Thresholds, critical levels, and generalized sufficientarian principles

This paper provides an axiomatic analysis of sufficientarian social evaluation. Sufficientarianism has emerged as an increasingly important notion of distributive justice. We propose a class of principles that we label generalized critical-level sufficientarian orderings. The distinguishing feature of our new class is that its members exhibit constant critical levels of well-being that are allowed to differ from the threshold of sufficiency. Our basic axiom assigns priority to those below the threshold, a property that is shared by numerous other sufficientarian approaches. When combined with the well-known strong Pareto principle and the assumption that there be a constant critical level, the axiom implies that the critical level cannot be below the threshold. The main results of the paper are characterizations of our new class and an important subclass. As a final observation, we identify the generalized critical-level sufficientarian orderings that permit us to avoid the repugnant conclusion and the sadistic conclusion, which are known as two fundamental challenges in population ethics.


Introduction
This paper provides an axiomatic foundation of a class of social-evaluation orderings that are based on sufficientarian principles. These orderings generalize the criticallevel sufficientarian orderings that we characterized in an earlier contribution (Bossert et al. 2021a) and that are inspired by Blackorby and Donaldson's (1984) critical-level generalized-utilitarian population principles. The notion of sufficiency is based on the view that individuals should be at or above a given threshold of utility, which we interpret as an indicator of lifetime well-being. The seminal contribution in this area is that of Frankfurt (1987) who advocates the maximization of the number of individuals whose utilities are above the threshold. Frankfurt's proposal is closely related to the head-count ratio, a well-know measure of poverty. However, Frankfurt's sufficientarian account exhibits several fundamental shortcomings. The first difficulty is that his proposed ranking of distributions of utility is not consistent with the Pareto principle. Moreover, his criterion is not distributionally sensitive to an adequate degree; this latter observation does not come as surprise because it shares this property with the head-count ratio that it resembles. Indeed, Frankfurt's ranking may recommend a very unequal distribution when the resources available are insufficient for allowing everyone to be above the threshold. Moreover, Frankfurt's account of sufficiency is in conflict with the well-established Pigou-Dalton transfer principle (see Pigou 1912;Dalton 1920). Crisp (2003) recommends an interesting refinement of Frankfurt's (1987) proposal. In addition to giving priority to those below the threshold, his principle pays due attention to distributional concerns below the threshold but assumes a neutral position towards redistributions that take place above the threshold; see Crisp (2003, p. 758) for details. Crisp's sufficientarian account is extended by Brown (2005), Huseby (2010Huseby ( , 2012, and Hirose (2016). As Crisp's proposal makes clear, sufficientarianism is related to but distinct from poverty measurement. A sufficientarian approach to social evaluation focuses primarily on those whose well-being is below the threshold level that represents sufficiency but it can be supplemented by additional criteria so that it is consistent with the Pareto principle.
To assess population consequences, an essential criterion is that of a critical level of lifetime well-being, which is familiar from the literature on population ethics. If a person at this level is added to a utility-unaffected population, the expanded society is as good as the original. Phrased in the context of sufficientarian social evaluation, a crucial question arises immediately: what is an ethically appealing relationship between the sufficiency threshold and this critical level? Although all of the existing sufficientarian theories assume (at least implicitly) that the two coincide, there is no a priori reason why such fundamentally different values should be the same. Indeed, allowing them to be distinct gives us considerably more flexibility when it comes to the choice of policies that may affect population size. This possibility of differing values for a critical level and threshold is a central feature of our class.
As a first result, we show that the critical level must be greater than or equal to the threshold of sufficiency. This is the case because if a constant critical level exists and is located below the threshold, strong Pareto is violated. Since the threshold for sufficiency is assumed to be associated with the fundamental needs of a human being (Braybrooke 1987;Wiggins 1998), it is very natural to consider a critical utility level that is higher than the threshold.
The main contribution of the paper is a characterization of our new class of generalized critical-level sufficientarian orderings. These criteria are compatible with the properties advocated by Crisp (2003) and, in addition, they approximate Frankfurt's (1987) in the limit. An additional result narrows down the class to a subclass that no longer contains Frankfurt's (1987) proposal as a limiting case. This is accomplished by adding a principle of transfers across the threshold. Each member of the resulting subclass has the attractive feature of respecting a transfer principle that is on as sound an ethical footing as the variants that are restricted to transfers below or above the threshold.
A notable feature of our orderings is that some of them can avoid two well-known undesirable attributes that have been studied in population ethics by authors such as Parfit (1976Parfit ( , 1982Parfit ( , 1984, Blackorby and Donaldson (1984), Arrhenius (2000), Blackorby et al. (2004Blackorby et al. ( , 2005, and Spears and Budolfson (2021). If the critical level is higher than the utility level that represents a neutral life, Parfit's repugnant conclusion can be avoided. Moreover, if the threshold for sufficiency is equal to neutrality, then Arrhenius's sadistic conclusion does not materialize. It is worth emphasizing that both of these unattractive conclusions can be avoided in our setting. This is in stark contrast to the standard formulation of critical-level generalized utilitarianism in population ethics-it does not leave any room to avoid both. The price to be paid for the possibility result is that Parfit's (1984) mere addition principle cannot be satisfied by any of our principles if the repugnant conclusion is to be avoided. Prior observations suggest that principles that circumvent the traditional population-ethics dilemmas have some normatively undesirable features. We argue that this view can be successfully challenged by providing an escape route in the form of a defensible approach that evades both the repugnant conclusion and the sadistic conclusion. A more detailed discussion follows in Sect. 5 once the requisite definitions have been introduced.
Section 2 introduces the setting and basic definitions. Our axioms are defined in Sect. 3, and Sect. 4 presents our central theorems and their proofs. Section 5 examines some population-ethics properties of our orderings and compares them with other well-known classes of orderings. Section 6 concludes. The appendix establishes the independence of the axioms used in our main result.

