Before proceeding to the survey of the fairnessrelevant properties of aggregating functions, we state a condition that is desirable, but not strictly necessary, from the perspective of outcome fairness.
4.1. FairnessRelevant Properties of Aggregating Functions
Impartiality is one of the three minimal conditions a welfarist social ranking of outcomes must satisfy, according to condition (i) stated in
Section 2. It requires that such a ranking must not be influenced by the identity of agents. In terms of properties of aggregating functions, this condition may be stated as follows.
Definition 2. Impartiality: An aggregating function $g$ satisfies the condition of impartiality if and only if, for any permutation $\pi $ of the set $\left\{1,\dots ,N\right\}$, Permutation invariance of an aggregating function
$g$ ensures that the social rank
$g\left(u\left(x\right)\right)$ of outcome
$x$ depends only on the pattern of agent utilities in
$x$ and not on the indices
$i=1,\dots ,N$ through which individual agents are identified. This implies that equals, i.e., agents with the same utilities, are treated equally, which fulfills condition (1) of outcome fairness stated in
Section 1.
Sensitivity to the patterns of utility distribution within outcomes allows relevant utility differences among agents to be recognized, which, in turn, is a prerequisite for condition (2) of outcome fairness stated in
Section 1, which requires “unequals” to be treated according to their relevant differences. The property of an aggregating function implying its sensitivity to patterns of distribution most often encountered in the literature is the Pigou–Dalton condition [
11,
19,
26,
29,
35].
Definition 3. Pigou–Dalton (PD) condition: An aggregating function $g$ satisfies the PD condition if and only iffor any $i,j=1,\dots ,N$ such that ${u}_{i}>{u}_{j}$ and any $0<\epsilon <{u}_{i}{u}_{j}$.
The PD condition being fulfilled by an aggregating function $g$ means that any transfer of utility from a betteroff agent to a worseoff agent increases the value of the social utility function $g\left(u(\cdot )\right)$ as long as the beneficiary of such a transfer does not end up being betteroff than its benefactor. Any such transfer of utility, which results in a more even utility distribution while preserving total utility and the relative utility ranking of agents, is called a Pigou–Dalton (PD) transfer. The PD condition can thus be expressed by stating that all PD transfers increase $g\left(u(\cdot )\right)$.
As discussed in
Section 3, certain types of aggregationbased social rankings of outcomes (axiological–egalitarian, Rawlsian–egalitarian, and complaintswithinoutcome) are prone to levelingdown effects, i.e., to disagreements between the social preferences and interests of all agents. Such disagreements are problematic from the perspective of outcome fairness, which is a question of how agents perceive outcomes. Indeed, if all agents perceive an outcome
$x$ to be at least as good as an outcome
$y$, i.e., if they attain at least as high a utility in
$x$ as in
$y$, while some agents fare better in
$x$ than in
$y$, it is difficult to justify why choosing
$y$ over
$x$ should be fair or socially more desirable. It is therefore important to recognize that a ranking of outcomes generated by a social utility function
$g\left(u(\cdot )\right)$ does not suffer from levelingdown effects if the aggregating function
$g$ is strongly increasing.
Definition 4. Strong increase (SI): An aggregating function $g$ is strongly increasing if and only if $g\left({u}^{\prime}\right)>g\left({u}^{\u2033}\right)$ whenever ${u}_{i}^{\prime}\ge {u}_{i}^{\u2033}$ for all $i=1,\dots ,N$ and ${u}_{j}^{\prime}>{u}_{j}^{\u2033}$ for some $j=1,\dots ,N$.
The SI property of an aggregating function $g$ implies that the ranking of outcomes generated by the social utility function $g\left(u(\cdot )\right)$ is Paretoinclusive. Indeed, if outcome $x$ is Paretosuperior to outcome $y$, i.e., if ${u}_{i}\left(x\right)\ge {u}_{i}\left(y\right)$ for all $i=1,\dots ,N$ and ${u}_{j}\left(x\right)>{u}_{j}\left(y\right)$ for some $j=1,\dots N$, then $g\left(u\left(x\right)\right)>g\left(u\left(y\right)\right)$, and $x$ is thus socially more desirable than $y$. Consequently, the outcome deemed the most socially desirable according to such a ranking must be Paretooptimal.
Another property enhancing the perception of agents of being treated fairly is the possibility of demonstrating to each of them that their individual interests contribute to the formation of the social ranking of outcomes without being entangled or confused with the interests of other agents. A social ranking of outcomes generated by a social utility function $g\left(u(\cdot )\right)$ has this property if the aggregating function $g$ is additive.
Definition 5. Additivity: An aggregating function $g$ is additive if and only if it is of the form $g\left(u\right)={{\displaystyle \sum}}_{i=1}^{N}{h}_{i}\left(u\right)$.
For each $i=1,\dots ,N$, the function ${h}_{i}(\cdot )$ describes how the interests of agent $i$ contribute to the social ranking generated by $g\left(u(\cdot )\right)$. Importantly, this contribution may depend not only on the utility attained by agent $i$ but also on other information contained in the utility profile $u$, e.g., on how well agent $i$ fares in comparison to other agents.
The latter dependence may lead to situations in which the interests of agent $i$ are served equally well in two outcomes $x$ and $y$, i.e., ${u}_{i}\left(x\right)={u}_{i}\left(y\right)$, but the contributions of its interests to the social rankings of $x$ and $y$ differ, i.e., ${h}_{i}\left(u\left(x\right)\right)\ne {h}_{i}\left(u\left(y\right)\right)$, due to differences in the utilities of other agents. The following tighter property of separability, which implies additivity and is not to be confused with the separateness of persons, is sufficient to prevent such effects.
Definition 6. Separability: An aggregating function $g$ is separable if and only if it is of the form $g\left(u\right)={{\displaystyle \sum}}_{i=1}^{N}h\left({u}_{i}\right)$, where $h$ is an increasing function.
Separability of an aggregating function
$g$ implies that the ranking of outcomes generated by the social utility function
$g\left(u(\cdot )\right)$ is independent of unconcerned agents ([
11], p. 67), i.e., the social ranks generated by
$g\left(u\left(x\right)\right)$ and
$g\left(u\left(y\right)\right)$ for outcomes
$x$ and
$y$ are not influenced by agents whose utilities do not change when
$x$ is changed to
$y$, since
${u}_{i}\left(x\right)={u}_{i}\left(y\right)$ implies
