these are indifferent, then there is nothing to make an affirmative moral case for x over y, or vice versa. By a further bit of reasoning, we see that the two outcomes are equally rather than incomparably good.Footnote ²¹

As for Well-Being Strong Pareto: if at least one person is better off in y than x, and everyone is at least as well off, then there is at least one claim in favour of y, and all other claims either also point this way or are null. Surely, then, y is better than x. If morality supervenes on the collection of individual perspectives, then: (a) the fact that y is ranked higher from at least one individual perspective should surely have pro tanto moral weight in favour of y, and moreover (b) only another such consideration, the fact that y is ranked lower than x from someone else's viewpoint, should be able to override this pro tanto weight.

The chain of reasoning from the claims framework to Pigou–Dalton is, I believe, equally compelling. Why? Higher has a claim in favour of x; Lower has a claim in favour of y; and everyone else has null claims. In this case of two conflicting claims, the outcome favoured by the stronger claim will be, on balance, better. But who does have the stronger claim? Surely it is Lower. Consider the plausible factors that, in general, might affect the strength of an affirmative claim: (a) the claimant's well-being difference between the two outcomes; (b) the claimant's well-being levels in the two outcomes; (c) her desert. We are assuming undifferentiated desert, and so factor (c) drops away. Crucially, by virtue of the antecedent conditions for the Pigou–Dalton principle, factor (a) drops away too. What Lower stands to gain in welfare, were outcome y to obtain rather than x, is exactly what Higher stands to lose. Higher cannot argue (as she might in a different case) that the magnitude of the change in her well-being, between the two outcomes, is larger than the magnitude of the change in Lower's – and thus that her claim is stronger. By the antecedent conditions, these magnitudes are the same.

And so we are left with factor (b). But, surely, this factor cuts for Lower. Lower can say to Higher: ‘The well-being level to which you assert a claim (your level in x) is higher than the well-being level to which I assert a claim (my level in y). Indeed, if my claim rather than yours is honoured, your level (in y) would be no lower than mine. Surely these facts about our well-being levels have some relevance to the comparative strength of our claims.’ And Lower can continue: ‘What can you say, in response, to show that x is on balance better than y? Nothing – not that I am less deserving, nor (as you might in another case) that the difference the x/y choice makes to your life is larger than the difference it makes to mine.’

A parenthetical note: a significant contribution of the claims framework (or so I believe) is to provide a unified defence of the well-being Pareto principles and the Pigou–Dalton principle. Often, these are seen as arising from two distinct moral concerns: efficiency or overall well-being, in the first case, and equity, in the latter case. To be sure, the well-being Pareto principles and the Pigou–Dalton principle are logically distinct: neither implies the other. But the claims framework shows how they are linked as a matter of justification.

The Anonymity axiom is defended as follows. Let's say that two outcomes, x and x*, are related by a ‘two-person permutation’ if there are two individuals who switch well-being levels (the level of one in x is the same as the level of the other in x*, and vice versa), and everyone else has the same well-being level in x as she does in x*. If x and x* are related by a two-person permutation, then only the two ‘switched’ individuals can have non-null claims. By symmetry the two claims are equally strong, and so x and x* are equally good.

Further, it can be shown that whenever the well-being levels in y are a permutation of the well-being levels in x, y can be reached from x by a finite series of two-person permutations.Footnote ²² Denote this series as x, x*, x**, x***, . . ., y. By transitivity, x is equally good as x* is equally good as x** . . . is equally good as y.

B. From the fundamental principles to continuous prioritarianism

What characterizes a prioritarian goodness ranking of outcomes? It is reasonably consistent with the literature to define ‘prioritarianism’ as the class of outcome rankings that satisfy the well-being Pareto principles, the Pigou–Dalton principle, and Anonymity – the three clusters of principles set out above – plus an axiom of separability.Footnote ²³ Separability says that if some individuals in the population of concern are ‘unaffected’ by a pair of outcomes, x and y – each such individual has the same well-being level in x as he does in y – then the comparative goodness of x versus y is independent of the specific well-being level of each such individual.Footnote ²⁴

But what justifies the Separability axiom? This axiom can be defended by direct appeal to the concept of claimsFootnote ²⁵ – although that defence is, admittedly, less compelling than the argument from the claims framework to the well-being Pareto principles, Pigou–Dalton, and Anonymity. A second defence of Separability is pragmatic. Dropping Separability would mean that even if the obtaining of one or another outcome in a pair has only a local impact (such that the only individuals affected by this pair are those within some region, or jurisdiction, or those with a certain social role, or living at a particular time), we couldn't ascertain the better outcome without determining the effects of the outcomes on the local group and the well-being levels of unaffected individuals in other regions, jurisdictions, generations, etc.

