2.1. The SWF framework: a quick overview

The SWF framework, as described here,Footnote ¹ is a methodology for implementing welfarism. Footnote ² Welfarism is a family of ethical views that includes utilitarianism as its most prominent member, but reaches more broadly (Adler, Reference Adler2012). Welfarists are consequentialists. Roughly speaking, consequentialists believe that the ethical status of an action depends upon what might happen were the action to be performed, and how good or bad these possible consequences might be. The philosophical notion of a “possible world” (a complete description of a possible history of the world) helps make this rough formulation more precise. At the core of any consequentialist ethical view is a world-ranking, which specifies for any pair of worlds d and $ {{d^{\ast }}} $ whether the first is better, worse, or equally good as the second – and does so in a well-behaved (transitive) fashion.

Welfarists take the world-ranking to be determined by individual well-being. This is captured in the Pareto axioms, Pareto Indifference and Strong Pareto. Pareto Indifference: If each person is equally well off in $ {d}^{\ast } $ as she is in d, then $ {d}^{\ast } $ and d are equally good. Strong Pareto: If each person is at least as well off in $ {d}^{\ast } $ as she is in d, and at least one person is better off in $ {d}^{\ast } $ than d, then $ {d}^{\ast } $ is better than d. Footnote ³

Consequentialists and, specifically, welfarists derive the ethical status of actions from the world-ranking. Let P be a set of choices (possible actions) facing a decision-maker. Welfarists aim to rank the choices in P as better or worse than each other, and to do so in light of the world ranking. But this is not straightforward. First, possible worlds are not cognitively tractable objects. A human decision-maker (even aided by computers) cannot write down a full description of even a single possible world, let alone each world in a set of worlds. Second, a human decision-maker, who is not omniscient, will not know for certain which world a given choice will lead to.

In short, welfarism needs a decision procedure: a methodology that is grounded in the world-ranking but implementable by human decision-makers, and that yields choice guidance. The SWF framework, in turn, is the most systematic such decision procedure. It is most appropriate for large-scale choices – in particular, governmental policies. I will therefore refer to P as a set of policies. A policy is some course of action that government might take: enacting a particular regulation, building infrastructure, disseminating information, etc. $ \mathbf{P}=\left\{P,{P}^{\ast },\dots \right\} $ , with P, $ {P}^{\ast } $ , etc. each a possible policy.

The SWF framework has the following components: a model population $ \mathbf{I}=\left\{1,\dots, N\right\} $ ; a set of outcomes $ \mathbf{O}=\left\{x,y,\dots \right\} $ ; a well-being measure $ w\left(\cdot \right) $ : the SWF proper, abbreviated as $ \succcurlyeq $ , which is a ranking of well-being vectors; the set P of policies, each represented as a probability distribution over outcomes; and an “uncertainty module” for the SWF, which is a rule for arriving at a ranking $ {\succcurlyeq}^{\mathbf{P}} $ of the policy set in light of the well-being measure and the SWF. I will briefly discuss each component in turn. See Adler (Reference Adler2019, Reference Adler, Adler and Norheim2022) for a much fuller presentation of the SWF framework. These works, in turn, build upon a long-standing literature in welfare economics (Bossert & Weymark, Reference Bossert, Weymark, Barberà, Hammond and Seidl2004; Weymark, Reference Weymark, Adler and Fleurbaey2016).

I is a representation of the population of interest: those individuals whose well-being the decision-maker takes into account in choosing among policies. $ \mathbf{I}=\left\{1,2,\dots, N\right\} $ is a set of N numbers, each denoting an individual. For simplicity, I will present the SWF framework using a fixed-population setup: each individual exists in all of the outcomes.Footnote ⁴

An outcome x is a simplified and cognitively tractable model of a possible world. An outcome describes some of the features of worlds that are relevant to individual well-being. More specifically, each outcome x is a list of attribute bundles, one for each member of the population. $ x=\left({b}_1(x),\dots, {b}_i(x),\dots {b}_N(x)\right) $ . A bundle describes some of the types of individual attributes that determine well-being: attributes such as income, health, environmental quality, and leisure.