$h\left({u}_{i}\left(x\right)\right)=h\left({u}_{i}\left(y\right)\right)$.
4.2. Classification of FairnessRelevant Aggregating Functions
The fairnessrelevant properties of an aggregating function
$g$ imply certain features of the ranking of outcomes generated by the social utility function
$g\left(u(\cdot )\right)$, which we discuss in this subsection. For each approach to the aggregationbased social ranking of outcomes described in
Section 3, some of the properties of aggregating functions described in
Section 4 are strictly required, while others are too strong or unnecessary. Thus, with each of the approaches discussed in
Section 3 we can associate a collection of properties described in
Section 4.1. These collections of properties are specific enough for a taxonomy of aggregating functions to be built, which we present in
Figure 2. Moreover, in
Table 1,
Table 2,
Table 3,
Table 4 and
Table 5, we provide an overview of several useful types of aggregating functions together with information on their fairnessrelevant properties and on how they can be associated with the approaches described in
Section 3.
Figure 2.
Taxonomy of fairness relevant aggregating functions. Boxes outlined in color (with bold labels) indicate fairnessrelevant properties of aggregating functions described in
Section 4.1. Boxes outlined in black (with italic labels) indicate approaches to utility aggregation described in
Section 3 and shown in
Figure 1. Notice that the sufficientist approach is not included here since it cannot be operationalized with a single aggregating function Bullet points (with regular labels) indicate exemplary types of aggregating functions described in
Section 4.2. Function types shown within an approach can be associated with that approach. Approaches shown within a property can be associated with function types possessing that property. Accordingly, approaches can be characterized by the collections of properties within which they are shown. Formulas and details specifying the shown exemplary types of aggregating functions are provided in
Table 1,
Table 2,
Table 3,
Table 4 and
Table 5. Notice that, for consistency with
Figure 1, the sum, representing the utilitarian approach, is shown here as part of the presented taxonomy. This helps to place the shown fairnessrelevant aggregating functions in relation to the sum of agent utilities, which is commonly used in IAMs. However, the sum itself is not a fairnessrelevant aggregating function. As discussed in
Section 3, the utilitarian approach, represented by the sum of agent utilities, has serious deficiencies from the perspective of outcome fairness.
Figure 2.
Taxonomy of fairness relevant aggregating functions. Boxes outlined in color (with bold labels) indicate fairnessrelevant properties of aggregating functions described in
Section 4.1. Boxes outlined in black (with italic labels) indicate approaches to utility aggregation described in
Section 3 and shown in
Figure 1. Notice that the sufficientist approach is not included here since it cannot be operationalized with a single aggregating function Bullet points (with regular labels) indicate exemplary types of aggregating functions described in
Section 4.2. Function types shown within an approach can be associated with that approach. Approaches shown within a property can be associated with function types possessing that property. Accordingly, approaches can be characterized by the collections of properties within which they are shown. Formulas and details specifying the shown exemplary types of aggregating functions are provided in
Table 1,
Table 2,
Table 3,
Table 4 and
Table 5. Notice that, for consistency with
Figure 1, the sum, representing the utilitarian approach, is shown here as part of the presented taxonomy. This helps to place the shown fairnessrelevant aggregating functions in relation to the sum of agent utilities, which is commonly used in IAMs. However, the sum itself is not a fairnessrelevant aggregating function. As discussed in
Section 3, the utilitarian approach, represented by the sum of agent utilities, has serious deficiencies from the perspective of outcome fairness.
Impartiality is a property inherent to all approaches to utility aggregation described in
Section 3. Approaches that reflect certain intrinsic ethical values of outcomes are impartial because these values exist independently of any particular group of agents and their identities and because the moral judgements based on these values do not change when applied to another group of agents with different identities experiencing the same circumstances. Impartiality is also one of the requirements of the separateness of persons, which calls for equal and individual attention to be given to the interests of all agents according to condition (i) for the separateness of persons as stated in
Section 3. Consequently, approaches to operationalizing the notion of the separateness of persons are impartial. Thus, all aggregating functions implementing any of the approaches to utility aggregation described in
Section 3 satisfy the condition of impartiality.
The PD condition is often interpreted in the literature as an aversion to inequality ([
11], p. 67). It is also one of the defining features of aggregate inequality measures. Because the reduction of inequalities in utility distribution is the sole objective of axiological–egalitarian social rankings of outcomes, various types of inequality measures can be used to define social utility functions that generate this type of ranking. Importantly, values of any aggregate inequality measure are bounded from below by zero, and this lower bound is attained for utility profiles which are perfectly equitable, and thus also socially most desirable in the axiologicalegalitarian ranking. Therefore, to be consistent with the convention applied throughout this paper that a social utility function
$g\left(u(\cdot )\right)$ takes higher values for socially more desirable outcomes, aggregating functions
g we associate with the axiological–egalitarian approach are defined as negatives of inequality measures. Examples of types of such aggregating functions are presented in
Table 1. The selection of inequality measures used there is not exhaustive but contains the most popular options.
Table 1.
Examples of types of aggregating functions that can be associated with the axiological–egalitarian approach.
Table 1.
Examples of types of aggregating functions that can be associated with the axiological–egalitarian approach.
Type of Function  Properties  Comments and References 