Let's now add some additional axioms to the mix – the four further axioms of Measurability, Consistency, Completeness, and Continuity.

These four further axioms can be justified on pragmatic grounds,Footnote ²⁷ and Consistency can also be defended on substantive grounds.Footnote ²⁸

The following can now be demonstrated. (See Appendix.)Footnote ²⁹ The fundamental axioms (the well-being Pareto principles, Pigou–Dalton, and Anonymity), plus Separability, plus the four further axioms, imply the following: the goodness ranking of outcomes is mirrored by the sum of concavely Footnote ³⁰ transformed individual well-being. We start with each individual well-being number in the vector corresponding to a given outcome, and ‘transform’ that well-being number using a strictly increasing and strictly concave function g(.) – as in Figure 1. Outcome x is at least as good as outcome y iff the sum of the concavely transformed well-being numbers corresponding to x is at least as large as the sum of the concavely transformed well-being numbers corresponding to y.

Figure 1.

Continuous prioritarianism

Explanation: The figure displays a strictly increasing and strictly concave g(.) and specifically illustrates why the ∑g(wi) formula satisfies the Pigou–Dalton principle. A change in well-being by amount Δw that occurs at a higher well-being level (wh as opposed to wl) produces a smaller change in transformed well-being.

For short, let's refer to a goodness ranking of this sort as ‘continuous prioritarianism’, and let's abbreviate the formula for the sum of concavely transformed well-being as ∑g(w_i). In so far as the extant literature on prioritarianism employs a mathematical representation, it often does use the formula ∑g(w_i). To repeat: this formula is a tractable specification of prioritarianism that emerges by combining the principles that capture the core of prioritarianism (the well-being Pareto principles, Pigou–Dalton, Anonymity, and Separability) with four further axioms.

A. Fundamental principles: Well-Being Pareto, DM Pigou–Dalton, DM Anonymity, Priority for the More Deserving

Let's now posit that individuals have an attribute, ‘desert’, which works as follows. First, for any person and any outcome, that person has some level of desert in that outcome. Second, these desert levels are intra- and interpersonally comparable, perhaps with some incompleteness. Finally, desert levels figure into the strength of claims, independent of well-being. A higher desert level tends to strengthen an individual's claim.

There are various plausible conceptions of desert, thus understood. Desert might be degree of prudence.Footnote ³¹ The more carefully I have attended to my own interests, the more powerful my claim, ceteris paribus. Alternatively, desert might be moral conscientiousness: the more I have worked to advance both my well-being and everyone else's, the stronger my claim, ceteris paribus.Footnote ³² Finally, desert might be some hybrid of prudence and moral conscientiousness.Footnote ³³

However, the analysis to follow does not rely upon adopting one or another specific conception of desert. Rather, desert is anything about an individual, other than her well-being, that modulates the strength of her claims.

With desert in play, we can use the claims framework to argue for four fundamental clusters of axioms, not three: the well-being Pareto principles, a desert-modulated (DM) Pigou–Dalton principle, desert-modulated (DM) Anonymity, and Priority for the More Deserving. (The well-being Pareto principles are the same as above, but are repeated for convenience.)

The argument for the well-being Pareto principles is the same as above.

Above, in arguing for the ordinary Pigou Dalton principle (for the case of undifferentiated desert), we observed that three factors might affect the strength of someone's claim between two outcomes: her well-being levels in the two outcomes, her difference in well-being between the two, and her desert levels in the two. We can use this observation to defend both DM Pigou–Dalton and Priority for the More Deserving. Consider, first, DM Pigou–Dalton. We need to show that Lower's claim to y over x is stronger than Higher's claim to x over y; if so, y will be better than x, since everyone else has null claims. The well-being-level factor tends to give Lower the stronger claim: he is worse off than Higher in at least one of the outcomes, and no better off in either. The well-being-difference factor drops away, since the differences are equal. Finally, the desert factor does not cut in favour of Higher (since Lower's desert level in either outcome is at least as great as Higher's in either), and may affirmatively weigh in favour of Lower (if Lower's desert level is strictly greater). On balance, then, Lower's claim to y over x is stronger than Higher's claim to x over y.

