# Exploring infinite population utilitarianism under strong anonymity

Geir B Asheim, Kohei Kamaga, Stéphane Zuber

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Geir B Asheim, Kohei Kamaga, Stéphane Zuber. Exploring infinite population utilitarianism under strong anonymity. Economic Theory, 2024, 10.1007/s00199-024-01620-6. halshs-04828788

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# Exploring Infinite Population Utilitarianism under
Strong Anonymity*

Geir B. Asheim¹†, Kohei Kamaga²†, Stéphane Zuber³,⁴†

¹*Department of Economics, University of Oslo, Oslo, Norway.

²Department of Economics, Sophia University, Tokyo, Japan.

³Université Paris 1 Panthéon Sorbonne and CNRS, Centre d'Économie
de la Sorbonne, 106 bd de l'Hopital, Paris, 75013, France.

⁴Paris School of Economics, Paris, France.

Contributing authors: g.b.asheim@econ.uio.no;
kohei.kamaga@sophia.ac.jp; Stephane.Zuber@univ-paris1.fr;

†These authors contributed equally to this work.

## Abstract

We examine utilitarian criteria for evaluating profiles of well-being among infinitely many individuals. Motivated by the non-existence of a natural 1-to-1 correspondence between people when alternatives have different population structures, with a different number of people in each generation, we impose equal treatment in the form of *Strong Anonymity*. We demonstrate how a novel criterion, *Strongly Anonymous Utilitarianism*, can be applied in the Ramsey model, leading to an efficient and sustainable stream. We show how the criterion is the result of combining *Strong Anonymity* with other regularity axioms (*Monotonicity*, *Finite Completeness*, and continuity axioms) as well as axioms of equity, population ethics, sensitivity, and separability. We relate it to other strongly anonymous utilitarian criteria.

**Keywords:** Utilitarianism, Intergenerational equity, Population ethics

**JEL Classification:** D63, D71, Q01

*This is the preprint version of the following paper:
Asheim, G.B., Kamaga, K. & Zuber, S. Exploring infinite population utilitarianism under strong anonymity. Econ Theory (2024). https://doi.org/10.1007/s00199-024-01620-6
**Acknowledgements.** We thank Marc Fleurbaey, Adam Jonsson, Marcus Pivato and two anonymous referees as well as seminar participants at Kobe University, Université de Montréal (CIREQ), Université de Paris 1 (CES), Sophia University, PET 2019 in Strasbourg, and the Online Social Choice and Welfare Seminar Series for many helpful comments. This paper has earlier been circulated under the title "Infinite population utilitarian criteria". Asheim thanks IMéRA (Institut d'études avancées – Aix Marseille Université) for providing an environment where this research was started and CIREQ (Centre Interuniversitaire de Recherche en Économie Quantitative) for facilitating his visits to Montreal. Kamaga thanks University of Oslo, CES (Centre d'Économie de la Sorbonne) at Université Paris 1 and CIREQ in Montreal for their hospitality and acknowledges financial support from JSPS KAKENHI Grant Nos. 20K01565 and 23H00814. Zuber acknowledges support by the Agence nationale de la recherche through the Fair-ClimPop project (ANR-16-CE03-0001-01), the Investissements d'Avenir program (PGSE-ANR-17-EURE-01), and the Riksbankens Jubileumsfond (the Swedish Foundation for Humanities and Social Sciences). The paper is part of the research activities at the Centre for the Study of Equality, Social Organization and Performance (ESOP) at the Department of Economics at the University of Oslo.

**Statements and Declarations**

**Competing interests.** The authors declare that they do not have any competing interests.

**Availability of data and material (data transparency).** Not applicable

**Code availability (software application or custom code).** Not applicable
# 1 Introduction

Since the seminal works by Ramsey (1928), Koopmans (1960), and Diamond (1965), social evaluation criteria for distributions of well-being among infinitely many generations or infinitely many individuals have been analyzed by using preorders for infinite well-being profiles, i.e., preorders defined for countably infinite-dimensional real vectors.<fnref n="1" /> There has been a particular interest in social evaluation criteria that are (generalized) utilitarian and anonymous, in the sense of being based on the undiscounted or unweighted sum of transformed well-being,<fnref n="2" /> so that the gain of one generation or individual might make up for the loss of another, and so that the generations or individuals are treated equally. Prominent examples are the utilitarian criterion that combines the finite-horizon application of utilitarianism with the Pareto dominance for the tails of profiles (Basu and Mitra, 2007a) and utilitarian overtaking or catching-up procedures with respect to time (Atsumi, 1965; von Weizsäcker, 1965; Gale, 1967).

As suggested by Fleurbaey and Michel (2003, p. 795), infinite well-being profiles can, however, be interpreted in at least three different ways. A common interpretation is that there are representative individuals, one for each generation. That is, in well-being profiles x and y given by

$$x = (x_1, \dots, x_i, \dots) \quad \text{and} \quad y = (y_1, \dots, y_i, \dots),$$

the ith component is the well-being of the representative individual of ith generation, where it is normally left unspecified whether the individual in each generation i is actually identical in the two profiles (Jonsson, 2021). This interpretation is based on two underlying assumptions. One is that there is a natural 1-to-1 correspondence between the individuals of the two streams so that the bearer of well-being at the ith component of profile x can be associated with the bearer of well-being at the ith component of profile y. The second assumption is that there is a natural order of counting the individuals or generations, namely according to time. Both the Basu-Mitra utilitarian criterion and the utilitarian overtaking or catching-up criteria are based on the existence of a natural 1-to-1 correspondence between the components of compared profiles. In particular, although such criteria are invariant to permutations that reorder a finite number of the components of one stream (corresponding to the axiom of Finite Anonymity), they are not invariant to all permutations (corresponding to the axiom of Strong Anonymity), implying that the natural 1-to-1 correspondence plays an essential role. Moreover, the utilitarian overtaking or catching-up criteria are sensitive for the order that the individuals are counted. In particular, by the utilitarian catching-up criterion two profiles are equally good if and only if the sum of the difference of transformed well-being converges conditionally to zero as time goes to

<footnote n="1">¹Other than preorders—that is, reflexive and transitive binary relations—complete and quasi-transitive relations have been also analyzed. See, for instance, Fleurbaey and Michel (2003) and Sakai (2010).</footnote>

<footnote n="2">²Throughout this paper we will use the term utilitarian to refer to generalized utilitarian or prioritarian criteria, based on the sum of transformed well-being. Our notion of well-being need not directly correspond to the notion of utility as used by philosophers; if not, utility will equal a particular transformation of well-being. A strictly concave transformation of utility yields prioritarianism. We will allow any transformations of well-being that have bounded non-concavity, as defined in Section 2.1.</footnote>
infinity.<fnref n="3" /> Also, normative significance can be assigned to temporal utilitarian averages, based on an expansionist view of infinite aggregation (Vallentyne and Kagan, 1997; Van Liedekerke and Lauwers, 1997; Arntzenius, 2014; Khan and Stinchcombe, 2018; Wilkinson, 2020; Pivato, 2022, 2023).

The second interpretation is that there exists a fixed (exogenously determined) population profile, although population size may vary from one generation to another. For instance, the well-being profiles x and y given by

$$x = \underbrace{(x_1, x_2, x_3, \dots, x_{10}, \dots)}_{\text{1st generation}} \quad \text{and} \quad y = \underbrace{(y_1, y_2, y_3, \dots, y_{10}, \dots)}_{\text{2nd generation}}$$

represent a situation where there exist three individuals alive in the 1st generation, seven individuals alive in the 2nd generation, and so on. In this interpretation there is no natural order for counting individuals; in particular, there is no natural order for counting individuals within each generation. This lessens the appeal of the overtaking and catching-up criteria mentioned above. On the other hand, at least if the identities of individuals alive are the same across profiles, then there exists a natural 1-to-1 correspondence between the components of two profiles. The Basu-Mitra utilitarian criterion utilizing this 1-to-1 correspondence might thus still be appealing. Moreover, we can impose that social evaluation of profiles be invariant to the order in which the individuals are counted when the same reordering is used for both streams. Such invariance—referred to as *Isomorphism Invariance* (Lauwers and Vallentyne, 2004), *Strong Relative Anonymity* (Asheim, d’Aspremont, and Banerjee, 2010), or simply *Relative Anonymity* (Pivato and Fleurbaey, 2024), and related to *Qualitativeness* (Askell, 2018)—implies Finite Anonymity.<fnref n="4" /> However, it rules out evaluation based on temporal averages. Utilitarian criteria—like the *Time-Invariant Overtaking* criterion (Asheim, d’Aspremont, and Banerjee, 2010) as well as the *Weak Catching-Up* criterion of Lauwers and Vallentyne (2004, p. 322)—that satisfy Relative Anonymity must be based on *unconditional convergence*: two profiles are equally good only if the sum of differences of transformed well-being converges to zero for each possible reordering of components.

Finally, the third interpretation, which this paper allows for, is that there exists a variable number of individuals alive in each generation and that the identities of individuals alive are not necessarily the same across distributions. For instance, well-being profiles x and y given by

$$x = \underbrace{(x_1, x_2, x_3, \dots, x_{10}, \dots)}_{\text{1st generation}} \quad \text{and} \quad y = \underbrace{(y_1, \dots, y_4, \dots, y_9, \dots)}_{\text{2nd generation}}$$

represent, respectively, a situation where there exist three individuals alive in the 1st generation, seven individuals alive in the 2nd generation, and so on, and a situation

<footnote n="3">³A series is conditionally convergent if it converges, but it need not converge for all possible reorderings.</footnote>

<footnote n="4">⁴Relative Anonymity implies Finite Anonymity under the axiom of *Finite Completeness* (completeness of a preorder for pairs of profiles which differ only for a finite number of components), see d'Aspremont and Gevers (1977, Lemma 4), Shinotsuka (1998, Proposition 1), and Asheim, d'Aspremont, and Banerjee (2010, Lemma 1).</footnote>
where there exist four individuals alive in the 1st generation, five individuals alive in the 2nd generation, and so on. Such pairs of well-being profiles may arise for example in economic growth models involving fertility choice. Moreover, component 4 might not be the same person in the two profiles, illustrating how policy choice may change the identity of people living in the future (for example, by delaying child-bearing). In this case, not only is there no natural order for counting individuals, there is also no natural 1-to-1 correspondence between the components of the two profiles since we cannot assume that the identities are the same. Even the Basu-Mitra utilitarian criterion might not seem appealing for comparing such pairs of profiles. One possibility, which we consider in this paper, is to impose that social evaluation of profiles be invariant to the order in which the individuals are counted even if the reordering applies to one stream only. Indeed, we will impose the axiom of Strong Anonymity by requiring that a well-being profile is deemed equally good as any permutation of it, permitting also permutations that reorder infinitely many components.

The axiom of Strong Anonymity implies the axioms of Finite Anonymity and Relative Anonymity, and it is compatible also with the third interpretation. However, this scope of applicability is not by itself a sufficient argument for imposing such a strong impartiality axiom; rather, it requires separate justification. One position, adopted by Zuber and Asheim (2012) and Asheim and Zuber (2013), is to argue that Strong Anonymity is needed in order to properly capture equal treatment.<fnref n="5" /> Another position is investigated in our companion paper (Asheim, Kamaga, and Zuber, 2024), where we show how Strong Anonymity for future people follows if future identities are unobservable. Combining such strong impartiality for future people with equal treatment of existing and future people yields Strong Anonymity for all, as imposed in the present paper.

The purpose of this paper is to propose and axiomatize utilitarian criteria for ranking infinite well-being profiles satisfying the axiom of Strong Anonymity, thus being neutral with respect to the order in which we compute the utilitarian sum. This implies that the criteria are applicable when considering the third interpretation where population within each generation might differ. We define a new criterion, the *Strongly Anonymous (Generalized) Utilitarian* social welfare relation, and demonstrate its applicability by showing how it leads to an efficient and sustainable stream in the Ramsey model.

We establish as our main result how Strongly Anonymous Utilitarianism is the result of combining Strong Anonymity with other regularity axioms (*Monotonicity*, *Finite Completeness*, and continuity axioms), an equity axiom, a population ethics axiom, a sensitivity axiom, and a separability axiom. By imposing these axioms we restore a 1-to-1 correspondence between reordered streams derived from the profiles as follows: Their components are rank-ordered starting with the worst-off and, if there is only a finite number of well-being components that can be rank-ordered in this manner, completed with well-beings equal to the smallest cluster point. The smallest cluster point of the set of components in a well-being profile is the smallest point such that, for every neighborhood, there are infinitely many components of the set within

<footnote n="5">⁵Lauwers (1997) discusses whether Finite Anonymity is strong enough to properly capture equal treatment.</footnote>
the neighborhood. The smallest cluster point equals the *limit inferior* of the stream obtained from the profile when its components are listed, independently of in which order this listing is made.

However, the axioms contradict that the sequence of rank-order is a natural order of counting the components of the streams that are derived in this manner. Rather, they imply that Strongly Anonymous Utilitarianism satisfies the principles of Differential Betterness and Differential Indifference (Lauwers and Vallentyne, 2004) applied to the derived streams. This means that two well-being profiles are equally good according to Strongly Anonymous Utilitarianism if and only if the sum of the difference of transformed well-being of the derived rank-ordered streams converges *unconditionally* to zero. In particular, conditional convergence according to rank does not suffice.

In a setting with an infinite, but countable, number of well-being components, it is impossible to combine Strong Anonymity with the strong Pareto principle, as shown by Van Liedekerke (1995) and Van Liedekerke and Lauwers (1997, p. 163). For example, the stream $(1, 0, 1, 0, \dots, 1, 0, \dots)$ can be transformed to the stream $(1, 1, 1, 0, \dots, 1, 0, \dots)$ by reordering infinitely many components, implying by Strong Anonymity that the two are equally good. However, by applying the strong Pareto principle component-by-component, the latter is strictly better. Strong Anonymity is even incompatible with the weak Pareto principle (Fleurbaey and Michel, 2003). Asheim, Kamaga, and Zuber (2022) show that, when Strong Anonymity and Monotonicity are combined with the continuity axioms and the equity axiom, there can be sensitivity for an increase of the well-being at a particular component of an infinite well-being profile if and only if the well-being at this component is below the limit inferior. Consequently, our sensitivity axiom is a version of the strong Pareto principle restricted to well-being levels below the limit inferior. Thus, it allows the full force of the strong Pareto principle for comparison of profiles where an earlier component never has higher well-being than a later. It will be shown, however, that this is sufficient for obtaining efficiency when Strongly Anonymous Utilitarianism is applied in the Ramsey model.

We study how Strongly Anonymous Utilitarianism relates to four other social welfare relations. First, we compare with a *Strongly Anonymous (Generalized) Utilitarian Catching-Up* criterion, which is a strongly anonymous variant of the utilitarian caching-up criterion in Gale (1967). Second, we turn to a *Strongly Anonymous (Generalized) Utilitarian Dominance* criterion, which is a strongly anonymous variant of the Basu-Mitra utilitarian criterion. Third, we take the limit of *Rank-Discounted Utilitarianism*, as proposed by Zuber and Asheim (2012), when the discount factor approaches one, leading to the *Limit of Rank-Discounted (Generalized) Utilitarianism*. Last, we consider a *Strongly Anonymous (Generalized) Utilitarian Cesàro Summation* criterion, which is a modification of the comparison of Cesàro summation of transformed well-beings so as to satisfy the Strong Anonymity axiom.

In Section 2, we present the framework of our analysis and introduce the Strongly Anonymous Utilitarian social welfare relation. In Section 3, we demonstrate the applicability of Strongly Anonymous Utilitarianism in the Ramsey model. In Section 4, we present an axiomatic characterization of Strongly Anonymous Utilitarianism and relate it to the Strongly Anonymous Utilitarian Dominance criterion (which it
extends). In Section 5, we complete the comparison of Strongly Anonymous Utilitarianism with the three other strongly anonymous utilitarian criteria by showing the links between them in terms of the notions of subrelation and weak extension. In Section 6, we offer some concluding discussion. Several proofs are relegated to an appendix.

# 2 Strongly Anonymous Utilitarianism

As mentioned in the introduction, there are settings with infinitely many people where there is no natural 1-to-1 correspondence between the components of different well-being profiles. This is true in particular if population structure depends on the choice of alternatives. Then there is no natural way of computing a utilitarian sum, which is a problem for extending utilitarian criteria to that context. We seek ways of computing the sum that are neutral with respect to how we compute the sum. Our exploration in subsection 2.2 starts by imposing Relative Anonymity on Utilitarian Catching-Up, leading to Weak Utilitarian Catching-Up, followed by requiring Strong Anonymity, resulting in Strongly Anonymous Utilitarian Catching-Up. However, as we show in Section 4, our axioms require us, in addition, to introduce Relative Anonymity on rank-ordered profiles. Thereby, we obtain the novel criterion of Strongly Anonymous Utilitarianism. Introducing these social welfare relations requires notation introduced in the following subsection 2.1.

## 2.1 Framework and notation

Let $\mathbb{N}$ denote the set of natural numbers $\{1, \dots, i, \dots\}$. Let $\mathbb{R}$ denote the set of real numbers, $\mathbb{R}_+$ the set of nonnegative real numbers, and $\mathbb{R}_{++}$ the set of positive real numbers (and likewise, write $\mathbb{R}_-$ for the set of non-positive real numbers). Let $\mathbf{x} = (x_1, \dots, x_i, \dots)$ denote an infinite stream (or profile), where $x_i \in \mathbb{R}$ is a one-dimensional indicator of the well-being of the individual at component $i \in \mathbb{N}$. Throughout, we refer to an infinite stream as a profile since we do not commit a specific order of its components in light of what Strong Anonymity requires.

We assume that well-being is cardinally measurable and fully comparable. We restrict attention to profiles with bounded well-being, and let

$$ \mathbf{X} = \{ \mathbf{x} = (x_1, \dots, x_i, \dots) \in \mathbb{R}^\mathbb{N} : \sup_{i \in \mathbb{N}} |x_i| < +\infty \} $$

denote the set of possible profiles. For $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, write $\mathbf{x} \ge \mathbf{y}$ whenever $x_i \ge y_i$ for all $i \in \mathbb{N}$; write $\mathbf{x} > \mathbf{y}$ if $\mathbf{x} \ge \mathbf{y}$ and $\mathbf{x} \neq \mathbf{y}$; and write $\mathbf{x} \gg \mathbf{y}$ whenever $x_i > y_i$ for all $i \in \mathbb{N}$.

For any $n \in \mathbb{N}$ and any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, let $\mathbf{x}_n \mathbf{y}$ denote the profile $\mathbf{z}$ such that $z_i = x_i$ for all $i \le n$ and $z_i = y_{i-n}$ for all $i > n$, that is, $\mathbf{x}_n \mathbf{y} = (x_1, \dots, x_n, y_1, y_2, \dots)$. For any $z \in \mathbb{R}$, write $z 1_{\mathbb{N}} = (z, z, \dots) \in \mathbf{X}$ for a constant profile. For any $z \in \mathbb{R}$ and $\mathbf{x} \in \mathbf{X}$, let $(z, \mathbf{x})$ denote the profile $(z, x_1, x_2, \dots)$ and write $(z 1_{\mathbb{N}}, \mathbf{x}) = (z, x_1, z, x_2, \dots)$.<fnref n="6" /> For any

<footnote n="6">⁶Our analysis does not lose any generality by the use of this particular form of combination of two infinite profiles because, under our assumption of Strong Anonymity, all possible forms of combination are equivalently good.</footnote>
$n \in \mathbb{N}$ and $\ell \in \mathbb{R}_+$, write

$$ \mathbf{X}_n^\ell = \{ \mathbf{x} \in \mathbf{X} : x_i \le \ell \text{ for all } i \le n \text{ and } x_j = \ell \text{ for all } j \ge n+1 \}. $$

A permutation $\pi$ of $\mathbb{N}$ is a bijection on $\mathbb{N}$. Let $\Pi$ denote the set of all permutations of $\mathbb{N}$. For any $\mathbf{x} \in \mathbf{X}$ and any $\pi \in \Pi$, write $\mathbf{x}_\pi = (x_{\pi(1)}, x_{\pi(2)}, \dots) \in \mathbf{X}$. Two subsets of $\mathbf{X}$ will be of particular interest. First, we introduce the set $\mathbf{X}^+$ of non-decreasing streams in $\mathbf{X}$. Formally,

$$ \mathbf{X}^+ = \{ \mathbf{x} \in \mathbf{X} : x_i \le x_{i+1} \text{ for all } i \in \mathbb{N} \}. $$

The other subset of $\mathbf{X}$ playing a key role in the remainder of the paper is the set of profiles which can be reordered into non-decreasing streams. This set is denoted $\mathbf{X}^\uparrow$ and is defined by

$$ \mathbf{X}^\uparrow = \{ \mathbf{x} \in \mathbf{X} : \text{ there exists } \pi \in \Pi \text{ such that } \mathbf{x}_\pi \in \mathbf{X}^+ \}. $$

By way of illustration, note that $\mathbf{x} = (1, 0, \dots, 0, \dots)$ is not non-decreasing and cannot be reordered into a non-decreasing stream, while $\mathbf{y} = (1, 0, 1, \dots, 1, \dots)$ is not non-decreasing but can be reordered into the non-decreasing stream $\mathbf{y}_\pi = (0, 1, 1, \dots, 1, \dots)$ by letting $\pi$ permute the first and second components. Hence, of these two profiles, only $\mathbf{y}$ is in $\mathbf{X}^\uparrow$.

For any profile $\mathbf{x} \in \mathbf{X}$, write $\ell(\mathbf{x}) := \liminf_{i \to +\infty} x_i$ for the limit inferior of $\mathbf{x}$, implying that $\ell(\mathbf{x})$ is the smallest cluster point of $\mathbf{x}$. Define the set $L(\mathbf{x})$ by $L(\mathbf{x}) := \{i \in \mathbb{N} : x_i < \ell(\mathbf{x})\}$.

We say that a transformation function $u : \mathbb{R} \to \mathbb{R}$ has bounded non-concavity whenever$^7$

$$ G_u = \sup_{x_1 < x_2 \le x_3 < x_4} \left[ \frac{u(x_4) - u(x_3)}{x_4 - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1} \right] < +\infty. $$

Note that this definition does not preclude that a transformation function $u$ with bounded non-concavity is actually concave. Let $U$ denote the set of continuous and increasing transformation functions with bounded non-concavity.

For fixed $u \in U$, define the sequence of functions $v_n^u : \mathbf{X} \to \mathbb{R}$, for $n \in \mathbb{N}$, by, for all $\mathbf{x} \in \mathbf{X}$,

$$ v_n^u(\mathbf{x}) = \sum_{i=1}^{n} u(x_i). $$

Hence, $v_n^u$ determines the sum of the transformed well-being of the first $n$ components of $\mathbf{x}$. For fixed $u \in U$, define the sequence of functions $w_n^u : \mathbf{X} \to \mathbb{R}$, for $n \in \mathbb{N}$, by, for all $\mathbf{x} \in \mathbf{X}$,

$$ w_n^u(\mathbf{x}) = \inf_{\pi \in \Pi} \sum_{i=1}^{n} u(x_{\pi(i)}). $$

<footnote>$^7$ $G_u$ is an index of non-concavity introduced by Chateauneuf, Cohen, and Meilijson (2005).</footnote>
It is actually the case that $w_n^u$ is the sum of the transformed well-being of the n smallest components of $\mathbf{x}$ if $|L(\mathbf{x})| \ge n$, and the sum of the transformed well-being of the $|L(\mathbf{x})|$ smallest components of $\mathbf{x}$ plus $(n - |L(\mathbf{x})|)u(\ell(\mathbf{x}))$ otherwise.

For any profile $\mathbf{x} \in \mathbf{X}$, define the derived rank-ordered stream $\mathbf{x}_{[\cdot]} = (x_{[1]}, \dots, x_{[r]}, \dots) \in \mathbf{X}^+$ as follows: Fix $u \in U$ and set $w_0^u(\mathbf{x}) = 0$. Let, for all $r \in \mathbb{N}$,

$$x_{[r]} = u^{-1}(w_r^u(\mathbf{x}) - w_{r-1}^u(\mathbf{x}))$$

be the well-being level of the individual at rank $r$.<fnref n="8" /> Applied to $\mathbf{x} = (1, 0, \dots, 0, \dots)$ and $\mathbf{y} = (1, 0, 1, \dots, 1, \dots)$, we obtain $\mathbf{x}_{[\cdot]} = (0, 0, \dots, 0, \dots)$ since $L(\mathbf{x})$ is empty, while $\mathbf{y}_{[\cdot]} = (0, 1, 1, \dots, 1, \dots)$. For any $\mathbf{x} \in \mathbf{X}$ and any $\pi \in \Pi$, write $\mathbf{x}_{[\pi]} = (x_{[\pi(1)]}, x_{[\pi(2)]}, \dots) \in \mathbf{X}^{\uparrow}$; i.e., $\mathbf{x}_{[\pi]}$ is the reordering of $\mathbf{x}_{[\cdot]}$ obtained by using the permutation $\pi$.

Component $[r]$ of the derived stream $\mathbf{x}_{[\cdot]}$ can be interpreted as individual with 'rank' $r$ when well-being is ordered from the lowest level to the highest (below the limit inferior). Hence, $\mathbf{x}_{[\cdot]}$ is a reordered stream of well-being, but where people above the limit inferior are discarded, as these have 'infinite' rank (in the sense that there are infinitely many people with lower well-being). Note that, for fixed $u \in U$ and any $\mathbf{x} \in \mathbf{X}$, we have $w_n^u(\mathbf{x}) = v_n^u(\mathbf{x}_{[\cdot]}) = \sum_{r=1}^n u(x_{[r]})$ for all $n \in \mathbb{N}$.