Bundles are lifetime bundles: individual i’s lifetime well-being in x is determined by her bundle there.Footnote ⁵ The most simplified application of the SWF methodology employs one-period bundles to model lifetime well-being: each individual exists for a single period in a given outcome, with his lifetime well-being determined by his attributes during this single period. A more complicated implementation employs multi-period bundles. Each bundle describes the individual’s longevity (the number of periods that she is alive) and, for each period alive, her period attributes (income, health, leisure, etc.). The multi-period setup provides insights into tradeoffs between longevity and other well-being determinants, and will be the fulcrum for my analysis of health policy in Section 3.

The well-being measure $ w\left(\cdot \right) $ maps bundles onto well-being numbers. These numbers represent admissible well-being comparisons, according to some theory of well-being. “Admissible” comparisons are those that the theory allows – that it sees as meaningful. Imagine that the theory allows for intra- and interpersonal comparisons of well-being levels and differences. Then $ w\left(\cdot \right) $ will mirror such comparisons, as follows. (1) Intrapersonal level comparisons. $ w\left({b}_i\right)\ge w\left({b}_i^{\ast}\right)\hbox{---} {b}_i $ and $ {b}_i^{\ast } $ two possible bundles for the same individual (i) – iffFootnote ⁶ i is at least as well off with $ {b}_i $ as $ {b}_i^{\ast } $ . (2) Intrapersonal difference comparisons. $ w\left({b}_i\right)-w\left({b}_i^{\ast}\right)\ge w\left({b}_i^{\ast \ast}\right)-w\left({b}_i^{\ast \ast \ast \hskip-0.3em }\right)\hbox{---} {b}_i $ , $ {b}_i^{\ast } $ , $ {b}_i^{\ast \ast } $ , $ {b}_i^{\ast \ast \ast } $ four possible bundles for the same individual (i) – iff the difference in i’s well-being between having $ {b}_i $ and having $ {b}_i^{\ast } $ is at least as large as the difference in i’s well-being between having $ {b}_i^{\ast \ast } $ and having $ {b}_i^{\ast \ast \ast } $ . (3) Interpersonal level comparisons. $ w\left({b}_i\right)\ge w\left({b}_j\right)\hbox{---} {b}_i $ and $ {b}_j $ two possible bundles for different individuals (i and j, $ i\ne j)\hbox{---} \mathrm{iff} $ i with $ {b}_i $ is at least as well off as j with $ {b}_j $ . (4) Interpersonal difference comparisons. $ w\left({b}_i\right)-w\left({b}_j\right)\ge w\left({b}_k\right)-w\left({b}_l\right)\hbox{---} {b}_i $ , $ {b}_j $ , $ {b}_k $ , $ {b}_l $ four possible bundles for individuals who are not all identical (i, j, k, l, and not $ i=j=k=l $ ) – iff the difference in well-being between i’s having $ {b}_i $ and j’s having $ {b}_j $ is at least as large as the difference in well-being between k’s having $ {b}_k $ and l’s having $ {b}_j $ .

The philosophical literature on well-being recognizes a wide range of types of well-being theories (see Adler, Reference Adler2019, chap. 3, and sources cited therein). These can be grouped into three families: “experientialist” theories, according to which an individual’s well-being is reducible to the mental states that he experiences (pains, pleasures, feelings of happiness, etc.); preference-based theories, according to which an individual’s well-being depends on the extent to which her preferences are satisfied; and objective-good theories, according to which well-being consists in the realization of various goods that are not reducible to either experiences or preference-satisfaction. These three theories correspond to different strands in economics: experientialism, to the burgeoning field of happiness economics; preference-based theories, to the view of well-being in neoclassical economics; and objective-good theories, to the literature on “capabilities.”