Consider, next, Priority for the More Deserving. We need here to show that Desi's claim to y over x is stronger than Lesi's to x over y. The well-being-level factor, now, does not weigh in favour of either individual with respect to claim strength. (Desi is better off than Lesi in y, but Lesi is better off than Desi in x, and indeed the two just swap well-being levels.) The well-being-difference factor, too, does not favour either individual, since the differences are equal. Finally, since Desi is unambiguously more deserving than Lesi (Desi's desert level in each of the outcomes is higher than Lesi's in each), the desert factor tends to give Desi the stronger claim. On balance, then, Desi's claim to y over x is stronger than Lesi's claim to x over y.

Finally, the argument for DM Anonymity just generalizes the argument above for Anonymity (in the case of undifferentiated desert). Assume that x* is related to x by a two-person permutation of combinations of well-being and desert levels. Call the two individuals Able and Bob. Able's well-being level in x* is the same as Bob's in x, and Able's desert level in x* is the same as Bob's in x. Conversely, Bob's well-being level in x* is the same as Able's in x, and Bob's desert level in x* is the same as Able's in x. Everyone else's well-being level and desert level does not vary between the two outcomes. Then, by symmetry, Able and Bob have equally strong claims between x and x*, and since everyone else has null claims, the two outcomes are equally good. Since every permutation of combinations of desert and well-being levels is a series of two-person permutations, DM Anonymity follows by transitivity.

To be sure, the fact that persuasive arguments can be separately mounted for each of the four clusters of principles does not mean that we should, on balance, endorse all of them. In particular, if the principles turn out to be logically inconsistent with respect to some set of outcomes S – that is, no goodness ranking of S satisfies all of them – then we will be forced to abandon the combination of the principles, at least with respect to S.

However, in the case of intrapersonally fixed desert, the principles are logically consistent. Let's say that a set of outcomes is characterized by ‘intrapersonally fixed desert’ if each individual's desert level in any outcome in the set is the same as her desert level in any other outcome. In any such set (at least if we add a further axiom regarding measurability), there is a ranking that satisfies the well-being Pareto Principles, DM Pigou–Dalton, DM Anonymity, and Priority for the More Deserving.Footnote ³⁶

B. From the fundamental principles to desert-modulated (DM) continuous prioritarianism

Recall that, in the discussion of undifferentiated desert, we defined ‘continuous prioritarianism’ as the ranking of outcomes using the formula ∑g(w_i). And we observed that this formula follows from the combination of the well-being Pareto principles, the Pigou–Dalton principle and Anonymity together with Separability and four further axioms.

This discussion generalizes to the case of differentiated desert. Separability and the further axioms are reworked for that case, as follows:

The upshot of the new, expanded group of fundamental axioms (Pareto, DM Pigou–Dalton, DM Anonymity, Priority for the More Deserving), plus DM Separability, plus the reworked additional axioms (DM Measurability, Consistency, Completeness, DM Continuity) is an approach that I'll term ‘desert-modulated continuous prioritarianism’. (See Appendix.) There is a two-place function f(.), which takes as its inputs both an individual well-being number and an individual desert number. This function has the shape displayed in Figure 2. First, holding constant desert, f(.) increases as well-being does, with a strictly concave arc. Second, at a given level of well-being, the rate of increase of f(.) with respect to well-being, i.e. its slope with respect to well-being, increases as the level of desert increases (‘the slope condition’).

Figure 2.

Desert-modulated continuous prioritarianism

Explanation: The figure illustrates f(.) as a function of well-being w for two different levels of desert, with d* > d. Note that f(.) is not merely strictly increasing and strictly concave in w for each given desert level, but satisfies the slope condition; at each level of w, f(w, d*) has a greater slope than f(w, d). The dashed lines illustrate how the ∑f(wi, di) formula satisfies DM Pigou–Dalton and Priority for the More Deserving.