Setting
A utility distribution for n ∈ N individuals is given by an n-dimensional vector u = (u 1 , . . . , u n ) ∈ R n . Each u i represents the level of lifetime well-being of individual i. The set of all possible distributions is given by = n∈N R n . We use 1 r to denote the vector that consists of r ∈ N ones. For u = (u 1 , . . . , u n ) ∈ R n , let u1 r denote the r -fold replica of u.
A neutral life is a life that, from the viewpoint of the person leading it, is as good as a life without any experiences. We follow the standard convention in population ethics and normalize the level of utility that corresponds to neutrality to zero; see Blackorby and Donaldson (1984, p. 14) or Blackorby et al. (2005, p. 25), for example. Thus, a life is worth living if lifetime utility is positive-above neutrality.
We employ a notion of sufficientarianism that can be captured by means of an ordering (that is, a reflexive, complete, and transitive binary relation) R defined on the set of distributions . For notational convenience, we write u Rv instead of (u, v) ∈ R with the interpretation that distribution u is at least as good as distribution v from a sufficientarian perspective. The asymmetric and symmetric parts corresponding to R are denoted by P and I .
The fundamental ingredient of a sufficientarian analysis is an exogenously given threshold level of utility, denoted by θ ∈ R. Its interpretation is that individuals who experience at least this level of well-being have enough, whereas those whose utility is below θ do not. For all n ∈ N and for all u ∈ R n , we define These are the sets of those whose lifetime utility is lower than, equal to, and higher than the threshold level θ . Let R n L = (−∞, θ) n , R n L E = (−∞, θ] n , and R n H E = [θ, ∞) n , and define L = ∪ n∈N R n L , L E = ∪ n∈N R n L E , and H E = ∪ n∈N R n H E . For notational simplicity, we write R L , R L E , and R H E instead of R 1 L , R 1 L E , and R 1 H E . Taking the threshold θ of sufficiency of well-being as exogenously given, an ordering R on is intended to evaluate utility distributions from a sufficientarian perspective. In particular, this means that the ordering compares utility distributions with priority being given to those below the threshold.
Let G be the set of all functions g : R → R that are increasing and strictly concave on R L , and continuous, increasing, and concave on R H E . As is well-known, because R L is an open interval, the (strict) concavity of g on R L implies that g is continuous on R L . An analogous observation does not apply to the half-open interval R H E -it is possible that an increasing and concave function is not continuous at the boundary point θ , which is why the requisite continuity property does not follow. The class of orderings that we focus on in this paper is defined as follows.
Definition 1 An ordering R on is generalized critical-level sufficientarian if there exist g ∈ G and α ∈ R H E with sup a∈R L g(a) ≤ g(α) such that, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u Rv if and only if i∈L n (u) The value α ∈ R H E is a critical level for this ordering. That is, it holds that, for all u ∈ , u I (u, α). Therefore, generalized critical-level sufficientarianism evaluates the value of a life as the shortfall of the transformed lifetime utility g(u i ) from the transformed critical level g(α) for a person i below the threshold, and as the difference between the transformed lifetime utility g(u j ) and the transformed critical level g(α) for a person j at or above the threshold. The principle ranks utility distributions by applying a lexical criterion to the resulting sums of values. First, the sums of the values of people below the threshold are compared. If this results in a strict ranking, this ranking is adopted as the sufficientarian ranking of the two distributions in question. If the requisite sums are equal, the tie is broken by comparing the sums that correspond to those at or above the threshold. Thus, the tie-breaking step of evaluation by a generalized critical-level sufficientarian ordering consists of the application of the critical-level generalized-utilitarian ordering associated with a critical level α to the lifetime utilities of those at or above the threshold. Note that the value of a life of a person at or above the threshold for sufficiency θ , g(u j )−g(α), is negative if α ∈ R H E is strictly higher than the threshold and u j is less than α because g is increasing on [θ, ∞), while the value of a life of a person below the threshold, g(u i ) − g(α), is always negative since g is increasing on (−∞, θ) and g(u i ) < sup a∈R L g(a) ≤ g(α).
We note that g is allowed to be discontinuous at θ because the limit of g as we approach θ from below does not have to be equal to the value of g at θ . For that reason, the function g need not be increasing on its entire domain R-it is possible that g(u j ) > g(θ ) for some values of u j below the threshold; see Fig. 1. Consequently, as Fig. 1 shows, the value of a life of a person at or above the threshold, g(u j ) − g(α), could be less than that of a life of a person below the threshold, g(u i ) − g(α), i.e., g(u j ) − g(α) < g(u i ) − g(α) < 0. However, this does not mean that generalized critical-level sufficientarianism evaluates a life of a person who has enough as worse than that of a person who does not have enough because the values of people are aggregated separately among people below the threshold and among people at or above the threshold. Figure 2 presents an example of a permissible function g that is continuous at θ and, thus, continuous on its entire domain. As the figure shows, the continuity of g on its entire domain does not imply the concavity of g on its entire domain. If θ = α and g is continuous on its entire domain (that is, if sup a∈R L g(a) = g(θ ) = g(α)), R reduces to a critical-level sufficientarian ordering-the case examined in detail by Bossert et al. (2021a); see Fig. 3. Note that the coincidence of θ and α does not imply the continuity of g ∈ G on its entire domain; see Fig. 4.
Clearly, whether a function g ∈ G is permissible depends on the value of the critical level of utility α ∈ R H E . Specifically, given a critical level α, every function g ∈ G that is continuous at θ is permissible, whereas if g ∈ G is discontinuous at θ , it must satisfy sup a∈R L g(a) ≤ g (α). Suppose that θ ≥ 0, and let a ∈ R ++ , b, c ∈ R with c ≤ a, and α ∈ R H E . The parameterized function g defined by provides a specific example of those functions illustrated in Figs. 1, 3, and 4. Note that

Axioms
Our first axiom is a core principle of sufficientarianism, which states that if (a) the distribution that corresponds to the subgroup of those who are below the threshold is better for one distribution than another, then the former distribution is better than the latter overall; and (b) if there is equal goodness below the threshold, then the relative ranking of the two distributions overall is determined by the relative aggregate values assigned to those at and above the threshold.
Priority below the threshold For all n, m, , r ∈ N, for all u ∈ R n L , for all v ∈ R m L , for all w ∈ R H E , and for all s There is a subtle difference between this axiom and the stronger axiom of absolute priority that we employ in Bossert et al. (2021a). Absolute priority extends the range of possible values of u and v to R n L E and R m L E rather than restricting attention to R n L and R m L , respectively. This difference is associated with the definition of the disadvantaged; in particular, it needs to be determined whether someone at the threshold is considered to be among those who are in need of special attention. This is parallel to the question of whether the weak or the strong definition of the poor is to be employed in the measurement of poverty. As shown by Donaldson and Weymark (1986), if people located on the poverty line are treated as poor, it is impossible to construct a poverty measure that satisfies some normatively desirable axioms. From this perspective, the weak definition of the poor is a more acceptable definition, and our definition of the disadvantaged conforms to this observation.
The following anonymity requirement states that, for any distribution, all of its permuted distributions are as good as the original. Thus, the axiom represents a fundamental equal-treatment property in a welfarist framework.
Anonymity For all n ∈ N and for all u, v ∈ R n , if v is obtained by applying a permutation to the components of u, then v I u.
Another axiom that needs little explanation is the strong Pareto principle. According to this property, a change is normatively desirable if it makes all individuals weakly better off and at least one individual strictly better off.
Strong Pareto For all n ∈ N and for all u, v ∈ R n , if u i ≥ v i for all i ∈ {1, . . . , n} with at least one strict inequality, then u Pv.
The following condition is a strong independence axiom familiar from the literature on population ethics; see, for example, Blackorby, Bossert, and Donaldson (2005, p. 159). For all u, v, w ∈ , u Rv ⇔ (u, w)R(v, w).