Negative Gini index $g\left(u\right)=\frac{{{\displaystyle \sum}}_{i=1}^{N}{{\displaystyle \sum}}_{j=1}^{N}\left{u}_{i}{u}_{j}\right}{2N{{\displaystyle \sum}}_{j=1}^{N}{u}_{j}}$  RRI, Impartiality, PD, Additivity  Gini index can be interpreted geometrically as the ratio between the area under the Lorenz curve of the utility profile
$u$ (given by the cumulative utility of the
$k$ worstoff agents as a function of $k=1,\dots ,N)$ and the area under the Lorenz curve representing a perfectly equal distribution (given by the 45° line).

Negative generalized entropy index ${g}_{\alpha}\left(u\right)=\{\begin{array}{ll}\frac{1}{N}{\displaystyle \sum _{i=1}^{N}}\mathrm{ln}\frac{{u}_{i}}{\mu \left(u\right)}\hfill & \mathrm{for}\alpha =0\hfill \\ \frac{1}{N}{\displaystyle \sum _{i=1}^{N}}\frac{{u}_{i}}{\mu \left(u\right)}\mathrm{ln}\frac{{u}_{i}}{\mu \left(u\right)}\hfill & \mathrm{for}\alpha =1\hfill \\ \frac{1}{N\alpha \left(\alpha 1\right)}{\displaystyle \sum _{i=1}^{N}}\left({\left(\frac{{u}_{i}}{\mu \left(u\right)}\right)}^{\alpha}1\right)\hfill & \mathrm{for}\alpha \ne 0,1\hfill \end{array}$ where $\mu \left(u\right)=\frac{1}{N}{{\displaystyle \sum}}_{i=1}^{N}{u}_{i}$  RRI, Impartiality, PD, Additivity  The parameter $\infty \le \alpha \le \infty $ determines the sensitivity to changes in different segments of the utility profile: lower values imply higher sensitivity to changes in the lower tail of the utility profile, while higher values imply higher sensitivity to changes in the upper tail. Special cases: ${g}_{0}(\cdot )$ is the mean log deviation or Theil $L$ index ${g}_{1}(\cdot )$ is the Theil $T$ index, and ${g}_{2}(\cdot )$ is half the squared coefficient of variation.