The SWF framework is agnostic about the nature of well-being. Any well-being theory can be plugged into the framework – with one big caveat. The theory must allow for interpersonal well-being comparisons. The SWF framework founders if combined with a well-being theory that eschews interpersonal comparisons – that counts as admissible only intrapersonal well-being level comparisons, or only intrapersonal well-being level and difference comparisons. I will explain why in a moment.

A well-being vector (denoted here with a bold lower-case letter such as w or v) is a list of well-being numbers, one for each individual in the population. $ \mathbf{w}=\left({\mathbf{w}}_1,\dots, {\mathbf{w}}_i,\dots, {\mathbf{w}}_N\right) $ , with $ {\mathbf{w}}_i $ the well-being number of individual i. The well-being measure $ w\left(\cdot \right) $ converts a given outcome x into a well-being vector. With $ \mathbf{w}(x) $ the well-being vector corresponding to x, we have $ \mathbf{w}(x)=\left(w\left({b}_1(x)\right),\dots, w\left({b}_i(x)\right),\dots w\left({b}_N(x)\right)\right) $ .

The SWF proper is a rule for ranking well-being vectors. Many such rules are possible, but four predominate in the literature: the utilitarian SWF, the prioritarian family of SWFs, the leximin SWF, and the rank-weighted family of SWFs. (In what follows, “ $ \mathbf{w}\succcurlyeq \mathbf{v} $ ” means that w is ranked by the SWF at least as good as v; “ $ \mathbf{w}\succ \mathbf{v} $ ,” that w is ranked better than v; and “ $ \mathbf{w}\sim \mathbf{v} $ ,” that the two are ranked equally good.)

The utilitarian SWF: $ \mathbf{w}\succcurlyeq \mathbf{v}\hskip0.4em \mathrm{iff}\hskip0.4em \sum \limits_{i=1}^N{\mathbf{w}}_i\ge \sum \limits_{i=1}^N{\mathbf{v}}_i. $

An SWF’s vector ranking (whatever it may be) immediately generates a ranking of the outcome set: outcome x at least as good as outcome $ y\hskip0.4em \mathrm{iff}\hskip0.4em \mathbf{w}(x)\succcurlyeq \mathbf{w}(y) $ .

The choice between these different types of SWFs is an ethical choice. The question how to compare possible arrangements of well-being among the population is a matter for ethical debate. In a given legal system, the legal authority to make this ethical choice – to select the SWF methodology as opposed to a different approach (e.g. BCA), and if so to use one SWF rather than others – will be vested in certain governmental officials or institutions (e.g. an elected President, a legislature).

“Prioritarianism” has that name because it gives priority to well-being changes affecting worse-off individuals, as can be seen in Figure 1. The degree of such priority depends upon the concavity of the transformation function. I have argued at length in favor of prioritarianism, as against utilitarianism, rank-weighted SWFs, and leximin (Adler, Reference Adler2012; Adler & Holtug, Reference Adler and Holtug2019).

Figure 1. A prioritarian transformation function. Explanation: Let w _L be the well-being level of a worse-off person, and let w _H be the well-being level of a better-off person. An increment Δw to the worse-off person’s well-being produces a bigger change in transformed well-being (as seen on the y-axis) than the same increment to the better-off person’s well-being.

The key difference between utilitarian and prioritarian SWFs concerns the Pigou–Dalton axiom. Pigou–Dalton is an equity axiom. It captures, formally, whether the SWF is sensitive to the distribution of well-being.

Every prioritarian SWF satisfies the Pigou–Dalton axiom. The utilitarian SWF does not; a pure transfer of well-being from a better- to a worse-off person does not change the sum total of well-being.