A social welfare relation (SWR) on a set $\mathbf{X}$ is a preorder (a reflexive and transitive binary relation) $\preceq$ where, for any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, $\mathbf{x} \preceq \mathbf{y}$ entails $\mathbf{x}$ is deemed socially at least as good as $\mathbf{y}$. The symmetric and asymmetric parts of $\preceq$ are denoted by $\sim$ and $\succ$, respectively. An SWR $\preceq'$ weakly extends an SWR $\preceq$ if, for all $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, $\mathbf{x} \preceq' \mathbf{y}$ implies $\mathbf{x} \preceq' \mathbf{y}$. An SWR $\preceq$ is a subrelation to an SWR $\preceq'$ if $\preceq'$ weakly extends $\preceq$ and, for all $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, $\mathbf{x} \succ \mathbf{y}$ implies $\mathbf{x} \succ' \mathbf{y}$.

## 2.2 The Strongly Anonymous (Generalized) Utilitarian SWR

To motivate Strongly Anonymous Utilitarianism, consider first the class of (Generalized) Utilitarian Catching-Up SWRs (Gale, 1967) (where every $u \in U$ corresponds to a different SWR):<fnref n="9" /> There exists $u \in U$ such that, for any profiles $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, $\mathbf{x}$ is at least as good as $\mathbf{y}$ if and only if

$$\liminf_{n \to +\infty} (v_n^u(\mathbf{x}) - v_n^u(\mathbf{y})) \ge 0.$$

The utilitarian catching-up criterion is not invariant to the order in which the individuals are counted. For a version of the utilitarian catching-up criterion that satisfies Relative Anonymity, consider the following definition of the class of Weak (Generalized) Utilitarian Catching-Up SWRs (Lauwers and Vallentyne, 2004, p. 322): There exists $u \in U$ such that, for any profiles $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, $\mathbf{x}$ is at least as good as $\mathbf{y}$ if and only

<footnote n="8">⁸Equivalently, we can derive $\mathbf{x}_{[\cdot]}$ as the non-decreasing permuted infinite subsequence of the elements in $L(\mathbf{x})$ if $L(\mathbf{x})$ is infinite, and the non-decreasing permuted finite subsequence of the elements in $L(\mathbf{x})$, followed by $x_{[r]} = \ell(\mathbf{x})$ for $r > |L(\mathbf{x})|$, if $L(\mathbf{x})$ is finite (Zuber and Asheim, 2012, p. 1576).</footnote>

<footnote n="9">⁹Throughout we consider transformation functions $u$ in the set $U$, implying that $u$ is not only continuous and increasing, but also has bounded non-concavity. Bounded non-concavity follows from our axiomatization of Strongly Anonymous Utilitarianism, defined in Definition 2. However, we might also define the various utilitarian criteria without insisting on $u$ having bounded non-concavity.</footnote>
if there exists $\Delta \in \mathbb{R}_+ \cup \{+\infty\}$ such that,

$$ \text{for all } \pi \in \Pi, \lim_{n \to +\infty} (v_n^u(\mathbf{x}_\pi) - v_n^u(\mathbf{y}_\pi)) = \Delta. $$

Through its unconditional convergence, the Weak Utilitarian Catching-Up criterion satisfies the principles of Differential Betterness and Differential Indifference (Lauwers and Vallentyne, 2004). The Differential Betterness principle states that, when the sum of the positive differences of transformed well-being $\sum_{i \le n: x_i > y_i} (u(x_i) - u(y_i))$ converges to some finite number $\Delta_+ > 0$ and the sum of the negative differences of transformed well-being $\sum_{i \le n: y_i > x_i} (u(x_i) - u(y_i))$ converges to some finite number $\Delta_- < 0$, then we should deem $\mathbf{x}$ better than $\mathbf{y}$ whenever $\Delta_+ + \Delta_- > 0$. We should also deem $\mathbf{x}$ better than $\mathbf{y}$ when the sum of positive difference is infinite and the sum of negative difference converges to some finite number. Similarly, the Differential Indifference principle states that, when $\Delta_+$ and $|\Delta_-|$ are both finite and equal to each other, we should be indifferent between $\mathbf{x}$ and $\mathbf{y}$. The Weak Utilitarian Catching-Up criterion satisfies these principles since, by Riemann's series theorem, unconditional and absolute convergence are equivalent.

To illustrate the effect of Relative Anonymity in this setting, consider $\mathbf{x} = (1, 0, 1, 0, \dots)$ and $\mathbf{y} = (0, 1, 0, 1, \dots)$. Then $\liminf_{n \to +\infty} (v_n^u(\mathbf{x}) - v_n^u(\mathbf{y})) = 0$ and $\liminf_{n \to +\infty} (v_n^u(\mathbf{y}) - v_n^u(\mathbf{x})) = u(0) - u(1) < 0$, implying that $\mathbf{x}$ is better than $\mathbf{y}$ according to the Utilitarian Catching-Up criterion. However, there exists $\pi \in \Pi$ such that $\mathbf{x}_\pi = (1, 1, 0, 1, 1, 0, \dots)$ and $\mathbf{y}_\pi = (0, 0, 1, 0, 0, 1, \dots)$ so that $v_n^u(\mathbf{x}_\pi) - v_n^u(\mathbf{y}_\pi) \to +\infty$ as $n \to +\infty$. Also, there exists $\pi' \in \Pi$ such that $\mathbf{x}_{\pi'} = (0, 0, 1, 0, 0, 1, \dots)$ and $\mathbf{y}_{\pi'} = (1, 1, 0, 1, 1, 0, \dots)$ so that $v_n^u(\mathbf{x}_{\pi'}) - v_n^u(\mathbf{y}_{\pi'}) \to -\infty$ as $n \to +\infty$. Hence, $\mathbf{x}$ and $\mathbf{y}$ are not comparable according to the Weak Utilitarian Catching-Up criterion. The mathematical reason for why Relative Anonymity matters is that the sum of difference between the transformed well-being of $\mathbf{x}$ and $\mathbf{y}$ does not unconditionally converge (nor diverge to one of $+\infty$ or $-\infty$). Conceptually, this means that the comparison of $\mathbf{x}$ and $\mathbf{y}$ made by the Utilitarian Catching-Up criterion depends on the order in which the individuals are counted.

Relative Anonymity and Weak Utilitarian Catching-Up rely on the idea that a person at a given component in $\mathbf{x}$ is the same as the person at that component in $\mathbf{y}$, as we apply the same permutation to the profiles. Hence, they assume a fixed population structure. But in the third interpretation of profiles in the introduction, there is no natural 1-to-1 correspondence between the components of different profiles. This interpretation can be permitted by considering strongly anonymous criteria. As we noted earlier, Asheim, Kamaga, and Zuber (2022) show that Strong Anonymity, when combined with Monotonicity, the continuity axioms, and the equity axiom, leads us to disregard well-being levels above the limit inferior. This motivates criteria defined on the derived rank-ordered profiles $\mathbf{x}_{[1]}$ and $\mathbf{y}_{[1]}$, as exemplified by the following class of SWRs.
**Definition 1.** The Strongly Anonymous (Generalized) Utilitarian Catching-Up SWR $\gtrsim^C$. There exists $u \in U$ such that, for any profiles $\mathbf{x}, \mathbf{y} \in \mathbf{X}, \mathbf{x} \gtrsim^C \mathbf{y}$ if and only if

$$ \liminf_{n \to +\infty} (w_n^u(\mathbf{x}) - w_n^u(\mathbf{y})) = \liminf_{n \to +\infty} (v_n^u(\mathbf{x}_{[n]}) - v_n^u(\mathbf{y}_{[n]})) \ge 0. $$

By the definition of the sequence of functions $w_n^u: \mathbf{X} \to \mathbb{R}$, for $n \in \mathbb{N}$, $\gtrsim^C$ satisfies the axiom of Strong Anonymity as $\mathbf{x}_{[n]}$ is not only the rank-ordered profile derived from $\mathbf{x}$ but also the rank-ordered profile derived from $\mathbf{x}_\pi$ for any $\pi \in \Pi$. If $\mathbf{x} = (1, 0, 1, 0, \dots)$ and $\mathbf{y} = (0, 1, 0, 1, \dots)$, which are equally good by Strong Anonymity, then we obtain $\mathbf{x} \sim^C \mathbf{y}$ as $\mathbf{x}_{[n]} = \mathbf{y}_{[n]} = (0, 0, 0, 0, \dots)$.

However, our axiomatization will lead us to a criterion that is not only strongly anonymous, but which does not even depend on the order of ranks. In particular, we should not give priority to component [1] by always listing this component first, as $\gtrsim^C$ does. In analogy to the difference between Utilitarian Catching-Up and Weak Utilitarian Catching-Up, this leads us naturally to the following class of SWRs that add Relative Anonymity to the 1-to-1 correspondence between the ranks of the derived streams $\mathbf{x}_{[n]}$ and $\mathbf{y}_{[n]}$.

**Definition 2.** The Strongly Anonymous (Generalized) Utilitarian SWR $\gtrsim^U$. There exists $u \in U$ such that, for any profiles $\mathbf{x}, \mathbf{y} \in \mathbf{X}, \mathbf{x} \gtrsim^U \mathbf{y}$ if and only if there exists $\Delta \in \mathbb{R}_+ \cup \{+\infty\}$ such that

$$ \text{for all } \pi \in \Pi, \quad \lim_{n \to +\infty} (v_n^u(\mathbf{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) = \Delta. $$

If, for all $\pi \in \Pi$, $\lim_{n \to +\infty} (v_n^u(\mathbf{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) = \Delta$, then $\lim_{n \to +\infty} (v_n^u(\mathbf{x}_{[n]}) - v_n^u(\mathbf{y}_{[n]})) = \Delta$. Hence, the following result is an immediate consequence of Definitions 1 and 2.

**Observation 1.** For any $u \in U$, the Strongly Anonymous (Generalized) Utilitarian SWR $\gtrsim^U$ associated with $u$ is a subrelation to the Strongly Anonymous (Generalized) Utilitarian Catching-Up SWR $\gtrsim^C$ associated with $u$.

Let $\mathbf{D} = \{(\mathbf{x}, \mathbf{y}) \in \mathbf{X} \times \mathbf{X} : \text{ there exists } \delta \in \mathbb{R}_+ \text{ such that } \sum_{i: y_i > x_i} (y_i - x_i) = \delta\}$. Then, as we prove in Appendix A.3, we can characterize Strongly Anonymous Utilitarianism as follows.

**Lemma 1.** Let the Strongly Anonymous (Generalized) Utilitarian SWR $\gtrsim^U$ be associated with the transformation function $u \in U$. For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}, \mathbf{x} \gtrsim^U \mathbf{y}$ if and only if $(\mathbf{x}_{[n]}, \mathbf{y}_{[n]}) \in \mathbf{D}$ and

$$ \lim_{n \to +\infty} (w_n^u(\mathbf{x}) - w_n^u(\mathbf{y})) \ge 0. $$

Strongly Anonymous Utilitarianism can thus be viewed as an application of the principles of Differential Betterness and Differential Indifference, but to the derived rank-ordered profiles $\mathbf{x}_{[n]}$ and $\mathbf{y}_{[n]}$.

To illustrate why Strongly Anonymous Utilitarian Catching-Up must be modified so as to satisfy Relative Anonymity for derived rank-ordered profiles—which we add
when moving from this criterion to Strongly Anonymous Utilitarianism, having the effect that the order of ranks does not matter—we use an example that is a bit more complicated. Consider the SWRS $\succsim^C$ and $\succsim^U$ defined for the identity transformation function $u(z) = z$ and let the profiles $\mathbf{x}$ and $\mathbf{y}$ be given by (see Example 1 of subsection 5.2 for formal definitions of the two profiles):

$$ 
\begin{aligned}
\mathbf{x} &= (z, \underbrace{\frac{2}{3}, \frac{2}{3}}_{2 \text{ components}}, \underbrace{\frac{2}{3}, \dots, \frac{2}{3}}_{3 \text{ components}}, \underbrace{\frac{4}{5}, \dots, \frac{4}{5}}_{4 \text{ components}}, \underbrace{\frac{4}{5}, \dots, \frac{4}{5}}_{5 \text{ components}}, \dots) \\
\mathbf{y} &= (\frac{1}{2}, \underbrace{\frac{1}{2}, \frac{1}{2}}_{\frac{1}{2}}, \underbrace{\frac{3}{4}, \dots, \frac{3}{4}}_{\frac{3}{4}}, \underbrace{\frac{3}{4}, \dots, \frac{3}{4}}_{\frac{3}{4}}, \underbrace{\frac{5}{6}, \dots, \frac{5}{6}}_{\dots}, \dots)
\end{aligned}
$$

where $z \le \frac{2}{3}$ so that both $\mathbf{x}$ and $\mathbf{y}$ are non-decreasing. Note that

$$ \mathbf{x} - \mathbf{y} = (z - \frac{1}{2}, \underbrace{\frac{1}{6}, \frac{1}{6}}_{2 \text{ components}}, \underbrace{-\frac{1}{12}, \dots, -\frac{1}{12}}_{3 \text{ components}}, \underbrace{\frac{1}{20}, \dots, \frac{1}{20}}_{4 \text{ components}}, \underbrace{-\frac{1}{30}, \dots, -\frac{1}{30}}_{5 \text{ components}}, \dots). $$

Hence, the first component of $\mathbf{x} - \mathbf{y}$ equals $z - \frac{1}{2}$, the two next $\frac{1}{6}$ summing to $\frac{1}{3}$, the three next $-\frac{1}{12}$ summing to $-\frac{1}{4}$, the four next $\frac{1}{20}$ summing to $\frac{1}{5}$, and so on. Hence,

$$ \sum_{i=1}^{\infty} (x_i - y_i) = z - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots = z - 1 + \ln 2, $$

implying that $\mathbf{x} \succsim^C \mathbf{y}$ if and only if $\frac{2}{3} \ge z \ge 1 - \ln 2 \approx 0.30685$, since both $\mathbf{x}$ and $\mathbf{y}$ are non-decreasing so that $\mathbf{x} = \mathbf{x}_{[1]}$ and $\mathbf{y} = \mathbf{y}_{[1]}$. However, such an alternating harmonic series is not absolute convergent, as each of the series $1, \frac{1}{3}, \frac{1}{5}, \dots$ and $-\frac{1}{2}, -\frac{1}{4}, -\frac{1}{6}, \dots$ diverges, and thus not unconditionally convergent by the Riemann series theorem. Hence, $\succsim^U$ deems $\mathbf{x}$ and $\mathbf{y}$ non-comparable for any $z \le \frac{2}{3}$.

Another version of strongly anonymous utilitarianism is given by the following class of SWRSs, which combines a finite application of $w_n^u$ and dominance for the tails of the derived rank-ordered profiles.

**Definition 3.** The Strongly Anonymous (Generalized) Utilitarian Dominance SWR $\succsim^D$. There exists $u \in U$ such that, for any profiles $\mathbf{x}, \mathbf{y} \in \mathbf{X}, \mathbf{x} \succsim^D \mathbf{y}$ if and only if there exists $n \in \mathbb{N}$ such that

$$ w_n^u(\mathbf{x}) - w_n^u(\mathbf{y}) \ge 0 \quad \text{and} \quad x_{[r]} \ge y_{[r]} \quad \text{for all } r > n. $$

Strongly Anonymous Utilitarian Dominance is related to the Basu-Mitra utilitarian criterion for infinite well-being profiles. Basu and Mitra (2007a) define their criterion by combining a finite application of classical utilitarianism and dominance for the tails of the profiles $\mathbf{x}$ and $\mathbf{y}$.<fnref n="10" /> The main difference between Strongly Anonymous Utilitarian Dominance and the Basu-Mitra utilitarian criterion is that the former satisfies Strong Anonymity as a strong impartiality requirement.

<footnote n="10">¹⁰See also the transfer-sensitive criteria proposed by Bossert, Sprumont, and Suzumura (2007) and a generalized representation presented by d'Aspremont (2007).</footnote>
Lemma 1 implies that when, for some $n \in \mathbb{N}$, $w_n^u(\mathbf{x}) \ge w_n^u(\mathbf{y})$ ($w_n^u(\mathbf{x}) > w_n^u(\mathbf{y})$ respectively) and $x_{[r]} \ge y_{[r]}$ for all $r > n$, we have $\mathbf{x} \succsim^U \mathbf{y}$ ($\mathbf{x} \succsim^U \mathbf{y}$ respectively) for $\succsim^U$ associated with $u \in U$. We therefore obtain the following result.

**Observation 2.** For any $u \in U$, the Strongly Anonymous (Generalized) Utilitarian Dominance SWR $\succsim^D$ associated with $u$ is a subrelation to the Strongly Anonymous (Generalized) Utilitarian SWR $\succsim^U$ associated with $u$.

However, Strongly Anonymous Utilitarian Dominance does not satisfy an intuitive minimal equity condition. In particular, if $\mathbf{x} = (0,0,\frac{1}{2},\frac{3}{4},\frac{7}{8},...)$ and $\mathbf{y} = (z,1,\frac{1}{2},\frac{3}{4},\frac{7}{8},...)$, with $z < -1$, and with $u$ being the identity function, we do not have $\mathbf{x} \succsim^D \mathbf{y}$ even though $z$ can be arbitrarily low.<fnref n="11" /> This is so because $x_{[1]} = (0,0,\frac{1}{2},\frac{3}{4},\frac{7}{8},...)$ and $y_{[1]} = (z,\frac{1}{2},\frac{3}{4},\frac{7}{8},...)$, so that $w_n^u(\mathbf{x}) > w_n^u(\mathbf{y})$ for all $n \in \mathbb{N}$ but $y_{[r]} > x_{[r]}$ for all $r > 1$. This problem does not arise with Strongly Anonymous Utilitarianism; indeed, with $u$ being the identity function, $\mathbf{x} \succsim^U \mathbf{y}$ since

$$ \text{for all } \pi \in \Pi, \lim_{n \to +\infty} (v_n^u(\mathbf{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) = 0 - z - \sum_{r=2}^{\infty} (\frac{1}{2})^{r-1} = -z - 1 > 0. $$

It is useful to remark that, independently of $u$, if $\ell(\mathbf{x}) > \ell(\mathbf{y})$, then $\mathbf{x} \succsim^D \mathbf{y}$ and, therefore, $\mathbf{x} \succsim^U \mathbf{y}$ because $\succsim^D$ is a subrelation to $\succsim^U$. Indeed, if $\ell(\mathbf{x}) > \ell(\mathbf{y})$, then there exists $n \in \mathbb{N}$ such that, for all $r \ge n$ $x_{[r]} > \ell(\mathbf{y}) \ge y_{[r]}$ so that, for some $m \ge n$ $w_m^u(\mathbf{x}) \ge w_m^u(\mathbf{y})$ and $x_{[r]} > y_{[r]}$ for all $r \ge m$. We also have that $\mathbf{x} \succsim^U \mathbf{y}$ implies $\ell(\mathbf{x}) \ge \ell(\mathbf{y})$.

## 3 Application to the Ramsey model

We demonstrate the applicability of Strongly Anonymous Utilitarianism by showing that it gives rise to a nonempty set of optimal streams in the Ramsey model of economic growth. Furthermore, if the initial stock of reproducible capital does not exceed the one corresponding to the Golden Rule, then any optimal stream satisfies that well-being cannot be increased at some time $i$ without being decreased at some other time $i'$. Hence, an optimal stream is in this sense efficient, even though the Strongly Anonymous Utilitarian SWR does not satisfy the strong Pareto principle.

In the Ramsey model, the law of motion governing the stock of reproducible capital, $k$, is given by a standard increasing, concave gross production function, $f$. Following Basu and Mitra (2007a, Section 5), the function $f: \mathbb{R}_+ \to \mathbb{R}_+$ is assumed to satisfy:

(i) $f(0) = 0$,

(ii) $f$ is continuous, increasing and strictly concave on $\mathbb{R}_+$,

(iii) $\lim_{k \to 0} \frac{f(k)}{k} > 1$ and $\lim_{k \to \infty} \frac{f(k)}{k} < 1$.

<footnote n="11">¹¹The worst-off component can thus face a very large loss when going from $\mathbf{x}$ to $\mathbf{y}$ while the second component has a small gain, and everyone else is indifferent, without $\succsim^D$ deeming $\mathbf{x}$ at least as good as $\mathbf{y}$. Hence, the Strongly Anonymous Utilitarian Dominance SWR does not satisfy the axiom of *Limited Inequality*, defined in subsection 4.3.</footnote>
Then there exists a unique number $\bar{k} > 0$ such that $f(\bar{k}) = \bar{k}$ and $f(k) > k$ for $k \in (0, \bar{k})$.

A feasible stream from $k \in [0, \bar{k}]$ is a sequence of capital stocks $\mathbf{k} = (k_1, \dots, k_i, \dots)$ satisfying:

$$k_1 = k, \quad 0 \le k_{i+1} \le f(k_i) \quad \text{for } i \in \mathbb{N}.$$

It follows from the definition of $\bar{k}$ that $\mathbf{k} \in [0, \bar{k}]^N$. Hence, $\bar{k}$ is the maximal attainable capital stock if one starts from an initial stock in $[0, \bar{k}]$. Associated with a feasible stream $\mathbf{k}$ from $k \in [0, \bar{k}]$ is a feasible well-being stream $\mathbf{x} = (x_1, \dots, x_i, \dots)$ from $k$, defined by

$$x_i = f(k_i) - k_{i+1} \quad \text{for } i \in \mathbb{N}.$$

For any $k \in [0, \bar{k}]$, the set of well-being streams associated with feasible capital streams from $k$ is contained in $[0, \bar{k}]^N$, keeping in mind that $f(\bar{k}) = \bar{k}$. We say that a feasible well-being stream $\hat{\mathbf{x}}$ from $k$ is optimal if, for all feasible well-being streams $\mathbf{x}$ from $k$, it is the case that $\hat{\mathbf{x}} \stackrel{U}{\sim} \mathbf{x}$.

It follows from the continuity and strict concavity of $f$ that there exists a unique number $k^* \in [0, \bar{k}]$ such that $f(k) - k$ is increasing on $[0, k^*]$ and decreasing on $[k^*, \bar{k}]$, implying that $f(k^*) - k^* \ge f(k) - k$ for all $k \in [0, \bar{k}]$. Since, for any $k \in (0, \bar{k})$, $f(k) - k > 0 = f(0) - 0 = f(\bar{k}) - \bar{k}$, we have that $0 < k^* < \bar{k}$. We write $x^* = f(k^*) - k^*$. By keeping the capital stock constant at $k^*$, a maximum sustainable well-being equal to $x^*$ is attained; this corresponds to the Golden Rule.

For any $\mathbf{x} \in [0, \bar{k}]^N$, let $\ell(\mathbf{x})$ denote the limit infimum of $\mathbf{x}$. Let $\xi: [0, \bar{k}] \to [0, \bar{k}]$ be a stationary well-being strategy that, for any $i \in \mathbb{N}$ and any $k_i \in [0, \bar{k}]$, determines $x_i = \xi(k_i)$, where $\xi(k) = 0$ if $k < k^*$ and $\xi(k) = x^*$ if $k \ge k^*$. Let $\mathbf{x}^\xi(k)$ be the feasible well-being stream from $k$ determined by the strategy $\xi$. If $k \ge k^*$, then $\mathbf{x}^\xi(k) = x^* \mathbb{1}_N$. If $0 < k < k^*$, then there exists $n \in \mathbb{N}$ such that $x_i^\xi(k) = 0$ for $i = 1, \dots, n$, and $x_i^\xi(k) = x^*$ for $i > n$. If $k = 0$, then $\mathbf{x}^\xi(k) = 0 \mathbb{1}_N$. Note that, $\ell(\mathbf{x}^\xi(k)) = x^*$ for $k \in (0, \bar{k}]$, while $\ell(\mathbf{x}^\xi(0)) = 0$.

**Lemma 2.** For any $k \in (0, \bar{k}]$, if $\mathbf{x}$ is a feasible well-being stream from $k$, then $\ell(\mathbf{x}) \le x^*$.

*Proof.* Fix $k \in (0, \bar{k}]$. It is sufficient to show that there is no feasible $\mathbf{x}$ from $k$ with $\ell(\mathbf{x}) > x^*$. By way of contradiction, suppose that there exists some feasible $\mathbf{x}'$ from $k$ with $\ell(\mathbf{x}') > x^*$. Write $\varepsilon = (\ell(\mathbf{x}') - x^*)/2$. Then there exists $n \in \mathbb{N}$ such that, for all $i > n$, $x_i' - x^* \ge \varepsilon > 0$. Since $f(k^*) - k^* \ge f(k) - k$ for all $k \in [0, \bar{k}]$, this implies:

$$0 < \varepsilon \le x_i' - x^* = f(k_i') - k_{i+1}' - f(k^*) + k^* \le f(k_i') - k_{i+1}' - f(k_i') + k_i' = k_i' - k_{i+1}'$$

Hence, the capital stock will be depleted in a finite time, contradicting the feasibility of $\mathbf{x}'$. $\square$

In his seminal work on optimal growth, Ramsey (1928) introduced an undiscounted utilitarian criterion which computes the total amount by which utility falls short of the maximum possible utility level, referred to as “bliss.” To show Proposition 1 below, we introduce a modified Ramsey criterion where we instead associate bliss with the maximum possible sustainable level of utility $u(x^*)$, given the production function
f and the transformation function $u \in U$. Hence, consider the complete preorder represented by $v^{\bar{u}}: X \to \mathbb{R}_- \cup \{-\infty\}$ where, for all $x \in X$,

$$v^{\bar{u}}(\mathbf{x}) = \lim_{n \to \infty} v_n^{\bar{u}}(\mathbf{x}) = \lim_{n \to \infty} \sum_{i=0}^{n} \bar{u}(x_i),$$

with the non-decreasing transformation function $\bar{u}: \mathbb{R} \to \mathbb{R}_-$ being defined by $\bar{u}(x) = u(x) - u(x^*)$ for all $x < x^*$ and $\bar{u}(x) = 0$ for all $x \ge x^*$. As an aside, $v^{\bar{u}}$ is well-defined and always takes a non-positive value (it may be $-\infty$); indeed it is an infinite sum of non-positive real numbers.