Negative Atkinson inequality index ${I}_{\gamma}\left(u\right)=\{\begin{array}{ll}\left(\frac{N}{{{\displaystyle \sum}}_{i=1}^{N}{u}_{i}}\right){\left({\displaystyle \prod _{i=1}^{N}}{u}_{i}\right)}^{\frac{1}{N}}1\hfill & \mathrm{for}\gamma =1\hfill \\ \left(\frac{N}{{{\displaystyle \sum}}_{i=1}^{N}{u}_{i}}\right){\left(\frac{{{\displaystyle \sum}}_{i=1}^{N}{u}_{i}^{1\gamma}}{N}\right)}^{\frac{1}{1\gamma}}1\hfill & \mathrm{for}\gamma \ne 1,\gamma \ge 0\hfill \end{array}$  RRI, Impartiality, PD  The parameter $0\le \gamma \le \infty $ determines the social aversion to inequality, with $\gamma =0$ corresponding to no aversion and $\gamma =\infty $ to infinite aversion. The utility profile that maximizes this function may be neither unique nor Paretooptimal.

Strong increase (SI) is a necessary condition for a social utility function to generate a Paretoinclusive ranking of outcomes. SI also implies that such a ranking gives preference to outcomes that are overall more efficient, in the sense that they deliver a higher total utility. Indeed, if outcome
$x$ is Paretosuperior to outcome
$y$, i.e., if
${u}_{i}\left(x\right)\ge {u}_{i}\left(y\right)$ for all
$i=1,\dots ,N$ and
${u}_{j}\left(x\right)\ge {u}_{j}\left(y\right)$ for some
$1\le j\le N$, then
${{\displaystyle \sum}}_{i=1}^{N}{u}_{i}\left(x\right)>{{\displaystyle \sum}}_{i=1}^{N}{u}_{i}\left(y\right)$. Consequently, if an aggregating function
$g$ satisfies both the PD and the SI conditions, the social utility function
$g\left(u(\cdot )\right)$ generates a ranking of outcomes that balances social preferences for equality (according to the PD condition) and for efficiency (according to the SI condition). Thus, aggregating functions possessing both the PD and the SI properties can be associated with the Paretian–egalitarian approach to building aggregationbased social rankings of outcomes. Examples of types of aggregating functions we can associate with the Paretian–egalitarian approach are presented in
Table 2. While these types of functions satisfy only the PD and the SI conditions, other types that additionally are additive or separable can also be considered as realizing the Paretian–egalitarian approach. Accounting for these additional properties, however, we associate such functions with other approaches to building aggregationbased social rankings of outcomes, as shown in
Figure 2.
Table 2.
Examples of types of aggregating functions that can be associated with the Paretian–egalitarian approach.
Table 2.
Examples of types of aggregating functions that can be associated with the Paretian–egalitarian approach.
Type of Function  Properties  Comments and References 

Generalized mean ${g}_{p}\left(u\right)=\{\begin{array}{ll}{\left(\frac{1}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}{u}_{i}^{p}\right)}^{1/p}\hfill & \mathrm{for}p\ne 0\hfill \\ {\left({\displaystyle {\displaystyle \prod}_{i=1}^{N}}{u}_{i}\right)}^{1/N}\hfill & \mathrm{for}p=0\hfill \end{array}$  RRI, Impartiality, PD (for $p<1)$, SI  The parameter $\infty \le p\le 1$ determines the tradeoff between equality (egalitarianism for $p=\infty $) and efficiency (utilitarianism for $p=1$). For $p>1$, this function is convex and does not satisfy the PD condition. For $p=0$, this function is the Nth root of the exponent of the Nash social welfare function, maximization of which by a utility profile $u$ yields the solution to the Nash bargaining problem [ 39, 40]. Maximization of this function with parameter $p$ is equivalent to maximization of the Atkinson function ( Table 5) with parameter $\gamma =1p$.

