1. Defining Prioritarianism

Let $N$ be the set of the $n$ individuals of concern and let $U \subseteq {{\cal R}^n}$ be the set of all possible allocations of well-being to these individuals. ^{Footnote 4} For any individual $i \in N$ and any allocation $u = \left( {{u_1}, \ldots, {u_n}} \right) \in U$ , the well-being of $i$ in $u$ , denoted ${u_i}$ , is assumed to be fully cardinal and interpersonally comparable. Moreover, let ${\rm{\Delta }}U$ be the set of simple lotteries over $U$ (i.e. the set of all probability distributions $p$ over $U$ for which the support, $supp\left( p \right)$ , is finite). Let ⪰ be the social ordering on ${\rm{\Delta }}U$ , so that p ⪰ $p{\rm{'}}$ denotes that ‘ $p$ is socially at least as good as $p{\rm{'}}$ ’. Similarly, in line with the standard abuse of notation, u ⪰ $\ u{\rm{'}}$ denotes that ‘the degenerate lottery yielding $u$ with certainty is socially at least as good as the degenerate lottery yielding $u{\rm{'}}$ with certainty’. Finally, let $ \succ $ and $\sim $ be the (strict) social betterness and social indifference relations (i.e. the asymmetric and symmetric parts of ⪰).

Given the notation above, the standard definition of Prioritarianism can be formally stated as follows:

Definition 1. (Prioritarianism) A social ordering ⪰ is prioritarian if, for any $u,u' \in U$ , u ⪰ $u'$ if and only if ${V_P}\left( u \right) \ge {V_P}\left( {u'} \right)$ , where

$${V_P}\left( u \right) = \mathop \sum \limits_{i \in N} f\left( {{u_i}} \right),$$

for some strictly increasing and strictly concave priority-weighting function $f$ .

Under Definition 1, the class of prioritarian social orderings can be shown to be equivalent to the class of welfarist (i.e satisfying Strong Pareto and Anonymity) and continuous social orderings that satisfy the following two principles: ^{Footnote 5}

Pigou-Dalton. A transfer of a given amount of well-being from an individual to another individual who is worse off (even after transfer) yields a socially better allocation.

Separability. The comparative goodness of two well-being allocations is invariant to the well-being levels of unaffected individuals.

Pigou-Dalton reflects the idea that Prioritarianism (unlike Utilitarianism) is sensitive to the distribution of well-being, whereas Separability reflects the idea that Prioritarianism (unlike Egalitarianism) evaluates each individual’s well-being independently of other individuals’ well-being.

Three clarifications are in order. First, for the purpose of this paper, I take Prioritarianism to be a view about the ethics of distribution in fixed-population settings. Difficult issues arise when Prioritarianism is applied in population ethics (Brown Reference Brown2007; Nebel Reference Nebel2021; Francis Reference Francis2022). Since these issues fall outside the scope of this paper, I assume throughout that $n$ is fixed. Second, Prioritarianism is not a single social ordering, but a family of such orderings: one for each strictly increasing and strictly concave function $f$ . For simplicity, I sometimes illustrate my arguments in the main text using the example $f\left( \cdot \right) = \sqrt \cdot $ , so that Prioritarianism implies that u ⪰ $u{\rm{'}}$ if and only if $\mathop \sum \nolimits_{i \in N} \sqrt {{u_i}} \ge \mathop \sum \nolimits_{i \in N} \sqrt {{u_{i{\rm{'}}}}} $ . ^{Footnote 6} My arguments can however be generalized to any other strictly increasing and strictly concave priority-weighting function (as I show in the Appendix). Third, I treat Prioritarianism as an axiological view (that is, a view about what is socially good) rather than a deontic view (that is, a view about what is right). Thus, nothing in this paper hinges on whether consequentialism is correct.