The goodness ranking of outcomes corresponds to the sum of these f values: outcome x is at least as good as y iff the sum of individual f values for the well-being/desert vector corresponding to x is at least as large as the sum of individual f values for the well-being/desert vector corresponding to y.

Let's use the symbol ∑f(w_i, d_i) to denote desert-modulated continuous prioritarianism. This is indeed an intuitive generalization of continuous prioritarianism, ∑g(w_i). If desert is intrapersonally fixed and each person has the same desert level as every other person, the ∑f(w_i, d_i) formula reduces to ∑g(w_i) – since f(.) is strictly increasing and strictly concave in well-being, just as g(.) is.

If desert is intrapersonally fixed but there is interpersonal variation in individual desert levels, the axiom Priority for the More Deserving comes into play and ∑f(w_i, d_i) satisfies this axiom by virtue of the slope condition. Further, as long as desert is intrapersonally fixed, ∑f(w_i, d_i) satisfies DM Pigou Dalton (by virtue of the slope condition plus the fact that f(.) is strictly concave in well-being) and the well-being Pareto principles.Footnote ³⁷ It is also straightforward to see that ∑f(w_i, d_i) satisfies DM Anonymity and DM Separability.

Incompatibilities between Priority for the More Deserving and the Well-Being Pareto Principles

Assume that the set of outcomes S is characterized by intrapersonally variable desert.

1. (1) It is possible that Priority for the More Deserving and Well-Being Pareto Indifference are inconsistent with respect to S. That is, there may be no goodness ranking of S which satisfies both of these axioms.

2. (2) If we assume, further, that the goodness ranking of S satisfies DM Measurability, DM Anonymity, and DM Continuity, it is possible that Priority for the More Deserving and Well-Being Strong Pareto are inconsistent with respect to S.

The first incompatibility (between Priority for the More Deserving and Well-Being Pareto Indifference) is illustrated by Table 2. The second is illustrated by Table 3.

Table 2.

The conflict between Priority for the More Deserving and Well-Being Pareto Indifference

Explanation: The symbols W_Jim and W_Jim* are not numbers. Rather, each denotes a possible well-being basis for Jim: some possible combination of facts that, if it obtains, suffices to determine how well off Jim is. Similarly, W_Sally and W_Sally* are possible well-being bases for Sally. Further, the intra- and interpersonal comparisons to which these well-being bases give rise are as follows. W denotes one and the same level of well-being, whether subscripted to Jim or Sally. (For example, in the above table, Jim in outcome y is equally well off as Jim in outcome zz, and equally well off as Sally in outcomes x and z.) Similarly, W* denotes one and the same level of well-being. Finally, W* denotes a higher level of well-being than W.

Similarly, the symbols D_Jim, D_Jim*, D_Sally, D_Sally* are not numbers, but denote a possible desert basis for Jim and Sally, respectively. Further, D denotes one and the same level of desert, whether subscripted to Jim or Sally; and D* denotes a higher level of desert, whether subscripted to Jim or Sally.

Well-being Pareto Indifference requires (1) that y be ranked equally good as zz and that x be ranked equally good as z. Priority for the More Deserving requires (2) that y be ranked better than x and that z be ranked better than zz. However, given transitivity of the goodness ranking, (1) and (2) can't both be true.

Table 3.

The conflict between Priority for the More Deserving and Well-Being Strong Pareto

Explanation: w* and w are well-being numbers, with w* > w. d* and d are desert numbers, with d* > d. And ε > 0. These numbers can be used in the table because we are now assuming DM Measurability.

Priority for the More Deserving requires that y be ranked better than x. By DM Continuity, y+ is also better than x for ε sufficiently small. By DM Anonymity, z+ is equally good as y+. By transitivity, then, z+ is better than x. But this contradicts Well-Being Strong Pareto: note that each individual's well-being level with z+ is less than his or her well-being level with x.

What happens if we drop DM Continuity? In that case, at least if we assume DM Measurability, Well-Being Strong Pareto and Priority for the More Deserving are logically consistent in all sets of outcomes (even with intrapersonally variable desert).Footnote ³⁹

However, even with DM Continuity dropped, Well-Being Strong Pareto creates tight constraints on the role of desert in modulating claim strength. Let's say that ‘Minimal Significance for Desert’ holds true if desert functions only as a tiebreaker.