Existence independence
We note that our orderings do not necessarily satisfy continuity on the entire domain. If we consider the subdomain of the utilities of those below the threshold, the continuity requirement can be applied.
Continuity below the threshold For all n ∈ N and for all u ∈ R n L , the sets are open in R n L . The following counterpart of this property requires that the continuity requirement applies to the subdomain of the utilities of those who have enough.
Continuity above and at the threshold For all n ∈ N and for all u ∈ R n H E , the sets are closed in R n H E . Some additional properties are required for our purposes. The first of these is very weak-it merely requires that a critical level exists for at least one distribution. The axiom appears in Blackorby et al. (2005, p. 160).
Weak existence of critical levels There exist u ∈ and α ∈ R such that u I (u, α).
The next axiom is a strengthening of this property, which first appeared in Blackorby and Donaldson (1984). It requires the existence of a utility value that can be applied to any distribution as a critical level.

Existence of constant critical levels
There exists α ∈ R such that, for all u ∈ , u I (u, α).
Our next axiom is new; it is one of the core axioms of our characterization of the class of generalized critical-level sufficientarian orderings. Consider any two population sizes n and m and two distributions u ∈ R n L and v ∈ R m L . The condition requires the existence of two utility levels λ and μ below the threshold such that if multiple individuals with utility λ are added to any u ∈ R n L and multiple individuals with utility μ are added to any v ∈ R m L , the relative ranking of the augmented distributions is the same as the relative ranking of u and v. The multiples are chosen so that the population sizes in the two augmented distributions are the same.
Expansion independence below the threshold For all n, m ∈ N, there exist k ∈ N with k > max{n, m} and λ, μ ∈ R L such that, for all u ∈ R n L and for all v ∈ R m L , Strictly speaking, the values k, λ, and μ may depend on n and m and, therefore, the use of the symbols k n,m , λ n,m , and μ n,m would be more accurate. For simplicity of presentation, however, we suppress this dependence whenever possible without ambiguity. Expansion independence below the threshold has a similar normative implication to existence of constant critical levels. In the presence of transitivity, existence of constant critical levels implies that, for all n, m ∈ N and for all k ∈ N with k > max{n, m}, there exists α ∈ R such that, for all u ∈ R n and for all v ∈ R m , Critical levels can be used to adjust population sizes without changing the relative ranking between the two distributions. We note that the axiom of existence of constant critical levels encompasses the entire domain, while expansion independence below the threshold focuses on the subdomain associated with those who are below the threshold. In a generalized critical-level sufficientarian ordering, the critical level α cannot be below the threshold of sufficiency. Thus, existence of constant critical levels cannot be applied to that subdomain and, therefore, an axiom such as expansion independence below the threshold is required. This also means that existence of constant critical levels is not implied even if attention is restricted to utilities below the sufficiency threshold (and even if transitivity is invoked). For that reason, we exercise appropriate caution when referring to similar rather than identical normative implications in the above discussion.
Finally, we introduce three axioms that capture a concern for distributional equity below, above, and across the threshold; see Pigou (1912) and Dalton (1920) for the principle of transfers in the context of income distributions. The first of these requires that a progressive transfer between two individuals below the threshold is socially beneficial. The second applies to transfers that take place above the threshold and merely demands that the post-transfer distribution be at least as good as the pre-transfer utility allocation. The reason why we only impose a weak variant of the axiom is that we want to allow the resulting class of orderings to include Crisp's (2003) proposal.

Weak principle of transfers above and at the threshold
As is well-known, these transfer principles are responsible for the curvature properties of the function g. The principle of transfers below the threshold leads to the strict concavity of g below the threshold, whereas the concavity of g on R H E follows from the weak principle of transfers above and at the threshold.
In a result that identifies an important subclass of the generalized critical-level sufficientarian orderings, we also employ the following principle of transfers across the threshold that applies to situations in which the donor of a transfer drops below the threshold as a consequence of the associated utility loss.

Axiomatic analysis
We begin by showing that a constant critical level must be greater than or equal to the threshold of sufficiency in the presence of priority below the threshold and strong Pareto.
Theorem 1 If an ordering R satisfies priority below the threshold, strong Pareto, and existence of constant critical levels, then α ≥ θ , where α is the constant critical level.
Proof By existence of constant critical levels, there exists α ∈ R such that, for all u ∈ , u I (u, α). By way of contradiction, suppose that α < θ. Let β ∈ (α, θ ), and consider v ∈ L and w ∈ H E . By strong Pareto, we obtain This contradicts strong Pareto. Blackorby et al. (2005, Theorem 6.8) show that if an ordering R satisfies anonymity, strong Pareto, existence independence, and weak existence of critical levels, then there exists a constant critical level α that applies to all utility distributions in . This observation immediately gives us the following corollary.
Corollary 1 If an ordering R satisfies priority below the threshold, anonymity, strong Pareto, existence independence, and weak existence of critical levels, then there exists a constant critical level α ∈ R H E .
A generalized critical-level sufficientarian ordering satisfies strong Pareto because it is associated with a critical level α that is greater than or equal to the threshold θ . As a simple illustration, suppose that we have a two-person situation with utilities Because at least one of these inequalities is strict, (v 1 , v 2 ) is better than (u 1 , u 2 ). Suppose now that θ ≤ v 1 and θ ≤ u 2 so that the first individual is deemed to have enough. For the distribution (v 1 , v 2 ), the aggregate utility of those below the threshold is zero, and the aggregate utility of those at or above the threshold is and u 1 is less than θ . Therefore, (v 1 , v 2 ) is better than (u 1 , u 2 ) according to a generalized critical-level sufficientarian ordering with a critical level that is greater than or equal to the threshold.
Our main result is the following characterization of the class of generalized criticallevel sufficientarian orderings.
Theorem 2 An ordering R satisfies priority below the threshold, anonymity, strong Pareto, existence independence, continuity below the threshold, continuity above and at the threshold, weak existence of critical levels, expansion independence below the threshold, the principle of transfers below the threshold, and the weak principle of transfers above and at the threshold if and only if R is a generalized critical-level sufficientarian ordering.
We establish three auxiliary results before presenting the proof of Theorem 2. The first of these shows that anonymity and existence independence together imply that any replications of two utility distributions must be ranked in the same way as the original distributions. Note that the original utility distributions are allowed to have different population sizes. This property appears in Zoli (2009), who labeled it strong population replication principle; see also Blackorby et al. (2005).

Lemma 1 If an ordering R satisfies anonymity and existence independence, then, for
By anonymity and the transitivity of R, it follows from (1) that Since (2) holds for all ∈ {0, . . . , k − 1}, we obtain, by the transitivity of R, that To show that By way of contradiction, we assume that u Rv does not hold. Since R is complete, we have v Pu. Using the argument employed to show (3), we obtain This is a contradiction. Thus, u Rv must hold.
The following lemma shows that if an ordering R satisfies some of the axioms of the theorem statement, it ranks utility distributions at or above the threshold by applying critical-level generalized utilitarianism.