2. The Challenge from Risky Non-Antiegalitarianism

Why believe that Prioritarianism violates Risky Non-Antiegalitarianism? Ord cleverly illustrates the idea by presenting a case where Heads and Tails are two equiprobable states of nature, $p$ and $p{\rm{'}}$ are two lotteries, and $i$ and $j$ are two individuals whose well-being levels in each state of nature under each lottery are given as in Table 1 (Ord Reference Ord2015: 301):

Table 1. Ord’s case

Since $p$ makes everyone better off in expectation and is perfectly equal both ex post and ex ante, Risky Non-Antiegalitarianism clearly implies that $p$ is socially better than $p{\rm{'}}$ in Ord’s case. Why would Prioritarianism make the opposite recommendation? Ord (Reference Ord2015) admits that the argument does not apply to the view known as Ex-Ante Prioritarianism, but focuses on a view known as Ex-Post Prioritarianism, which ‘coheres best with standard approaches to decision theory’ (2015: 299). ^{Footnote 7} The phrase ‘standard approaches to decision theory’ here refers to EUT, which in this context means that the social ordering can be represented by the expected value of some social utility function over well-being allocations. If the social ordering satisfies EUT and the social utility function is given by ${V_P}\left( u \right) = \mathop \sum \nolimits_{i \in N} f\left( {{u_i}} \right)$ for all $u \in U$ , ^{Footnote 8} then a view known as Ex-Post Prioritarianism follows:

Definition 2. ( Ex-Post Prioritarianism) A social ordering ⪰ is ex-post prioritarian if, for any $p,p' \in \Delta U$ , p ⪰ $p'$ if and only if ${V_{EPP}}\left( p \right) \ge {V_{EPP}}\left( {p'} \right)$ , where

$${V_{EPP}}\left( p \right) = \mathop \sum \limits_{u \in supp\left( p \right)} p\left( u \right)\left( {\mathop \sum \limits_{i \in N} f\left( {{u_i}} \right)} \right)$$

for some strictly increasing and strictly concave priority-weighting function $f$ .

That Ex-Post Prioritarianism violates Risky Non-Antiegalitarianism can be illustrated using the aforementioned simplifying assumption that the priority-weighting function is the square root. Under this assumption, Ex-Post Prioritarianism clearly implies that $p$ is socially worse than $p{\rm{'}}$ in Ord’s case. ^{Footnote 9} This is because:

$$\underbrace {{1 \over 2}\left[ {\sqrt 4 + \sqrt 4 \left] { + {1 \over 2}} \right[\sqrt {100} + \sqrt {100} } \right]}_{{{V}_{{EPP}}}\left( {p} \right)} \lt \underbrace {{1 \over 2}\left[ {\sqrt {36} + \sqrt {49} \left] { + {1 \over 2}} \right[\sqrt {36} + \sqrt {49} } \right]}_{{{V}_{{EPP}}}\left( {{p'}} \right)}.$$

Conceptually, Ex-Post Prioritarianism violates Risky Non-Antiegalitarianism because the concavity of the priority-weighting function does not only result in sensitivity to the distribution of well-being among people, but also in risk aversion. In other words, Ex-Post Prioritarianism does not only give more priority to those that are worst off in a given outcome: it also gives more priority to those that are in the worst outcomes (cf. Ord Reference Ord2015: 301). Ord’s case is constructed such that $p{\rm{'}}$ is risk-free whereas $p$ is sufficiently risky that the risk aversion of Ex-Post Prioritarianism renders $p{\rm{'}}$ to be socially better than $p$ .

3. A Prioritarian Escape Route

Risky Non-Antiegalitarianism is logically weaker than Weak Ex-Ante Pareto. Is violating Risky Non-Antiegalitarianism also more concerning than violating Weak Ex-Ante Pareto for someone with prioritarian sympathies? A negative answer to this question is given by Adler and Holtug (Reference Adler and Holtug2019: 138), who argue that ‘it is not clear why prioritarians (who place no intrinsic moral weight on inequality) should be more bothered by Ord’s demonstration [that Ex-Post Prioritarianism violates Risky Non-Antiegalitarianism] than by violations of ex ante Strong Pareto alone’.

It is worth distinguishing between at least two interpretations of Adler and Holtug’s reply. On the first interpretation, it invokes a principle saying something like, ‘if a theory places no intrinsic moral weight on feature $F$ , then advocates of the theory need not be bothered by counterexamples that rely on $F$ ’. Such a principle rules out too many counterexamples. For example, a critique against Utilitarianism is that it favours the perfectly unequal well-being distribution $\left( {2 + \varepsilon, 0} \right)$ over the perfectly equal well-being distribution $\left( {1,1} \right)$ . It would be unsatisfactory for the utilitarian to respond to this critique by simply pointing out that ‘it is not clear why utilitarians (who place no intrinsic weight on the distribution of well-being) should be bothered by this’. ^{Footnote 10}

On the other interpretation of Adler and Holtug’s reply, their complaint is rather that Risky Non-Antiegalitarianism somehow begs the question against Prioritarianism. However, this reply cannot be right. Adler and Holtug defines Prioritarianism in accordance with Definition 1 and argues for it on the basis of Strong Pareto, Anonymity, Separability, Continuity and Pigou-Dalton. I show below that Risky Non-Antiegalitarianism is compatible with Prioritarianism in this sense and therefore also with each of these axioms. Thus, Risky Non-Antiegalitarianism clearly does not beg the question against Prioritarianism under Definition 1.