The leximin SWF and rank-weighted SWFs also satisfy the Pigou–Dalton axiom, but they have significant disadvantages as compared to prioritarian SWFs – or so I have argued. The leximin SWF is absolutist. Consider a leaky transfer of well-being between a transferor (the individual initially better off) and a transferee (the individual initially worse off) which is such that the transferor ends up better off than where the transferee started; no one else is affected. The transfer is “leaky” in that the transferor’s well-being is reduced by $ \varDelta w $ , while the transferee’s increases by some fraction of the loss to the transferor – by $ \rho \varDelta w $ , with ρ between 0 and 1. Leximin sees every such transfer as an ethical improvement, regardless of how small ρ is. By contrast, every prioritarian SWF will favor some such transfers (if ρ is sufficiently large) but disapprove others (if ρ is too small).Footnote ⁸

Prioritarianism can be seen as filling the “space” of distribution sensitivity between utilitarianism, at one extreme, and leximin, at the other. Utilitarianism is wholly insensitive to equity: even a pure transfer of well-being from a better- to a worse-off person (affecting no one else), which shrinks the gap between them at no cost to overall well-being, is a matter of indifference to utilitarianism. Leximin is wholly insensitive to overall well-being. Any transfer from a better- to a worse-off individual (affecting no one else) that leaves the transferor better off than where the transferee started, is favored by leximin – regardless of the degree of leakage and the loss to overall well-being. Prioritarianism balances overall well-being against equity, and offers the decision-maker flexibility in doing so. As $ g\left(\cdot \right) $ becomes increasingly concave, less weight is given to overall well-being: prioritarianism approaches leximin. As $ g\left(\cdot \right) $ becomes less concave, and approaches linearity, less weight is given to equity; prioritarianism approaches utilitarianism.

Rank-weighted SWFs are less tractable than prioritarian SWFs. Consider once again the case of a leaky transfer that leaves the transferor better off than where the transferee started, and affects no one else. For prioritarians, knowing the starting and ending well-being levels of the two individuals, together with the transformation function $ g\left(\cdot \right) $ , is sufficient to determine whether the transfer is an ethical improvement. For those who endorse a rank-weighed SWF, this is not sufficient information. Whether the transfer is an ethical improvement will generally depend upon how the starting and ending well-being levels of transferee and transferor compare to the well-being levels of everyone else – even though they are unaffected.Footnote ⁹

Formally, prioritarian SWFs (as well as the utilitarian SWF and leximin SWF), by contrast with rank-weighted SWFs, satisfy an axiom of Separability:

The tractability benefits of Separability will come into clearer view in Section 2.2, when we consider the topic of uncertainty modules.

Let us turn now to the question of interpersonal comparability. A basic axiom of the SWF framework, which I have referred to as the “Fundamental Principle of Invariance” (Adler, Reference Adler2019, chap. 2), says: the SWF’s ranking of outcomes should be invariant to replacement of well-being measure $ w\left(\cdot \right) $ by an informationally equivalent well-being measure $ {w}^{+}\left(\cdot \right) $ . If $ w\left(\cdot \right) $ accurately represents all of the admissible well-being comparisons, according to whichever well-being theory is being used, and $ {w}^{+}\left(\cdot \right) $ does so as well, then the SWF’s outcome ranking should be the same whether $ w\left(\cdot \right) $ is used to map outcomes onto vectors or instead $ {w}^{+}\left(\cdot \right) $ is. For the SWF to rank outcomes one way with $ w\left(\cdot \right) $ , and a different way with $ {w}^{+}\left(\cdot \right) $ , would mean that the outcome ranking is arbitrary from the perspective of welfarism: the ranking depends on the choice between the two measures, a choice that cannot be justified on welfarist grounds because the measures contain the very same well-being information.

Assume now that our well-being theory sees only intrapersonal comparisons as admissible: either nothing more than intrapersonal level comparisons, or nothing more than intrapersonal level and difference comparisons. It permits neither interpersonal level comparisons, nor interpersonal difference comparisons. Note that if well-being measure $ w\left(\cdot \right) $ accurately represents the theory’s intrapersonal comparisons, then so does any other well-being measure $ {w}^{+}\left(\cdot \right) $ that is an individual-specific cardinal rescaling of $ w\left(\cdot \right) $ .Footnote ¹¹ The two measures are informationally equivalent if interpersonal comparisons are inadmissible.