Our strategy of proof is to show that there is a feasible well-being stream $\hat{\mathbf{x}}$ which satisfies that $v^{\bar{u}}(\hat{\mathbf{x}})$ is finite and $v^{\bar{u}}(\hat{\mathbf{x}}) \ge v^{\bar{u}}(\mathbf{x})$ for all feasible well-being streams $\mathbf{x}$, followed by a demonstration that $\hat{\mathbf{x}}$ is optimal under Strongly Anonymous Utilitarianism. For any $k \in (0, \bar{k}]$, let $\mathbf{X}^*(k)$ denote the set of feasible well-being streams $\mathbf{x}$ from $k$ which satisfy that $x_i \in [0, x^*]$ for all $i \in \mathbb{N}$ and $\ell(\mathbf{x}) = x^*$. The following existence result is proven in Appendix A.1.

**Lemma 3.** For any $k \in (0, k^*]$, there exists a feasible well-being stream $\hat{\mathbf{x}}$ from $k$ such that $v^{\bar{u}}(\hat{\mathbf{x}}) > -\infty$ and $v^{\bar{u}}(\hat{\mathbf{x}}) \ge v^{\bar{u}}(\mathbf{x})$ for all feasible well-being streams $\mathbf{x}$ from $k$. Furthermore, $\hat{\mathbf{x}} \in \mathbf{X}^*(\mathbf{x})$, it is non-decreasing, and there is no feasible well-being stream $\mathbf{x}'$ from $k$ such that $\mathbf{x}' \ge \hat{\mathbf{x}}$.

By Lemma 2, we know that, for any $k \in (0, \bar{k}]$ and any feasible stream $\mathbf{x}$ from $k$, we have $\ell(\mathbf{x}) \le x^*$. This implies that $\mathbf{x}_{[1]} \le x^* \mathbb{1}_{\mathbb{N}}$. Furthermore, for any two feasible streams from $k$, $\mathbf{x}$ and $\mathbf{x}'$, we have that $v^{\bar{u}}(\mathbf{x}_{[1]}) \ge v^{\bar{u}}(\mathbf{x}'_{[1]}) > -\infty$ implies $\mathbf{x} \succsim^U \mathbf{x}'$.<fnref n="12" /> Likewise, $v^{\bar{u}}(\mathbf{x}_{[1]}) > v^{\bar{u}}(\mathbf{x}'_{[1]})$ implies $\mathbf{x} \succ^U \mathbf{x}'$, from which it follows that $\mathbf{x} \succsim^U \mathbf{x}'$ implies $v^{\bar{u}}(\mathbf{x}_{[1]}) \ge v^{\bar{u}}(\mathbf{x}'_{[1]})$.

The main result of this section is that there exists an optimal Strongly Anonymous Utilitarian well-being stream in the Ramsey model.<fnref n="13" />

**Proposition 1.** Fix $u \in U$ and consider the Strongly Anonymous (Generalized) Utilitarian SWR $\succsim^U$ associated with $u$.

1. For any $k \in (k^*, \bar{k}]$, any feasible well-being stream $\hat{\mathbf{x}}$ from $k$ with $\hat{\mathbf{x}} \ge x^* \mathbb{1}_{\mathbb{N}}$ satisfies that $\hat{\mathbf{x}} \succsim^U \mathbf{x}$ for all feasible well-being streams $\mathbf{x}$ from $k$. The stream $\mathbf{x}^\xi(k)$ is one of these.

2. For any $k \in (0, k^*]$, there exists a feasible well-being stream $\hat{\mathbf{x}}$ from $k$ which satisfies that $\hat{\mathbf{x}} \succsim^U \mathbf{x}$ for all feasible well-being streams $\mathbf{x}$ from $k$. Furthermore, any such optimal stream is non-decreasing, and there is no feasible well-being stream $\mathbf{x}'$ from $k$ such that $\mathbf{x}' > \hat{\mathbf{x}}$.

<footnote n="12">¹²Recall that $\bar{u}(x) = u(x) - u(x^*)$ for all $x \in (-\infty, x^*]$ and that $\mathbf{x}_{[1]} \in (-\infty, x^*]^N$. The series $\sum_{r=1}^n (u(x_{[r]}) - u(x^*))$ is absolutely convergent when it is finite, because $(u(x_{[r]}) - u(x^*)) = -|u(x_{[r]}) - u(x^*)|$ for any $r$. Since the difference of two absolutely convergent series is also absolutely convergent (and thus unconditionally convergent), this implies that, for all $\pi \in \Pi$, $\lim_{n \to +\infty} (v_n^u(\mathbf{x}_{[\pi]}) - v_n^u(\mathbf{x}'_{[\pi]}) = v^{\bar{u}}(\mathbf{x}_{[1]}) - v^{\bar{u}}(\mathbf{x}'_{[1]})$, when $v^{\bar{u}}(\mathbf{x}_{[1]})$ and $v^{\bar{u}}(\mathbf{x}'_{[1]})$ are finite.</footnote>

<footnote n="13">¹³We consider only streams for an initial stock $k \in (0, \bar{k}]$. If $k = 0$, then the only feasible stream is $\mathbf{x}^\xi(0) = \mathbb{1}_{\mathbb{N}}$, which is trivially optimal.</footnote>
*Proof. Part 1.* Assume that $k \in (k^*, \bar{k}]$. Let $\hat{\mathbf{x}}$ be feasible from $k$ with $\hat{\mathbf{x}} \ge x^* 1_N$. Then $\mathbf{x}[\cdot] = x^* 1_N$. By Lemma 2, we know that, for any feasible $\mathbf{x}$ from $k$, $\ell(\mathbf{x}) \le x^*$. Hence, for any feasible $\mathbf{x}$ from $k$, $\mathbf{x}[\cdot] \le x^* 1_N$, so that $\mathbf{x}[\cdot] \le \hat{\mathbf{x}}[\cdot]$ and $\hat{\mathbf{x}} \gtrsim_U \mathbf{x}$. There exist feasible $\hat{\mathbf{x}}$ from $k$ with $\hat{\mathbf{x}} \ge x^* 1_N$ since $\mathbf{x}^\xi(k) = x^* 1_N$ is feasible from $k$. Indeed, $\mathbf{x}^\xi(k) \gtrsim_U \mathbf{x}$ for all feasible $\mathbf{x}$ from $k$.

*Part 2.* Assume that $k \in (0, k^*]$. Statement 2 is proven through three steps.

*Step 1:* If $\mathbf{x} \notin \mathbf{X}^*(k)$ is a feasible well-being stream from $k$, then there exists a well-being stream $\mathbf{x}' \in \mathbf{X}^*(k)$ such that $\mathbf{x}' \succ_U \mathbf{x}$. By Lemma 2, we know that, for any feasible $\mathbf{x}$ from $k$, $\ell(\mathbf{x}) \le x^*$. If $\ell(\mathbf{x}) < x^*$, then $\mathbf{x}^\xi(k) \succ_U \mathbf{x}$ because $\ell(\mathbf{x}^\xi(k)) = x^* > \ell(\mathbf{x})$. If $\ell(\mathbf{x}) = x^*$ and $\mathbf{x} \notin \mathbf{X}^*(k)$, then $x_i > x^*$ for some $i \in \mathbb{N}$. Since no stream $\mathbf{x} > x^* 1_N$ is feasible from $k \le k^*$, there exists $i'$ such that $x_{i'} < x^*$. Furthermore, because $f$ is increasing, there exists $\varepsilon$ such that $\mathbf{x}'$ defined as follows is feasible from $k$: $x_i' = x^*$ for all $i \in \mathbb{N}$ such that $x_i \ge x^*$, $x_{i'}' = x_{i'} + \varepsilon \le x^*$, and $x_j' = x_j$ for all other $j$. By construction, $\mathbf{x}' \in \mathbf{X}^*(k)$. But $\mathbf{x}'[\cdot] > \mathbf{x}[\cdot]$, implying that $\mathbf{x}'[\cdot] \succ_U \mathbf{x}[\cdot]$.

*Step 2:* There exists an optimal well-being stream in $\mathbf{X}^*(k)$. Note that, for any $\mathbf{x} \in \mathbf{X}^*(k)$, we have $v^{\bar{u}}(\mathbf{x}[\cdot]) = v^{\bar{u}}(\mathbf{x})$. By Lemma 3, we know that there exists $\hat{\mathbf{x}} \in \mathbf{X}^*(k)$ such that $v^{\bar{u}}(\hat{\mathbf{x}}) > -\infty$ and $v^{\bar{u}}(\hat{\mathbf{x}}) \ge v^{\bar{u}}(\mathbf{x})$ for all feasible $\mathbf{x}$ from $k$. This implies $v^{\bar{u}}(\hat{\mathbf{x}}[\cdot]) \ge v^{\bar{u}}(\mathbf{x}[\cdot])$ and thus $\hat{\mathbf{x}} \gtrsim_U \mathbf{x}$ for all $\mathbf{x} \in \mathbf{X}^*(k)$. Combined with Step 1 and using the transitivity of $\gtrsim^U$, this implies that $\hat{\mathbf{x}} \gtrsim_U \mathbf{x}$ for all feasible $\mathbf{x}$ from $k$.

*Step 3:* Properties of an optimal well-being stream. Assume that $\hat{\mathbf{x}}$ feasible from $k$ satisfies that $\hat{\mathbf{x}} \gtrsim_U \mathbf{x}$ for all feasible $\mathbf{x}$ from $k$. Since, for any $\mathbf{x} \in \mathbf{X}^*(k)$, we have $v^{\bar{u}}(\mathbf{x}[\cdot]) = v^{\bar{u}}(\mathbf{x})$, it follows that $v^{\bar{u}}(\hat{\mathbf{x}}) \ge v^{\bar{u}}(\hat{\mathbf{x}}[\cdot]) \ge v^{\bar{u}}(\mathbf{x}[\cdot]) = v^{\bar{u}}(\mathbf{x})$ for all $\mathbf{x} \in \mathbf{X}^*(k)$. If $\mathbf{x} \notin \mathbf{X}^*(k)$ is feasible from $k$, then it follows from Step 3 of the proof of Lemma 3 that there exists $\mathbf{x}' \in \mathbf{X}^*(k)$ such that $v^{\bar{u}}(\mathbf{x}') > v^{\bar{u}}(\mathbf{x})$. Hence, an optimal well-being stream from $k$ satisfies the properties of $\hat{\mathbf{x}}$ of Lemma 3: An optimal $\hat{\mathbf{x}}$ from $k$ is non-decreasing, and there is no feasible $\mathbf{x}'$ from $k$ such that $\mathbf{x}' \ge \hat{\mathbf{x}}$. $\square$

A first implication of Proposition 1 is that, although the Strongly Anonymous Utilitarian SWR $\gtrsim^U$ cannot compare all well-being streams, it has sufficient comparability to yield an optimal stream that is at least as good as any feasible well-being stream in the Ramsey model.<fnref n="14" /> A second implication of Proposition 1 is that Strongly Anonymous Utilitarianism yields an optimal stream which is both efficient and non-decreasing if the initial capital stock $k$ does not exceed the one corresponding to the Golden Rule.<fnref n="15" /> This possibility of combining Strong Anonymity with efficiency is related to Asheim, Bossert, Sprumont, and Suzumura (2010, Theorem 2). They show that in the Ramsey model there exist infinite-horizon choice functions satisfying the Suppes-Sen principle, even when this principle allows for infinite permutations, when $f: \mathbb{R}_+ \to \mathbb{R}_+$ in addition to (i) and (ii) is differentiable and satisfies $\inf_{t \ge 0} f'(k) > 1$.

<footnote n="14">¹⁴Since, by Proposition 2 of subsection 5.2, the Strongly Anonymous Utilitarian SWR is a subrelation to $\gtrsim^C$ as well as the Limit of Rank-Discounted Utilitarian SWR $\gtrsim^R$ and the Strongly Anonymous (Generalized) Utilitarian Cesàro Summation SWR $\gtrsim^S$ introduced in subsection 5.1, the results of Proposition 1 hold also for these SWRs.</footnote>

<footnote n="15">¹⁵Proposition 1 makes a distinction between two cases. In the other case, with $k \in (k^*, \bar{k}]$, we have that any feasible stream with well-being never falling below the highest sustainable well-being level, $x^*$, is optimal. The constant well-being stream at the highest sustainable well-being level, $x^*$, is feasible. But although it is optimal, the stream $x^* 1_N$ is not efficient for $k > k^*$. Indeed, the well-being stream $\mathbf{x}'$ such that $x_1' = f(k) - k^*$ and $x_i' = x^*$ for all $i \ge 2$ is feasible. And $x_1' = f(k) - k^* > f(k^*) - k^* = x^*$, so that $\mathbf{x}' > x^* 1_N$. Note that $\mathbf{x}'$ is also optimal.</footnote>
Under this assumption, there exists no Golden Rule stock $k^*$ and only the second implication of Proposition 1 applies.<fnref n="16" />

# 4 Axiomatic characterization

In this section we provide an axiomatic characterization of the Strongly Anonymous (Generalized) Utilitarian SWR.

## 4.1 Regularity axioms

We impose five regularity axioms. Firstly, as we consider settings where there might not be any natural 1-to-1 correspondence between the components of different well-being profiles, we impose that any SWR must satisfy the conjunction of the following two axioms.

**Axiom (Strong Anonymity).** For any $\pi \in \Pi$ and $x \in X$, $x \sim x_\pi$.

**Axiom (Monotonicity).** For any $x, y \in X$, if $x > y$, then $x \gtrsim y$.

Jointly, they imply that a profile is at least as good as another if some reordering of the one profile makes its well-beings as high as the other when the comparison is made component-by-component.

On the domain of bounded infinite well-being profiles SWRs need not be complete. However, we impose that any SWR must be complete for profiles that have the same tail.

**Axiom (Finite Completeness).** For any $x, y, z \in X$ and any $n \in \mathbb{N}$, either $x_n z \gtrsim y_n z$, or $y_n z \gtrsim x_n z$, or both.

As is well-known, utilitarian SWRs over infinite well-being profiles are not continuous in the topology of uniform convergence. Instead, we impose two weaker continuity axioms, both of which are implied by continuity in the topology of uniform convergence for any monotone SWR (Asheim, Kamaga, and Zuber, 2022, Lemma 6). To present the first of these, we let $d_1: X \times X \to \mathbb{R}_+$ be the distance function given by, for all $x, y \in X$,

$$d_1(x, y) = \min \left\{ 1, \sum_{i=1}^{\infty} |x_i - y_i| \right\}.$$

Using the distance function $d_1$, we define weak continuity as follows; see Svensson (1980).

**Axiom (Weak Continuity).** For any $x \in X$, the sets $\{y \in X : y \gtrsim x\}$ and $\{y \in X : x \gtrsim y\}$ are closed in $(X, d_1)$.

The second continuity axiom applies when combining two profiles with a constant profile. It requires that indifference between two such profiles when combined with

<footnote n="16">¹⁶In their analyses of the Ramsey model, Dubey and Mitra (2013) and Banerjee (2017) show that non-monotone streams can be maximal under Finite Anonymity and the strong Pareto principle. Such non-monotone streams will not be optimal under Strongly Anonymous Utilitarianism, being non-sensitive for well-being above the Golden Rule level $x^*$, as the limit inferior of any profile cannot exceed $x^*$. Hence, any stream which involves the capital stock moving back and forth across the Golden Rule stock $k^*$, with well-being alternating between 0 and a level exceeding the Golden Rule level, as considered by Banerjee (2017), is not a Strongly Anonymous Utilitarian optimum.</footnote>
constant profiles in the neighborhood of this particular constant profile should be preserved in the limit.

**Axiom (Restricted Continuity).** For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, any $\ell \in \mathbb{R}$, and any $\underline{\ell} < \bar{\ell}$ such that $\underline{\ell} \le \ell \le \bar{\ell}$, if $(\ell' \mathbb{1}_{\mathbb{N}}, \mathbf{x}) \sim (\ell' \mathbb{1}_{\mathbb{N}}, \mathbf{y})$ for all $\ell' \in (\underline{\ell}, \bar{\ell}) \setminus \{\ell\}$, then $(\ell \mathbb{1}_{\mathbb{N}}, \mathbf{x}) \sim (\ell \mathbb{1}_{\mathbb{N}}, \mathbf{y})$.

**Definition 4.** An SWR $\gtrsim$ is regular whenever it satisfies Strong Anonymity, Monotonicity, Finite Completeness, Weak Continuity, and Restricted Continuity.

## 4.2 Axioms on equity and population ethics

Any SWR satisfying Strong Anonymity, Monotonicity, and Weak Continuity is insensitive to changes in components between the limit inferior and the limit superior, while permitting sensitivity to changes in components smaller than the limit inferior and greater than the limit superior (this follows by substituting Lemma 5 for Lemma 3 in the proof of Asheim, Kamaga, and Zuber, 2022, Proposition 1). Thus, preference for the combination of a tiny gain in a component above the limit superior and a large loss in a component between the limit inferior and the limit superior is allowed. The minimal equity axiom that we define below rules out such inequitable preferences.

For an SWR $\gtrsim$, let the set $\mathcal{E}$ be defined as

$$ \mathcal{E} = \{\varepsilon \in \mathbb{R}_{++} : \text{ for all } \mathbf{x}, \mathbf{y} \in \mathbf{X} \text{ with } y_i < x_i \le x_j < y_j \text{ for some } i, j \in \mathbb{N}, \text{ and } \\ x_k = y_k \text{ for all } k \in \mathbb{N} \setminus \{i, j\}, \mathbf{x} \gtrsim \mathbf{y} \text{ whenever } x_j - y_j = \varepsilon (y_i - x_i) \}, $$

The set $\mathcal{E}$ considers conditions under which a transfer from rich to poor people is acceptable. *Hammond equity* (Hammond, 1976) claims that any loss for the rich is acceptable, so that $\mathcal{E} = \mathbb{R}_{++}$. On the other hand, an SWR $\gtrsim$ satisfies the *Pigou-Dalton Transfer principle* (in its weak version) only if $1 \in \mathcal{E}$. Indeed, non-leaky transfers are always welfare improving according to the Pigou-Dalton Transfer principle. Last, utilitarian SWRs (with $u$ being the identity function) are such that $(0, 1] \in \mathcal{E}$. Hence, in all these cases, $\mathcal{E}$ is nonempty. Here we suggest a much weaker axiom than both Hammond equity and Pigou-Dalton, and which generalized utilitarianism (in the case of finite profiles) satisfies for any transformation function $u \in U$: We require that transfers from rich to poor people are acceptable if the well-being loss for the rich is sufficiently small.

**Axiom (Limited Inequity).** $\mathcal{E} \neq \emptyset$.

As discussed at the end of subsection 2.2, the Strongly Anonymous Utilitarian Dominance SWR does not satisfy this minimal equity condition. It is thus excluded from our discussion although it is a subrelation to the other criteria that we will study.

We impose also an axiom on population ethics, relating to the effect of adding one person (or infinitely many people) to a population. This kind of question has been addressed in the literature stemming from Parfit (1984) and discussed at length in Blackorby, Bossert, and Donaldson (2005). In particular, Blackorby and Donaldson (1984) and Blackorby, Bossert, and Donaldson (1995) argue in favor of the existence of a critical level, such that a person's life contributes positively to the value of a population if and only if the well-being of the additional person is above this critical
level. In general, the critical level may depend on how well-being is distributed. Our axiom imposes some regularity in the level of the critical level for infinite populations by asserting that, if it is acceptable to add one person at some level of well-being to a population, then it is also acceptable to add infinitely many people at this level.

**Axiom (Critical-Level Consistency).** For any $\mathbf{x} \in \mathbf{X}$, and any $z \in \mathbb{R}$, $\mathbf{x} \gtrsim (z, \mathbf{x})$ (resp. $\mathbf{x} \lesssim (z, \mathbf{x})$) if and only if $\mathbf{x} \gtrsim (z1_{\mathbb{N}}, \mathbf{x})$ (resp. $\mathbf{x} \lesssim (z1_{\mathbb{N}}, \mathbf{x})$).

By adding Critical-Level Consistency to Limited Inequity and the regularity axioms, it follows from the analysis of Asheim, Kamaga, and Zuber (2022) that there cannot be any sensitivity above the limit inferior: Any profile $\mathbf{x}$ is equally good as the derived reordered profile $\mathbf{x}_{[1]}$, where components that are rank-ordered, starting with the worst-off.

**Lemma 4.** If a regular SWR $\gtrsim$ satisfies Limited Inequity and Critical-Level Consistency, then, for all $\mathbf{x} \in \mathbf{X}$, $\mathbf{x} \sim \mathbf{x}_{[1]}$.

*Proof.* Assume that $\gtrsim$ is a regular SWR which satisfies Limited Inequity and Critical-Level Consistency. If $\mathbf{x} \in \mathbf{X}^{\uparrow}$, then the result follows from Strong Anonymity, because $\gtrsim$ is regular.

Let $\mathbf{x} \notin \mathbf{X}^{\uparrow}$, and consider $\tilde{\mathbf{x}}$ defined as follows: $\tilde{x}_i = x_i$ if $i \in L(\mathbf{x})$ and $\tilde{x}_i = \ell(\mathbf{x})$ if $i \in \mathbb{N} \setminus L(\mathbf{x})$. Since $\gtrsim$ is regular (and therefore satisfies Strong Anonymity, Monotonicity, Weak Continuity, and Restricted Continuity), and satisfies Limited Inequity and Critical-Level Consistency, Asheim, Kamaga, and Zuber (2022, Proposition 6) implies that, for all $\mathbf{y}, \tilde{\mathbf{y}} \in \mathbf{X}$ such that $\mathbf{y} \ge \tilde{\mathbf{y}}$, if $\ell(\mathbf{y}) = \ell(\tilde{\mathbf{y}}) = \ell$ and $\tilde{y}_i \ge \ell$ for all $i \in \mathbb{N}$ such that $y_i > \tilde{y}_i$, then $\mathbf{y} \sim \tilde{\mathbf{y}}$. Thus, we obtain that $\mathbf{x} \sim \tilde{\mathbf{x}}$. By transitivity of $\gtrsim$, it remains to be shown that $\tilde{\mathbf{x}} \sim \mathbf{x}_{[1]}$. There are three cases.

*Case 1:* $|L(\mathbf{x})| = K < +\infty$. By Strong Anonymity, $\tilde{\mathbf{x}} \sim \mathbf{x}_{[1]}$ since $\mathbf{x}_{[1]}$ is a permutation of $\tilde{\mathbf{x}}$.

*Case 2:* $|L(\mathbf{x})| = +\infty$ and $|\mathbb{N} \setminus L(\mathbf{x})| = +\infty$. The profile $(\ell 1_{\mathbb{N}}, \mathbf{x}_{[1]})$ is a permutation of $\tilde{\mathbf{x}}$, so that $(\ell 1_{\mathbb{N}}, \mathbf{x}_{[1]}) \sim \tilde{\mathbf{x}}$ by Strong Anonymity. Since $\gtrsim$ satisfies Strong Anonymity and Weak continuity, Asheim, Kamaga, and Zuber (2022, Lemma 5) implies that, for all $\mathbf{y} \in \mathbf{X}$, $\mathbf{y} \sim (\ell, \mathbf{y}) \sim (\ell 1_{\mathbb{N}}, \mathbf{y})$, where $\ell = \ell(\mathbf{y})$. Thus, $(\ell 1_{\mathbb{N}}, \mathbf{x}_{[1]}) \sim \mathbf{x}_{[1]}$ and, by transitivity of $\gtrsim$, $\tilde{\mathbf{x}} \sim \mathbf{x}_{[1]}$.

*Case 3:* $|L(\mathbf{x})| = +\infty$ and $|\mathbb{N} \setminus L(\mathbf{x})| = K < +\infty$. The following profile is a permutation of $\tilde{\mathbf{x}}$: $\mathbf{y}$ such that $y_i = \ell$ for $i = 1, \dots, K$ and $y_i = x_{[i-K]}$ for $i \ge K+1$. Therefore, by Strong Anonymity, $\mathbf{y} \sim \tilde{\mathbf{x}}$. Repeated application of Asheim, Kamaga, and Zuber (2022, Lemma 5) implies $\mathbf{y} \sim \mathbf{x}_{[1]}$ and, by transitivity of $\gtrsim$, $\tilde{\mathbf{x}} \sim \mathbf{x}_{[1]}$. $\square$

## 4.3 Sensitivity and separability axioms

The axiomatization of (generalized) utilitarian SWRs requires sensitivity and separability axioms.<fnref n="17" /> In view of Lemma 4 we can consider weak versions of such axioms, being concerned with components below the limit inferior, thus components having finite rank.

<footnote n="17">¹⁷Note that a completely insensitive SWR, deeming any profile as equally good as any other profile, is regular and satisfies Limited Equity and Critical-Level Consistency.</footnote>
The following sensitivity axiom, which is inspired by the weak dominance axiom introduced by Basu and Mitra (2007b) and examined by Alcantud (2012) and Sakamoto (2012), requires that SWRs are sensitive if a component below the limit inferior is increased.

**Axiom (Liminf-Restricted Dominance).** For any **x**, **y** ∈ **X** with $x_i > y_i$ for some $i \in \mathbb{N}$ and $x_j = y_j$ for all $j \in \mathbb{N} \setminus \{i\}$, **x** ≻ **y** whenever $\ell(\mathbf{x}) > y_i$.