Lemma 2
If an ordering R satisfies anonymity, strong Pareto, existence independence, continuity above and at the threshold, weak existence of critical levels, and the weak principle of transfers above and at the threshold, then there exist a continuous, increasing, and concave function g H E : R H E → R and α ∈ R H E such that, for all n, m ∈ N, for all u ∈ R n H E , and for all v ∈ R m H E , By the weak principle of transfers above and at the threshold, g H E is concave.
Our final auxiliary result establishes an observation that is parallel to that of the previous lemma. It states that the ordering R ranks utility distributions below the threshold by applying a variant of critical-level generalized utilitarianism.

Lemma 3
If an ordering R satisfies anonymity, strong Pareto, existence independence, continuity below the threshold, expansion independence below the threshold, and the principle of transfers below the threshold, then there exist an increasing and strictly concave function g L : R L → R and δ ∈ R such that, for all n, m ∈ N, for all u ∈ R n L , and for all v ∈ R m L , Proof The proof proceeds in five steps.
Step 1. We show that there exists an increasing and strictly concave (and, thus, continuous) function g L : R L → R such that, for all n ∈ N and for all u, v ∈ R n L , For all n ∈ N, let R n L be the restriction of R to R n L . Since R satisfies anonymity, strong Pareto, existence independence, and continuity below the threshold, R n L inherits these properties on R n L . Thus, for all n ≥ 3, there exists a continuous and increasing function g n L : see Blackorby et al. (2005, Theorem 4.7). Since g n L is unique up to increasing affine transformations and R satisfies existence independence, we may choose g L = g n L for all n ≥ 3 (see Blackorby et al. 2005, Chapter 6). Thus, for all n ≥ 3 and for all By existence independence, this result extends to population sizes 1 and 2. Since R satisfies the principle of transfers below the threshold, g L must be strictly concave.
Step 2. We show that, for all n, m ∈ N with n = m, there exists a unique δ n,m ∈ R such that, for all u ∈ R n L and for all v ∈ R m L , where g L : R L → R is an increasing and strictly concave function that satisfies (6). Let n, m ∈ N with n = m. Since R satisfies expansion independence below the threshold, there exist k n,m ∈ N with k n,m > max{n, m} and λ n,m , μ n,m ∈ R L such that, for all u ∈ R n L and for all v ∈ R m L , Thus, it follows from Step 1 that, for all n, m ∈ N, for all u ∈ R n L , and for all v ∈ R m L , Given n, m ∈ N with n = m, we define δ n,m ∈ R by By definition, δ n,m satisfies (7) for all u ∈ R n L and for all v ∈ R m L . To prove the uniqueness of δ n,m , we show that, for all n, m ∈ N with n = m, . Let n, m ∈ N with n = m. Without loss of generality, suppose that m > n. Using δ n,m given by (8), we define n,m by n,m = (m − n)δ n,m .
Since δ n,m satisfies (7) for all u ∈ R n L and for all v ∈ R m L , it follows that for all a, b ∈ R L . We distinguish two cases.
First, suppose that n,m > 0. Since g L is increasing and strictly concave on R L , g L is not bounded below. Thus, there exists b ∈ R L such that Since g L is not bounded below, there exists d ∈ (−∞, a) ⊂ R L such that By the continuity of g L , there exists b ∈ (d, a) ⊂ R L such that Therefore, we obtain By the continuity of g L , there exists a ∈ (d, b) ⊂ R L such that To complete Step 2, let n, m ∈ N with n = m and suppose, by way of contradiction, that there exists δ ∈ R with δ = δ n,m that satisfies (7) for all u ∈ R n L and for all v ∈ R m L . Without loss of generality, suppose that δ > δ n,m and m > n . Let (a, . . . , a) that (a, . . . , a)I (b, . . . , b). Since δ n,m satisfies (7) for all u ∈ R n L and for all v ∈ R m L , we obtain Since (m − n)δ n,m < (m − n)δ and δ satisfies (7) for all u ∈ R n L and for all v ∈ R m L , we have b1 m Pa1 n . This is a contradiction. Thus, only δ n,m satisfies (7) for all u ∈ R n L and for all v ∈ R m L .
Step 3. We show that there exists a unique δ ∈ R such that, for all n ∈ N, for all u ∈ R n L , and for all v ∈ R n+1 L , where g L : R L → R is an increasing and strictly concave function that satisfies (6). That is, δ satisfies (5) in the lemma statement for any n ∈ N and m = n + 1. For simplicity, we denote δ n,n+1 by δ n for all n ∈ N. We show that δ n = δ n+1 for all n ∈ N. Let n ∈ N, u ∈ R n L , v ∈ R n+1 L , and a ∈ R L . By existence independence, we obtain Thus, for each n ∈ N, δ n+1 satisfies (9). From Step 2, it follows that δ n = δ n+1 for all n ∈ N. Thus, δ = δ n satisfies (9).
Step 4. We show that, for all n, m, ∈ N with |m − n| = 1, for all u ∈ R n L , and for all v ∈ R m L , δ satisfies (5). Let n, m, ∈ N and suppose, without loss of generality, that m = n + 1. From Lemma 1 and Step 2, it follows that, for all u ∈ R n L and for all v ∈ R n+1 L , Thus, it follows from Step 3 that, for all n, ∈ N, Since δ n ,(n+1) satisfies (5) for all u ∈ R n L and for all v ∈ R m L , Step 4 is complete.
Step 5. We complete the proof. Let n, m ∈ N, u ∈ R n L , and v ∈ R m L . Without loss of generality, suppose that m > n. Let = m − n. Define h ∈ N by