Moreover, it seems reasonable to assume that many prioritarians would be more bothered by violating Risky Non-Antiegalitarianism than by violating Weak Ex-Ante Pareto. As Ord writes:

At least for prioritarians that fit Ord’s description, it does seem like violating Risky Non-Antiegalitarianism would be more concerning than violating Weak Ex-Ante Pareto. Fortunately for such prioritarians, I argue that there is a way for prioritarians inclined towards EUT to satisfy Risky Non-Antiegalitarianism.

The key observation underlying the argument is that Ex-Post Prioritarianism is not the only ordering of lotteries that both satisfies EUT and ranks degenerate lotteries (i.e. lotteries yielding one given well-being allocation with certainty) according to the prioritarian outcome-ranking formula ${V_P}\left( \cdot \right) = \mathop \sum \nolimits_{i \in N} f\left( \cdot \right)$ . This observation is often overlooked in the literature. ^{Footnote 11} I suspect that this oversight is caused by a failure to recognize that EUT only implies that the social ordering of lotteries can be represented by the expectation of some social utility function, and that the social utility function only needs to be a strictly increasing transformation of ${V_P}\left( \cdot \right)$ in order for the social ordering to cohere with the prioritarian outcome-ranking. ^{Footnote 12}

A social ordering is thus both prioritarian and satisfies EUT if and only if there exists some strictly increasing function $g$ such that the ordering can be represented by the expectation of $g\left( {{V_P}\left( \cdot \right)} \right)$ . Ex-Post Prioritarianism corresponds to the special case where $g$ is set as the identity function (i.e. $g = 1$ ). Another special case, known as Expected Equally Distributed Equivalent (EEDE) Prioritarianism, is obtained when $g$ is set as the inverse of the priority-weighting function $f$ (i.e. $g = {f^{ - 1}}$ ): ^{Footnote 13}

Definition 3. (EEDE Prioritarianism). A social ordering ⪰ is EEDE prioritarian if, for any $p,p' \in \Delta U$ , p ⪰ $p'$ if and only if ${V_{EPP}}\left( p \right) \ge {V_{EPP}}\left( {p'} \right)$ , where

$${V_{EPP}}\left( p \right) = \mathop \sum \limits_{u \in supp\left( p \right)} p\left( u \right){f^{ - 1}}\left( {{1 \over n}\mathop \sum \limits_{i \in N} f\left( {{u_i}} \right)} \right)$$

for some strictly increasing and strictly concave priority-weighting function $f$ (with ${f^{ - 1}}$ being its inverse function). ^{Footnote 14}

EEDE Prioritarianism ranks degenerate lotteries in the same way as Ex-Post Prioritarianism, but has substantially different implications for the ranking of non-degenerate lotteries. For the purpose of this paper, the important point is that EEDE Prioritarianism avoids Ord’s counterexample (cf. Fleurbaey Reference Fleurbaey2010): ^{Footnote 15}

Proposition 1. EEDE Prioritarianism satisfies Risky Non-Antiegalitarianism.

The formal proof of Proposition 1 is laid out in the Appendix. Here I instead illustrate the key intuition for the proposition using Ord’s case and our previous assumption that $f\left( \cdot \right) = \sqrt \cdot $ (so that ${f^{ - 1}}\left( \cdot \right) = {( \cdot )^2}$ ). In line with Risky Non-Antiegalitarianism, EEDE Prioritarianism implies that $p$ is socially better than $p{\rm{'}}$ in Ord’s case. This is because:

$$\underbrace {{1 \over 2}{{\left[ {{{\sqrt 4 + \sqrt 4 } \over 2}} \right]}^2} + {1 \over 2}{{\left[ {{{\sqrt {100} + \sqrt {100} } \over 2}} \right]}^2}}_{{{{V}}_{{{EEDE}}}}\left( {p} \right)} \gt \underbrace {{1 \over 2}{{\left[ {{{\sqrt {36} + \sqrt {49} } \over 2}} \right]}^2} + {1 \over 2}{{\left[ {{{\sqrt {36} + \sqrt {49} } \over 2}} \right]}^2}}_{{{V}}_{{{EEDE}}}\left( {{p'}} \right)}. $$