When an SWR satisfies Strong Anonymity, Monotonicity, and Liminf-Restricted Dominance, it satisfies a strong Pareto principle on the domain $\mathbf{X}^{\uparrow}$ of all profiles that can be permuted into non-decreasing streams. This is the principle that Zuber and Asheim (2012) named *Restricted Strong Pareto*.

Separability axioms guarantee that only affected people matter when comparing two profiles, implying that we can ignore the well-being of people not affected by a choice. This is particularly convenient in an intertemporal context where the available alternatives never affect the well-being of past people. The following separability axiom asserts that evaluation must be independent of unconcerned people, provided the well-being of affected people is below the limit inferior; in order words, when people that matter for the choice are at finite rank.

**Axiom (Liminf-Restricted Finite Separability).** For any **x**, **y**, $\tilde{\mathbf{x}}$, $\tilde{\mathbf{y}} \in \mathbf{X}$, $\mathbf{x} \succsim \mathbf{y}$ if and only if $\tilde{\mathbf{x}} \succsim \tilde{\mathbf{y}}$ whenever there exists a finite subset $N \subset \mathbb{N}$ such that (i) $x_i = \tilde{x}_i < \min\{\ell(\mathbf{x}), \ell(\tilde{\mathbf{x}})\}$ for all $i \in N$, (ii) $y_i = \tilde{y}_i < \min\{\ell(\mathbf{y}), \ell(\tilde{\mathbf{y}})\}$ for all $i \in N$, and (iii) $x_j = y_j$ and $\tilde{x}_j = \tilde{y}_j$ for all $j \in \mathbb{N} \setminus N$.

## 4.4 The characterization result

It is our main result that the four axioms of subsection 4.2–4.3, combined with the regularity axioms of subsection 4.1, are sufficient to obtain Strongly Anonymous Utilitarianism.

**Theorem 1.** *The Strongly Anonymous (Generalized) Utilitarian SWR $\succsim^U$ is regular. Consider any regular SWR $\succsim$. There exists $u \in U$ such that the Strongly Anonymous (Generalized) Utilitarian SWR $\succsim^U$ associated with $u$ is a subrelation to $\succsim$ if and only if $\succsim$ satisfies Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability.*

We establish in Appendix A.4 the result that $\succsim^U$ is regular as well as the only-if statement of the theorem. Hence, it remains to be shown that there exists $u \in U$ such that $\succsim^U$ associated with $u$ is a subrelation to any regular SWR satisfying our axioms. To do so, we need two lemmas. We begin with a lemma which shows that any regular SWR satisfying our axioms applies (generalized) utilitarianism to the heads of profiles if the profiles have the same constant tail.

**Lemma 5.** *If a regular SWR $\succsim$ satisfies Limited Inequity, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability, then there exists $u \in U$ such that for*
all $\ell \in \mathbb{R}$ and all $n \in \mathbb{N}$, and all $\mathbf{x}, \mathbf{y} \in \mathbf{X}_n^\ell$,

$$ \mathbf{x} \gtrsim \mathbf{y} \quad \text{if and only if} \quad \sum_{i=1}^{n} u(x_i) \geq \sum_{i=1}^{n} u(y_i). \qquad (1) $$

*Proof.* See Appendix A.5 for the proof, which is based on the application of the additive representation theorem of Debreu (1960) on $\mathbf{X}_n^\ell$. $\square$

The second lemma shows that the Strongly Anonymous Utilitarian Dominance SWR is a subrelation to any SWR satisfying all our axioms.

**Lemma 6.** Consider any regular SWR $\gtrsim$. There exists $u \in U$ such that the Strongly Anonymous (Generalized) Utilitarian Dominance SWR $\gtrsim^D$ associated with $u$ is a subrelation to $\gtrsim$ if $\gtrsim$ satisfies Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability.

*Proof.* Assume that $\gtrsim$ is a regular SWR which satisfies Limited Inequity, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability. By Lemma 5, there exists $u \in U$ such that for all $\ell \in \mathbb{R}$ and all $n \in \mathbb{N}$ and for all $\mathbf{x}, \mathbf{y} \in \mathbf{X}_n^\ell$, $\mathbf{x} \gtrsim \mathbf{y}$ if and only if $\sum_{i=1}^n u(x_i) \geq \sum_{i=1}^n u(y_i)$. Let $\gtrsim^D$ be the Strongly Anonymous (Generalized) Utilitarian Dominance SWR associated with $u$.

*Step 1:* Assume that $\mathbf{x} \succ^D \mathbf{y}$. Then there exists $n \in \mathbb{N}$ such that $\sum_{r=1}^n u(x_{[r]}) > \sum_{r=1}^n u(y_{[r]})$ and $x_{[r]} \geq y_{[r]}$ for all $r > n$. Define $\hat{\mathbf{x}}, \hat{\mathbf{y}}$, and $\tilde{\mathbf{y}} \in \mathbf{X}$ by

$$ \hat{\mathbf{x}} = (x_{[1]}, \dots, x_{[n]}, x_{[n+1]}, x_{[n+1]}, \dots), $$

$$ \hat{\mathbf{y}} = (y_{[1]}, \dots, y_{[n]}, x_{[n+1]}, x_{[n+1]}, \dots), $$

$$ \tilde{\mathbf{y}} = (y_{[1]}, \dots, y_{[n]}, x_{[n+1]}, x_{[n+2]}, \dots). $$

We obtain that $\hat{\mathbf{x}}, \hat{\mathbf{y}} \in \mathbf{X}_n^\ell$, where $\ell = x_{[n+1]}$ and $\sum_{i=1}^n u(\hat{x}_i) > \sum_{i=1}^n u(\hat{y}_i)$ so that $\hat{\mathbf{x}} \succ \hat{\mathbf{y}}$ by Lemma 5. Since $\gtrsim$ satisfies Liminf-Restricted Finite Separability, this is equivalent to $\mathbf{x}_{[1]} \succ \tilde{\mathbf{y}}$. Furthermore, by Monotonicity, $\tilde{\mathbf{y}} \gtrsim \mathbf{y}_{[1]}$ and, by transitivity, $\mathbf{x}_{[1]} \succ \mathbf{y}_{[1]}$. By Lemma 4 and transitivity of $\gtrsim$, this implies that $\mathbf{x} \succ \mathbf{y}$.

*Step 2:* Assume that $\mathbf{x} \sim^D \mathbf{y}$. Then there exists $n \in \mathbb{N}$ such that $\sum_{r=1}^n u(x_{[r]}) = \sum_{r=1}^n u(y_{[r]})$ and $x_{[r]} = y_{[r]}$ for all $r > n$. We can define $\hat{\mathbf{x}}$, and $\hat{\mathbf{y}} \in \mathbf{X}$ like in Step 1, so that $\hat{\mathbf{x}}, \hat{\mathbf{y}} \in \mathbf{X}_n^\ell$, where $\ell = x_{[n+1]}$, $\sum_{i=1}^n u(\hat{x}_i) = \sum_{i=1}^n u(\hat{y}_i)$, and $\hat{\mathbf{x}} \sim \hat{\mathbf{y}}$ by Lemma 5. Since $\gtrsim$ satisfies Liminf-Restricted Finite Separability, this is equivalent to $\mathbf{x}_{[1]} \sim \mathbf{y}_{[1]}$. By Lemma 4 and transitivity of $\gtrsim$, this implies that $\mathbf{x} \sim \mathbf{y}$. $\square$

**Lemma 6** shows that all regular SWRs satisfying our axioms must agree with the comparisons made by the Strongly Anonymous Utilitarian Dominance SWR $\gtrsim^D$. As discussed in subsection 2.2, $\gtrsim^D$ does not satisfies Limited Inequity. However, by Lemmas 4–6, and the results of Appendix A.4, we can characterize the Strongly Anonymous Utilitarian SWR $\gtrsim^U$ as the minimal regular SWR satisfying our axioms.

*Proof of the if-statement of Theorem 1.* Assume that $\gtrsim$ is a regular SWR which satisfies Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability. By Lemma 6, there exists $u \in U$ such that the
Strongly Anonymous (Generalized) Utilitarian Dominance SWR $\gtrsim^D$ associated with $u$ is a subrelation to $\gtrsim$. Let $\gtrsim^U$ be the Strongly Anonymous (Generalized) Utilitarian SWR associated with $u$. It suffices to show that $\gtrsim^U$ is a subrelation to $\gtrsim$.

**Part 1:** For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, $\mathbf{x} \gtrsim^U \mathbf{y}$ implies $\mathbf{x} \gtrsim \mathbf{y}$. Let $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ be such that $\mathbf{x} \gtrsim^U \mathbf{y}$. By Lemma 4, $\mathbf{y}_{[1]} \sim \mathbf{y}$. Hence, by transitivity, it suffices to show that $\mathbf{x} \gtrsim \mathbf{y}_{[1]}$. Our strategy of proof is to construct a sequence $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ such that $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ converges to $\mathbf{y}_{[1]}$ in $d_1$ and such that $\mathbf{x} \gtrsim \mathbf{y}^n$ for each $n \in \mathbb{N}$. Since $\gtrsim$ satisfies Weak Continuity, we thus obtain $\mathbf{x} \gtrsim \mathbf{y}_{[1]}$.

**Construction of the sequence $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$.** Since $\mathbf{x} \gtrsim^U \mathbf{y}$, it follows from Lemma 1 that $(\mathbf{x}_{[1]}, \mathbf{y}_{[1]}) \in \mathbf{D}$. Hence, there exists $\delta \in \mathbb{R}_+$ such that $\sum_{r=1}^{\infty} [y_{[r]} - \min\{x_{[r]}, y_{[r]}\}] = \sum_{r: y_{[r]} > x_{[r]}} (y_{[r]} - x_{[r]}) = \delta$. Let $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ be the sequence defined by, for each $n \in \mathbb{N}$,

$$ 
\begin{aligned}
y_1^n &= u^{-1}(u(y_{[1]}) + \min\{0, \inf_{k \ge n} (v_k^u(\mathbf{x}_{[1]}) - v_k^u(\mathbf{y}_{[1]}))\}), \\
y_r^n &= y_{[r]} \quad \text{for } r \in \{2, 3, \dots, n-1, n\}, \\
y_r^n &= \min\{x_{[r]}, y_{[r]}\} \quad \text{for } r \in \{n+1, n+2, \dots\}.
\end{aligned}
$$

Note that $\mathbf{y}^n$ is well-defined since $u \in \mathcal{U}$ has bounded non-concavity and is thus unbounbed below.

The sequence $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ converges to $\mathbf{y}_{[1]}$ in $d_1$. By the construction of $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$, we have that

$$ 
\begin{aligned}
d_1(\mathbf{y}_{[1]}, \mathbf{y}^n) &\le y_{[1]} - y_1^n + \sum_{r=n+1}^{\infty} [y_{[r]} - \min\{x_{[r]}, y_{[r]}\}] \\
&= y_{[1]} - y_1^n + \delta - \sum_{r=1}^{n} [y_{[r]} - \min\{x_{[r]}, y_{[r]}\}].
\end{aligned}
$$

Moreover, $\inf_{k \ge n} (v_k^u(\mathbf{x}_{[1]}) - v_k^u(\mathbf{y}_{[1]})) \le v_n^u(\mathbf{x}_{[1]}) - v_n^u(\mathbf{y}_{[1]})$ and, since $\mathbf{x} \gtrsim^U \mathbf{y}$,

$$ \lim_{n \to \infty} \left( \inf_{k \ge n} (v_k^u(\mathbf{x}_{[1]}) - v_k^u(\mathbf{y}_{[1]})) \right) = \lim_{n \to \infty} (v_n^u(\mathbf{x}_{[1]}) - v_n^u(\mathbf{y}_{[1]})) \ge 0. $$

Hence, $\lim_{n \to \infty} y_1^n = y_{[1]}$. By the definition of $\delta$ we obtain that $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ converges to $\mathbf{y}_{[1]}$ in $d_1$.

$\mathbf{x} \gtrsim^D \mathbf{y}^n$ for each $n \in \mathbb{N}$. Consider any $n \in \mathbb{N}$. Note first that

$$ 
\begin{aligned}
v_n^u(\mathbf{x}_{[1]}) - v_n^u(\mathbf{y}^n) &= v_n^u(\mathbf{x}_{[1]}) - u(y_1^n) - \sum_{r=2}^n u(y_{[r]}) \\
&\ge v_n^u(\mathbf{x}_{[1]}) - v_n^u(\mathbf{y}_{[1]}) - \inf_{k \ge n} (v_k^u(\mathbf{x}_{[1]}) - v_k^u(\mathbf{y}_{[1]})) \ge 0.
\end{aligned}
$$

Also remark that, by definition, $y_r^n = \min\{x_{[r]}, y_{[r]}\} \le x_{[r]}$ for all $r > n$. Hence, $\mathbf{x} \gtrsim^D \mathbf{y}^n$. By Lemma 6, this implies $\mathbf{x} \gtrsim \mathbf{y}^n$.

**Part 2:** For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, $\mathbf{x} \succ^U \mathbf{y}$ implies $\mathbf{x} \succ \mathbf{y}$. If $\mathbf{x} \succ^U \mathbf{y}$, then there exists $\Delta \in \mathbb{R}_{++} \cup \{\infty\}$ such that $\lim_{n \to \infty} (v_n^u(x_{[1]}) - v_n^u(y_{[1]})) = \Delta$. Hence, we can find $\tilde{\Delta} \in \mathbb{R}_{++}$ with $\tilde{\Delta} < \Delta$. Define $\tilde{x} \in \mathbb{R}$ by $\tilde{x} = u^{-1}(u(x_{[1]}) - (\Delta - \tilde{\Delta}))$ if $\Delta \in \mathbb{R}_{++}$ and $\tilde{x} = u^{-1}(u(x_{[1]}) - \tilde{\Delta})$ if $\Delta = \infty$ so that, in any case, $\tilde{x} < x_{[1]}$. If $\min_{i \in \mathbb{N}} x_i$ is well-defined and $j \in \mathbb{N}$ is such that $x_j = \min_{i \in \mathbb{N}} x_i$, define $\tilde{x} \in \mathbf{X}$ by $\tilde{x}_j = \tilde{x}$ and $\tilde{x}_i = x_i$
for all $i \neq j$. Alternatively, if $\min_{i \in \mathbb{N}} x_i$ is not well-defined, define $\tilde{\mathbf{x}} \in \mathbf{X}$ by $\tilde{x}_1 = \tilde{x}$ and $\tilde{x}_i = x_i$ for all $i \ge 2$. In both cases, $\mathbf{x} \succ \tilde{\mathbf{x}}$ since $\succsim$ satisfies Liminf-Restricted Dominance. Hence, by transitivity of $\succsim$ and part 1 of this proof, to establish $\mathbf{x} \succ \mathbf{y}$ it suffices to show that $\tilde{\mathbf{x}} \succsim^U \mathbf{y}$. This follows since, by construction, either, for all $\pi \in \Pi$, $\lim_{n \to \infty} (v_n^u(\tilde{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) = \tilde{\Delta}$ or, for all $\pi \in \Pi$, $\lim_{n \to \infty} (v_n^u(\tilde{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) = \infty$. $\square$

Strongly Anonymous Utilitarianism is based on the sum of transformed well-being, where $u$ is the transformation function. If well-being corresponds to utility and $u$ is strictly concave, then the Strongly Anonymous Utilitarian SWR $\succsim^U$ is what philosophers call *prioritarian*, having the property of assigning more weight to increases in utility for poor than to increases in utility for rich and thus satisfying the Pigou-Dalton transfer principle.<fnref n="18" /> When there are finitely many people, it is known that if we consider a sequence of functions $u \in U$ with increasing concavity, the sequence of associated prioritarian social orderings tends to converge to the conclusions of a kind of Rawlsian criterion (actually a Leximin criterion) that gives infinite priority to the worst-off. One could conjecture that, similarly, Strongly Anonymous Utilitarianism will converge to a version of infinite population Leximin when we make $u$ more and more concave.

This conjecture is actually false. This is seen by considering any real numbers $x < y < z$ and the infinite streams $(x, z1_\mathbb{N})$ and $y1_\mathbb{N}$. A Rawlsian criterion (considering the infinimum) or a Leximin criterion (for instance the version proposed by Asheim and Zuber, 2013) would deem $y1_\mathbb{N}$ strictly better than $(x, z1_\mathbb{N})$. But for any $u \in U$, however concave we choose it to be, it is the case that $(x, z1_\mathbb{N}) \succ^U y1_\mathbb{N}$ for $\succsim^U$ associated with $u$. This is because the limit inferior of $(x, z1_\mathbb{N})$ is larger than the limit inferior of $y1_\mathbb{N}$. Remark that Basu-Mitra utilitarian criterion and utilitarian overtaking or catching-up procedures with transformed well-being yield the same conclusion. This is in contrast with Rank-Discounted Utilitarianism as proposed by Zuber and Asheim (2012), also defined in subsection 5.1 below. In that case, a Leximin criterion can be seen as the limit case when the rank-discount factor goes to zero (Asheim and Zuber, 2013).

# 5 Extensions of Strongly Anonymous Utilitarianism

The Strongly Anonymous Utilitarian SWR is related to strongly anonymous versions of utilitarian criteria proposed in the literature. As we observed (Observation 1), it is a subrelation to the Strongly Anonymous Utilitarian Catching-Up SWR, as given in Definition 1 of subsection 2.2. In Appendix A.6, we provide an axiomatization of the Strongly Anonymous Utilitarian Catching-Up SWR, based on just one additional axiom compared to that of the Strongly Anonymous Utilitarian SWR. In this section, we show that the Strongly Anonymous Utilitarian SWR is also a subrelation to the Limit of Rank-Discounted Utilitarian SWR and the Strongly Anonymous Utilitarian Cesàro Summation SWR to be defined in subsection 5.1. We also establish the links between the three extensions of the Strongly Anonymous Utilitarian SWR.

<footnote n="18">¹⁸Note that much of the literature on infinite population utilitarian criteria combine anonymity with translation scale invariance to obtain utilitarianism in terms of untransformed well-being; Basu and Mitra (2007a) is an example. Here we obtain generalized utilitarianism by imposing a separability axiom instead of translation scale invariance.</footnote>
## 5.1 Limit of rank-discounting and Cesàro summation

The class of Rank-Discounted Utilitarian SWRs introduced by Zuber and Asheim (2012) is designed to satisfy the axiom of Strong Anonymity while retaining the usual discounted utilitarian form. For any transformation function $u \in U$ and any rank-discount factor $\beta \in (0, 1)$, the Rank-Discounted (Generalized) Utilitarian SWR associated with $u$ is characterized by the social welfare function $\rho_{\beta}^u: X \to \mathbb{R}$ defined by, for all $x \in X$,<fnref n="19" />

$$ \rho_{\beta}^{u}(\mathbf{x}) = \inf_{\pi \in \Pi} \sum_{i=1}^{\infty} \beta^{i-1} u(x_{\pi(i)}) = \sum_{r=1}^{\infty} \beta^{r-1} u(x_{[r]}). $$

By letting the discount factor $\beta$ approach 1, Rank-Discounted Utilitarianism can be used to define the following class of strongly anonymous (generalized) utilitarian SWRs.

**Definition 5.** The Limit of Rank-Discounted (Generalized) Utilitarian SWR $\gtrsim^R$. There exists $u \in U$ such that, for any profiles $\mathbf{x}, \mathbf{y} \in X$, $\mathbf{x} \gtrsim^R \mathbf{y}$ if and only if

$$ \liminf_{\beta \to 1^{-}} (\rho_{\beta}^{u}(\mathbf{x}) - \rho_{\beta}^{u}(\mathbf{y})) \geq 0. $$

This SWR is related to other existing criteria. Asheim and Zuber (2013) consider the limit of rank-discounting when $\beta$ approaches 0, while Jonsson and Voorneveld (2018) consider the limit of time-discounting when $\beta$ approaches 1.

We next introduce a class of strongly anonymous (generalized) utilitarian SWRs based on the notion of Cesàro summation. For any $u \in U$ and any $n \in \mathbb{N}$, define strongly anonymous (generalized) utilitarian partial Cesàro summation $\sigma_n^u: X \to \mathbb{R}$ by, for all $\mathbf{x} \in X$,

$$ \sigma_n^u(\mathbf{x}) = \inf_{\pi \in \Pi} \frac{1}{n} \sum_{k=1}^{n} \sum_{i=1}^{k} u(x_{\pi(i)}) = \frac{1}{n} \sum_{k=1}^{n} \sum_{r=1}^{k} u(x_{[r]}) = \sum_{r=1}^{n} \frac{n-r+1}{n} u(x_{[r]}). $$

**Definition 6.** The Strongly Anonymous (Generalized) Utilitarian Cesàro Summation SWR $\gtrsim^S$. There exists $u \in U$ such that, for any profiles $\mathbf{x}, \mathbf{y} \in X$, $\mathbf{x} \gtrsim^S \mathbf{y}$ if and only if

$$ \liminf_{n \to +\infty} (\sigma_n^u(\mathbf{x}) - \sigma_n^u(\mathbf{y})) \geq 0. $$

In the context of discrete dynamic programming, Veinott (1966, p. 1293) proposes a similar criterion, which is used by Jonsson and Voorneveld (2018) for comparisons with their Limit of Time-Discounted Utilitarianism.

Note that when, for some $n \in \mathbb{N}$, $w_n^u(\mathbf{x}) \ge w_n^u(\mathbf{y})$ ($w_n^u(\mathbf{x}) > w_n^u(\mathbf{y})$ respectively) and $x_{[r]} \ge y_{[r]}$ for all $r > n$, we have $\lim_{\beta \to 1^{-}} (\rho_{\beta}^u(\mathbf{x}) - \rho_{\beta}^u(\mathbf{y})) \ge 0$ and $\lim_{n \to +\infty} (\sigma_n^u(\mathbf{x}) - \sigma_n^u(\mathbf{y})) \ge 0$.

<footnote n="19">¹⁹Zuber and Asheim (2012) do not require $u$ to have bounded non-concavity. However, they show in their Proposition 6 that $u$ must have bounded non-concavity to satisfy the Pigou-Dalton transfer principle.</footnote>
$\sigma_n^u(\mathbf{y})) \ge 0 (\lim_{\beta \to 1^-} (\rho_\beta^u(\mathbf{x}) - \rho_\beta^u(\mathbf{y}))) > 0$ and $\lim_{n \to +\infty} (\sigma_n^u(\mathbf{x}) - \sigma_n^u(\mathbf{y})) > 0$ respectively), implying that $\mathbf{x} \succsim^R \mathbf{y}$ and $\mathbf{x} \succsim^S \mathbf{y}$ ($\mathbf{x} \succ^R \mathbf{y}$ and $\mathbf{x} \succ^S \mathbf{y}$ respectively) for $\succsim^R$ and $\succsim^S$ associated with $u \in U$. We therefore obtain the following result.

**Observation 3.** For any $u \in U$, the Strongly Anonymous (Generalized) Utilitarian Dominance SWR $\succsim^D$ associated with $u$ is a subrelation to the Limit of Rank-Discounted (Generalized) Utilitarian SWR $\succsim^R$ associated with $u$ and the Strongly Anonymous (Generalized) Utilitarian Cesàro Summation SWR $\succsim^S$ associated with $u$.

## 5.2 Links between the SWRs

The following proposition relates the two SWRs that we have introduced in this section as well as the Strongly Anonymous Utilitarian Catching-Up SWR $\succsim^C$ introduced in Section 2 to each other and to the Strongly Anonymous Utilitarian SWR $\succsim^U$. We do so by showing that they satisfy all our axioms. The proof is relegated to Appendix A.7.

**Proposition 2. (i)** The Strongly Anonymous (Generalized) Utilitarian Catching-Up SWR $\succsim^C$, the Limit of Rank-Discounted (Generalized) Utilitarian SWR $\succsim^R$, and the Strongly Anonymous (Generalized) Utilitarian Cesàro Summation SWR $\succsim^S$ are regular.

(ii) For any $u \in U$, $\succsim^C$ associated with $u$ is weakly extended by $\succsim^S$ associated with $u$, which in turn is weakly extended by $\succsim^R$ associated with $u$.

(iii) For any $u \in U$, the Strongly Anonymous (Generalized) Utilitarian SWR $\succsim^U$ associated with $u$ is a subrelation to $\succsim^C$ associated with $u$, $\succsim^R$ associated with $u$, and $\succsim^S$ associated with $u$.

Proposition 2 shows that there are strong links between the SWRs that we have introduced. It is also important to show that they are distinct. In the following example already discussed in subsection 2.2, we present a pair of well-being profiles that are indifferent according to $\succsim^C$, $\succsim^R$, and $\succsim^S$ but non-comparable by $\succsim^U$.