Fairness ratio $g\left(u\right)=\underset{1\le i\le N}{\mathrm{min}}\frac{{\overline{\theta}}_{i}\left(u\right)}{{P}_{i}^{*}}$
where ${\overline{\theta}}_{i}\left(u\right)={{\displaystyle \sum}}_{k=1}^{i}{u}_{\left(k\right)}$ for $i=1,\dots ,N$, ${u}_{\left(k\right)}$ is the $k$thsmallest element of u, ${P}_{i}^{*}=\underset{u\in \mathcal{U}}{\mathrm{max}}{\overline{\theta}}_{i}\left(u\right)$ for $i=1,\dots ,N$, and $\mathcal{U}$ is the set of all attainable utility profiles $u$  RRI, Impartiality, PD, SI  Maximization of this function leads to the most even distribution of agent utilities with respect to their maximal attainable utilities. A utility profile that maximizes this function is Lorenznondominated.

Underachievement function ${g}_{\alpha}\left(u\right)=\mu \left(u\right)\alpha \rho \left(u\right)$ where $\mu \left(u\right)=\frac{1}{N}{{\displaystyle \sum}}_{i=1}^{N}{u}_{i}$ and $\rho \left(u\right)$ is an inequality measure  RRI, Impartiality, PD, SI  As a mixture of the average utility $\mu \left(u\right)$ and an inequality measure $\rho \left(u\right)$, this function explicitly represents a Paretian–egalitarian tradeoff between efficiency and equality. The parameter $0<\alpha <{\overline{\alpha}}_{\rho}$ determines the balance in this tradeoff, with smaller values of $\alpha $ favoring efficiency and larger values of $\alpha $ favoring equality. The upper bound ${\overline{\alpha}}_{\rho}$ depends on the choice of the inequality measure $\rho \left(u\right)$. For $\alpha \ge {\overline{\alpha}}_{\rho}$ this function may not satisfy the PD or SI conditions.

Distance to reference point ${g}_{p,z}\left(u\right)={\left({\displaystyle \sum _{i=1}^{N}}{\left{u}_{i}A\right}^{p}\right)}^{1/p}$  RRI, Impartiality, PD, SI  The parameter $A<\infty $ determines an aspiration level for agent utilities, although it may potentially be unattainable for some or all agents. The parameter $1\le p\le \infty $ determines the type of distance, which also determines the tradeoff between equality (egalitarianism for $p=\infty $) and efficiency (utilitarianism for $p=1$).

Logsumexp function $g\left(u\right)=\mathrm{ln}\left({\displaystyle \sum _{i=1}^{N}}{e}^{{u}_{i}}\right)$  Impartiality, PD, SI  This function is also known as “soft min”, as its value is predominantly determined by the smallest element of the utility profile. This function generates a social ranking of outcomes that is close in spirit to Rawlsian–egalitarian ranking, but which is also Paretoinclusive.