As between individuals at the same well-being level, Priority for the More Deserving says that it is morally better to give a fixed benefit to the more deserving individual rather than to the less deserving one. Conversely, Minimal Significance for Desert says that it is not morally better to give a smaller benefit to the more deserving individual in preference to a larger benefit for the less deserving one.

We can now show that (assuming only DM Anonymity), if Minimal Significance for Desert does not hold true, there will be some set of outcomes with intrapersonally variable desert in which Well-Being Strong Pareto is violated. See Table 4.

Table 4.

Dropping Minimal Significance for Desert can yield a violation of Well-Being Strong Pareto

Explanation: W and D are not numbers, but rather indicate the well-being or desert basis of Sally or Jim, depending on the subscript. Let W* denote a greater well-being level than W, and W** a yet greater well-being level. (Thus Jim's well-being difference between outcomes x and y is greater than Sally's well-being difference between outcomes y and x.) D* indicates a higher level of desert than D.

If Minimal Significance is dropped, then it is possible for there to be a pair of outcomes x and y as displayed here such that y is better than x. Consider now a set of outcomes that includes x, y, and z. By DM Anonymity, z is equally good as y. Because y is better than x, it follows by transitivity that z is better than x. But this violates Well-Being Strong Pareto, since one individual (Jim) is worse off with z than x, while the other (Sally) is equally well off.

It is important to be clear about the nature of the conflicts between Priority for the More Deserving and the well-being Pareto principles illustrated by Tables 2 and 3. These tables do not show that Priority for the More Deserving conflicts with the well-being Pareto principles in every set of outcomes that has intrapersonally variable desert. Rather, these tables demonstrate that there are some sets, with intrapersonally variable desert, in which Priority for the More Deserving conflicts with the well-being Pareto principles. Similarly, Table 4 does not show that dropping Minimal Significance yields a conflict with Well-Being Strong Pareto in every set of outcomes with intrapersonally variable desert. Rather, it shows that doing so yields a conflict in some such sets.

A. Well-Being retained as the ‘currency’ for individual claims

Assume, first, that the valence of individual claims remains fixed in terms of well-being. For short, well-being remains the ‘currency’ of claims. If an individual is equally well off in x and y, she has a null claim between them. If she is better off in x than y, she has a claim in favour of x over y. If so, the well-being Pareto principles follow immediately.

Might we also endorse Priority for the More Deserving, notwithstanding Conflict? (1) Since Conflict occurs only with intrapersonally variable desert, we might insist on intrapersonally fixed desert. Every set of outcomes should be such that no person's desert varies across outcomes. But this is absurd. Desert, whatever exactly it might be, is surely not ‘built into’ a person's identity. If so, it is possible that any given person might find herself at any one of a plurality of desert levels; and thus a decision-maker should be free to count as possible a set of outcomes in which desert levels vary intrapersonally.

(2) A different thought is that any set of outcomes in which desert varies intrapersonally should be divided into subsets within which desert is intrapersonally fixed; and we should consider outcomes in different such subsets as incomparably good. But a moment's reflection shows that this proposal conflicts with the well-being Pareto principles. If x and y are such that some are better off in x than y, and everyone is at least as well off, then Well-Being Strong Pareto requires that x be ranked better than y – and doesn't allow that they be incomparable.

(3) Yet another strategy is to limit the applicability of Priority for the More Deserving to certain sets. As already emphasized, Priority for the More Deserving does not conflict with the well-being Pareto principles in all sets. Rather, the conflict arises in some (not all) sets of outcomes with intrapersonally variable desert. Thus we might preserve the full force of the well-being Pareto principles, and apply Priority for the More Deserving only in sets of outcomes where no conflict with those principles arises. For example, consider the outcomes described in Table 2. Let the set of outcomes be S = {x, y, z, zz}. As Table 2 illustrates, there is no ranking of this set that satisfies both Priority for the More Deserving and Well-Being Pareto Indifference. But now consider a different set, namely S⁺ = {x, y, z}. There is a ranking of S⁺ that satisfies both axioms.Footnote ⁴⁰