**Example 1.** Assume that $\succsim^U$, $\succsim^C$, $\succsim^S$, and $\succsim^R$ are associated with the identity transformation function $u(z) = z$. For all $n \in \mathbb{N}$, we define function $g: \mathbb{N} \to \mathbb{N}$ by $g(n) = n(n-1)/2$. Letting $z = 1 - \ln 2 \approx 0.30685$, consider the following profiles $\mathbf{x}$, $\mathbf{y} \in \mathbf{X}^+$:

$$ 
\begin{aligned}
\mathbf{x} &= (z, \underbrace{\frac{2}{3}, \frac{2}{3}, \frac{2}{3}, \dots, \frac{2}{3}}_{2}, \underbrace{\frac{4}{5}, \dots, \frac{4}{5}, \dots}_{3}, \underbrace{\frac{4}{5}, \dots, \frac{4}{5}, \dots}_{5}, \dots) \\
\mathbf{y} &= (\frac{1}{2}, \underbrace{\frac{1}{2}, \frac{1}{2}, \dots}_{3}, \underbrace{\frac{3}{4}, \dots, \frac{3}{4}, \dots}_{3}, \underbrace{\frac{3}{4}, \dots, \frac{3}{4}, \dots}_{5}, \underbrace{\frac{5}{6}, \dots, \frac{5}{6}, \dots}_{\dots}).
\end{aligned}
$$

Formally, $\mathbf{x}$ is defined by $x_1 = z$ and, for each $n \in \mathbb{N}$ and for all $i \in \{g(2n) + 1, \dots, g(2(n+1))\}$,

$$ x_i = 1 - \frac{1}{2n+1}, $$

and $\mathbf{y}$ is defined by, for each $n \in \mathbb{N}$ and for all $i \in \{g(2n-1) + 1, \dots, g(2n+1)\}$,

$$ y_i = 1 - \frac{1}{2n}. $$
Note that

$$ \mathbf{x} - \mathbf{y} = \underbrace{(z - \frac{1}{2}, \frac{1}{6}, \frac{1}{6}}_{2}, \underbrace{-\frac{1}{12}, \dots, -\frac{1}{12}}_{3}, \underbrace{\frac{1}{20}, \dots, \frac{1}{20}}_{4}, \underbrace{-\frac{1}{30}, \dots, -\frac{1}{30}}_{5}, \dots). $$

More precisely, for each $n > 1$ and for all $i \in \{g(n) + 1, \dots, g(n + 1)\}$,

$$ x_i - y_i = (-1)^n \frac{1}{n(n+1)}. $$

Thus, for each $n > 1$,

$$ \sum_{i=g(n)+1}^{g(n+1)} (x_i - y_i) = (-1)^n \frac{n}{n(n+1)} = (-1)^n \frac{1}{n+1}. $$

The alternating harmonic series converges to $\ln 2$. Hence:

$$ \begin{aligned} \sum_{i=1}^{\infty} (x_i - y_i) &= (x_1 - y_1) + \sum_{n=2}^{\infty} \sum_{i=g(n)+1}^{g(n+1)} (x_i - y_i) \\ &= 1 - \ln 2 - \frac{1}{2} + \sum_{n=2}^{\infty} (-1)^n \frac{1}{n+1} = -\ln 2 + \sum_{n=0}^{\infty} (-1)^n \frac{1}{n+1} = 0. \end{aligned} $$

Since $\mathbf{x}_{[1]} = \mathbf{x}$ and $\mathbf{y}_{[1]} = \mathbf{y}$, we obtain $\mathbf{x} \sim^C \mathbf{y}$ and, from Proposition 2, $\mathbf{x} \sim^S \mathbf{y}$ and $\mathbf{x} \sim^R \mathbf{y}$.

However, for all $n > 1$:

$$ \sum_{\substack{i=1 \\ y_i > x_i}}^{g(2n)} (y_i - x_i) = (y_1 - x_1) + \sum_{k=2}^{n} \left( \sum_{i=g(2k-1)+1}^{g(2k)} (y_i - x_i) \right) = -1 + \ln 2 + \frac{1}{2} \left( \sum_{k=1}^{n-1} \frac{1}{k} \right). $$

Therefore, $\lim_{n \to \infty} \sum_{i \le n: y_i > x_i} (y_i - x_i) = \infty$ and $(\mathbf{x}, \mathbf{y}) \notin \mathbf{D}$. Likewise, by considering all even $u \in \mathbb{N}$, we obtain that $\lim_{n \to \infty} \sum_{i \le n: x_i > y_i} (x_i - y_i) = \infty$ and $(\mathbf{y}, \mathbf{x}) \notin \mathbf{D}$. Since $\mathbf{x}_{[1]} = \mathbf{x}$ and $\mathbf{y}_{[1]} = \mathbf{y}$, it follows from Lemma 1 that $\gtrsim^U$ cannot compare $\mathbf{x}$ and $\mathbf{y}$.

The next example presents a pair of well-being profiles that are indifferent according to $\gtrsim^S$ and $\gtrsim^R$ but non-comparable by $\gtrsim^C$.

**Example 2.** Assume that $\gtrsim^C$, $\gtrsim^S$, and $\gtrsim^R$ are associated with the identity transformation function $u(z) = z$. For all $n \in \mathbb{N}$, we define function $\bar{g}: \mathbb{N} \to \mathbb{N}$ by $\bar{g}(n) = \sum_{i=1}^{n} 2^i$.

Consider the following profiles $\mathbf{x}, \mathbf{y} \in \mathbf{X}^+$:

$$ \begin{aligned} \mathbf{x} &= (0, \underbrace{0, \dots, 0}_{2^2=4}, \underbrace{\frac{3}{4}, \dots, \frac{3}{4}}_{2^3=8}, \underbrace{\frac{3}{4}, \dots, \frac{3}{4}}_{2^4=16}, \underbrace{\frac{15}{16}, \dots, \frac{15}{16}}_{2^5=32}, \underbrace{\frac{15}{16}, \dots, \frac{15}{16}}_{2^6=64}, \dots) \\ \mathbf{y} &= (-1, \underbrace{\frac{1}{2}, \dots, \frac{1}{2}}_{2^2=4}, \underbrace{\frac{1}{2}, \dots, \frac{1}{2}}_{2^3=8}, \underbrace{\frac{7}{8}, \dots, \frac{7}{8}}_{2^4=16}, \underbrace{\frac{7}{8}, \dots, \frac{7}{8}}_{2^5=32}, \underbrace{\frac{31}{32}, \dots, \frac{31}{32}}_{2^6=64}, \dots). \end{aligned} $$

<footnote n="20">²⁰Noting that there are $n$ elements in the set $\{g(n) + 1, \dots, g(n + 1)\}$.</footnote>
Formally, $\mathbf{x}$ is defined by $x_i = 0$ for $i \in \{1, \dots, 5\}$ and for each $n \in \mathbb{N}$ and for all $i \in \{\bar{g}(2n), \dots, \bar{g}(2n+2)-1\}$,

$$x_i = \sum_{k=1}^{2n} \frac{1}{2^k},$$

and $\mathbf{y}$ is defined by $y_1 = -1$ and for each $n \in \mathbb{N}$ and for all $i \in \{\bar{g}(2n-1), \dots, \bar{g}(2n+1)-1\}$,

$$y_i = \sum_{k=1}^{2n-1} \frac{1}{2^k}.$$

Note that

$$\mathbf{x} - \mathbf{y} = \left(1, \underbrace{-\frac{1}{2}, \dots, -\frac{1}{2}}_{2^2=4}, \underbrace{\frac{1}{4}, \dots, \frac{1}{4}}_{2^3=8}, \underbrace{-\frac{1}{8}, \dots, -\frac{1}{8}}_{2^4=16}, \underbrace{\frac{1}{16}, \dots, \frac{1}{16}}_{2^5=32}, \dots\right).$$

More precisely, for each $n \in \mathbb{N}$ and for all $i \in \{\bar{g}(n), \dots, \bar{g}(n+1)-1\}$,

$$x_i - y_i = (-1)^n \frac{1}{2^n}.$$

Note that $\mathbf{x}_{[1]} = \mathbf{x}$ and $\mathbf{y}_{[1]} = \mathbf{y}$. We show that $\gtrsim_C^S$ cannot compare $\mathbf{x}$ and $\mathbf{y}$, while $\gtrsim_R$ and $\gtrsim_S$ conclude these profiles are indifferent. To this end, we define $\mathbf{s} \in \mathbb{R}^N$ by $s_n = \sum_{i=1}^n (x_i - y_i)$ for all $n \in \mathbb{N}$. Thus, $\mathbf{s}$ is given by

$$\mathbf{s} = \left(1, \underbrace{\frac{1}{2}, 0, -\frac{1}{2}, -1}_{2^2+1}, \underbrace{-\frac{3}{4}, -\frac{2}{4}, -\frac{1}{4}, 0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}}_{2^3-1}, \underbrace{1, \frac{7}{8}, \dots, \frac{1}{8}, 0, -\frac{1}{8}, \dots, -\frac{7}{8}, -1}_{2^4+1=17}, \dots\right).$$

Note that, for each $n \in \mathbb{N}$, $s_i = 1$ if $i = \sum_{j=1}^{2n+1} 2^j - 1$ and $s_i = -1$ if $i = \sum_{j=1}^{2n} 2^j - 1$. Thus, $\mathbf{x}$ and $\mathbf{y}$ are non-comparable by $\gtrsim^C$. On the other hand, for each $n \in \mathbb{N}$:

$$\frac{1}{\bar{g}(2n)-1} \sum_{k=1}^{\bar{g}(2n)-1} s_k = 0 \quad \text{and} \quad \frac{1}{\bar{g}(2n+1)-2} \sum_{k=1}^{\bar{g}(2n+1)-2} s_k = 0.$$

Also, it can be checked that

$$\lim_{n \to +\infty} \frac{1}{n} \sum_{k=1}^{n} s_k = 0.$$

Thus, we obtain $\mathbf{x} \sim^S \mathbf{y}$. By Proposition 2, $\mathbf{x} \sim^R \mathbf{y}$ follows as well.

# 6 Discussion

We have proposed and characterized SWRs over infinite well-being profiles which are generalized utilitarian and which treat people equally in the strong form of Strong Anonymity. They are generalized utilitarian in the sense that they depend on the function $u$ that maps well-being into the transformed well-being, which is additive and thereby enables us to compare the gain of one to the loss of another. Indeed, the Strongly Anonymous Utilitarian SWR $\gtrsim^U$ characterized in Theorem 1 is a different
criterion for each $u$ in the set $U$ of continuous and increasing transformation functions
with bounded non-concavity.

All of these SWRs deem one profile as better than another if the limit inferior of the former strictly exceeds that of the latter. Also, even for profiles with the same limit inferior, differences in well-being above the common limit inferior do not influence the evaluation. Hence, the distinction between the various SWRs concerns how they compare profiles with the same limit inferior on the basis of the components of well-being that do not exceed this common smallest cluster point.

One might argue that these features severely limit the applicability of the SWRs
that we have analyzed. We beg to differ. The common limit inferior might be associated
with bliss, as originally suggested by Ramsey (1928), in which case well-being above
this level is not even feasible, or with the Golden Rule, as we have illustrated in Section
3. Moreover, if the technology has positive net productivity and people are treated
equally, then the time profile of well-being will be non-decreasing and tend to bliss or
the Golden Rule as time goes to infinity (Asheim, Buchholz, and Tungodden, 2001).
Thus, the proposed SWRs are applicable and provide evaluation in line with already
existing utilitarian criteria in the context of such technologies.

Our SWRs can also be used if the rate of population growth is endogenous. Since,
by assumption, total population is always infinite, there cannot be a trade-off between
the size of the total population and the level of well-being. This implies that there
cannot be a *Repugnant Conclusion* (Parfit, 1984), whereby a large population with
acceptable but poor lives is preferred to a smaller population with excellent lives, in
terms of the total number of people that will ever live. The well-being of people might
be affected by the spatiotemporal structure of the total population, in the sense that
the existence of other people in the proximity might yield benefits or costs. However,
having a large population simultaneously at a given point in time does not by itself
have value, as evaluated by our SWRs.<fnref n="21" /> Hence, if a lower rate of population growth
can provide room for higher current well-being and faster growth of well-being towards
bliss, then the lower population growth is preferable. In this sense, our SWRs escape
the Repugnant Conclusion when interpreted in terms of the number of people living
at a given point in time.

# Appendix A Proofs

## A.1 Proof of Lemma 3

*Step 1: Existence of a maximum for $v^{\bar{u}}$. The Ramsey model, as described in Section 3, satisfies Assumptions 1 and 2 of Brock (1970). It therefore follows from Theorem 1 of Brock (1970) that, for any $k \in (0, k^*]$, there exists a feasible $\hat{\mathbf{x}}$ from $k$ such that $\lim \sup_{n \to \infty} (v_n^{\bar{u}}(\hat{\mathbf{x}}) - v_n^{\bar{u}}(\mathbf{x})) \ge 0$ for all feasible $\mathbf{x}$ from $k$, where $\hat{\mathbf{x}} = x^* 1_N$ if $k = k^*$. Recall that $v^{\bar{u}}: \mathbf{X} \to \mathbb{R}_+ \cup \{-\infty\}$ where, for all $\mathbf{x} \in \mathbf{X}$, $v^{\bar{u}}(\mathbf{x}) = \lim_{n \to \infty} v_n^{\bar{u}}(\mathbf{x})$ is well-defined (it may be $-\infty$) and always takes a non-positive value; indeed it is an infinite

<footnote n="21">²¹Indeed, adding a person at some time without influencing the well-being of other people cannot improve the profile, as it does not influence the evaluation if the added person's well-being is as large as the limit inferior and, otherwise, it reduces the well-being of finitely ranked people with well-being exceeding that of the added person.</footnote>
sum of non-positive real numbers. Furthermore, since $\mathbf{x}^{\xi}(k)$ is feasible and satisfies $v^{\bar{u}}(\mathbf{x}^{\xi}(k)) > -\infty$, it follows that $v^{\bar{u}}(\hat{\mathbf{x}}) \ge v^{\bar{u}}(\mathbf{x}^{\xi}(k)) > -\infty$. Therefore, Theorem 1 of Brock (1970) implies that there exists a feasible $\hat{\mathbf{x}}$ from $k$ such that $v^{\bar{u}}(\hat{\mathbf{x}}) \ge v^{\bar{u}}(\mathbf{x})$ for all feasible $\mathbf{x}$ from $k$, where $\hat{\mathbf{x}} = x^* 1_{\mathbb{N}}$ if $k = k^*$.

**Step 2:** If a well-being stream $\hat{\mathbf{x}}$ is maximum for $v^{\bar{u}}$, then $\hat{k}_i \le k^*$ for all $i \in \mathbb{N}$. By definition, $\hat{k}_1 = k \le k^*$. For $n \ge 2$, by way of contradiction, suppose $\hat{k}_n > k^*$ and $\hat{k}_i \le k^*$ for $i \in \{1, \dots, n-1\}$. Then $\mathbf{x}'$ defined by $x'_i = \hat{x}_i$ for $i \in \{1, \dots, n-2\}$, $x'_{n-1} = f(\hat{k}_{n-1}) - k^* > f(\hat{k}_{n-1}) - \hat{k}_n = \hat{x}_{n-1}$, and $x'_i = x^*$ for $i \ge n$ is feasible from $k$ since $k'_n = f(\hat{k}_{n-1}) - x'_{n-1} = k^*$. Since $\hat{x}_{n-1} < x'_{n-1} \le x^*$ and $x'_i = \hat{x}_i$ for $i \in \{1, \dots, n-2\}$, it follows that $\sum_{i=1}^{n-1} (\bar{u}(x'_i) - \bar{u}(\hat{x}_i)) > 0$. Furthermore, $\sum_{i=n}^{\infty} (\bar{u}(x'_i) - \bar{u}(\hat{x}_i)) = -\sum_{i=n}^{\infty} \bar{u}(\hat{x}_i) \ge 0$. Hence $v^{\bar{u}}(\mathbf{x}') > v^{\bar{u}}(\hat{\mathbf{x}})$, which contradicts that $v^{\bar{u}}(\hat{\mathbf{x}}) \ge v^{\bar{u}}(\mathbf{x})$ for all feasible $\mathbf{x}$ from $k$.

**Step 3:** If a well-being stream $\hat{\mathbf{x}}$ is maximum for $v^{\bar{u}}$, then $\hat{\mathbf{x}} \in \mathbf{X}^*(k)$. By way of contradiction, suppose that $\hat{\mathbf{x}} \notin \mathbf{X}^*(k)$. Note that $\ell(\hat{\mathbf{x}}) = x^*$ since $\ell(\hat{\mathbf{x}}) < x^*$ would contradict $v^{\bar{u}}(\hat{\mathbf{x}}) > -\infty$ and $\ell(\hat{\mathbf{x}}) > x^*$ would contradict Lemma 2. Hence, $\hat{x}_i > x^*$ for some $i \in \mathbb{N}$. Since no stream $\mathbf{x} > x^* 1_{\mathbb{N}}$ is feasible from $k \le k^*$, there exists $i'$ such that $\hat{x}_{i'} < x^*$. Furthermore, because $f$ is increasing, there exists $\varepsilon$ such that $\mathbf{x}'$ defined as follows is feasible from $k: x'_i = x^*$ for all $i \in \mathbb{N}$ such that $\hat{x}_i \ge x^*$, $x'_{i'} = \hat{x}_{i'} + \varepsilon \le x^*$, and $x'_j = \hat{x}_j$ for all other $j$. By construction, $\mathbf{x}' \in \mathbf{X}^*(k)$ and $v^{\bar{u}}(\mathbf{x}') > v^{\bar{u}}(\hat{\mathbf{x}})$. This contradicts that $\hat{\mathbf{x}}$ is maximum for $v^{\bar{u}}$.

**Step 4:** If a well-being stream $\hat{\mathbf{x}}$ is maximum for $v^{\bar{u}}$, then $\hat{\mathbf{x}}$ is non-decreasing. By way of contradiction, suppose that $\hat{x}_n > \hat{x}_{n+1}$ for some $n \in \mathbb{N}$. By Step 2, $\hat{k}_n \le k^*$, while, by Step 1, $\hat{k}_n = k^*$ implies that $\hat{x}_i = x^*$ for all $i \ge n$, contradicting $\hat{x}_n > \hat{x}_{n+1}$. Hence, $\hat{k}_n < k^*$. By Step 3, $\hat{x}_{n+1} < \hat{x}_n \le x^*$. There are two subcases:

(i) If $f(\hat{k}_n) - \hat{x}_{n+1} > k^*$, then $\mathbf{x}'$ with $x'_i = \hat{x}_i$ for $i \in \{1, \dots, n-1\}$, $x'_n = f(\hat{k}_n) - k^* > \hat{x}_{n+1}$, and $x'_i = x^*$ for $i \ge n+1$ is feasible from $k$. Since $\min\{x^*, x'_n\} > \hat{x}_{n+1}$, it follows that $\bar{u}(x'_n) > \bar{u}(\hat{x}_{n+1})$, which implies $\sum_{i=1}^{n-1} (\bar{u}(x'_i) - \bar{u}(\hat{x}_i)) = 0$ and $\sum_{i=n}^{\infty} (\bar{u}(x'_i) - \bar{u}(\hat{x}_i)) > 0$. Hence, $v^{\bar{u}}(\mathbf{x}') > v^{\bar{u}}(\hat{\mathbf{x}})$, which contradicts that $\hat{\mathbf{x}}$ is maximum for $v^{\bar{u}}$.

(ii) If $f(\hat{k}_n) - \hat{x}_{n+1} \le k^*$, then $\mathbf{x}'$ with $x'_i = \hat{x}_i$ for $i \notin \{n, n+1\}$, $x'_{n+1} = \hat{x}_n$, and $x'_n = \sup X'_n$, where $X'_n = \{x \in [0, f(\hat{k}_n)] : f(f(\hat{k}_n) - x) - x'_{n+1} \ge \hat{k}_{n+2}\}$, is feasible from $k$ since (a) $\hat{x}_{n+1} \in X'_n$ implying that $X'_n$ is nonempty, (b) $f$ is continuous implying that $X'_n$ is compact and contains its supremum, and (c) by construction, $k'_{t+2} = f(f(\hat{k}_n) - x'_n) - x'_{n+1} = \hat{k}_{t+2}$. To verify that $\hat{x}_{n+1} \in X'_n$, write $k''_{n+1} = f(\hat{k}_n) - \hat{x}_{n+1}$ and $k''_{n+2} = f(\hat{k}_{n+1}) - \hat{x}_n$, implying that $\hat{x}_{n+1} \in X'_n$ is equivalent to $k''_{n+2} \ge \hat{k}_{n+2}$ since $x'_{n+1} = \hat{x}_n$. To show this, note that

$$f(\hat{k}_n) - \hat{k}_{n+1} + f(\hat{k}_{n+1}) - \hat{k}_{n+2} = \hat{x}_n + \hat{x}_{n+1} = \hat{x}_{n+1} + \hat{x}_n = f(\hat{k}_n) - k''_{n+1} + f(k''_{n+1}) - k''_{n+2}.$$

$$\text{Hence, } k''_{n+2} - \hat{k}_{n+2} = f(k''_{n+1}) - k''_{n+1} - (f(\hat{k}_{n+1}) - \hat{k}_{n+1}) > 0 \text{ since}$$

$$\hat{k}_{n+1} = f(\hat{k}_n) - \hat{x}_n < f(\hat{k}_n) - \hat{x}_{n+1} = k''_{n+1} \le k^*$$
and $f(k) - k$ is increasing on $[0, k^*]$. Since $k''_{n+2}$ is strictly greater than $\hat{k}_{n+2}$, it follows from the continuity of $f$ that $\min\{x^*, x'_n\} > \hat{x}_{n+1}$ and $\bar{u}(x'_n) > \bar{u}(\hat{x}_{n+1})$, which implies $\sum_{i=1}^{n-1} (\bar{u}(x'_i) - \bar{u}(\hat{x}_i)) = 0$ and $\sum_{i=n}^{\infty} (\bar{u}(x'_i) - \bar{u}(\hat{x}_i)) > 0$. Hence, $v^{\bar{u}}(\mathbf{x}') > v^{\bar{u}}(\hat{\mathbf{x}})$, which contradicts that $\hat{\mathbf{x}}$ is maximum for $v^{\bar{u}}$.

**Step 5:** If a well-being stream $\hat{\mathbf{x}}$ is maximum for $v^{\bar{u}}$, then there is no feasible well-being stream $\mathbf{x}'$ from $k$ such that $\mathbf{x} > \hat{\mathbf{x}}$. If $k = k^*$, then $\hat{\mathbf{x}} = x^*1_N$ is efficient. If $k < k^*$, then $x^*1_N$ is not feasible. Furthermore, $\hat{\mathbf{x}}$ is a nondecreasing stream in $\mathbf{X}^*(k)$, implying that $\hat{x}_1 < x^*$. By way of contradiction, suppose there is $\mathbf{x}'$ feasible from $k$ such that $\mathbf{x}' > \hat{\mathbf{x}}$. If $x'_1 > \hat{x}_1$ then $v^{\bar{u}}(\mathbf{x}') > v^{\bar{u}}(\hat{\mathbf{x}})$ contradicting that $\mathbf{x}$ is maximum for $v^{\bar{u}}$. If $x'_n > \hat{x}_n$ for $n \ge 2$, then we can construct a feasible stream $\mathbf{x}''$ such that $x''_1 > x'_1$, $x''_n = \hat{x}_n$ and $x''_i = x'_i$ for all $i \notin \{1, n\}$; indeed $f$ is increasing and we just need to accumulate less capital in period 1. Then $v^{\bar{u}}(\mathbf{x}'') > v^{\bar{u}}(\hat{\mathbf{x}})$, which contradicts that $\hat{\mathbf{x}}$ is maximum for $v^{\bar{u}}$.

## A.2 Preliminary results

We first prove some preliminary results that we need to prove our main results, where Lemma A2 below is a mathematical result that might have independent interest.

**Proposition A1.** Consider any SWR $\gtrsim$. If there exists $u \in U$ such that the Strongly Anonymous (Generalized) Utilitarian Dominance SWR $\gtrsim^D$ associated with $u$ is a subrelation to $\gtrsim$, then $\gtrsim$ satisfies Strong Anonymity, Monotonicity, Finite Completeness, Restricted Continuity, Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability.

*Proof.* Let the SWR $\gtrsim$ have the property that there exists $u \in U$ such that the Strongly Anonymous (Generalized) Utilitarian Dominance SWR $\gtrsim^D$ associated with $u$ is a subrelation to $\gtrsim$.

$\gtrsim$ satisfies Strong Anonymity. For any $\pi \in \Pi$ and $\mathbf{x} \in \mathbf{X}$, if $\mathbf{y} = \mathbf{x}_{\pi}$, then $\mathbf{x}_{[1]} = \mathbf{y}_{[1]}$. Hence, $\mathbf{x} \sim^D \mathbf{y}$ and $\mathbf{x} \sim \mathbf{y}$ since $\gtrsim^D$ is a subrelation to $\gtrsim$.

$\gtrsim$ satisfies Monotonicity. For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, if $\mathbf{x} > \mathbf{y}$, then $\mathbf{x}_{[1]} \ge \mathbf{y}_{[1]}$. Since $u \in U$ is increasing, $u(x_{[1]}) \ge u(y_{[1]})$ and $x_{[r]} \ge y_{[r]}$ for all $r \ge 2$. Hence, $\mathbf{x} \gtrsim^D \mathbf{y}$ and $\mathbf{x} \gtrsim \mathbf{y}$ since $\gtrsim^D$ is a subrelation to $\gtrsim$.

$\gtrsim$ satisfies Finite Completeness. For any $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbf{X}$ and for any $N \in \mathbb{N}$, let $\hat{\mathbf{z}} = \mathbf{x}_n\mathbf{z}$ and $\hat{\mathbf{y}} = \mathbf{y}_n\mathbf{z}$. Profiles $\hat{\mathbf{z}}$ and $\hat{\mathbf{y}}$ differ by a finite number of components, so there exists $k \in \mathbb{N}$ such that $\hat{z}_{[r]} = \hat{z}_{[r]}$ for all $r \ge k$. And, for any $u \in U$, we necessarily have that either $v_k^u(\hat{z}_{[r]}) - v_k^u(\hat{z}_{[r]}) \ge 0$ or $v_k^u(\hat{z}_{[r]}) - v_k^u(\hat{z}_{[r]}) \le 0$ (or both). This implies that either $\mathbf{x}_n\mathbf{z} \gtrsim^D \mathbf{y}_n\mathbf{z}$ or $\mathbf{y}_n\mathbf{z} \gtrsim^D \mathbf{x}_n\mathbf{z}$ (or both). Because $\gtrsim^D$ is a subrelation to $\gtrsim$, either $\mathbf{x}_n\mathbf{z} \gtrsim \mathbf{y}_n\mathbf{z}$ or $\mathbf{y}_n\mathbf{z} \gtrsim \mathbf{x}_n\mathbf{z}$ (or both).