Additivity is an indispensable but not a defining feature of aggregating functions associated with approaches to utility aggregation that operationalize the notion of the separateness of persons. Indeed, some inequality measures representing the axiological–egalitarian approach as well as the utilitarian total utility are additive (
Figure 2). In the context of the separateness of persons, however, an additive form of the aggregating function makes it possible to demonstrate to each agent not only that their interests have received individual consideration but also how they have influenced the social ranking of outcomes, which fulfills conditions (i) and (ii) of the separateness of agents in
Section 3.
Rawls’s idea for operationalizing the separateness of persons is based on the concept of original position in which all agents agree to act in a way that benefits the worstoff one among them. Accordingly, in the Rawlsian–egalitarian approach, the utility of each agent is considered individually; however, it determines the social rank of an outcome only if it is the lowest utility attained in that outcome. Formally, the contribution of the interests of agent
$i$ to the Rawlsian–egalitarian rank of outcome
$x$ can be expressed as
${h}_{i}\left(u\left(x\right)\right)={u}_{i}\left(x\right){\delta}_{ij}$, where
$j={\mathrm{argmin}}_{i=1,\dots ,N}{u}_{i}\left(x\right)$ is the index of the agent with the lowest utility in the utility profile
$u\left(x\right)$ and
${\delta}_{ij}$ is the Kronecker delta, which equals 1 for
$i=j$ and 0 otherwise. The social rank of
$x$ can then be generated by an additive aggregating function
$g\left(u\left(x\right)\right)={{\displaystyle \sum}}_{i=1}^{N}{h}_{i}\left(u\left(x\right)\right)={\mathrm{min}}_{i=1,\dots ,N}{u}_{i}\left(x\right)$. Thus, we can associate the minimum function with the Rawlsian–egalitarian approach to building a social ranking of outcomes (
Table 3).
Table 3.
Example of an aggregating function that can be associated with the Rawlsian–egalitarian approach.
Table 3.
Example of an aggregating function that can be associated with the Rawlsian–egalitarian approach.
Type of Function  Properties  Comments and References 

Minimum function $g\left(u\right)=\underset{i=1,\dots ,N}{\mathrm{min}}{u}_{i}$  RRI, Impartiality, Additivity  
In the complaintswithinoutcome approach to operationalizing the separateness of persons, an outcome is considered socially more desirable if it gives rise to fewer complaints among all agents. The additive form of the social utility function
$g\left(u(\cdot )\right)={{\displaystyle \sum}}_{i=1}^{N}{h}_{i}\left(u(\cdot )\right)$ representing such social preferences has a natural interpretation. For each outcome
$x$, the value
${h}_{i}\left(u\left(x\right)\right)$ represents the strength of the complaint agent
$i$ may have regarding outcome
$x$ together with the weight given to agent
$i$’s complaint. Different choices can be made regarding what constitutes a legitimate basis for complaints. For instance, an agent may have a complaint regarding outcome
$x$ if its utility
${u}_{i}\left(x\right)$ is lower than the average utility among agents realized in
$x$. Alternatively, an agent may have a complaint against all agents who are betteroff within outcome
$x$ or against the bestoff agent. The weight given to a complaint depends on a decision on how complaints are to be aggregated. For example, in the case of a summative rule, all complaints have equal weights, e.g., a weight of 1. If all complaints are collapsed into the single strongest complaint, all weaker complaints receive a weight of 0. Examples of types of additive aggregating functions we can associate with the complaintswithinoutcome approach are presented in
Table 4.
Table 4.
Examples of types of aggregating functions that can be associated with the complaintswithinoutcome approach.
Table 4.
Examples of types of aggregating functions that can be associated with the complaintswithinoutcome approach.
Type of Function  Properties  Comments and References 