This strategy violates the axiom of Consistency. Assume that two outcomes x and y are as described by Priority for the More Deserving – so that the axiom requires y to be ranked better than x. The strategy now under discussion has the upshot that y is better than x if the further outcomes whose moral goodness is under consideration meet certain conditions.Footnote ⁴¹ But shouldn't the goodness comparison of x versus y depend only on what each person's well-being and desert would be if x were to obtain, and what her well-being and desert would be if y were to obtain? These facts about x and y themselves, not the further outcomes, are sufficient to fix the valence and strength of each person's claims between the two outcomes.

B. A non-well-being currency for claims?

Recall that the claims framework is meant to operationalize the idea, originating with Nagel, that the moral comparison of two outcomes depends upon how they compare from the perspective of each of the individuals in the population. I have hitherto assumed that each person-centred comparison is a comparison in terms of well-being. If so, the valence of someone's claims is fixed by her well-being.

However, we might believe that some non-well-being item is the appropriate currency for individual claims. That is, we might think that the pattern of individual comparisons in terms of this non-well-being item is actually the supervenience base for morality. For example, if we believe that each individual, as an autonomous agent, should be entitled to determine what matters from her perspective, then we might adopt all-things-considered preferences, not well-being, as the currency for claims. If we are moved by the ‘luck egalitarian’ insight that individuals cannot complain of a well-being shortfall if they had an adequate opportunity to avoid that shortfall, then we might adopt opportunity-for-well-being, rather than well-being simpliciter, as the currency for claims.

For short, call the new currency (whatever it might be) ‘claimfare’. The valence of an individual's claim between two outcomes will now be fixed by her level of claimfare in the two. And we will now endorse the Claimfare Pareto principles: (a) Claimfare Pareto Indifference (if each person has the same level of claimfare in x as he does in y, the two outcomes are equally good); and (b) Claimfare Strong Pareto (if at least one person has more claimfare in y than he does in x, and everyone else has at least as much claimfare in y as she does in x, then y is better).

Yet the shift from well-being to claimfare hardly salvages the project of desert-modulated claims. Suppose that individual desert is taken as an ingredient in claim strength, apart from an individual's level of claimfare. This yields Priority for the More Deserving*, with the asterisk indicating that the principle is framed now in terms of claimfare rather than well-being. But now Priority for the More Deserving* will come into conflict with the claimfare Pareto principles, in a manner isomorphic to the conflict between Priority for the More Deserving and the well-being Pareto principles.

Perhaps, however, the thought of salvaging the project by shifting currency is meant to be taken in a different way. ‘Let's not use desert as an extra factor that bears on claim strength, above and beyond individual holdings of some currency. Rather, let's incorporate desert into the currency itself.’

In particular, imagine that the currency for claims is a hybrid of well-being and desert. Whether x is ranked more highly than y from the standpoint of i depends both on individual i’s well-being level in each outcome, and on her desert level in each outcome. This posit of a hybrid currency, if sound, would warrant desert-modulated Pareto principles (principles framed in terms of a mixture of desert and well-being). Such principles, if appropriately framed, would not conflict with Priority for the More Deserving even with intrapersonally variable desert.

For example, consider the following principles, ‘DM Pareto Indifference’ and ‘DM Strong Pareto’:

The DM Pareto Principles are consistent with Priority for the More Deserving, even with intrapersonally variable desert. To see this, note that the desert-modulated continuous prioritarian formula, ∑f(w_i, d_i), satisfies both the DM Pareto Principles and Priority for the More Deserving in ranking any set S even if an individual's desert number (d_i) is not the same in all of the outcomes in S.Footnote ⁴²

However, the strategy of incorporating desert into the currency for claims turns out to be problematic. Consider two possibilities. (1) Desert and well-being are both positive contributors to the hybrid currency. If x and y are such that individual i is equally well off in the two outcomes but has a higher desert level in y than x, y is ranked higher from i’s perspective. Conversely, if x and y are such that individual i is equally deserving in the two outcomes but has a higher well-being level in y than x, y is ranked higher from i’s perspective.