From existence independence and
Step 4, it follows that and the proof is complete.
We are now in a position to prove our characterization result.
Proof of Theorem 2 'If.' Suppose that R is a generalized critical-level sufficientarian ordering associated with a function g ∈ G and α ∈ R H E such that g(α) ≥ sup a∈R L g(a). We show that R satisfies the axioms of the theorem. To see that R satisfies priority below the threshold, let n, m, , r ∈ N, u ∈ R n L , v ∈ R m L , w ∈ R H E , and s ∈ R r H E . To prove part (a) of the property, suppose that u Pv.
By the definition of R, u Pv implies By the definition of R, we obtain (u, w)P(v, s). The proof of part (b) is analogous That R satisfies anonymity follows immediately by definition. We next show that R satisfies strong Pareto. Let n ∈ N and u, v ∈ R n , and suppose that u i ≥ v i for all i ∈ {1, . . . , n} with at least one strict inequality. Note that L n (u) ⊆ L n (v).
First, suppose that L n (u) = L n (v). Since g is increasing on R L and R H E , we obtain i∈L n (u) In either case, we obtain u Pv.
Next, suppose that L n (u) ⊂ L n (v). Since g is increasing on R L and g(α) ≥ sup a∈R L g(a), it follows that and, therefore, u Pv. Now we show that R satisfies existence independence. Let n, m, ∈ N, u ∈ R n , v ∈ R m , and w ∈ R . Letū = (u, w) andv = (v, w). It follows that Thus, by the definition of R, u Rv ⇔ (u, w)R(v, w). Now we show that R satisfies continuity below the threshold. Let n ∈ N and u ∈ R n L .
Let b ∈ R L be such that b = min{v * 1 , . . . , v * n }. Since g is continuous and increasing on R L and is not bounded below, there exists ε ∈ R ++ such that b + ε < θ and Since g is strictly concave, it follows that, for all i ∈ {1, . . . , n}, Thus, we obtain Let B ε (v * ) ⊂ R n L be the open ball with radius ε and center v * . Note that, for all v ∈ B ε (v * ) and for all i ∈ {1, . . . , n}, Thus, by (10), it follows that, for all and, thus, To prove that R satisfies continuity above and at the threshold, let n ∈ N and u ∈ R n H E . To show that {v ∈ R n H E | v Ru} is closed in R n H E , let v t t∈N be a sequence in {v ∈ R n H E | v Ru} and assume that v t t∈N converges to v * ∈ R n H E . By way of contradiction, suppose that We define ∈ R ++ by Since g is continuous on R n H E and v t t∈N converges to v * ∈ R n H E , there exists t * ∈ N such that, for all t ≥ t * , which means that u Pv t for all t ≥ t * . This is a contradiction since v t Ru for all t ∈ N.
The proof that {v ∈ R n H E | u Rv} is closed in R n H E is analogous. It is straightforward that R satisfies weak existence of critical levels because α is a constant critical level for R.
We now show that R satisfies expansion independence below the threshold. Let n, m ∈ N. If n = m, then the requisite condition is satisfied for any k > max{m, n} and λ, μ ∈ R L such that λ = μ. Thus, we consider the case in which n = m. Without loss of generality, suppose that m > n. Let k = m + 1. We show that there exist λ, μ ∈ R L such that Since g is increasing on R L and g(α) ≥ sup a∈R L g(a), g(a) < g(α) for all a ∈ R L . Furthermore, g is not bounded below because the function is strictly concave. Thus, there exist λ, a ∈ R L with λ > a such that The function g is continuous on R L as a consequence of strict concavity. Thus, it follows from the intermediate value theorem that there exists μ ∈ (a, λ) ⊂ R L such that that is, We show that u Rv ⇔ū Rv.
Since λ and μ satisfy (11), we obtain The proof that v Ru if and only ifv Rū is analogous. Finally, R satisfies the principle of transfers below the threshold and the weak principle of transfers above and at the threshold because g is strictly concave on R L and concave on R H E .
'Only if.' Suppose that R satisfies the axioms of the theorem statement. From Lemma 2, there exist a continuous, increasing, and concave function g H E : R H E → R and α ∈ R H E such that (4) is satisfied for all n, m ∈ N, for all u ∈ R n H E , and for all v ∈ R m H E . Furthermore, from Lemma 3, there exist an increasing and strictly concave (and thus continuous) function g L : R L → R and δ ∈ R such that (5) is satisfied for all n, m ∈ N, for all u ∈ R n L , and for all v ∈ R m L . Using the functions g H E and g L , we define the function g : R → R by Note that, for all x ∈ R L , g(x) is defined by adding the constant g H E (α) − δ to g L (x). Because g inherits the corresponding properties of g L and g H E , g is increasing and strictly concave (and therefore continuous) on R L , and continuous, increasing, and concave on R H E . We show that R is the generalized critical-level sufficientarian ordering associated with g and α. Let n, m ∈ N, u ∈ R n , and v ∈ R m . First, suppose that i∈L n (u) or i∈L n (u) We show that u Pv follows in either case. First, suppose that (12) is true. From the definition of g, it follows that i∈L n (u) By (5), it follows that Thus, by part (a) of priority below the threshold, we obtain Since R satisfies anonymity, we obtain Thus, by the transitivity of R, it follows that u Pv. Now suppose that (13) is true. By the definition of g, From (5), it follows that Furthermore, by (4), we obtain By part (b) of priority below the threshold, it follows that Since R is transitive and satisfies anonymity, we obtain u Pv. Next, suppose that i∈L n (u) and i∈E n (u)∪H n (u) We show that u I v. By (5), Analogously, by (4), By part (b) of priority below the threshold, we obtain Because R is transitive and satisfies anonymity, it follows that u I v.
Since R is complete, the above argument implies that R is the generalized criticallevel sufficientarian ordering associated with g and α. To complete the proof, we show that g(α) ≥ sup a∈R L g(a). By way of contradiction, suppose that g(α) < sup a∈R L g(a). Since g is continuous and increasing on R L , there exists c ∈ R L such that g(c) = g(α). Let a ∈ R H E and b ∈ R L be such that c < b < θ < a. Note that g(b) > g(c) = g(α). We define u, v ∈ R 3 by u = (b, b, a) and v = (b, a, a).

We obtain
Since R is the generalized critical-level sufficientarian ordering associated with g and α, we obtain u Pv. However, this is a contradiction because R satisfies strong Pareto. Thus, g(α) ≥ sup a∈R L g(a).
We now examine some distributional implications of the generalized critical-level sufficientarian orderings. Assume that a society is in possession of a sufficiently large amount of a resource that allows everyone to be at or above the sufficiency threshold. In the absence of population change, the optimal distribution with respect to a generalized critical-level sufficientarian ordering is identical to the solution of maximizing the total sum of transformed utility levels g(u i ) subject to the constraints that require everyone's utility be at least as large as the threshold of sufficiency. This implies that generalized critical-level sufficientarian orderings have the same distributional implication as the (generalized) utilitarian orderings with a floor constraint, a problem that has been examined extensively in experiments; see, for instance, Frohlich and Oppenheimer (1992), Faravelli (2007), and Gaertner and Schokkaert (2012). In these contributions, subjects are asked to choose what they consider the normatively most attractive option among a range of social orderings. According to Frohlich and Oppenheimer (1992), the utilitarian orderings with a floor constraint receive more support than each of the maximin and utilitarian orderings. Our axiomatic analysis provides a theoretical foundation of utilitarianism with a floor.
One difficulty of utilitarianism with a floor is that it does not provide rankings when there is not enough of the resource to ensure that everyone reaches the threshold. An obvious advantage of our orderings is that they are capable of providing distributional judgments in these cases as well. Clearly, the distribution favored by our orderings is crucially dependent on the shortfall of the value of a function g from g(α) on its subdomain R L of utilities below the threshold. If the shortfall becomes infinitely large, more precisely, if | sup a∈R L g(a)− g(α)| approaches infinity, the resulting generalized critical-level sufficientarian ordering approaches Frankfurt's proposal as the primary criterion. However, any value of g(α) above sup a∈R L g(a) leads to a violation of the principle of transfers across the threshold. To identify the requisite subclass of our orderings, we add this principle to our list of axioms to obtain the following result.
Theorem 3 An ordering R satisfies priority below the threshold, anonymity, strong Pareto, existence independence, continuity below the threshold, continuity above and at the threshold, weak existence of critical levels, expansion independence below the threshold, the principle of transfers below the threshold, the weak principle of transfers above and at the threshold, and the principle of transfers across the threshold if and only if R is a generalized critical-level sufficientarian ordering associated with g ∈ G and α ∈ R H E such that sup a∈R L g(a) = g(α).
Proof 'If.' Let R be a generalized critical-level sufficientarian ordering associated with g ∈ G and α ∈ R H E such that g(α) = sup a∈R L g(a). All axioms other than the principle of transfers across the threshold follow from Theorem 2. To prove that the remaining property is satisfied, suppose that n ∈ N, u, v ∈ R n , i, j ∈ {1, . . . , n}, and ε > 0 are such that u i = v i + ε ≤ v j − ε = u j < θ ≤ v j and u k = v k for all k ∈ {1, . . . , n} \ {i, j}. By definition of R, we have to show that u Pv, that is, or, equivalently, Define the functionḡ : R L ∪ {θ } → R by lettingḡ(a) = g(a) for all a ∈ R L and g(θ ) = sup a∈R L g(a). Because g is strictly concave, so isḡ. Thus, (14) is equivalent to which follows from the strict concavity ofḡ because 'Only if.' Clearly, R must be a generalized critical-level sufficientarian ordering by virtue of Theorem 2. By way of contradiction, suppose that g(α) > sup a∈R L g(a). Consider a two-person distribution (θ, θ − t), where t > 0 and a sufficiently small transfer from individual 1 to individual 2 in the amount of ε > 0. Then, the resulting distribution is (θ − ε, θ − t + ε). We note that provided that ε is sufficiently small. This implies (θ, θ − t)P(θ − ε, θ − t + ε), a violation of the principle of transfers across the threshold.
Thus, if the principle of transfers across the threshold is added to our list of axioms, the resulting orderings are such that there exists a function g ∈ G with g(θ ) ≤ sup a∈R L g(a) < sup a∈R H E g(a) such that, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u Rv if and only if and i∈E n (u)∪H n (u) Examples of transformations g ∈ G that satisfy g(θ ) ≤ sup a∈R L g(a) < sup a∈R H E g(a) are given by those illustrated in Figs. 1 and 3 in Sect. 2. The transformations of Figs. 2 and 4 are no longer permissible in the subclass of orderings characterized in Theorem 3. At this point, an ethical implication of the shape of g becomes clearer. A gap between sup a∈R L g(a) and g(α) represents a deviation from the principle of transfers across the threshold.