However, as Table 1 illustrates, none of the major types of SWFs are invariant to individual-specific cardinal rescalings of the well-being measure. For each such SWF, swapping one well-being measure $ w\left(\cdot \right) $ for another measure $ {w}^{+}\left(\cdot \right) $ that makes exactly the same intrapersonal comparisons of levels and differences leads to a change in the outcome ranking. In short, if interpersonal comparisons are inadmissible, then all these SWFs violate the Fundamental Principle of Invariance. Indeed, it can be shown that any non-dictatorial SWF will violate the Fundamental Principle of Invariance if interpersonal comparisons are inadmissible.Footnote ¹²

Table 1. An individual-specific cardinal rescaling of the well-being measure

Explanation: The left of the table displays well-being vectors assigned to outcomes x, y, and z by a well-being measure w(∙), while the right displays vectors per an individual-specific cardinal rescaling of $ w\left(\cdot \right) $ , $ {w}^{+}\left(\cdot \right) $ such that $ {w}^{+}\left({b}_i\right)={c}_iw\left({b}_i\right)+{d}_i $ , $ {c}_i>0 $ . The table illustrates that neither the utilitarian SWF, nor a prioritarian SWF, nor a rank-weighted SWF, nor the leximin SWF are invariant to an individual-specific cardinal rescaling of well-being numbers. The square root transformation function is used for the prioritarian SWF, and the rank-weighted SWF uses integer weights $ \left({k}_1=N,{k}_2=N-1,\dots, {k}_N=1\right) $ . Source: Adler (Reference Adler, Adler and Norheim2022, 78).

Should we, in fact, endorse a well-being theory that rejects interpersonal comparisons? Some economists will say “yes” and therefore reject the SWF framework. But interpersonal comparisons are a matter of common sense (Adler, Reference Adler, Adler and Norheim2022, 75–80). To be sure, how to make interpersonal well-being comparisons for purposes of the SWF framework implicates contested ethical questionsFootnote ¹³ – but the decision to adopt any policy-analysis methodology implicates contested ethical questions.

Special puzzles do arise in explaining how a preference-based well-being theory allows for interpersonal comparisons. Adler (Reference Adler, Adler and Fleurbaey2016b, Reference Adler2019) shows how to construct a well-being measure that respects individual preferences and makes both intrapersonal and interpersonal comparisons of levels and differences, by using individuals’ von Neumann/Morgenstern utility functions. Another fairly widespread methodology for preference-based interpersonal comparisons is the so-called “equivalence approach” (as implemented, e.g., in the use of “equivalent income” as the measure of an individual’s well-being given her preferences) (Adler & Decancq, Reference Adler, Decancq, Adler and Norheim2022; Cookson et al., Reference Cookson, Norheim, Skarda, Adler and Norheim2022; Fleurbaey, Reference Fleurbaey, Adler and Fleurbaey2016).

2.2. The uncertainty module

In what follows, I focus on utilitarianism (the most influential version of welfarism, historically and up through the present) and prioritarianism (which I take to be utilitarianism’s strongest competitor).

The SWF framework uses an “uncertainty module” to capture the decision-maker’s uncertainty about which outcome would result were a given policy to be implemented. Each policy in P is associated with a probability distribution over O, the set of outcomes. $ {\unicode{x03C0}}_P(x) $ is the probability of outcome x, were policy P to be chosen.Footnote ¹⁴ $ {\succcurlyeq}^{\mathbf{P}} $ , recall, is the ranking of the policy set. An uncertainty module for a particular SWF ( $ \hskip0.15em \succcurlyeq \hskip0.15em $ , a ranking of well-being vectors) is a rule for arriving at $ {\succcurlyeq}^{\mathbf{P}} $ in light of the probabilities over outcomes for each P; the well-being measure $ w\left(\cdot \right) $ ; and the SWF. Each SWF has multiple uncertainty modules. Every such module must satisfy a consistency constraint, which applies to the ranking of “degenerate” policies (policies that assign probability 1 to one outcome and 0 to all others): if P leads with probability 1 to outcome x, and $ {P}^{\ast } $ with probability 1 to outcome $ {x}^{\ast } $ , then the policies must be ranked according to the SWF’s ranking of the well-being vectors associated with x and $ {x}^{\ast } $ . Once nondegenerate policies come into the picture, however, the different modules for a given SWF may diverge.