A moment's reflection shows why this variant of the incorporationist strategy misfires. Imagine that Desi and Lesi are equally well off, and Desi has a higher desert level than Lesi. Thus (given the premise that desert makes a positive contribution to currency level) Desi has more of the hybrid currency than Lesi. Imagine, now, that we can increase Desi's holdings of the hybrid currency by Δh, or increase Lesi's holdings of the hybrid currency by the same amount, Δh. Because Desi has more of the hybrid currency, she has a weaker claim to the increase than Lesi. In short, this variant of the incorporationist strategy would not justify the axiom of Priority for the More Deserving, but instead would support a principle of Priority for the Less Deserving! Such a principle is very counterintuitive.

The other possibility is that: (2) Desert is a negative contributor to the hybrid currency, while well-being is a positive contributor. If x and y are such that individual i is equally well off in the two outcomes but has a higher desert level in y, y is ranked lower from i’s perspective. Conversely, if x and y are such that individual i is equally deserving in the two outcomes but has a higher well-being level in y than x, y is ranked higher from i’s perspective.

The negative-contribution proposal has the troubling implication that it is pro tanto morally better to make individuals less deserving. If individual i is less deserving in x than y and equally well off, then (on the negative contribution view) x is ranked higher from i’s perspective; and strong Pareto formulated in terms of the hybrid currency favours x over y if no one else is affected. But surely morality doesn't counsel a lowering of desert. Intuition says just the opposite. For example, Arneson writes: ‘[I]t is better from the moral point of view that persons be more deserving rather than less deserving.’Footnote ⁴³

C. Non-claim-based arguments for prioritarianism

Prioritarianism (here understood as a quasi-ordering of outcomes that satisfies the well-being Pareto principles, Pigou–Dalton, Anonymity, and Separability) and continuous prioritarianism (prioritarianism together with further axioms yielding the ∑g(w_i) formula) might well be justifiable without reference to the claims framework. For example, Derek Parfit's original presentation of prioritarianism appeals to two separate ideas that support, respectively, the Well-Being Strong Pareto principle and the Pigou–Dalton principle, namely (1) that ‘benefits are good’ and levelling down is to be avoided (strong Pareto); and (2) that ‘benefits to the worse off matter more’ (Pigou–Dalton).Footnote ⁴⁴ Arneson's argument for prioritarianism is similar. Unlike the claims framework, this approach offers no unified rationale for the well-being Pareto principles and Pigou–Dalton.

Nils Holtug defends prioritarianism with reference to a certain type of intrinsic value:

David McCarthy, in a very different approach, uses axioms regarding the ranking of lotteries to characterize continuous prioritarianism.Footnote ⁴⁶ The core axioms are ‘Anteriority’, ‘Two-Stage Anonymity’, and the ‘Priority Principle’ – the last requiring a divergence between the moral and individual ranking of lotteries over well-being levels. (McCarthy doesn't accept the ‘Priority Principle’, but his analysis outlines a justificatory route to continuous prioritarianism based upon that principle.)

Space constraints preclude a detailed discussion of whether these rationales for prioritarianism support some kind of desert-modulated prioritarianism where individuals differ in desert. The following brief observations will have to suffice. Assume, first, that the rationale does indeed support the well-being Pareto principles regardless of interpersonal variation in desert, as long as desert is intrapersonally fixed. (This appears to be true, for example, of Arneson, who writes: ‘The position thus taken is that it is morally desirable that everyone, even Hitler (the least deserving person on earth, let's say) should have . . . more rather than less well-being.’Footnote ⁴⁷) Assume, next, that the rationale supports Priority for the More Deserving. (Again, this is true of Arneson.Footnote ⁴⁸) The question then will be whether the rationale can justify desert-modulated Pareto principles that are equivalent to the well-being Pareto principles in the case of intrapersonally fixed desert, but are logically consistent with Priority for the More Deserving even with intrapersonally variable desert. This is true, for example, of the DM Pareto principles stated above. (Arneson's position on this issue is not clear.)

I argued above that the claim-based rationale for prioritarianism doesn't justify desert-modulated Pareto principles, since a hybrid well-being/desert currency for claims is implausible. Thus the project of desert-modulated prioritarianism collapses. Whether such a collapse occurs on other rationales for prioritarianism remains to be seen.