Negative maximum downside semideviation $g\left(u\right)=\underset{i=1,\dots ,N}{\mathrm{max}}\left(\mu \left(u\right){u}_{i}\right)$ where
$\mu \left(u\right)=\frac{1}{\mathrm{N}}{{\displaystyle \sum}}_{i=1}^{N}{u}_{i}$  RRI, Impartiality, Additivity  
Negative mean downside semideviation $g\left(u\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}}{\left(\mu \left(u\right){u}_{i}\right)}_{+}$ where $\mu \left(u\right)=\frac{1}{\mathrm{N}}{{\displaystyle \sum}}_{i=1}^{N}{u}_{i}$ and
${(\cdot )}_{+}=\mathrm{max}\left(\cdot ,0\right)$  RRI, Impartiality, Additivity  
Negative total downside difference $g\left(u\right)={\displaystyle \sum _{i=1}^{N}}{\displaystyle \sum _{j=1,j\ne i}^{N}}{\left({u}_{j}{u}_{i}\right)}_{+}$ where
${(\cdot )}_{+}=\mathrm{max}\left(\cdot ,0\right)$  RRI, Impartiality, PD, Additivity  
In the claimsacrossoutcomes approach to operationalizing the separateness of persons, a social ranking of outcomes is built on pairwise comparisons: outcome
$x$ is considered socially more desirable than outcome
$y$ if the claims of agents in favor of
$x$ over
$y$ outweigh the claims in favor of
$y$ over
$x$. If the social utility function
$g\left(u(\cdot )\right)$ is additive, then the balance of claims in favor of
$x$ and claims in favor of
$y$ is conveniently expressed by the difference
$g\left(u\left(x\right)\right)g\left(y\left(y\right)\right)={{\displaystyle \sum}}_{i=1}^{N}\left({h}_{i}\left(u\left(x\right)\right){h}_{i}\left(u\left(y\right)\right)\right)$, where the value
${h}_{i}\left(u\left(x\right)\right){h}_{i}\left(u\left(y\right)\right)$ is interpreted, if positive, as the strength of the claim of agent
$i$ in favor of
$x$ over
$y$ and, if negative, as the strength of the claim of agent
$i$ in favor of
$y$ over
$x$. What distinguishes additive aggregating functions that can be associated with the claimsacrossoutcomes approach from aggregating functions that can be associated with the Rawlsian–egalitarian and complaintswithinoutcome approaches is the SI property. Indeed, as explained in
Section 3, any social ranking of outcomes based on a claimsacrossoutcomes approach is Paretoinclusive, while rankings of the other two types may disagree with Pareto ordering, and the SI property of an aggregating function
$g$ ensures that the social utility function
$g\left(u(\cdot )\right)$ generates a Paretoinclusive ranking of outcomes. Examples of types of additives and strictly increasing aggregating functions we can associate with the claimsacrossoutcomes approach are presented in
Table 5.
Table 5.
Examples of types of aggregating functions that can be associated with the claimsacrossoutcomes approach. Notice that the sufficientist approach is not included here since it cannot be operationalized with a single aggregating function.
Table 5.
Examples of types of aggregating functions that can be associated with the claimsacrossoutcomes approach. Notice that the sufficientist approach is not included here since it cannot be operationalized with a single aggregating function.
Type of Function  Properties  Comments and References 

Type2 achievement function ${g}_{\alpha ,A}\left(u\right)=\underset{1\le i\le N}{\mathrm{min}}\left({u}_{i}A\right)+\alpha {\displaystyle \sum _{i=1}^{N}}({u}_{i}A)$  RRI, Impartiality, SI, Additivity  The parameter $A<\infty $ determines an aspiration level for agent utilities, in relation which to agents are formulating their claims, although it may potentially be unattainable for some or all agents. To keep inequality in check, a preferential relative weight of $1+1/\alpha $ is given to the claim of the worstoff agent.

Ordered weighted average (OWA) ${g}_{w}\left(u\right)={\displaystyle \sum _{i=1}^{N}}{w}_{i}{u}_{\left(i\right)}$ where
${w}_{1}>{w}_{2}>\dots >{w}_{N}>0$ and ${u}_{\left(i\right)}$ is the ithsmallest element of $u$  RRI, Impartiality, PD, SI, Additivity  This function represents a claimsacrossoutcomes approach in which the strength of an agent’s claim is determined by their utility multiplied by a weight depending on the position of their utility in the ordered utility profile. The parameter $w=\left({w}_{1},\dots ,{w}_{N}\right)$ with ${{\displaystyle \sum}}_{i=1}^{N}{w}_{i}=1$ is a vector of decreasing weights ${w}_{1}>{w}_{2}>\dots >{w}_{N}>0$ determining how decreasing priority is given to the utilities of betteroff agents. In some definitions of OWA functions, the weights ${w}_{1},\dots ,{w}_{N}$ must belong to the interval $\left[0,1\right]$, need not be positive, and/or need not be arranged in a strictly decreasing sequence. However, OWA functions with some weights equal to zero fail to satisfy the SI condition, while OWA functions with weights not arranged in a strictly decreasing sequence fail to satisfy the PD condition.

Atkinson function ${g}_{\gamma}\left(u\right)={\displaystyle \sum _{i=1}^{N}}{h}_{\gamma}\left({u}_{i}\right)$ where ${h}_{\gamma}\left({u}_{i}\right)=\{\begin{array}{ll}\mathrm{ln}{u}_{i}\hfill & \mathrm{for}\gamma =1\hfill \\ {\left(1\gamma \right)}^{1}{u}_{i}^{1\gamma}\hfill & \mathrm{for}\gamma \ne 1,\gamma 0\hfill \end{array}$  RRI, Impartiality, PD, SI, Additivity, Separability  
Negative exponential function $g\left(u\right)={\displaystyle \sum _{i=1}^{N}}{e}^{{u}_{i}}$ 
Impartiality, PD, SI, Additivity, Separability  This function is a prioritarian aggregating function giving a very high strength to the claim of the worstoff agent. This function is particularly sensitive to small utility values, with sensitivity quickly diminishing for larger utility values.