In the literature on SWFs under uncertainty (for overviews, see Adler, Reference Adler2012 chap. 7, Reference Adler2019, Reference Adler, Adler and Norheim2022; Mongin & Pivato, Reference Mongin, Pivato, Adler and Fleurbaey2016), the dominant uncertainty module for the utilitarian SWF is a rule that I will term “simple utilitarianism” (SU). How to apply prioritarianism under uncertainty is more contested. Three modules predominate: “ex post prioritarianism” (EPP), “ex ante prioritarianism” (EAP), and expected equally-distributed-equivalent prioritarianism (EEDEP).Footnote ¹⁵ Each of these assigns numerical scores to policies and ranks them according to the scores.Footnote ¹⁶ I will denote the scores as, respectively, $ {S}^{SU}(P) $ , $ {S}^{EPP}(P) $ , $ {S}^{EAP}(P) $ , and $ {S}^{EEDEP}(P) $ ; “S” indicates “social welfare” and the superscript the particular module at hand. The scores are calculated as follows. ( $ g\left(\cdot \right) $ , as above, is the transformation function that defines a specific prioritarian SWF; EPP, EAP, and EEDEP are all modules for that SWF.)

Simple Utilitarianism: $ {S}^{SU}(P)=\sum \limits_x{\pi}_p(x)\sum \limits_{i=1}^N{\mathbf{w}}_i(x)=\sum \limits_{i=1}^N\sum \limits_x{\pi}_p(x){\mathbf{w}}_i(x). $

Ex Post Prioritarianism: $ {S}^{EPP}(P)=\sum \limits_x{\pi}_p(x)\sum \limits_{i=1}^Ng\left({\mathbf{w}}_i(x)\right)=\sum \limits_{i=1}^N\sum \limits_x{\pi}_P(x)g\left({\mathbf{w}}_i(x)\right). $

Ex Ante Prioritarianism: $ {S}^{EAP}(P)=\sum \limits_{i=1}^Ng\left(\sum \limits_x{\pi}_P(x){\mathbf{w}}_i(x)\right). $

Expected Equally Distributed Equivalent Prioritarianism: $ {S}^{EEDEP}(P)=\sum \limits_x{\pi}_P(x){g}^{-1}\left(\sum \limits_{i=1}^Ng\left({\mathbf{w}}_i(x)\right)/N\right). $

In a nutshell: The SU score is the expected sum of individuals’ well-being or, equivalently, the sum across individuals of expected well-being. The EPP score is the expected sum of individuals’ transformed well-being or, equivalently, the sum across individuals of expected transformed well-being. The EAP score is the sum, across individuals, of transformed expected well-being. The prioritarian “equally distributed equivalent” for a given well-being vector w is that well-being level w such that a vector with everyone at w is ranked equally good by the prioritarian SWF as w. The EEDEP score is the expected value of these equally distributed equivalents.

Uncertainty axioms are constraints that, it might be proposed, the uncertainty module should satisfy. There are a range of such axioms that seem ethically plausible, but I will focus on three: Ex Ante Pareto, Dominance, and Policy Separability. Footnote ¹⁷ These axioms will help clarify what is at stake in the choice between EPP, EAP, and EEDEP, and why a single dominant module has emerged for utilitarianism (SU) but not so for prioritarianism. And Policy Separability will be the foundation for the analysis of health policy in Section 3.

Table 2 states how the various modules fare with respect to these three uncertainty axioms.

Table 2. Uncertainty modules and axioms

Table 3 displays a basic dilemma with respect to a prioritarian module. No such module can satisfy both Ex Ante Pareto and Dominance. EPP and EEDEP satisfy Dominance, at the inevitable cost of violating Ex Ante Pareto; EAP satisfies Ex Ante Pareto, at the inevitable cost of violating Dominance.