Resolving population-ethics dilemmas
As noted in Sect. 2, a generalized critical-level sufficientarian ordering applies, as a tiebreaking device, a critical-level generalized-utilitarian ordering to the lifetime utilities of those at or above the threshold. In the context of population ethics, it is known that critical-level generalized utilitarianism cannot escape a well-known population-ethics dilemma: none of the members of this class allows us to avoid both Parfit's (1976;1982; repugnant conclusion and Arrhenius's (2000) sadistic conclusion. As we illustrate later, this population-ethics dilemma also applies to other well-established utilitarian population principles. In contrast to these difficulties, generalized criticallevel sufficientarianism has a subclass that allows us to avoid the dilemma.
An ordering R implies the repugnant conclusion if, for all n ∈ N, for all ξ ∈ R ++ , and for all ε ∈ (0, ξ), there exists m ∈ N with m > n such that ε1 m Pξ 1 n .
According to a principle that implies the repugnant conclusion, population size can always be substituted for quality of life-any population with an arbitrarily high common level of lifetime well-being is declared inferior to some larger population in which everyone's utility is positive but arbitrarily close to neutrality. Critical-level generalized utilitarianism implies the repugnant conclusion if and only if the critical level α is non-positive. A prominent example of a class of principles that imply the repugnant conclusion is given by total generalized utilitarianism, which is the subclass of the critical-level generalized-utilitarian principles that results from setting the critical level equal to zero-the level of utility that represents a neutral life; see Parfit (1976Parfit ( , 1982Parfit ( , 1984.
The axiom that requires an ordering R to avoid the repugnant conclusion is defined as the negation of the repugnant conclusion.
Avoidance of the repugnant conclusion There exist n ∈ N, ξ ∈ R ++ , and ε ∈ (0, ξ) such that, for all m ∈ N with m > n, To define the sadistic conclusion and the axiom that requires avoiding it, we need additional notation. Let ++ and −− denote the sets of all positive and all negative distributions, that is, An ordering R implies the sadistic conclusion (Arrhenius 2000) if there exist u ∈ , v ∈ ++ , and w ∈ −− such that (u, w)P (u, v). According to this conclusion, it is possible that adding a number of people with negative utilities to an existing population is better than adding a (possibly different) number of people with positive utilities to the same existing population. Critical-level generalized utilitarianism implies the sadistic conclusion if and only if the critical level α is non-zero. Again, the axiom that requires an ordering R to avoid the sadistic conclusion is defined as the negation of the sadistic conclusion.
Avoidance of the sadistic conclusion For all u ∈ , for all v ∈ ++ , and for all w ∈ −− , (u, v)R(u, w).
Avoidance of the sadistic conclusion is shown to relate to Parfit's (1984) mere addition principle by Franz and Spears (2020); see also Greaves (2017). The mere addition principle requires that if any number of people with utility levels above zero (neutrality) are added to an otherwise unaffected population, the augmented utility distribution cannot be worse than the original.
Mere addition For all u ∈ and for all v ∈ ++ , (u, v)Ru. Franz and Spears (2020) show that avoidance of the sadistic conclusion and the mere addition principle become equivalent under some axioms. The axioms they use include the extended continuity axiom that is defined on the entire domain of R and applies even to the comparison of well-being distributions with different population sizes. Thus, as we will see below, the equivalence between avoidance of the sadistic conclusion and the mere addition principle does not apply to the class of generalized critical-level sufficientarian orderings.
The following theorem examines the properties of avoidance of the repugnant conclusion, avoidance of the sadistic conclusion, and mere addition in the context of the generalized critical-level sufficientarian orderings. For each of the three axioms, we determine the members of this class of orderings that satisfy it. In addition, we identify the members of our class that satisfy any two of the properties.