This example illustrates that EEDE Prioritarianism does not violate Risky Non-Antiegalitarianism in Ord’s case when $f\left( \cdot \right) = \sqrt \cdot $ . However, as Proposition 1 states (and as I prove in the Appendix), EEDE Prioritarianism satisfies Risky Non-Antiegalitarianism more generally. Conceptually, the reason for this is that applying the inverse of the priority-weighting function to the mean of priority-weighted well-being preserves the sensitivity to the distribution of well-being associated with Ex-Post Prioritarianism without retaining its risk aversion. In Ord’s case, this means that EEDE Prioritarianism is neutral about the risk associated with $p$ , but still averse to the inequality in $p{\rm{'}}$ . In line with Risky Non-Antiegalitarianism, EEDE Prioritarianism therefore makes the plausible verdict that $p$ is socially better than $p{\rm{'}}$ .

To summarize, EEDE Prioritarianism satisfies both EUT and Risky Non-Antiegalitarianism. Insofar as EEDE Prioritarianism counts as a form of Prioritarianism (more on this below), this shows that Risky Non-Antiegalitarianism is not convincing by itself as an objection against Prioritarianism, but should rather be viewed as a reason for prioritarians with an inclination towards EUT to adopt EEDE Prioritarianism instead of Ex-Post Prioritarianism.

4. Sharpening the Challenge to Prioritarianism

The argument of this paper has so far been that, because EEDE Prioritarianism satisfies both EUT and Risky Non-Antiegalitarianism, prioritarians can hold on to both of these principles. This crucially relies on the taxonomic assumption that EEDE Prioritarianism counts as a form of Prioritarianism. Rejecting this assumption requires that one gives up Definition 1.

The appropriateness of adopting any given definition of Prioritarianism depends on the context in which it is invoked. In the context of this paper, there are two strong prima facie reasons for adopting Definition 1. First, Definition 1 is the definition that is employed in the aforementioned papers by Ord (Reference Ord2015) and by Adler and Holtug (Reference Adler and Holtug2019). Second, Definition 1 is the most widely adopted explicit definition of Prioritarianism in the literature more broadly. In light of this, the choice of Definition 1 can hardly be viewed as particularly contentious or question-begging.

It is nonetheless worth commenting on the main alternative definition of Prioritarianism, which has been developed by McCarthy (Reference McCarthy2017). ^{Footnote 16} In order to do this, I first need to state two normative principles introduced by McCarthy (Reference McCarthy2017). The first of these is the Priority Principle: ^{Footnote 17}

Priority Principle. Let ${U^e} = \{ \left( {{u_1}, \ldots, {u_n}} \right) \in {{\cal R}^n}\;|\;\left( {\forall i,j \in N:{u_i} = {u_j}} \right)\} $ be the subset of well-being allocations with perfect equality. For any $u,u',u'' \in {U^e}$ ,

1. (i) $u \succ u'$ iff ${u_i} \gt u{'_i}$ for all $i \in N$ , and

2. (ii) if ${u_i} \gt u{'_i} \gt u'{'_i}$ and $u{'_i} = {1 \over 2}{u_i} + {1 \over 2}u'{'_i}$ for all $i \in N$ , then $u' \succ \left[ {{1 \over 2},u;{1 \over 2},u''} \right]$ .

Part (i) of the Priority Principle simply amounts to the claim that the social betterness relation between any two well-being allocations with perfect equality must always align with the individual betterness relations between these allocations. Part (ii) of the Priority Principle amounts to the much more controversial claim that the social betterness relation between any two lotteries over well-being allocations with perfect equality must sometimes deviate from the individual betterness relations between these lotteries. In particular, part (ii) implies that the social ordering is generally more risk averse than the individual ordering.

The second principle that must be introduced before McCarthy’s (Reference McCarthy2017) definition of Prioritarianism can be stated is the following:

Anteriority. For any lotteries $p,p' \in \Delta U$ , if, for every $i \in N$ , the probability that $i$ obtains any given welfare level is the same under $p$ as under $p'$ , then $p\sim p'$ .

Anteriority states that if each individual faces the same probability of obtaining any given welfare level under one lottery as under another, then these two lotteries are equally socially valuable. This could be viewed as reflecting the prioritarian-sounding idea that the social ordering should evaluate lotteries solely on the basis of each individual’s prospect of well-being taken separately (cf. Broome Reference Broome2015).