Table 3. Dominance and Ex Ante Pareto

Explanation: Each of the policies (P, P ⁺, P ^*, and P ^**) leads to some outcome with probability 0.5 and some other outcome with probability 0.5. The table displays the well-being vectors corresponding to the outcomes.

In the top half of the table, for any prioritarian SWF, there is some cutoff value c > 0 (which depends on the transformation function) such that the well-being vector $ \left(50\hbox{--} \unicode{x025B}, 50\hbox{--} \unicode{x025B} \right) $ is preferred by the SWF to (70, 30) and (30, 70) for every ε, $ 0<\unicode{x025B} <c $ . Dominance requires that the module for that SWF rank $ {P}^{+} $ over P. But note that Ex Ante Strong Pareto requires that P be ranked above $ {P}^{+} $ . Dominance requires that the module for the utilitarian SWF rank P over $ {P}^{+} $ , since the well-being vectors (70, 30) and (30, 70) are preferred by the utilitarian SWF to $ \left(50\hbox{--} \unicode{x025B}, 50\hbox{--} \unicode{x025B} \right) $ .

SU ranks P over $ {P}^{+} $ , as does EAP. EPP and EEDEP rank $ {P}^{+} $ over P.

In the bottom half of the table, Dominance requires that the module for any prioritarian SWF rank $ {P}^{\ast \ast } $ over $ {P}^{\ast } $ , since any prioritarian SWF prefers the well-being vector (50, 50) to (70, 30) and (30, 70). However, Ex Ante Pareto Indifference requires that $ {P}^{\ast \ast } $ and $ {P}^{\ast } $ be ranked equally good. Dominance does not here constrain the module for a utilitarian SWF, since that SWF is indifferent between (50, 50) and both (70, 30) and (30, 70).

SU ranks $ {P}^{\ast } $ and $ {P}^{\ast \ast } $ equally good, as does EAP. EPP and EEDEP rank $ {P}^{\ast \ast } $ over $ {P}^{\ast } $ .

By contrast, as Table 3 also illustrates, utilitarians face no such dilemma. SU satisfies both Ex Ante Pareto and Dominance.

In order to grasp the difference between utilitarianism and prioritarianism with respect to Ex Ante Pareto and Dominance, it is critical to understand that what Dominance demands depends upon the SWF. Assume that P has probability 0.5 of yielding well-being vector $ \mathbf{w}=\left(25,81\right) $ and probability 0.5 of yielding well-being vector $ {\mathbf{w}}^{\ast }=\left(81,25\right) $ , while policy $ {P}^{\ast } $ has probability 1 of yielding vector v = (50, 50). Consider the prioritarian SWF using a square-root transformation function. Dominance requires that the uncertainty module for this SWF prefer $ {P}^{\ast } $ to P; according to this SWF, v is better than both $ {\mathbf{w}} $ and $ {\mathbf{w}}^{\ast } $ . By contrast, Dominance requires that the uncertainty module for the utilitarian SWF prefer P to $ {P}^{\ast } $ ; according to that SWF, both w and $ {\mathbf{w}}^{\ast } $ are better than v.Footnote ¹⁹

It is also important to stress that the prioritarian SWF satisfies the Pareto axiom at the level of well-being vectors.Footnote ²⁰ Problems emerge only with uncertainty. Finally, it should be observed that the conflict between Dominance and Ex Ante Pareto transcends prioritarianism. It can be shown that any SWF which satisfies Pigou–Dalton faces this conflict.Footnote ²¹

In my own view, the Dominance axiom is compelling (if one policy is certain to yield a better outcome than a second policy, then surely the first policy is better), as are the Pareto axioms at the level of well-being vectors, while the Ex Ante Pareto axioms are less persuasive. For example, in the kind of case illustrated by the top half of Table 3, Ex Ante Strong Pareto requires a preference for one policy over a second even though the decision-maker can be sure that, if the individuals were fully informed, at least one would prefer the second policy. But how to handle conflicts between Dominance and Ex Ante Pareto is ethically contestable. Some readers may believe that the inevitability of such conflict given a concern for equity (Pigou–Dalton) is a powerful argument for the utilitarian SWF. Others may reject this point of view and endorse prioritarianism, but with the caveat that it should be applied under uncertainty with the EAP module (so as to satisfy the Ex Ante Pareto axiom, at the cost of Dominance).