$\gtrsim$ satisfies Restricted Continuity. Consider any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ and any $\ell \in \mathbb{R}$. Assume that there exist $\underline{\ell} < \bar{\ell}$ such that $\underline{\ell} \le \ell \le \bar{\ell}$ and $(\ell'1_N, \mathbf{x}) \sim (\ell'1_N, \mathbf{y})$ for all $\ell' \in (\underline{\ell}, \bar{\ell}) \setminus \{\ell\}$. It suffices to show the conclusion $(\ell1_N, \mathbf{x}) \sim (\ell1_N, \mathbf{y})$ when $\ell = \underline{\ell}$ and $\ell = \bar{\ell}$. Note that $\gtrsim^D$ is complete on

$$ \{(\mathbf{x}, \mathbf{y}) \in \mathbf{X} \times \mathbf{X} \mid \exists \ell \in \mathbb{R} \text{ and } n \in \mathbb{N} \text{ such that } \mathbf{x}_{[n]}, \mathbf{y}_{[n]} \in \mathbf{X}_{n}^{\ell}\}. $$
Hence, since $\succeq^D$ is a subrelation to $\succeq$, we obtain that, for each $\ell \in \mathbb{R}$ and $n \in \mathbb{N}$, for all $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ such that $\mathbf{x}_{[\ell]}, \mathbf{y}_{[\ell]} \in \mathbf{X}_n^\ell, \mathbf{x} \succeq \mathbf{y}$ is equivalent to $\mathbf{x} \succeq^D \mathbf{y}$. There are three cases.

**Case 1:** $\ell > \max\{\ell(\mathbf{x}), \ell(\mathbf{y})\}$. There exists $\ell' \in (\ell, \bar{\ell})$ such that $\ell' \ge \max\{\ell(\mathbf{x}), \ell(\mathbf{y})\}$. Write $\mathbf{x}' = (\ell'1_{\mathbb{N}}, \mathbf{x})$ and $\mathbf{y}' = (\ell'1_{\mathbb{N}}, \mathbf{y})$. Then $\mathbf{x}'_{[\ell]} = \mathbf{x}_{[\ell]}$ and $\mathbf{y}'_{[\ell]} = \mathbf{y}_{[\ell]}$. By definition, $(\ell'1_{\mathbb{N}}, \mathbf{x}) \sim^D \mathbf{x}$ and $(\ell'1_{\mathbb{N}}, \mathbf{y}) \sim^D \mathbf{y}$, so that $(\ell'1_{\mathbb{N}}, \mathbf{x}) \sim \mathbf{x}$ and $(\ell'1_{\mathbb{N}}, \mathbf{y}) \sim \mathbf{y}$ since $\succeq^D$ is a subrelation to $\succeq$. Since, by assumption, $(\ell'1_{\mathbb{N}}, \mathbf{x}) \sim (\ell'1_{\mathbb{N}}, \mathbf{y})$, we have $\mathbf{x} \sim \mathbf{y}$ by transitivity. By the same argument, $(\ell1_{\mathbb{N}}, \mathbf{x}) \sim \mathbf{x}$ and $(\ell1_{\mathbb{N}}, \mathbf{y}) \sim \mathbf{y}$, since also $\ell \ge \max\{\ell(\mathbf{x}), \ell(\mathbf{y})\}$. As $\mathbf{x} \sim \mathbf{y}$, we obtain $(\ell1_{\mathbb{N}}, \mathbf{x}) \sim (\ell1_{\mathbb{N}}, \mathbf{y})$ by transitivity.

**Case 2:** $\ell < \min\{\ell(\mathbf{x}), \ell(\mathbf{y})\}$. If $\ell = \bar{\ell}$, then there does not exist $r \in \mathbb{N}$ such that $\max\{y_{[r]}, \underline{\ell}\} < x_{[r]} \le \ell$ (or $\max\{x_{[r]}, \underline{\ell}\} < y_{[r]} \le \ell$). To show this, suppose to the contrary that $\ell = \bar{\ell}$ and there exists $r \in \mathbb{N}$ such that $\max\{y_{[r]}, \underline{\ell}\} < x_{[r]} \le \ell$. Choose $\ell'$ and $\ell''$ such that $\max\{y_{[r]}, \underline{\ell}\} < \ell' < \ell'' < x_{[r]}$. Since $x_{[r]} \le \bar{\ell} \le \ell(\mathbf{y})$, there exists $n > r$ such that $y_{[n]} \ge \ell''$. Write $\mathbf{x}' = (\ell'1_{\mathbb{N}}, \mathbf{x})$, $\mathbf{y}' = (\ell'1_{\mathbb{N}}, \mathbf{y})$, $\mathbf{x}'' = (\ell''1_{\mathbb{N}}, \mathbf{x})$, and $\mathbf{y}'' = (\ell''1_{\mathbb{N}}, \mathbf{y})$. By definition, $\mathbf{x}'_{[\ell]} \in \mathbf{X}_n^{\ell'}$ on which $\succeq^D$ and $\succeq$ coincide, so that $\mathbf{x}' \sim^D \mathbf{y}'$ is equivalent to $\mathbf{x}' \sim \mathbf{y}'$. Likewise, $\mathbf{x}''_{[\ell]} \in \mathbf{X}_n^{\ell''}$ on which $\succeq^D$ and $\succeq$ coincide, so that $\mathbf{x}'' \sim^D \mathbf{y}''$ is equivalent to $\mathbf{x}'' \sim \mathbf{y}''$. By assumption, $\mathbf{x}' \sim \mathbf{y}'$ and $\mathbf{x}'' \sim \mathbf{y}'$, implying that $\mathbf{x}' \sim^D \mathbf{y}'$ and $\mathbf{x}'' \sim^D \mathbf{y}''$. However, by construction

$$w_n^u(\mathbf{x}^{\prime\prime}) - w_n^u(\mathbf{x}^{\prime}) \ge (n-r)(\ell^{\prime\prime} - \ell^{\prime}) > w_n^u(\mathbf{y}^{\prime\prime}) - w_n^u(\mathbf{y}^{\prime}).$$

Since $w_n^u(\mathbf{x}') = w_n^u(\mathbf{y}')$ and $w_n^u(\mathbf{x}'') = w_n^u(\mathbf{y}'')$ are needed for $\mathbf{x}' \sim^D \mathbf{y}'$ and $\mathbf{x}'' \sim^D \mathbf{y}'$, we obtain a contradiction. Thus, since there does not exist $r \in \mathbb{N}$ such that $\max\{y_{[r]}, \underline{\ell}\} < x_{[r]} \le \ell$ (or $\max\{x_{[r]}, \underline{\ell}\} < y_{[r]} \le \ell$), we obtain that $(\ell'1_{\mathbb{N}}, \mathbf{x}) \sim (\ell'1_{\mathbb{N}}, \mathbf{y})$ for all $\ell' \in (\underline{\ell}, \ell)$ implies $(\ell'1_{\mathbb{N}}, \mathbf{x}) \sim^D (\ell'1_{\mathbb{N}}, \mathbf{y})$ for all $\ell' \in (\underline{\ell}, \ell]$. Therefore $(\ell1_{\mathbb{N}}, \mathbf{x}) \sim (\ell1_{\mathbb{N}}, \mathbf{y})$ since $\succeq^D$ is a subrelation to $\succeq$.

Likewise, if $\ell = \bar{\ell}$, in which case it suffices to let $\bar{\ell} \in (\ell, \min\{\ell(\mathbf{x}), \ell(\mathbf{y})\})$, there does not exist $r \in \mathbb{N}$ such that $\ell \le x_{[r]} < \min\{y_{[r]}, \bar{\ell}\}$ (or $\ell \le y_{[r]} < \min\{x_{[r]}, \bar{\ell}\}$). Similarly, this can be used to show that $(\ell'1_{\mathbb{N}}, \mathbf{x}) \sim (\ell'1_{\mathbb{N}}, \mathbf{y})$ for all $\ell' \in (\underline{\ell}, \bar{\ell})$ implies $(\ell1_{\mathbb{N}}, \mathbf{x}) \sim (\ell1_{\mathbb{N}}, \mathbf{y})$ also in this subcase.

**Case 3:** $\min\{\ell(\mathbf{x}), \ell(\mathbf{y})\} \le \ell \le \max\{\ell(\mathbf{x}), \ell(\mathbf{y})\}$. Suppose that $\ell(\mathbf{x}) > \ell(\mathbf{y})$ (the case $\ell(\mathbf{x}) < \ell(\mathbf{y})$ is similar). We first show that $(\underline{\ell}, \bar{\ell}) \cap (\ell(\mathbf{y}), \ell(\mathbf{x})) = \emptyset$. Because if there exists $\ell' \in (\underline{\ell}, \bar{\ell})$ such that $\ell(\mathbf{y}) < \ell' < \ell(\mathbf{x})$, then, by assumption, $\mathbf{x}' \sim \mathbf{y}'$, where $\mathbf{x}' = (\ell'1_{\mathbb{N}}, \mathbf{x})$ and $\mathbf{y}' = (\ell'1_{\mathbb{N}}, \mathbf{y})$. However, $\mathbf{x}' \succ^D \mathbf{y}'$ since $\ell(\mathbf{x}') > \ell(\mathbf{y}')$, and thus $\mathbf{x}' \succ \mathbf{y}'$ since $\succeq^D$ is a subrelation to $\succeq$, leading to a contradiction. Hence, either $\ell = \bar{\ell} = \max\{\ell(\mathbf{x}), \ell(\mathbf{y})\}$, which allows the proof of Case 1 to be adapted, or $\ell = \bar{\ell} = \min\{\ell(\mathbf{x}), \ell(\mathbf{y})\}$, which allows the proof of Case 2 to be adapted.

$\succeq$ satisfies Critical-Level Consistency. Consider any $\mathbf{x} \in \mathbf{X}$ and $z \in \mathbb{R}$. If $z \ge \ell(\mathbf{x})$ then $\mathbf{x}_{[\ell]} = \mathbf{y}_{[\ell]}$ whenever $\mathbf{y} = (z, \mathbf{x})$; similarly, $\mathbf{x}_{[\ell]} = \tilde{\mathbf{y}}_{[\ell]}$ whenever $\tilde{\mathbf{y}} = (z1_{\mathbb{N}}, \mathbf{x})$. Thus, if $z \ge \ell(\mathbf{x})$, $\mathbf{x} \sim^D (z, \mathbf{x}) \sim^D (z1_{\mathbb{N}}, \mathbf{x})$. Because $\succeq^D$ is a subrelation to $\succeq$, we obtain that $\mathbf{x} \sim^D (z, \mathbf{x}) \sim^D (z1_{\mathbb{N}}, \mathbf{x})$. If $z < \ell(\mathbf{x})$ and $\mathbf{y} = (z, \mathbf{x})$, then there exists $n = |\{j \in \mathbb{N}: y_j \le z\}|$ such that $x_{[r]} = y_{[r]}$ for all $r < n$, $x_{[n]} = y_{[n+1]} > y_{[n]} = z$, and $x_{[r]} \ge x_{[r-1]} = y_{[r]}$ for all $r > n$. Hence, $w_n^u(\mathbf{x}) - w_n^u(\mathbf{y}) > 0$ (since $u \in U$ is increasing) and $x_{[r]} \ge y_{[r]}$ for all $r > n$. Therefore, $\mathbf{x} \succ^D \mathbf{y}$ and $\mathbf{x} \succ \mathbf{y}$ since $\succeq^D$ is a subrelation to
```
~. If $z < \ell(\mathbf{x})$ and $\tilde{\mathbf{y}} = (z1_{\mathbb{N}}, \mathbf{x})$, $\ell(\tilde{\mathbf{y}}) < \ell(\mathbf{x})$ so that $\mathbf{x} \succ^D (z1_{\mathbb{N}}, \mathbf{x})$ and $\mathbf{x} \succ (z1_{\mathbb{N}}, \mathbf{x})$ since $\succeq^D$ is a subrelation to $\succeq$.
```

```
~ satisfies Liminf-Restricted Dominance. Consider any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ with $x_i > y_i$ for some $i \in \mathbb{N}$, $\liminf_{j \in \mathbb{N}} x_j > y_i$, and $x_j = y_j$ for all $j \in \mathbb{N} \setminus \{i\}$. Write $n = |\{j \in \mathbb{N} | y_j \le y_i\}|$. There are two cases.
```

Case 1: $x_i \ge \ell(\mathbf{x})$. Write $n = |\{j \in \mathbb{N} : y_j \le y_i\}|$. Then $x_{[r]} = y_{[r]}$ for all $r < n$, $x_{[n]} = y_{[n+1]} > y_{[n]} = y_i$, and $x_{[r]} \ge x_{[r-1]} = y_{[r]}$ for all $r > n$. Hence, $w_n^u(\mathbf{x}) - w_n^u(\mathbf{y}) > 0$ (since $u \in U$ is increasing) and $x_{[r]} \ge y_{[r]}$ for all $r > n$. Therefore, $\mathbf{x} \succ^D \mathbf{y}$ and $\mathbf{x} \succ \mathbf{y}$ since $\succeq^D$ is a subrelation to $\succeq$.

Case 2: $x_i < \ell(\mathbf{x})$. Write $m = |\{j \in \mathbb{N} : x_j \le x_i\}|$ and $n = |\{j \in \mathbb{N} : y_j \le y_i\}|$. Note that $m \ge n$ since $x_i > y_i$ and $x_j = y_j$ for all $j \in \mathbb{N} \setminus \{i\}$. Then $x_{[r]} = y_{[r]}$ for all $r < n$, $x_{[n]} = \min\{x_i, y_{[n+1]}\} > y_{[n]} = y_i$, $x_{[r]} \ge x_{[r-1]} = y_{[r]}$ for all $r \in \{n+1, n+2, \dots, m\}$, and $x_{[r]} = y_{[r]}$ for all $r > m$. Hence, $w_m^u(\mathbf{x}) - w_m^u(\mathbf{y}) > 0$ (since $u \in U$ is increasing) and $x_{[r]} = y_{[r]}$ for all $r > m$. Therefore, $\mathbf{x} \succ^D \mathbf{y}$ and $\mathbf{x} \succ \mathbf{y}$ since $\succeq^D$ is a subrelation to $\succeq$.

```
~ satisfies Liminf-Restricted Finite Separability. Consider any $\mathbf{x}, \mathbf{y}, \tilde{\mathbf{x}}, \tilde{\mathbf{y}} \in \mathbf{X}$ for which there exists a finite subset $N \subset \mathbb{N}$, such that:
```

(i) $x_i = \tilde{x}_i < \min\{\ell(\mathbf{x}), \ell(\tilde{\mathbf{x}})\}$ for all $i \in N$;

(ii) $y_i = \tilde{y}_i < \min\{\ell(\mathbf{y}), \ell(\tilde{\mathbf{y}})\}$ for all $i \in N$;

(iii) $x_j = y_j$ and $\tilde{x}_j = \tilde{y}_j$ for all $j \in \mathbb{N} \setminus N$.

Write $\bar{x} = \max_{i \in N} x_i$, $m = |\{j \in \mathbb{N} : x_j \le \bar{x}\}|$, $\tilde{m} = |\{j \in \mathbb{N} : \tilde{x}_j \le \bar{x}\}|$, and $\bar{m} = \max\{m, \tilde{m}\}$. Also, write $\bar{y} = \max_{i \in N} y_i$, $n = |\{j \in \mathbb{N} : y_j \le \bar{y}\}|$, $\tilde{n} = |\{j \in \mathbb{N} : \tilde{y}_j \le \bar{y}\}|$, and $\bar{n} = \max\{n, \tilde{n}\}$. By definition, we have $x_{[r]} = y_{[r]}$ for $r > \bar{m}$, so that $\mathbf{x} \succ^D \mathbf{y}$ if and only if $w_m^u(\mathbf{x}) - w_m^u(\mathbf{y}) \ge 0$, that is if and only if $\sum_{i \in N} u(x_i) \ge \sum_{i \in N} u(y_i)$. Likewise, we have $\tilde{x}_{[r]} = \tilde{y}_{[r]}$ for $r > \bar{n}$, so that $\tilde{\mathbf{x}} \succ^D \tilde{\mathbf{y}}$ if and only if $w_m^u(\tilde{\mathbf{x}}) - w_m^u(\tilde{\mathbf{y}}) \ge 0$, that is if and only if $\sum_{i \in N} u(\tilde{x}_i) \ge \sum_{i \in N} u(\tilde{y}_i)$. But since $\sum_{i \in N} u(x_i) = \sum_{i \in N} u(\tilde{x}_i)$ and $\sum_{i \in N} u(y_i) = \sum_{i \in N} u(\tilde{y}_i)$, we have that $\mathbf{x} \succ^D \mathbf{y}$ if and only if $\tilde{\mathbf{x}} \succ^D \tilde{\mathbf{y}}$. Because $\succeq^D$ is a subrelation to $\succeq$, this implies that $\mathbf{x} \succ^D \mathbf{y}$ if and only if $\tilde{\mathbf{x}} \succ^D \tilde{\mathbf{y}}$. $\square$

**Lemma A1.** Let $a, b \in \mathbb{R}$ be such that $a < b$. If $u \in U$, then there exist $0 < \Gamma < \Lambda$ satisfying that, for any $c, d \in \mathbb{R}$ such that $a \le c < d \le b$,

$$ \Gamma|d-c| < |u(d) - u(c)| < \Lambda|d-c|. $$

*Proof.* Assume that $u \in U$; thus, $u : \mathbb{R} \to \mathbb{R}$ is continuous and increasing and has bounded non-concavity. Let $a, b \in \mathbb{R}$ be such that $a < b$. Consider any $c, d \in \mathbb{R}$ such that $a \le c < d \le b$ and let $e, f \in \mathbb{R}$ be such that $e < a$ and $f > b$. Because $u$ has bounded non-concavity, there exists $\Theta > 0$ such that

$$ \left[ \frac{u(d) - u(c)}{d-c} \middle/ \frac{u(a) - u(e)}{a-e} \right] < \Theta \quad \text{and} \quad \left[ \frac{u(f) - u(b)}{f-b} \middle/ \frac{u(d) - u(c)}{d-c} \right] < \Theta. $$

Write

$$ \Gamma = \frac{1}{\Theta} \frac{u(f) - u(g)}{f-b} \quad \text{and} \quad \Lambda = \Theta \frac{u(a) - u(e)}{a-e}. $$
Given that $u$ is increasing, $\Gamma > 0$. Furthermore,

$$ \left[ \frac{u(d)-u(c)}{d-c} \middle/ \frac{u(a)-u(e)}{a-e} \right] < \Theta \quad \text{implies that} \quad u(d)-u(c) < \Lambda(d-c), $$

which is equivalent to $|u(d) - u(c)| < \Lambda|d - c|$. And

$$ \left[ \frac{u(f)-u(b)}{f-b} \middle/ \frac{u(d)-u(c)}{d-c} \right] < \Theta \quad \text{implies that} \quad u(d)-u(c) > \Gamma(d-c) $$

which is equivalent to $|u(d) - u(c)| > \Gamma|d - c|$. $\square$

**Lemma A2.** For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ and any $\pi \in \Pi$, if $\sum_{i=1}^{\infty} |x_{\pi(i)} - y_i| < +\infty$, then

$$ \sum_{r=1}^{\infty} |x_{[r]} - y_{[r]}| \le \sum_{i=1}^{\infty} |x_{\pi(i)} - y_i|. $$

*Proof. Step 1.* We show that for any $n \in \mathbb{N}\{1\}$, for any $(x_1, \dots, x_n)$, $(y_1, \dots, y_n) \in \mathbb{R}_+^n$ with $x_1 \le \dots \le x_n$ and $y_1 \le \dots \le y_n$, and for any permutations $\pi_1$ and $\pi_2$ on $\{1, \dots, n\}$,

$$ \sum_{i=1}^{n} |x_i - y_i| \le \sum_{i=1}^{n} |x_{\pi_1(i)} - y_{\pi_2(i)}|. \quad (A1) $$

We begin by showing that (A1) holds if $n = 2$. Let $(x_1, x_2)$, $(y_1, y_2) \in \mathbb{R}_+^2$ with $x_1 \le x_2$ and $y_1 \le y_2$. Also, let $\pi_1$ and $\pi_2$ be permutations on $\{1, 2\}$. Without loss of generality, we assume $\pi_1$ is the identity mapping. Thus, we show that

$$ |x_1 - y_1| + |x_2 - y_2| \le |x_1 - y_2| + |x_2 - y_1|. \quad (A2) $$

If $x_1 = x_2$ or $y_1 = y_2$, then (A2) holds with equality. Suppose that $x_1 < x_2$ and $y_1 < y_2$. We distinguish two cases. First, suppose that $x_1 \le y_1 < y_2 \le x_2$. Then,

$$ |x_1 - y_1| + |x_2 - y_2| < |x_1 - y_2| + |x_2 - y_2| < |x_1 - y_2| + |x_2 - y_1|. $$

Thus, (A2) holds. Next, suppose that $x_1 \le y_1$ and $x_2 \le y_2$. Then,

$$ \begin{aligned} |x_1 - y_1| + |x_2 - y_2| &= |y_1 - x_1 + y_2 - x_2| \\ &\le |y_1 - x_2| + |y_2 - x_1| = |x_1 - y_2| + |x_2 - y_1|. \end{aligned} $$

Thus, (A2) holds.

We now consider the case where $n > 2$. Let $(x_1, \dots, x_n)$, $(y_1, \dots, y_n) \in \mathbb{R}_+^n$ with $x_1 \le \dots \le x_n$ and $y_1 \le \dots \le y_n$. Consider any permutations $\pi_1$ and $\pi_2$ on $\{1, \dots, n\}$. Again, without loss of generality, we assume $\pi_1$ is the identity mapping. For each $k \in \{0, 1, \dots, n\}$, we define $(y_1^k, \dots, y_n^k) \in \mathbb{R}_+^n$ as follows. If $k = 0$, $(y_1^0, \dots, y_n^0) = (y_{\pi_2(1)}, \dots, y_{\pi_2(n)})$. For each $k \in \{1, \dots, n\}$, $(y_1^k, \dots, y_n^k)$ is constructed from $(y_1^{k-1}, \dots, y_n^{k-1})$ by exchanging $y_k^{k-1}$ and $y_j^{k-1} = y_k$, where $j$ is the first
component in $(y_1^{k-1}, \dots, y_n^{k-1})$ in which $y_k$ appears; thus, $j \ge k$. Formally, we define

$$y_i^k = \begin{cases} y_k & \text{if } i = k \\ y_k^{k-1} & \text{if } i = \min\{j \in \{k, \dots, n\} : y_j^{k-1} = y_k\} \\ y_i^{k-1} & \text{otherwise.} \end{cases}$$

Note that $(y_1^n, \dots, y_n^n) = (y_1, \dots, y_n)$. Further, for each $k \in \{1, \dots, n\}$, either $(y_1^k, \dots, y_n^k) = (y_1^{k-1}, \dots, y_n^{k-1})$ or $(y_1^k, \dots, y_n^k)$ is constructed by exchanging two components of $(y_1^{k-1}, \dots, y_n^{k-1})$ so as to make them rank-ordered. Thus, from (A2), it follows that for each $k \in \{1, \dots, n\}$,

$$\sum_{i=1}^{n} |x_i - y_i^k| \le \sum_{i=1}^{n} |x_i - y_i^{k-1}|.$$

Thus, we obtain

$$\sum_{i=1}^{n} |x_i - y_i| \le \sum_{i=1}^{n} |x_i - y_{\pi_2(i)}|.$$

**Step 2.** Let $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ and $\pi \in \Pi$. Assume $\sum_{i=1}^{\infty} |x_{\pi(i)} - y_i| < +\infty$. We show that $\ell(\mathbf{x}) = \ell(\mathbf{y})$. By way of contradiction, suppose $\ell(\mathbf{x}) \ne \ell(\mathbf{y})$. Without loss of generality, we assume $\ell(\mathbf{x}) > \ell(\mathbf{y})$. Let $\varepsilon = (\ell(\mathbf{x}) - \ell(\mathbf{y})) / 3$. From the definition of $\ell(\mathbf{x})$, there exists $N \in \mathbb{N}$ such that, for all $n > N$, $x_n > \ell(\mathbf{x}) - \varepsilon$. We define $M \subseteq \mathbb{N}$ by $M = \{n \in \mathbb{N} : y_n < \ell(\mathbf{y}) + \varepsilon\}$. From the definition of $\ell(\mathbf{y})$, it follows that $|M| = +\infty$. Let $\mathbf{z} = \mathbf{x}_{\pi}$ and define $\bar{N} = \max\{\pi(n) \in \mathbb{N} : n \in \{1, \dots, N\}\}$. Then, for all $n > \bar{N}$, $z_n > \ell(\mathbf{x}) - \varepsilon$. Letting $\bar{M} = \{n \in M : n \ge \bar{N}\}$, we obtain $|z_i - y_i| = z_i - y_i > \varepsilon > 0$ for all $i \in \bar{M}$ and, consequently, $\sum_{i \in \bar{M}} |z_i - y_i| = +\infty$. Thus,

$$\sum_{i=1}^{\infty} |x_{\pi(i)} - y_i| \ge \sum_{i \in \bar{M}} |z_i - y_i| = +\infty.$$

This is a contradiction since $\sum_{i=1}^{\infty} |x_{\pi(i)} - y_i| < +\infty$.