Having introduced the Priority Principle and Anteriority, it is now possible to present McCarthy’s (Reference McCarthy2017) alternative definition of Prioritarianism:

Definition 1* (McCarthy’s definition of Prioritarianism) An anonymous social ordering ⪰ that conforms to EUT is prioritarian if it satisfies the Priority Principle and Anteriority. ^{Footnote 18}

EEDE Prioritarianism does not count as a version of Prioritarianism under Definition 1*. In fact, EEDE Prioritarianism violates both the Priority Principle and Anteriority (as is implied by Proposition 2 and 3 below). More broadly, the Priority Principle and Anteriority can be used to sharpen the challenge that Risky Non-Antiegalitarianism poses to Prioritarianism. This is most straightforward in the case of the Priority Principle, as the following proposition shows (proof in Appendix):

Proposition 2. Let ⪰ be any social ordering that satisfies Continuity. If ⪰ satisfies the Priority Principle, then ⪰ must violate Risky Non-Antiegalitarianism.

Continuity requires that if a well-being allocation $u$ is socially better than a well-being allocation $u{\rm{'}}$ , then any well-being profile that is sufficiently close (in the Euclidean sense) to $u$ must also be socially better than $u{\rm{'}}$ , and any well-being allocation that is sufficiently close to $u{\rm{'}}$ must be socially worse than $u$ . Proposition 2 states that, under Continuity, the Priority Principle rules out Risky Non-Antiegalitarianism. Avoiding this conclusion by rejecting Continuity would amount to admitting that the strength of the prioritarian concern formalized by the Priority Principle only matters infinitesimally. ^{Footnote 19} Thus, in practise, Proposition 2 implies that social orderings that are prioritarian under Definition 1* must violate Risky Non-Antiegalitarianism.

A similar result can be stated in terms of Anteriority, though it requires somewhat stronger assumptions than Continuity (proof in Appendix):

Proposition 3. Let ⪰ be any social ordering that satisfies EUT (or just Stochastic Equivalence ^{Footnote 20} ), Continuity, and is prioritarian as defined by Definition 1. If ⪰ satisfies Anteriority, then ⪰ must violate Risky Non-Antiegalitarianism.

Proposition 3 thus shows that, under assumptions that prioritarians tend to accept, Anteriority rules out Risky Non-Antiegalitarianism.

The upshot of the discussion so far concerning the Priority Principle and Anteriority is threefold. First, EEDE does not count as a version of Prioritarianism under Definition 1*. Second, Definition 1* implies that prioritarians must violate Risky Non-Antiegalitarianism (given Continuity). Third, even under Definition 1, prioritarians that accept EUT (or just Stochastic Equivalence) must choose between rejecting the Priority Principle and Anteriority, or rejecting Risky Non-Antiegalitarianism.

In the next section, I provide some reasons against the view that the Priority Principle and Anteriority should be considered core principles of Prioritarianism.

5. Assessing the Priority Principle and Anteriority

How should we determine whether a condition is a core principle of Prioritarianism? McCarthy points out that ‘it was [Derek] Parfit who introduced the term “the priority view” (as well as its synonym “prioritarianism”), so any definition of it has to largely defer to what he said about it’ (2017: 219). In discussing whether the Priority Principle and Anteriority are indeed core principles of Prioritarianism, I will at least partly adopt this methodology. I start by discussing the Priority Principle and turn to Anteriority thereafter. Readers with no interest in the semantic debate regarding the definition of Prioritarianism may skip this section as it is mainly focused on taxonomy.

The Priority Principle plays a similar role for Definition 1* as Pigou-Dalton does for the standard definition of Prioritarianism: it is meant to distinguish Prioritarianism from Utilitarianism. However, while Pigou-Dalton is directly about the distribution of well-being across people, the Priority Principle is instead about the distribution of well-being across outcomes under risk. Given that Parfit’s (Reference Parfit1991) seminal paper that introduced Prioritarianism is focused entirely on the distribution of well-being across people and does not even mention risk, there is a prima facie case for thinking that Pigou-Dalton is a better candidate for a core principle of Prioritarianism compared with the Priority Principle.