Let us turn now to Policy Separability. Policy Separability is a tractability axiom. Note that each policy is associated with a list of bundle lotteries, one for each person in the population. Let us say that an individual is “unaffected” if she faces the same lottery over bundles for each policy in P. If Policy Separability holds true of an uncertainty module, knowing how each policy in P maps onto a list of bundle lotteries, one for each affected person, is sufficient information for ranking the policy set. Policies need not be characterized as probability distributions over whole outcomes. Instead, it is sufficient to ascertain how policies endow individuals with bundle lotteries – and not all individuals, just the affected ones. In other words, information about the joint probability distribution of bundles among all N individuals in the population, or among the affected group, is not needed to rank policies. See Table 4, illustrating Policy Separability.

Table 4. Policy Separability

Explanation: Each of the policies (P, $ {P}^{+} $ , P ^*, P ^**, P′, P″) leads to some outcome with probability 0.5 and some other outcome with probability 0.5. The table displays the bundles that each person in the population receives in these outcomes.

The top part of the table illustrates Prong 1 of Policy Separability. Note that the probability distribution of outcomes for policy P is different from that for policy $ {P}^{+} $ . However each individual in the population faces the same bundle lottery with P as he does with $ {P}^{+} $ : Juan a lottery with equal chances of b and $ {b}^{\ast } $ , Mitch a lottery with equal chances of $ {b}^{\prime } $ and $ {b}^{{\prime\prime} } $ . Therefore, each faces the same lottery over well-being levels with P as he does with $ {P}^{+} $ . Prong 1 of Policy Separability thus requires the two policies to be ranked equally good.

The bottom part of the table illustrates Prong 2 of Policy Separability. James is unaffected in the $ {P}^{\ast }/{P}^{\ast \ast } $ comparison: he faces the same bundle lottery with the two policies. Therefore, he faces the same well-being lottery with the two policies. Prong 2 of Policy Separability requires that the $ {P}^{\ast }/{P}^{\ast \ast } $ ranking be invariant to which well-being lottery James faces. Note that $ {P}^{\prime } $ is the same as $ {P}^{\ast } $ , and $ {P}^{{\prime\prime} } $ the same as $ {P}^{\ast \ast } $ , with respect to the bundle lotteries faced by Amaya and by Frank. The only difference is that James faces a different bundle lottery with $ {P}^{\prime } $ and $ {P}^{{\prime\prime} } $ than he does with $ {P}^{\ast } $ and $ {P}^{\ast \ast } $ . Thus, Prong 2 of Policy Separability requires that $ {P}^{\prime }/{P}^{{\prime\prime} } $ ranking be the same as the $ {P}^{\ast }/{P}^{\ast \ast } $ ranking.

The relation between Separability (an axiom regarding the ranking of well-being vectors) and Policy Separability (an uncertainty axiom) is subtle. If an SWF fails Separability, none of its modules will satisfy Policy Separability.Footnote ²² If an SWF satisfies Separability, some of its modules will satisfy Policy Separability, while others will not. This is illustrated by prioritarianism. The prioritarian SWF satisfies Separability; its EPP and EAP modules satisfy Policy Separability; its EEDEP module does not.

EEDEP satisfies Ex Ante Pareto when individuals are identically situatedFootnote ²³ although not in general (see Table 3) or even when some are identically situated and everyone else is unaffected;Footnote ²⁴ it purchases this limited compliance with Ex Ante Pareto at the cost of violating Policy Separability. EPP does not satisfy Ex Ante Pareto even when individuals are identically situated, but satisfies Policy Separability.Footnote ²⁵ To my mind, EPP is on balance more attractive than EEDEP, given the large tractability benefits of Policy Separability, but that conclusion is certainly contestable.