**Step 3.** Let $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ and $\pi \in \Pi$. Suppose that $\sum_{i=1}^{\infty} |x_{\pi(i)} - y_i| < +\infty$. It follows from Step 2 that $\ell(\mathbf{x}) = \ell(\mathbf{y})$. We define $\tilde{\mathbf{x}} \in \mathbf{X}$ by, for each $i \in \mathbb{N}$,

$$\tilde{x}_i = \begin{cases} x_{\pi(i)} & \text{if } x_{\pi(i)} < \ell(\mathbf{x}), \\ \ell(\mathbf{x}) & \text{if } x_{\pi(i)} \ge \ell(\mathbf{x}). \end{cases}$$

Analogously, define $\tilde{\mathbf{y}} \in \mathbf{X}$ by, for each $i \in \mathbb{N}$,

$$\tilde{y}_i = \begin{cases} y_i & \text{if } y_i < \ell(\mathbf{y}), \\ \ell(\mathbf{y}) & \text{if } y_i \ge \ell(\mathbf{y}). \end{cases}$$

Note that for all $i \in \mathbb{N}$, $|\tilde{x}_i - \tilde{y}_i| \le |x_{\pi(i)} - y_i|$. Thus,

$$\sum_{i=1}^{\infty} |\tilde{x}_i - \tilde{y}_i| \le \sum_{i=1}^{\infty} |x_{\pi(i)} - y_i|. \quad (A3)$$
Note that for any $n \in \mathbb{N}$, there exists $N \in \mathbb{N}$ with $N > n$ such that $(\tilde{x}_1, \dots, \tilde{x}_N)$ and $(\tilde{y}_1, \dots, \tilde{y}_N)$ contain all the components of $(x_{[1]}, \dots, x_{[n]})$ and $(y_{[1]}, \dots, y_{[n]})$, respectively. Hence, by (A1),

$$ \sum_{i=1}^{n} |x_{[i]} - y_{[i]}| \le \sum_{i=1}^{N} |\tilde{x}_i - \tilde{y}_i| \le \sum_{i=1}^{\infty} |\tilde{x}_i - \tilde{y}_i|. $$

Thus, from (A3), the sequence $\{D_n\}_{n \in \mathbb{N}}$ in $\mathbb{R}_+$ defined by $D_n = \sum_{i=1}^{n} |x_{[i]} - y_{[i]}|$ is bounded above. Since $\{D_n\}_{n \in \mathbb{N}}$ is non-decreasing, it is convergent and we obtain

$$ \lim_{n \to +\infty} D_n = \sum_{i=1}^{\infty} |x_{[i]} - y_{[i]}| \le \sum_{i=1}^{\infty} |x_{\pi(i)} - y_i|. $$

Thus, the proof is completed. □

### A.3 Proof of Lemma 1

Let the Strongly Anonymous (Generalized) Utilitarian SWR $\succeq_U$ be associated with the transformation function $u \in U$.

**Step 1:** For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, if $\mathbf{x} \succeq_U \mathbf{y}$, then $(\mathbf{x}_{[1]}, \mathbf{y}_{[1]}) \in \mathbf{D}$ and $\lim_{n \to \infty} (w_n^u(\mathbf{x}) - w_n^u(\mathbf{y})) \ge 0$. By Definition 2, there exists $\Delta \in \mathbb{R}_+ \cup \{\infty\}$ such that, for all $\pi \in \Pi$, $\lim_{n \to \infty} (v_n^u(\mathbf{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) = \Delta$. In particular, $\lim_{n \to \infty} (v_n^u(\mathbf{x}_{[1]}) - v_n^u(\mathbf{y}_{[1]})) = \lim_{n \to \infty} (w_n^u(\mathbf{x}) - w_n^u(\mathbf{y})) = \Delta \ge 0$. Suppose by way of contradiction that $(\mathbf{x}_{[1]}, \mathbf{y}_{[1]}) \notin \mathbf{D}$. This implies that

$$ \lim_{n \to \infty} \sum_{r \le n: y_{[r]} > x_{[r]}} (y_{[r]} - x_{[r]}) = +\infty. $$

Let $a = \min\{\inf(\mathbf{x}_{[1]}), \inf(\mathbf{y}_{[1]})\}$ and $b = \max\{\sup(\mathbf{x}_{[1]}), \sup(\mathbf{y}_{[1]}))\}$. By Lemma A1, there exists $\Gamma > 0$ satisfying that for any $c, d \in \mathbb{R}$ such that $a \le c < d \le b$, $\Gamma|d-c| < |u(d) - u(c)|$. Hence, for all $n \in \mathbb{N}$,

$$ \Gamma \sum_{r \le n: y_{[r]} > x_{[r]}} (y_{[r]} - x_{[r]}) < \sum_{i \le n: y_{[i]} > x_{[i]}} (u(y_{[i]}) - u(x_{[i]})), $$

so that

$$ \lim_{n \to \infty} \sum_{r \le n: y_{[r]} > x_{[r]}} (u(y_{[r]}) - u(x_{[r]})) = +\infty. $$

There exists $\varepsilon > 0$ and $\pi \in \Pi$ which alternates between selecting elements from $\{r \in \mathbb{N} : y_{[r]} > x_{[r]}\}$ and $\{r \in \mathbb{N} : x_{[r]} > y_{[r]}\}$ and which always selects a sufficient number of elements of $\{r \in \mathbb{N} : y_{[r]} > x_{[r]}\}$ before selecting additional elements from $\{r \in \mathbb{N} : x_{[r]} > y_{[r]}\}$ such that, for all $n \in \mathbb{N}$, $w_n^u(\mathbf{x}_{[\pi]}) - w_n^u(\mathbf{y}_{[\pi]}) \le -\varepsilon$. This contradicts that $\Delta \in \mathbb{R}_+ \cup \{\infty\}$.

**Step 2:** For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, if $(\mathbf{x}_{[1]}, \mathbf{y}_{[1]}) \in \mathbf{D}$ and $\lim_{n \to \infty} (w_n^u(\mathbf{x}) - w_n^u(\mathbf{y})) \ge 0$, then $\mathbf{x} \succeq_U \mathbf{y}$. Since $(\mathbf{x}_{[1]}, \mathbf{y}_{[1]}) \in \mathbf{D}$, there exists $\delta \in \mathbb{R}_+$ such that $\sum_{i: y_{[i]} > x_{[i]}} (y_{[i]} - x_{[i]}) = \delta$. By Lemma A1, the sum $\sum_{i \le n: y_{[i]} > x_{[i]}} (u(y_{[i]}) - u(x_{[i]}))$ is bounded above by $\Lambda\delta$, and it is by definition increasing. Hence, there exists $\Delta_- \in \mathbb{R}_-$ such that $\sum_{i: y_{[i]} > x_{[i]}} (u(x_{[i]}) - u(y_{[i]})) = \Delta_-.$
Write $\Delta = \lim_{n \to \infty} (w_n^u(\mathbf{x}) - w_n^u(\mathbf{y})) = \sum_{r=1}^{\infty} (u(x_{[r]}) - u(y_{[r]}))$. If $\Delta \in \mathbb{R}_+$, then we have $\Delta_+ = \sum_{i \le n: x_{[i]} > y_{[i]}} (u(x_{[i]}) - u(y_{[i]})) = \Delta - \Delta_- < +\infty$. Hence, $\sum_{r=1}^{\infty} |u(x_{[r]}) - u(y_{[r]})| = \Delta_+ - \Delta_- < +\infty$. We obtain $\mathbf{x} \gtrsim^U \mathbf{y}$ since absolute convergence implies unconditional convergence. Similarly, if $\Delta = +\infty$, then $\Delta_+ = +\infty$ and $\lim_{n \to \infty} (v_n^u(\mathbf{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) = +\infty$ for all $\pi \in \Pi$, thereby obtaining $\mathbf{x} \gtrsim^U \mathbf{y}$ also in this case.

## A.4 Completing the proof of Theorem 1

The proof of Theorem 1 is completed by means of the following result.

**Proposition A2.** The Strongly Anonymous (Generalized) Utilitarian SWR $\gtrsim^U$ is regular. Consider any SWR $\gtrsim$. If there exists $u \in U$ such that the Strongly Anonymous (Generalized) Utilitarian SWR $\gtrsim^U$ associated with $u$ is a subrelation to $\gtrsim$, then $\gtrsim$ satisfies Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability.

*Proof.* Let $u \in U$, and let $\gtrsim$ be a SWR to which the Strongly Anonymous Utilitarian SWR $\gtrsim^U$ associated with $u$ is subrelation. By Observation 2, the Strongly Anonymous Utilitarian Dominance SWR $\gtrsim^D$ associated with $u$ is a subrelation to $\gtrsim^U$. Hence, since $\gtrsim^U$ associated with $u$ is a subrelation to the SWR $\gtrsim$, also $\gtrsim^D$ associated with $u$ is a subrelation to $\gtrsim$. So, by Proposition A1, $\gtrsim$ satisfies Strong Anonymity, Monotonicity, Finite Completeness, Restricted Continuity, Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability. It remains to be shown that $\gtrsim^U$ satisfies Weak Continuity and that $\gtrsim$ satisfies Limited Inequity.

*Step 1:* $\gtrsim^U$ satisfies Weak Continuity. Consider any $\mathbf{y} \in \mathbf{X}$. Let $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ be a sequence in $\{\mathbf{x} \in \mathbf{X}: \mathbf{x} \gtrsim^U \mathbf{y}\}$, and assume that $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ converges to $\mathbf{x}^*$ in $d_1$. We show that $\mathbf{x}^* \gtrsim^U \mathbf{y}$, which means that $\{\mathbf{x} \in \mathbf{X}: \mathbf{x} \gtrsim^U \mathbf{y}\}$ is closed in $(\mathbf{X}, d_1)$. Since $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ converges to $\mathbf{x}^*$ in $d_1$, it follows from Lemma A2 that

$$ \lim_{k \to +\infty} \sum_{r=1}^{\infty} \left| x_{[r]}^k - x_{[r]}^* \right| = 0. \qquad (A4) $$

Because $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ are bounded sequences that converge to $\mathbf{x}^*$ in $d_1$, then $a = \inf_{k \in \mathbb{N}} \{\inf_t \mathbf{x}_t^k\}$ and $b = \sup_{k \in \mathbb{N}} \{\sup_t \mathbf{x}_t^k\}$ are well-defined finite numbers. By Lemma A1, there exists $\Lambda \in \mathbb{R}_+$ such that for all $z, z' \in [a, b]$:

$$ |u(z) - u(z')| \le \Lambda |z - z'|. \qquad (A5) $$

Let $\varepsilon \in (0, \Lambda)$. By (A4), there exists $K \in \mathbb{N}$ such that

$$ \sum_{r=1}^{\infty} \left| x_{[r]}^{K} - x_{[r]}^{*} \right| < \frac{\varepsilon}{2\Lambda}. $$

From (A5), we obtain

$$ \sum_{r=1}^{\infty} \left| u(x_{[r]}^{K}) - u(x_{[r]}^{*}) \right| \le \Lambda \sum_{i=1}^{\infty} \left| x_{[i]}^{K} - x_{[i]}^{*} \right| < \frac{\varepsilon}{2}. \qquad (A6) $$
On the other hand, since $\mathbf{x}^K \gtrsim^U \mathbf{y}$, it follows that

$$ \lim_{n \to +\infty} \sum_{r=1}^{n} \left( u(x_{[r]}^{K}) - u(y_{[r]}) \right) \geq 0. $$

Thus, there exists $N \in \mathbb{N}$ such that for all $n > N$,

$$ \sum_{r=1}^{n} \left( u(x_{[r]}^{K}) - u(y_{[r]}) \right) \geq -\frac{\varepsilon}{2}. \qquad (A7) $$

Thus, for all $n > N$, we obtain by (A6) and (A7) that

$$ \begin{aligned} \sum_{r=1}^{n} \left( u(x_{[r]}^{*}) - u(y_{[r]}) \right) &= \sum_{r=1}^{n} \left( u(x_{[r]}^{*}) - u(x_{[r]}^{K}) \right) + \sum_{r=1}^{n} \left( u(x_{[r]}^{K}) - u(y_{[r]}) \right) \\ &\geq -\sum_{r=1}^{n} \left| u(x_{[r]}^{*}) - u(x_{[r]}^{K}) \right| + \sum_{r=1}^{n} \left( u(x_{[r]}^{K}) - u(y_{[r]}) \right) \\ &\geq -\sum_{r=1}^{\infty} \left| u(x_{[r]}^{*}) - u(x_{[r]}^{K}) \right| + \sum_{r=1}^{n} \left( u(x_{[r]}^{K}) - u(y_{[r]}) \right) \\ &> -\varepsilon. \end{aligned} $$

Since $\varepsilon \in (0, \Lambda)$ was chosen arbitrarily, we obtain

$$ \liminf_{n \to +\infty} \sum_{r=1}^{n} \left( u(x_{[r]}^{*}) - u(y_{[r]}) \right) = \liminf_{n \to +\infty} \left( v_{n}^{u}(\mathbf{x})_{[1]}^{*} - v_{n}^{u}(\mathbf{y})_{[1]} \right) = \liminf_{n \to +\infty} \left( w_{n}^{u}(\mathbf{x}^{*}) - w_{n}^{u}(\mathbf{y}) \right) \geq 0. $$

Note also that, for each $k \in \mathbb{N}$ and each $r \in \mathbb{N}$, $x_{[r]}^{*} + |x_{[r]}^{k} - x_{[r]}^{*}| \geq x_{[r]}^{k} \geq x_{[r]}^{*} - |x_{[r]}^{k} - x_{[r]}^{*}|$ and $y_{[r]} \geq y_{[r]} - |x_{[r]}^{k} - x_{[r]}^{*}|$. Hence, $\max\{x_{[r]}^{k}, y_{[r]}\} \geq \max\{x_{[r]}^{*}, y_{[r]}\} - |x_{[r]}^{k} - x_{[r]}^{*}|$ and $\max\{x_{[r]}^{k}, y_{[r]}\} - x_{[r]}^{k} \geq \max\{x_{[r]}^{*}, y_{[r]}\} - 2|x_{[r]}^{k} - x_{[r]}^{*}|$ for each $k \in \mathbb{N}$ and each $t \in \mathbb{N}$. We obtain that for each $N \in \mathbb{N}$ and $k \in \mathbb{N}$,

$$ \sum_{r=1}^{N} \left[ \max\{x_{[r]}^{k}, y_{[r]}\} - x_{[r]}^{k} \right] \geq \sum_{r=1}^{N} \left[ \max\{x_{[r]}^{*}, y_{[r]}\} - x_{[r]}^{*} \right] - 2 \sum_{r=1}^{N} |x_{[r]}^{k} - x_{[r]}^{*}|. $$

Because $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ are bounded sequences that converge to $\mathbf{x}^*$ in $d_1^S$, there exists $K \in \mathbb{N}$ such that $\sum_{r=1}^{\infty} |x_{[r]}^K - x_{[r]}^*|$ is finite. This implies that

$$ \sum_{r: y_{[r]} > x_{[r]}} (y_{[r]} - x_{[r]}) = \sum_{r=1}^{\infty} \left[ \max\{x_{[r]}^{*}, y_{[r]}\} - x_{[r]}^{*} \right] = \delta $$

where $\delta \in \mathbb{R}_+$. Hence $(\mathbf{x}_{[1]}^*, \mathbf{y}_{[1]} \in \mathbf{D}$.

Finally, the fact that $(\mathbf{x}_{[1]}^*, \mathbf{y}_{[1]} \in \mathbf{D}$ and $\liminf_{n \to +\infty} (w_n^u(\mathbf{x}^*) - w_n^u(\mathbf{y})) \geq 0$ imply that there exists $\Delta \in \mathbb{R}_+ \cup \{\infty\}$ such that $\lim_{n \to \infty} (w_n^U(\mathbf{x}^*) - w_n^U(\mathbf{y})) = \Delta$. Thus, $\mathbf{x}^* \gtrsim^U \mathbf{y}$ follows. The proof that $\{\mathbf{x} \in \mathbf{X} : \mathbf{y} \gtrsim^U \mathbf{x}\}$ is closed in $(\mathbf{X}, d_1)$ is analogous.

**Step 2: Any SWR $\gtrsim$ to which $\gtrsim^U$ is a subrelation satisfies Limited Inequality.** We first show that $\gtrsim^U$ associated with $u$ satisfies Limited Inequality. Since $u \in U$, $u$ has
bounded non-concavity. Hence, there exists $\Gamma \in \mathbb{R}_{++}$ such that:

$$ \Gamma > \left[ \frac{u(z_4)-u(z_3)}{z_4-z_3} / \frac{u(z_2)-u(z_1)}{z_2-z_1} \right] $$

for all $z_1 < z_2 \le z_3 < z_4$. Consider any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ with $y_i < x_i \le x_j < y_j$ for some $i, j \in \mathbb{N}$, $x_k = y_k$ for all $k \in \mathbb{N} \setminus \{i, j\}$, and $y_j - x_j = (1/\Gamma)(x_i - y_i)$. There are three cases:

1. If $y_j < \ell(\mathbf{y})$, then writing $n = |\{k \in \mathbb{N} : y_k \le y_j\}|$, $x_{[k]} = y_{[k]}$ for all $k > n$ and

$$ \begin{aligned} v_n^u(\mathbf{x}_{[k]}) - v_n^u(\mathbf{y}_{[k]}) &= u(x_i) + u(x_j) - u(y_i) - u(y_j) \\ &= \frac{u(x_i)-u(y_i)}{\Gamma} \left[ \Gamma - \frac{u(y_j)-u(x_j)}{y_j-x_j} / \frac{u(x_i)-u(y_i)}{x_i-y_i} \right] > 0 \end{aligned} $$

because $u$ is increasing. Hence, $\mathbf{x} \gtrsim_D \mathbf{y}$ and $\mathbf{x} \gtrsim^U \mathbf{y}$ since $\gtrsim_D$ is a subrelation to $\gtrsim^U$.

2. If $x_j < \ell(\mathbf{y}) \le y_j$, then, for any $n > |\{k \in \mathbb{N} : x_k \le x_j\}|$, $y_{[n]} \ge x_{[n]} > x_j$ and $v_n^u(\mathbf{x}_{[k]}) - v_n^u(\mathbf{y}_{[k]}) = u(x_i) + u(x_j) - u(y_i) - u(y_j)$. Hence, since $y_{[n]}$ is non-decreasing and converges to $\ell(\mathbf{y})$, we have, for all $\pi \in \Pi$,

$$ \begin{aligned} \lim_{n \to \infty} (v_n^u(\mathbf{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) &= u(x_i) + u(x_j) - u(y_i) - u(\ell(\mathbf{y})) \\ &\ge u(x_i) + u(x_j) - u(y_i) - u(y_j) \\ &= \frac{u(x_i)-u(y_i)}{\Gamma} \left[ \Gamma - \frac{u(y_j)-u(x_j)}{y_j-x_j} / \frac{u(x_i)-u(y_i)}{x_i-y_i} \right] > 0 \end{aligned} $$

because $u$ is increasing, implying that $\mathbf{x} \gtrsim^U \mathbf{y}$.

3. If $\ell(\mathbf{y}) \le x_j$, then, for all $\pi \in \Pi$,

$$ \lim_{n \to \infty} (v_n^u(\mathbf{x}_{[\pi]}) - v_n^u(\mathbf{y}_{[\pi]})) = u(\min\{x_i, \ell(\mathbf{y})\}) - u(\min\{y_i, \ell(\mathbf{y})\}) \ge 0 $$

because $x_i > y_i$ and $u$ is increasing, implying that $\mathbf{x} \gtrsim^U \mathbf{y}$.

Hence, $1/\Gamma$ is in the set $\mathcal{E}$ determined for $\gtrsim^U$, implying that $\gtrsim^U$ satisfies Limited Inequality.

However, any SWR $\gtrsim$ to which $\gtrsim^U$ is a subrelation has also the property that $\mathbf{x} \gtrsim \mathbf{y}$ for any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ as considered above. Hence, $1/\Gamma$ is also in the set $\mathcal{E}$ determined for $\gtrsim$, implying that also $\gtrsim$ satisfied Limited Inequality. $\square$

## A.5 Proof of Lemma 5

*Step 1: Representing a complete SWR $\gtrsim_n^\ell$ on $(-\infty, \ell]^n$. Let $\ell \in \mathbb{R}$ and $n \in \mathbb{N}$ with $n \ge 3$. We define the binary relation $\gtrsim_n^\ell$ on $(-\infty, \ell]^n$ as follows. For all $(x_1, \dots, x_n)$, $(y_1, \dots, y_n) \in (-\infty, \ell]^n$,

$$ (x_1, \dots, x_n) \gtrsim_n^\ell (y_1, \dots, y_n) \quad \text{if and only if} \quad \mathbf{x}_n \mathbf{z} \gtrsim \mathbf{y}_n \mathbf{z}, \qquad (A8) $$
where $z = \ell 1_N$. Since $\gtrsim$ satisfies Finite Completeness and Weak Continuity, $\gtrsim_n^\ell$ is a continuous complete preorder on $\mathbb{R}_+^N$. Since $\gtrsim$ satisfies Strong Anonymity and Liminf-Restricted Finite Separability, we have that $\gtrsim_n^\ell$ satisfies the independence condition of Debreu (1960). Furthermore, each coordinate of $\mathbb{R}_+^n$ is essential in the sense of Debreu (1960) because $\gtrsim$ satisfies Liminf-Restricted Dominance. Thus, it follows from Theorem 3 in Debreu (1960) that there exist $n$ real-valued functions $u_n^{1\ell}, \dots, u_n^{n\ell}$ on $\mathbb{R}_+$ such that, for all $(x_1, \dots, x_n)$, $(y_1, \dots, y_n) \in (-\infty, \ell]^n$,

$$ (x_1, \dots, x_n) \gtrsim_n^\ell (y_1, \dots, y_n) \quad \text{if and only if} \quad \sum_{i=1}^n u_n^{i\ell}(x_i) \ge \sum_{i=1}^n u_n^{i\ell}(y_i). $$

Also, the functions $u_n^{1\ell}, \dots, u_n^{n\ell}$ are determined up to a common positive affine transformation.

Since $\gtrsim_n^\ell$ is a continuous complete pre-order on $\mathbb{R}_+^n$ and that $\gtrsim$ satisfies Strong Anonymity and Liminf-Restricted Dominance, the functions $u_n^{1\ell}, \dots, u_n^{n\ell}$ must be the same continuous and increasing function. By defining $u_n^\ell = u_n^{i\ell}$ for all $i \in \{1, \dots, n\}$, we obtain the following representation of $\gtrsim_n^\ell$ on $(-\infty, \ell]^n$, for $\ell \in \mathbb{R}$ and $n \in \mathbb{N}$ with $n \ge 3$: There exists a continuous and increasing real-valued function $u_n^\ell$ on $(-\infty, \ell]$ such that, for all $(x_1, \dots, x_n)$, $(y_1, \dots, y_n) \in (-\infty, \ell]^n$,

$$ (x_1, \dots, x_n) \gtrsim_n^\ell (y_1, \dots, y_n) \quad \text{if and only if} \quad \sum_{i=1}^n u_n^\ell(x_i) \ge \sum_{i=1}^n u_n^\ell(y_i). \quad (A9) $$

Note that $u_n^\ell$ is determined up to a positive affine transformation.

*Step 2: Link between the complete SWRs $\gtrsim_n^\ell$ for different $n \in \mathbb{N}$. Fix $\ell \in \mathbb{R}$ and write $u^\ell = u_n^\ell$. We show that $u^\ell$ has the property, for all $n \ge 3$ it is the case that, for all $\mathbf{x}, \mathbf{y} \in \mathbf{X}_n^\ell$,

$$ \mathbf{x} \gtrsim \mathbf{y} \quad \text{if and only if} \quad \sum_{i=1}^{n} u^{\ell}(x_i) \geq \sum_{i=1}^{n} u^{\ell}(y_i). \quad (A10) $$

It is straightforward to prove this if $n=3$. Thus, let $n>3$. Then, it follows from (A8) and (A9) that for any $\mathbf{x}, \mathbf{y} \in \mathbf{X}_3^\ell$,

$$ \begin{align*} (x_1, x_2, x_3) \gtrsim_3^\ell (y_1, y_2, y_3) &\Leftrightarrow \mathbf{x} \gtrsim \mathbf{y} \Leftrightarrow (x_1, x_2, x_3, \ell, \dots, \ell) \gtrsim_n^\ell (y_1, y_2, y_3, \ell, \dots, \ell) \\ &\Leftrightarrow \sum_{i=1}^3 u_n^\ell(x_i) \ge \sum_{i=1}^3 u_n^\ell(y_i). \end{align*} $$

Recall that $u^\ell$ is determined up to a positive affine transformation. Thus, there exist $a_n^\ell \in \mathbb{R}_++$ and $b_n^\ell \in \mathbb{R}$ such that $u_n^\ell = a_n^\ell u^\ell + b_n^\ell$. Thus, it follows from (A8) and (A9) that for all $\mathbf{x}, \mathbf{y} \in \mathbf{X}_n^\ell$,

$$ \begin{align*} \mathbf{x} \gtrsim \mathbf{y} &\Leftrightarrow (x_1, \dots, x_n) \gtrsim_n^\ell (y_1, \dots, y_n) \Leftrightarrow \sum_{i=1}^n u_n^\ell(x_i) \ge \sum_{i=1}^n u_n^\ell(y_i) \\ &\Leftrightarrow \sum_{i=1}^n a_n^\ell u^\ell(x_i) + nb_n^\ell \ge \sum_{i=1}^n a_n^\ell u^\ell(y_i) + nb_n^\ell \Leftrightarrow \sum_{i=1}^n u^\ell(x_i) \ge \sum_{i=1}^n u^\ell(y_i). \end{align*} $$
Let $n = 2$ and $\ell \in \mathbb{R}$ and assume that $\mathbf{x}, \mathbf{y} \in \mathbf{X}_2^\ell$. Since $\mathbf{x}, \mathbf{y} \in \mathbf{X}_3^\ell$, (A8) and (A9) imply that

$$ \mathbf{x} \gtrsim \mathbf{y} \Leftrightarrow \sum_{i=1}^{3} u^{\ell}(x_i) \geq \sum_{i=1}^{3} u^{\ell}(y_i) \Leftrightarrow \sum_{i=1}^{2} u^{\ell}(x_i) \geq \sum_{i=1}^{2} u^{\ell}(y_i). $$

Likewise, for $n = 1$, we trivially have that $\mathbf{x} \gtrsim \mathbf{y}$ is equivalent to $u^\ell(x_1) \geq u^\ell(y_1)$ for $\mathbf{x}, \mathbf{y} \in \mathbf{X}_1^\ell$ because $u^\ell$ is increasing and $\gtrsim$ satisfies Liminf-Restricted Dominance.