Theorem 4 Suppose that R is a generalized critical-level sufficientarian ordering. (a) R satisfies avoidance of the repugnant conclusion if and only if (i) θ > 0 or (ii)
α > 0. Note that L n (u) = ∅ = L m (v). It follows that and we obtain u Pv. Thus, R satisfies avoidance of the repugnant conclusion. Next, suppose that α > 0. Since R satisfies avoidance of the repugnant conclusion if θ > 0, we assume that θ ≤ 0. To show that R avoids the repugnant conclusion, let ξ, ε ∈ R be such that Let n, m ∈ N with m > n, and define the distributions u and v by u = ξ 1 n and v = ε1 m .
and i∈E n (u)∪H n (u) This implies u Pv. Thus, R satisfies avoidance of the repugnant conclusion.
Thus, v Pu if m is sufficiently large. This means that R implies the repugnant conclusion.
'Only if.' Suppose first that θ > 0. Let a, b ∈ R be such that θ > a > 0 > b. Furthermore, let n ∈ N be such that This means that R implies the sadistic conclusion. Next, suppose that θ < 0 and that the corresponding critical level α is less than zero. Let a, b ∈ R be such that a > 0 > b > α. Furthermore, let n ∈ N be such that In analogy to the case θ > 0, it follows that the sadistic conclusion is implied.
Finally, suppose that the corresponding critical level α is greater than zero. Let a, b ∈ R be such that α > a > 0 > b > θ, and consider n ∈ N such that As in the previous case, the sadistic conclusion is implied. Thus, θ = 0 or θ < 0 and α = 0.
The possibility of avoiding both the repugnant conclusion and the sadistic conclusion is in stark contrast to two well-known utilitarian population principles other than critical-level generalized utilitarianism. Ng's (1986) number-dampened utilitarianism and its generalizations presented by Blackorby et al. (2005) and Asheim and Zuber (2014) cannot avoid both the repugnant and sadistic conclusions. Its most general form examined in Asheim and Zuber (2014) evaluates social states by means of the value f (n) n n i=1 [g(u i ) − g(α)], where α ∈ R + and f is a non-decreasing positive-valued function of population size. Rank-discounted utilitarianism proposed by Asheim and Zuber (2014) cannot escape the dilemma either. It employs the value n r =1 β r [g(u [r ] )−g(α)], where β ∈ (0, 1) and u [ ] is a non-decreasing rearrangement of u; see also Pivato (2020) and Asheim and Zuber (2022) for further generalized forms of rank-dependent utilitarianism. That these two classes of principles are subject to the dilemma is a consequence of an impossibility result established by Bossert et al. (2021b).
Avoidance of the repugnant conclusion is a relatively weak requirement because it is based on existential quantifiers. However, the proof of part (a) of Theorem 4 reveals that our orderings satisfy specific stronger variants. For instance, there exists ξ ∈ R ++ such that, for all n ∈ N, for all ε ∈ (0, ξ), and for all m ∈ N with m > n, we have ξ 1 n Rε1 m .
On the other hand, that avoidance of the sadistic conclusion is satisfied is a novel point of our new class of orderings because critical-level generalized utilitarian orderings do not avoid the conclusion as long as the critical level is positive; our orderings can avoid it even in this case. This observation essentially relies on the lexicographic structure of our orderings. The members of our class favor small improvements in the utilities of those below the threshold, even at a great expense on the part of those above the threshold.
There are alternative population principles that are capable of avoiding both the repugnant and sadistic conclusions. Prominent examples are given by the maximin and leximin principles; see, for instance, Bossert (1990), Blackorby et al. (1996Blackorby et al. ( , 2005, and Zuber (2018). Since these orderings give priority to the worst-off, they are subject to what Fleurbaey and Tungodden (2010) call the tyranny of non-aggregation. According to these egalitarian orderings, a small gain by the worst-off justifies large sacrifices by numerous better-off individuals as long as their relative well-being rankings remain the same. As a consequence of priority below the threshold, this argument applies to the generalized critical-level sufficientarian orderings when the worst-off is below the threshold and people at or above the threshold make sacrifices. However, unlike maximin and leximin, generalized critical-level sufficientarianism allows us to aggregate gains and losses among those below the threshold and among those at or above the threshold.
Another reconciliation of the dilemma is established by Pivato (2020) by means of the class of rank-additive orderings. He makes a distinction between possibilist and actualist settings in which population principles are to be examined. The possibilist framework considers the aggregation of utilities of infinitely many potential people who could exist by assigning a utility of zero to those who do not actually exist, while the actualist framework examines the aggregation of utilities of finitely many people who actually do exist. Note that the framework we and most of the existing literature on population ethics employ is actualist. Pivato (2020) shows that in the possibilist framework, the class of rank-additive orderings contains those orderings that avoid both the repugnant and sadistic conclusions. Unlike his reconciliation, our observation that both the repugnant and the sadistic conclusion can be avoided by generalized critical-level sufficientarian orderings illustrates that this reconciliation can be achieved without having to give priority to the worst-off and even within the actualist framework.
As shown in part (e) of Theorem 4, no generalized critical-level sufficientarian ordering can satisfy both avoidance of the repugnant conclusion and the mere addition principle. A related impossibility result established by Ng (1989) states that avoidance of the repugnant conclusion is incompatible with the mere addition principle and nonantiegalitarianism. Non-antiegalitarianism demands that if two distributions u and v are composed of the same set of individuals and all individuals in u have the same level of utility and, moreover, total utility in u is higher than total utility in v, then, all else being equal, u must be better than v according to R. All generalized criticallevel sufficientarian orderings identified in Theorem 3 satisfy non-antiegalitarianism because, as a consequence of the three transfer principles, an equal distribution cannot be worse than an unequal distribution for a given population and a given total utility. Because increasing total equal utility leads to a better distribution by definition of these principles, non-antiegalitarianism follows. However, some generalized criticallevel sufficientarian orderings do not satisfy non-antiegalitarianism-in particular, the orderings that fail to satisfy the principle of transfers across the threshold may suffer from this shortcoming. Therefore, the impossibility result established in part (e) of Theorem 4 is not a direct implication of Ng's (1989) result.