McCarthy’s (Reference McCarthy2017) argument for not regarding Pigou-Dalton as a core principle of Prioritarianism is that it is not ‘transparently plausible’ because prioritarians have not provided a ‘convincing backstory about why there is a sufficiently determinate welfare measure, and an explanation of why intuition about the distribution of welfare units is reliable’ (2017: 222). However, this strikes me as an argument against accepting Pigou-Dalton, but not necessarily as an argument against viewing Pigou-Dalton as a core principle of Prioritarianism. As Adler and Holtug (Reference Adler and Holtug2019: 112) point out, ‘[t]he substantive plausibility of Pigou–Dalton should not be conflated with the prior, taxonomic, question about what distinguishes utilitarianism from prioritarianism’.

The Priority Principle also raises conceptual issues of its own. In particular, as McCarthy (Reference McCarthy2017) himself has pointed out, the deviation between social and individual orderings implied by the Priority Principle raises difficult questions about how these orderings should be interpreted. It is therefore not clear that the Priority Principle is more ‘transparently plausible’ than Pigou-Dalton.

Let us now turn to Anteriority. Anteriority plays a similar role in Definition 1* as Separability plays in the standard definition of Prioritarianism: it is meant to distinguish Prioritarianism from Egalitarianism. In particular, McCarthy (Reference McCarthy2017) takes Anteriority to be implied by Parfit’s remark that ‘[a]ccording to the priority view, and in contrast with egalitarianism, it does not matter how well off people are relative to one another’ (Parfit Reference Parfit1991: 22–23). To illustrate this point, McCarthy (Reference McCarthy2017) invokes the case described in Table 2, which was first introduced by Myerson (Reference Myerson1981). ^{Footnote 21}

Table 2. Myerson’s case

In Myerson’s case, each of the two individuals have a 50% probability of obtaining a well-being level of 0 and a 50% probability of obtaining a well-being level of 1 – and this holds true under both $q$ and under $q{\rm{'}}$ . However, $q$ is guaranteed to result in a perfectly equal outcome whereas $q{\rm{'}}$ is guaranteed to result in an unequal outcome. McCarthy (Reference McCarthy2017) therefore argues that an egalitarian should prefer $q$ over $q{\rm{'}}$ , whereas a prioritarian should be indifferent between $q$ and $q{\rm{'}}$ . Anteriority is meant to capture this prioritarian judgement.

McCarthy’s argument for regarding Anteriority as a core principle of Prioritarianism has some force. However, as in the discussion of the Priority Principle above, the fact that risk does not feature prominently in Parfit’s (Reference Parfit1991) seminal paper (nor in a lot of other literature on Prioritarianism) weighs against viewing Anteriority as a core principle of Prioritarianism. Moreover, $q$ and $q{\rm{'}}$ do not only differ in terms of inequality; they also differ in the sense that $q$ (but not $q{\rm{'}}$ ) implies that there is a chance that no individual ends up in the bad outcome. As Fleurbaey (Reference Fleurbaey2015) has pointed out, ‘this difference may matter even without introducing the kind of comparisons between individuals that prioritarians want to avoid’. Thus, even if one accepts that prioritarians should not be concerned with inequality, it is not clear that prioritarians must be indifferent between $q$ and $q{\rm{'}}$ . ^{Footnote 22}

The assessment above suggests that there is a case for not regarding the Priority Principle and Anteriority as core principles of Prioritarianism. ^{Footnote 23} Insofar as this assessment is correct, prioritarians need not violate Risky Non-Antiegalitarianism. Instead, as I have shown, they can adopt EEDE Prioritarianism.

It is worth noting, however, that inasmuch as one does take the Priority Principle and Anteriority to be part of the definition of Prioritarianism (as Definition 1* purports), this may offer a better justification for biting the bullet with regards to violating Risky Non-Antiegalitarianism. In section 3, I criticized Adler and Holtug’s (Reference Adler and Holtug2019) bullet-biting reply to Ord (Reference Ord2015), partly because it does not show that Risky Non-Antiegalitarianism begs the question against the core principles that Adler and Holtug’s (Reference Adler and Holtug2019) take to define Prioritarianism (i.e. Pigou-Dalton, Separability, etc.). However, this critique relies on the fact that Adler and Holtug (Reference Adler and Holtug2019) explicitly endorse and defend Definition 1. In contrast, as Proposition 2 and 3 show, there is a very close link between accepting Prioritarianism under Definition 1* and rejecting Risky Non-Antiegalitarianism. Given this, a bullet-biting reply to the challenge from Risky Non-Antiegalitarianism seems more plausible under Definition 1* than under Definition 1.