**Step 3:** Extension to all $\ell \in \mathbb{R}$. We show that there exists a continuous and increasing function $u: \mathbb{R}_+ \to \mathbb{R}_+$ such that, for all $\ell \in \mathbb{R}$, all $n \in \mathbb{N}$, and all $\mathbf{x}, \mathbf{y} \in \mathbf{X}_n^\ell$,

$$ \mathbf{x} \gtrsim \mathbf{y} \quad \text{if and only if} \quad \sum_{i=1}^{n} u(x_i) \geq \sum_{i=1}^{n} u(y_i). \qquad (A11) $$

From Step 2, we know that, for any $\ell \in \mathbb{R}$, there exists a continuous and increasing function $u^\ell: (-\infty, \ell] \to \mathbb{R}$ such that (A10) holds for all $n \in \mathbb{N}$ and for all $\mathbf{x}, \mathbf{y} \in \mathbf{X}_n^\ell$. Since, for each $\ell \in \mathbb{R}$, $u^\ell$ is unique up to a positive affine transformation, we can without loss of generality normalize these functions for all $\ell \geq 1$ so that $u^\ell(0) = 0$ and $u^\ell(1) = 1$.

We show first that $u^\ell = u^{\ell'}$ on $(-\infty, \ell]$ for all $\ell, \ell' \in \mathbb{R}$ satisfying $\ell > \ell' \geq 1$. Let $\ell > \ell' \geq 1$ and consider any $\mathbf{x}, \mathbf{y} \in \mathbf{X}_3^\ell$ and $\mathbf{x}', \mathbf{y}' \in \mathbf{X}_3^{\ell'}$ such that $(x_1, x_2, x_3) = (x'_1, x'_2, x'_3)$ and $(y_1, y_2, y_3) = (y'_1, y'_2, y'_3)$, implying that none of the three first components exceeds $\ell'$. Since $\gtrsim$ satisfies Liminf-Restricted Finite Separability:

$$ \mathbf{x} \gtrsim \mathbf{y} \Leftrightarrow \mathbf{x}' \gtrsim \mathbf{y}' \Leftrightarrow \sum_{i=1}^{3} u^{\ell'}(x_i) \geq \sum_{i=1}^{3} u^{\ell'}(y_i). $$

Because $\mathbf{x}, \mathbf{y} \in \mathbf{X}_3^\ell$ and $u^\ell$ in representation (A10) is unique up to an affine transformation, there exists $a^\ell \in \mathbb{R}_+$ and $b^\ell$ such that $u^\ell = a^\ell u^{\ell'} + b^\ell$ on $(-\infty, 1]$. On the other hand, since by assumption $u^\ell$ and $u^{\ell'}$ are normalized with $u^\ell(0) = u^1(0) = 0$ and $u^\ell(1) = u^{\ell'}(1) = 1$, we obtain $b^\ell = 0$ and $a^\ell = 1$ so that $u^\ell(x) = u^{\ell'}(x)$ for all $x \in (-\infty, \ell]$.

Likewise, we obtain that, for all $\ell \in \mathbb{R}$ with $\ell < 1$, all $n \in \mathbb{N}$, and all $\mathbf{x}, \mathbf{y} \in \mathbf{X}_n^\ell$,

$$ \mathbf{x} \gtrsim \mathbf{y} \Leftrightarrow \sum_{i=1}^{n} u^1(x_i) \geq \sum_{i=1}^{n} u^1(y_i). \qquad (A12) $$

Define the function $u: \mathbb{R} \to \mathbb{R}$ as follows:

for all $z \leq 1$, $u(z) = u^1(z)$.

for all $z > 1$, $u(z) = u^\ell(z)$ for some $\ell \geq z$.

Since $u^\ell = u^{\ell'}$ on $(-\infty, \ell]$ for all $\ell, \ell' \in \mathbb{R}$ satisfying $\ell > \ell' \geq 1$, $u$ is a well-defined real-valued function on $\mathbb{R}$. Also, it follows that, for all $\ell \in \mathbb{R}$ and all $z \in (-\infty, \ell]$, $u(z) = u^\ell(z)$. Thus, $u$ is continuous and increasing on $\mathbb{R}$ since, for all $\ell \in \mathbb{R}$, $u^\ell$ in continuous and increasing on $(-\infty, \ell]$.

**Step 4:** $u$ has bounded non-concavity. Since $\gtrsim$ satisfies Limited Inequality, there exists $\varepsilon > 0$ such that, for all $z_1 < z_2 \leq z_3 < z_4$ with $z_4 - z_3 = \varepsilon(z_2 - z_1)$, we have that
x ≳ y if x and y are defined by $x_1 = z_2$, $x_2 = z_3$, $y_1 = z_1$, $y_2 = z_4$ and $x_i = y_i = z_4$ for all $i \ge 3$. Because $x, y \in \mathbf{X}_{z_4}^2$, Step 3 implies that $u(z_2) + u(z_3) \ge u(z_1) + u(z_4)$. This can be written:

$$ \varepsilon \frac{u(z_4)-u(z_3)}{z_4-z_3} = \frac{u(z_4)-u(z_3)}{z_2-z_1} \le \frac{u(z_2)-u(z_1)}{z_2-z_1}. $$

So, for all $z_1 < z_2 \le z_3 < z_4$ such that $z_4 - z_3 = \varepsilon(z_2 - z_1)$:

$$ \frac{1}{\varepsilon} \ge \frac{u(z_4)-u(z_3)}{z_4-z_3} \bigg/ \frac{u(z_2)-u(z_1)}{z_2-z_1}. $$

Writing $\mathbf{X}_{\varepsilon} = \{(z_1, z_2, z_3, z_4) \in \mathbb{R}^4 | z_1 < z_2 \le z_3 < z_4 \text{ and } z_4 - z_3 = \varepsilon(z_2 - z_1)\}$ and

$$ G_u(\varepsilon) = \sup_{(z_1, z_2, z_3, z_4) \in \mathbf{X}_{\varepsilon}} \frac{u(x_4)-u(x_3)}{x_4-x_3} / \frac{u(x_2)-u(x_1)}{x_2-x_1}, $$

Chateauneuf, Cohen, and Meilijson (2005, Lemma 1) show that $G_u = G_u(\varepsilon)$ for any $\varepsilon \in \mathbb{R}_{++}$. Our results above imply that $G_u(\varepsilon) < 1/\varepsilon$. Hence, $G_u < +\infty$ and $u$ has bounded non-concavity.

## A.6 Characterizing Strongly Anonymous Utilitarian Catching-Up

We characterize ≳C by applying an additional axiom.

**Axiom (Restricted Tail Continuity).** For any $\mathbf{x}, \mathbf{y} \in \mathbf{X}^+$, if there exists $m > 0$ such that, for all $k \ge m$, $\mathbf{x}_k(z_k \mathbb{1}_N) \gtrsim \mathbf{y}_k(z_k \mathbb{1}_N)$, where $z_k = \max\{x_k, y_k\}$, then $\mathbf{x} \gtrsim \mathbf{y}$.

Adding this axiom to the four axioms of Theorem 1 that characterizes the Strongly Anonymous Utilitarian SWR, we obtain a characterization of the Strongly Anonymous Utilitarian Catching-Up SWR in terms of set inclusion between SWRs. As the following result shows, the Strongly Anonymous Utilitarian Catching-Up SWR is the smallest element in terms of set inclusion among the regular SWRs that satisfy the five axioms. This result will be used to relate the Strongly Anonymous Utilitarian Catching-up SWR to the SWRs of subsection 5.1.

**Proposition A3.** The Strongly Anonymous (Generalized) Utilitarian Catching-Up SWR ≳C is regular and satisfies Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, Liminf-Restricted Finite Separability, and Restricted Tail Continuity. If ≳ is a regular SWR satisfying Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, Liminf-Restricted Finite Separability, and Restricted Tail Continuity, then there exists $u \in U$ such that ≳ weakly extends the Strongly Anonymous (Generalized) Utilitarian Catching-Up SWR ≳C associated with u.

*Proof. Part 1:* ≳C is regular and satisfies the axioms. By Observations 1 and 2, ≳C is a subrelation to ≳U and ≳D. Therefore, the proof of regularity of ≳C is very similar to the proof of regularity of ≳U in Proposition A2 of Appendix
A.4. Also, by Theorem 1, $\gtrsim^C$ satisfies Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability.

It remains to be shown that, for any $u \in U$, $\gtrsim^C$ associated with $u$ satisfies Restricted Tail Continuity. Consider any $\mathbf{x}, \mathbf{y} \in \mathbf{X}^+$ for which there exists $m > 0$ such that, for all $k \ge m$, $\mathbf{x}_k(z_k \mathbb{1}_N) \gtrsim^C \mathbf{y}_k(z_k \mathbb{1}_N)$, where $z_k = \max\{x_k, y_k\}$. By definition, $\mathbf{x}_k(z_k \mathbb{1}_N) \gtrsim^C \mathbf{y}_k(z_k \mathbb{1}_N)$ means that $\liminf_{n \to +\infty} (w_n^u(\mathbf{x}_k(z_k \mathbb{1}_N)) - w_n^u(\mathbf{y}_k(z_k \mathbb{1}_N))) = v_k^u(\mathbf{x}_k)] - v_k^u(\mathbf{y}_k)] \ge 0$. So there exists $m > 0$ such that, for all $k \ge m$, $v_k^u(\mathbf{x}_k)] - v_k^u(\mathbf{y}_k)] = w_k^u(\mathbf{x}) - w_k^u(\mathbf{y}) \ge 0$. Therefore, it is necessarily the case that $\liminf_{n \to +\infty} (w_n^u(\mathbf{x}) - w_n^u(\mathbf{y})) \ge 0$ and, thus, $\mathbf{x} \gtrsim^C \mathbf{y}$.

**Part 2:** A regular $\gtrsim$ satisfying the axioms weakly extends $\gtrsim^C$. Assume that $\gtrsim$ is a regular SWR satisfying Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, Liminf-Restricted Finite Separability, and Restricted Tail Continuity. By Lemma 6, there exists $u \in U$ such that the Strongly Anonymous (Generalized) Utilitarian Dominance SWR $\gtrsim^D$ associated with $u$ is a subrelation to $\gtrsim$. Let $\gtrsim^C$ be the Strongly Anonymous (Generalized) Utilitarian Catching-Up SWR associated with $u$, and let $\mathbf{x}, \mathbf{y} \in \mathbf{X}$ be such that $\mathbf{x} \gtrsim^C \mathbf{y}$. We want to prove that $\mathbf{x} \gtrsim \mathbf{y}$. By Lemma 4, $\mathbf{x}_1 \sim \mathbf{x}$ and $\mathbf{y}_1 \sim \mathbf{y}$. By transitivity, it suffices to show that $\mathbf{x}_1 \gtrsim \mathbf{y}_1$. Our strategy of proof is to construct a sequence $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ such that $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ converges to $\mathbf{y}_1$ in $d_1$ and such that $\mathbf{x}_1 \gtrsim \mathbf{y}^n$ for each $n \in \mathbb{N}$. Since $\gtrsim$ satisfies Weak Continuity, we thus obtain $\mathbf{x}_1 \gtrsim \mathbf{y}_1$.

*Construction of the sequence* $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$. Let $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ be defined by, for each $n \in \mathbb{N}$,

$$ 
\begin{aligned}
y_1^n &= u^{-1}(u(y_{[1]}) + \min\{0, \inf_{k \ge n} (v_k^u(\mathbf{x}_{[1]}) - v_k^u(\mathbf{y}_{[1]}))\}), \\
y_r^n &= y_{[r]} \quad \text{for all } r > 1.
\end{aligned}
$$

Note that $\mathbf{y}^n$ is well-defined since $u \in U$ has bounded non-concavity and is thus unbounded below.

*The sequence* $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ converges to $\mathbf{y}_1$ in $d_1$. By the construction of $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$, $d_1(\mathbf{y}_1, \mathbf{y}^n) \le y_1^n - y_1^n$. Moreover, $\inf_{k \ge n} (v_k^u(\mathbf{x}_1)] - v_k^u(\mathbf{y}_1)]) \le v_k^u(\mathbf{x}_1)] - v_k^u(\mathbf{y}_1)]$ for all $k \ge n$ and, since $\mathbf{x} \gtrsim^C \mathbf{y}$,

$$ \lim_{n \to \infty} \left( \inf_{k \ge n} (v_k^u(\mathbf{x}_1]) - v_k^u(\mathbf{y}_1]) \right) = \liminf_{n \to \infty} (v_n^u(\mathbf{x}_1]) - v_n^u(\mathbf{y}_1]) \ge 0. $$

Hence, $\lim_{n \to \infty} y_1^n = y_1$, and we obtain that $\{\mathbf{y}^n\}_{n \in \mathbb{N}}$ converges to $\mathbf{y}_1$ in $d_1$.

$\mathbf{x}_1 \gtrsim \mathbf{y}^n$ for each $n \in \mathbb{N}$. Consider any $n \in \mathbb{N}$. Note first that, for each $k \ge n$,

$$ 
\begin{aligned}
v_k^u(\mathbf{x}_1]) - v_k^u(\mathbf{y}^n) &= v_k^u(\mathbf{x}_1]) - u(y_1^n) - \sum_{r=2}^k u(y_{[r]}) \\
&\ge v_k^u(\mathbf{x}_1]) - v_k^u(\mathbf{y}_1]) - \inf_{k \ge n} (v_k^u(\mathbf{x}_1]) - v_k^u(\mathbf{y}_1]) \ge 0.
\end{aligned}
$$

For each $k \ge n$, write $z_k = \max\{x_{[k]}, y_k^n\}$, $\tilde{\mathbf{x}}^k = (\mathbf{x}_1)_k(z_k \mathbb{1}_N)$, and $\tilde{\mathbf{y}}^{n,k} = (\mathbf{y}^n)_k(z_k \mathbb{1}_N)$. Then

$$ v_k^u(\tilde{\mathbf{x}}^k) - v_k^u(\tilde{\mathbf{y}}^{n,k}) = v_k^u(\mathbf{x}_1]) - v_k^u(\mathbf{y}^n) \ge 0 $$
and $\tilde{x}_{[r]}^k = \tilde{y}_{[r]}^{n,k}$ for all $r > n$. Hence, $\tilde{x}^k \gtrsim^D \tilde{y}^{n,k}$. By Lemma 6, this implies $\tilde{x}^k \gtrsim \tilde{y}^{n,k}$. Since $\mathbf{x}_{[r]}$, $\mathbf{y}^n \in \mathbf{X}^+$ and, for each $k \ge n$, $(\mathbf{x}_{[r]})_k(z_k 1_N) \gtrsim (\mathbf{y}^n)_k(z_k 1_N)$, it follows from Restricted Tail Continuity that $\mathbf{x}_{[r]} \gtrsim \mathbf{y}^n$. $\square$

## A.7 Proof of Proposition 2

Let the Limit of Rank-Discounted Utilitarian SWR $\gtrsim^R$, the Strongly Anonymous Utilitarian Cesàro Summation SWR $\gtrsim^S$, the Strongly Anonymous Utilitarian Catching-Up SWR $\gtrsim^C$, and the Strongly Anonymous Utilitarian SWR $\gtrsim^U$ all be associated with $u$.

*Proof of part (i):* $\gtrsim^C$, $\gtrsim^R$, and $\gtrsim^S$ are regular SWRs. It follows from Proposition A3 that $\gtrsim^C$ is regular. Moreover, it follows from Observation 3 and Proposition A1 that $\gtrsim^R$ and $\gtrsim^S$ satisfy Strong Anonymity, Monotonicity, Finite Completeness, and Restricted Continuity. By Definition 4, it remains to be shown that $\gtrsim^R$ and $\gtrsim^S$ satisfy Weak Continuity.

$\gtrsim^S$ satisfies Weak Continuity. Consider any $\mathbf{y} \in \mathbf{X}$. Let $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ be a sequence in $\{\mathbf{x} \in \mathbf{X} : \mathbf{x} \gtrsim^S \mathbf{y}\}$, and assume that $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ converges to $\mathbf{x}^*$ in $d_1$. We show that $\mathbf{x}^* \gtrsim^S \mathbf{y}$, which means that $\{\mathbf{x} \in \mathbf{X} : \mathbf{x} \gtrsim^S \mathbf{y}\}$ is closed in $(\mathbf{X}, d_1)$. For any $\mathbf{x}_{[r]}^k$, $\mathbf{x}_{[r]}^* \in \mathbf{X}^+$, we obtain

$$ \lim_{n \to +\infty} \sum_{r=1}^{n} \frac{n-r+1}{n} \left| u(x_{[r]}^k) - u(x_{[r]}^*) \right| \le \sum_{r=1}^{\infty} \left| u(x_{[r]}^k) - u(x_{[r]}^*) \right|. $$

Thus, applying the same argument as that used in Step 1 of the proof of Proposition A2, it follows from the local Lipschitz continuity of $u$ (Lemma A1) and the convergence of $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ that there exists $K \in \mathbb{N}$ such that

$$ \lim_{n \to +\infty} \sum_{r=1}^{n} \frac{n-r+1}{n} \left| u(x_{[r]}^K) - u(x_{[r]}^*) \right| < \frac{\varepsilon}{2}, \quad (A13) $$

where $\varepsilon \in (0, \Lambda)$ and $\Lambda \in \mathbb{R}_{++}$ are obtained in Lemma A1. Furthermore, since

$$ \liminf_{n \to +\infty} \sum_{r=1}^{n} \frac{n-r+1}{n} \left( u(x_{[r]}^K) - u(y_{[r]}) \right) \ge 0 $$

follows from the fact that $\mathbf{x}^K \gtrsim^S \mathbf{y}$, there exists $N \in \mathbb{N}$ such that for all $n > N$,

$$ \sum_{r=1}^{n} \frac{n-r+1}{n} \left( u(x_{[r]}^{K}) - u(y_{[r]}) \right) \geq -\frac{\varepsilon}{2}. \quad (A14) $$

Thus, employing an argument analogous to that used in Step 1 of the proof of Proposition A2, and using (A13) and (A14) instead of (A6) and (A7), it follows that, for all $n > N$,

$$ \sum_{r=1}^{n} \frac{n-r+1}{n} \left( u(x_{[r]}^{*}) - u(y_{[r]}) \right) > -\varepsilon. $$

Hence, since $\varepsilon \in (0, \Lambda)$ was chosen arbitrarily, we obtain

$$ \liminf_{n \to +\infty} \sum_{r=1}^{n} \frac{n-r+1}{n} \left( u(x_{[r]}^{*}) - u(y_{[r]}) \right) = \liminf_{n \to +\infty} \left( \sigma_{n}^{u}(\mathbf{x}^{*}) - \sigma_{n}^{u}(\mathbf{y}) \right) \geq 0. $$

Thus, $\mathbf{x}^* \gtrsim^S \mathbf{y}$ follows. The proof that $\{\mathbf{x} \in \mathbf{X} : \mathbf{y} \gtrsim^S \mathbf{x}\}$ is closed in $(\mathbf{X}, d_1)$ is analogous.
``$\succeq^R$ satisfies Weak Continuity. Consider any $\mathbf{y} \in \mathbf{X}$. Let $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ be a sequence in $\{\mathbf{x} \in \mathbf{X} : \mathbf{x} \succeq^R \mathbf{y}\}$, and assume that $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ converges to $\mathbf{x}^*$ in $d_1$. We show that $\mathbf{x}^* \succeq^R \mathbf{y}$, which means that $\{\mathbf{x} \in \mathbf{X} : \mathbf{x} \succeq^R \mathbf{y}\}$ is closed in $(\mathbf{X}, d_1)$. For any $\mathbf{x}_{[\cdot]}^k$, $\mathbf{x}_{[\cdot]}^* \in \mathbf{X}^+$, we obtain, for all $\beta \in (0, 1)$,

$$ \sum_{r=1}^{\infty} \beta^{r-1} \left| u(x_{[r]}^k) - u(x_{[r]}^*) \right| \le \sum_{r=1}^{\infty} \left| u(x_{[r]}^k) - u(x_{[r]}^*) \right|. $$

Thus, applying the same argument as that used in Step 1 of the proof of Proposition A2, it follows from the local Lipschitz continuity of $u$ (Lemma A1) and the convergence of $\{\mathbf{x}^k\}_{k \in \mathbb{N}}$ that there exists $K \in \mathbb{N}$ such that for all $\beta \in (0, 1)$,

$$ \sum_{r=1}^{\infty} \beta^{r-1} \left| u(x_{[r]}^K) - u(x_{[r]}^*) \right| < \frac{\varepsilon}{2}, \quad (A15) $$

where $\varepsilon \in (0, \Lambda)$ and $\Lambda \in \mathbb{R}_{++}$ are obtained in Lemma A1. Furthermore, since

$$ \liminf_{\beta \to 1^{-}} (\rho_{\beta}^{u}(\mathbf{x}^{K}) - \rho_{\beta}^{u}(\mathbf{y})) \ge 0 $$

follows from the fact that $\mathbf{x}^K \succeq^R \mathbf{y}$, there exists $\beta^* \in (0, 1)$ such that for all $\beta \in (\beta^*, 1)$,

$$ \rho_{\beta}^{u}(\mathbf{x}^{K}) - \rho_{\beta}^{u}(\mathbf{y}) \geq -\frac{\varepsilon}{2}. \quad (A16) $$

Thus, employing an argument analogous to that used in Step 1 of the proof of Proposition A2, and using (A15) and (A16) instead of (A6) and (A7), it follows that, for all $\beta \in (\beta^*, 1)$,

$$ \begin{aligned} \rho_{\beta}^{u}(\mathbf{x}^{*}) - \rho_{\beta}^{u}(\mathbf{y}) &= \rho_{\beta}^{u}(\mathbf{x}^{*}) - \rho_{\beta}^{u}(\mathbf{x}^{K}) + \rho_{\beta}^{u}(\mathbf{x}^{K}) - \rho_{\beta}^{u}(\mathbf{y}) \\ &= \sum_{r=1}^{\infty} \beta^{r-1} \left( u(x_{[r]}^{*}) - u(x_{[r]}^{K}) \right) + \rho_{\beta}^{u}(\mathbf{x}^{K}) - \rho_{\beta}^{u}(\mathbf{y}) \\ &\geq -\sum_{r=1}^{\infty} \beta^{r-1} \left| u(x_{[r]}^{*}) - u(x_{[r]}^{K}) \right| + \rho_{\beta}^{u}(\mathbf{x}^{K}) - \rho_{\beta}^{u}(\mathbf{y}) > -\varepsilon. \end{aligned} $$

Hence, since $\varepsilon \in (0, \Lambda)$ was chosen arbitrarily, we obtain

$$ \liminf_{\beta \to 1^{-}} (\rho_{\beta}^{u}(\mathbf{x}^{*}) - \rho_{\beta}^{u}(\mathbf{y})) \ge 0. $$

Thus, $\mathbf{x}^* \succeq^R \mathbf{y}$ follows. The proof that $\{\mathbf{x} \in \mathbf{X} : \mathbf{y} \succeq^R \mathbf{x}\}$ is closed in $(\mathbf{X}, d_1)$ is analogous.

*Proof of part (iii):* $\succeq^U$ is a subrelation to $\succeq^C$, $\succeq^R$, and $\succeq^S$. By Observation 1, this holds for $\succeq^C$. In view of Theorem 1 it suffices to show that $\succeq^R$ and $\succeq^S$ are regular and satisfy Limited Inequity, Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability. It follows from part (i) that $\succeq^R$ and $\succeq^S$ are regular and, by Observation 3 and Proposition A1, satisfy Critical-Level Consistency, Liminf-Restricted Dominance, and Liminf-Restricted Finite Separability. The proof of
$\gtrsim^R$ and $\gtrsim^S$ satisfying Limited Inequity is omitted as it is similar to the proof that $\gtrsim^U$ satisfies this axiom in Step 2 of the proof of Proposition A2.

*Proof of part (ii):* $\gtrsim^S$ weakly extends $\gtrsim^C$. In view of Proposition A3 and the proof of part (iii) it suffices to show that $\gtrsim^S$ satisfies Restricted Tail Continuity. This proof is omitted because it is similar to the proof that $\gtrsim^C$ satisfies this axiom in part 2 of the proof of Proposition A3.

*Proof of part (ii):* $\gtrsim^R$ weakly extends $\gtrsim^S$. It follows from Jonsson and Voorn- eveld (2018, Lemma 1) that, for any $\mathbf{x}, \mathbf{y} \in \mathbf{X}$, $\liminf_{n \to +\infty} (\sigma_n^u(\mathbf{x}) - \sigma_n^u(\mathbf{y})) \le \liminf_{\beta \to 1^-} (\rho_\beta^u(\mathbf{x}) - \rho_\beta^u(\mathbf{y}))$. Thus, $\mathbf{x} \gtrsim^S \mathbf{y}$ implies $\mathbf{x} \gtrsim^R \mathbf{y}$, showing that $\gtrsim^R$ weakly extends $\gtrsim^S$.

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