Conclusion
This paper provides an axiomatic analysis of sufficientarianism. We examine the relationship between critical levels and thresholds for sufficiency that result from the strong Pareto principle. Moreover, we characterize a class of orderings with the property that the critical level need not be equal to the threshold of sufficiency. At least one issue remains to be addressed-namely, the possibility of several thresholds, a notion that is explored in some of the recent literature on sufficientarianism; see, for instance, Casal (2007) and Huseby (2010). In these approaches, the lower (or lowest) threshold is assumed to represent the basic needs for a decent human life, while higher thresholds are intended to identify levels of a flourishing life. Integrating multiple thresholds into generalized critical-level sufficientarian orderings is a challenge to be addressed in future work. For this purpose, our new axiom of expansion independence below the threshold may turn out to be helpful in extending the single-threshold method to multiple-threshold versions. A promising approach consists of combining the axiom with a plausible extension of priority below the threshold that respects multiple layers of thresholds.
We adopt a welfarist framework throughout this paper. In contrast, contributions such as those of Anderson (1999Anderson ( , 2007 apply sufficientarian ideas to the capability approach, according to which priority is to be given to improvements in the capabilities of those who do not have enough. Each person's capability is given as a set in the space of functionings and, thus, capability distributions are to be compared. See also Alcantud et al. (2019) who examine sufficientarianism in the context of opportunities or chances of success. Clearly, these are perfectly legitimate alternatives but we do not pursue them. It seems to us that welfarism rests on a solid normative foundation, which is why we focus on the variant of sufficientarianism explored here.
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Appendix: Independence of the axioms of Theorem 2
(i) Let g : R → R be an increasing and strictly concave (and, thus, continuous) function and define the ordering R 1 as follows. For all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , that is, R 1 is the critical-level generalized-utilitarian ordering associated with g and the critical level of utility given by θ . The ordering R 1 satisfies all of the axioms in the theorem except priority below the threshold. (ii) Let α 1 = θ + 1 and α i = θ for all i ∈ N \ {1}. Let g ∈ G and define, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 2 v if and only if i∈L n (u) The ordering R 2 satisfies all of our axioms except anonymity. Consider u = (θ + 1, θ − 1) and v = (θ − 1, θ + 1). Then, v P 2 u follows since g(u 2 ) = g(v 1 ) and g(θ + 1) − g(α 1 ) < g(θ + 1) − g(α 2 ). (iii) Let g : R → R be a continuous and increasing function that is strictly concave on R L and convex on R H E . Define the ordering R 3 as follows. For all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 3 v if and only if i∈L n (u) and i∈H n (u) The ordering R 3 satisfies all axioms except strong Pareto. (iv) Let g ∈ G and α ∈ R H E with sup a∈R L g(a) = g(α), and define, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 4 v if and only if i∈L n (u) or i∈L n (u) The ordering R 4 satisfies all of our axioms except existence independence.
(v) To present an example that shows that continuity below the threshold is not implied, we need some additional notation and definitions. For n ∈ N and u ∈ R n , let (u (1) , . . . , u (n) ) denote a permutation of u such that u (1) ≥ · · · ≥ u (n) ; that is, the utilities in such a permutation are ranked from highest to lowest, with ties being broken arbitrarily. For each n ∈ N, the leximin ordering R n lex on R n is defined by letting, for all u, v ∈ R n , u R n lex v if and only if u is a permutation of v or there exists j ∈ {1, . . . , n} such that u (i) = v (i) for all i > j and u ( j) > v ( j) . Given a threshold level θ , the corresponding critical-level leximin ordering R lex on is defined by letting, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , For all n ∈ N and for all u ∈ R n , define For all n ∈ N and for all u ∈ R n , if K n (u) = ∅, we let ω ∅ = (u i ) i∈K n (u) . Using the leximin orderings R n lex for n-dimensional utility distributions and the critical-level leximin ordering R lex associated with the threshold θ , we define the extended critical-level leximin ordering R * lex on ∪ n∈N (−∞, θ − 1) n ∪ {ω ∅ } as follows. For all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , Let g : R → R be increasing and strictly concave (and, thus, continuous). We define the ordering R 5 as follows. For all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 5 v if and only if (u i ) i∈K n (u) P * lex (v i ) i∈K m (v) or (u i ) i∈K n (u) I * lex (v i ) i∈K m (v) and i∈L n (u)\K n (u) or (u i ) i∈K n (u) I * lex (v i ) i∈K m (v) and i∈L n (u)\K n (u) and i∈H n (u) The ordering R 5 satisfies all the axioms except continuity below the threshold. Showing that expansion independence below the threshold is satisfied is not trivial and, thus, we provide a detailed proof. Let n, m ∈ N. We show that there exist k ∈ N with k > max{n, m} and λ, μ ∈ [θ − 1, θ) ⊂ R L that satisfy the requisite condition. Note that any k ∈ N with k > max{n, m} and any λ, μ ∈ [θ − 1, θ) satisfy that, for all u ∈ R n L and for all v ∈ R m L , whereū = (u, λ1 k−n ) andv = (v, μ1 k−m ). Thus, by the definition of R 5 , it suffices to show that there exist k ∈ N with k > max{n, m} and λ, μ ∈ [θ − 1, θ) such that, for all u ∈ R n L and for all v ∈ R m L , and i∈L n (u)\K n (u) whereū = (u, λ1 k−n ) andv = (v, μ1 k−m ). First, note that, for all u ∈ R n L and for all v ∈ R m L , If (u i ) i∈K n (u) is a permutation of (v i ) i∈K m (v) , then |K n (u)| = |K m (v)|. Thus, if n = m, conditions (16) and (17) are satisfied for any k > max{n, m} and λ, μ ∈ [θ − 1, θ) such that λ = μ. Next, we consider the case where n = m. Without loss of generality, suppose that m > n. We begin by showing that there exist k ∈ N with k > m and λ, μ ∈ [θ − 1, θ) such that Since g is increasing on R, g(a) < g(θ ) for all a ∈ R L . Furthermore, g is not bounded below because of strict concavity. Thus, there exist (sufficiently large) k ∈ N with k > m and λ ∈ (θ − 1, θ) such that Since g is continuous on R, it follows from the intermediate value theorem that there exists μ ∈ (θ − 1, λ) ⊂ [θ − 1, θ) such that g(μ) = g(λ) − m−n k−m [g(θ ) − g(λ)], that is, We now show that, for k, λ, and μ that satisfy (18), conditions (16) and (17) are satisfied. By (15), it is straightforward that (16) is satisfied. To show that (17) holds, let u ∈ R n L and v ∈ R m L . Since (u i ) i∈K n (u) I * lex (v i ) i∈K m (v) implies |K m (v)| = |K n (u)|, we can assume m −n = |L m (v)\ K m (v)|−|L n (u)\ K n (u)|. Letū = (u, λ1 k−n ) andv = (u, μ1 k−m ). Then, we obtain i∈L n (u)\K n (u) since |L k (ū) \ K k (ū)| = |L k (v) \ K k (v)|. Thus, R 5 satisfies expansion independence below the threshold. (vi) Let g ∈ G and α ∈ R H E with sup a∈R L g(a) = g(α), and define the ordering R 6 as follows. For all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 6 v ⇔ i∈L n (u) [g(u i ) − g(α)] > i∈L m (v) [g(v i ) − g(α)] or i∈L n (u) [g(u i ) − g(α)] = i∈L m (v) [g(v i ) − g(α)] where R lex is the critical-level leximin ordering associated with θ . The ordering R 6 satisfies all of the axioms except continuity above and at the threshold. (vii) Let g ∈ G and α ∈ R H E with sup a∈R L g(a) = g(α). Define, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 7 v if and only if i∈L n (u) or i∈L n (u) [g(u i ) − g(α))] = i∈L m (v) [g(v i ) − g(α)] and |H n (u) ∪ E n (u)| = |H m (v) ∪ E m (v)| and i∈H n (u)∪E n (u) The ordering R 7 does not satisfy weak existence of critical levels. All other axioms are satisfied. (viii) Let g ∈ G and define, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 8 v if and only if g(v i ) and i∈H n (u) [g(u i ) − g(θ )] ≥ i∈H m (v) [g(v i ) − g(θ )] .
Note that lim a→−∞ g(a) = −∞ since g ∈ G. Thus, there exists u ∈ R L that satisfies this inequality. This means that R 8 violates expansion independence below the threshold. All other axioms are satisfied. (ix) Define, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 9 v if and only if i∈L n (u) or i∈L n (u) [u i − θ ] = i∈L m (v) [v i − θ ] and i∈H n (u) The ordering R 9 does not satisfy the principle of transfers below the threshold. All other axioms are satisfied. (x) Finally, let g : R → R be a continuous and increasing function that is strictly concave on R L and strictly convex on R H E . Define, for all n, m ∈ N, for all u ∈ R n , and for all v ∈ R m , u R 10 v if and only if i∈L n (u) and i∈H n (u) The ordering R 10 does not satisfy the weak principle of transfers above and at the threshold. All other axioms are satisfied.