As mentioned above, if a social utility function is of the additive form
$g\left(u(\cdot )\right)={{\displaystyle \sum}}_{i=1}^{N}{h}_{i}\left(u(\cdot )\right)$, the strength of the claim of agent
$i$ in favor of
$x$ over
$y$ is given by the value
${h}_{i}\left(u\left(x\right)\right){h}_{i}\left(u\left(y\right)\right)$, with negative values quantifying the strength of claims in the opposite direction. In this general form, the strength of agent
$i$’s claim may depend not only on the utilities agent
$i$ attains in
$x$ and
y but also on the utilities attained by other agents. For instance, when the social utility function is given by an OWA function (
Table 5), the strength of each agent’s claim depends both on its utility in the considered outcome and on how well it fares relative to the other agents. This may lead to situations in which an agent has the same basis for two claims in favor of outcomes
$x$ and
$x\prime $ over an alternative outcome
$y$ because
${u}_{i}\left(x\right)={u}_{i}\left({x}^{\prime}\right)$ but in which the strengths of these two claims differ because of differences in the utilities attained by other agents in
$x$ and
$x\prime $. Consequently, a ranking of outcomes generated by an additive social utility function in its general form may be shaped not only by the utilities of agents who make claims but also by social preferences for certain properties of outcomes considered to have intrinsic ethical value, e.g., equality of utility distribution.
Promoting values other than the utility of agents is, however, contrary to the premise of both the sufficientist and the prioritarian approaches to operationalizing the separateness of persons, namely that the strength of the claim of any agent should be a function of that agent’s utility alone. Thus, any aggregating function we can associate with these two approaches must be separable. Indeed, if a social utility function is to be impartial, SI, and of additive form $g\left(u(\cdot )\right)={{\displaystyle \sum}}_{i=1}^{N}{h}_{i}\left({u}_{i}(\cdot )\right)$, it follows that ${h}_{1}=\dots ={h}_{N}=h$, where $h$ is a strictly increasing function.
The sufficientist ranking of outcomes is based on a twotier system of primary and secondary claims defined in relation to a reference utility
$C$, called the compassion threshold. Each agent is entitled to a primary claim for one of the alternative outcomes
$x$ and
$y$ over the other if, in at least one of these outcomes, the agent’s utility falls below the compassion threshold
$C$—and the farther below this threshold it falls, the stronger the agent’s claim is considered to be. Thus, the strength of the primary claim of agent
$i$ for
$x$ over
$y$ is given by the difference
$h\left(\mathrm{min}\left({u}_{i}\left(x\right),C\right)\right)h\left(\mathrm{min}\left({u}_{i}\left(y\right),C\right)\right)$, where
$h(\cdot )$ is a strictly increasing and strictly concave function. The total strength of primary claims of all agents for
$x$ over
$y$ is given by
${g}_{\mathrm{p}}\left(x\right){g}_{\mathrm{p}}\left(y\right)$, where
${g}_{\mathrm{p}}(\cdot )={{\displaystyle \sum}}_{i=1}^{N}h\left(\mathrm{min}\left({u}_{i}(\cdot ),C\right)\right)$ is the primary outcome score of separable form. In the sufficientist ranking, the outcome
$x$ is preferred over
$y$ if and only if
${g}_{\mathrm{p}}\left(x\right)>{g}_{\mathrm{p}}\left(y\right)$. When
${g}_{\mathrm{p}}\left(x\right)={g}_{\mathrm{p}}\left(y\right)$, each agent with a utility above the compassion threshold is entitled to a secondary claim for resolving the tie. The strength of the secondary claim of agent
$i$ for
$x$ over
$y$ is given by the difference
$\mathrm{max}\left({u}_{i}\left(x\right),C\right)\mathrm{max}\left({u}_{i}\left(y\right),C\right)$, and the total strength of secondary claims for
$x$ over
$y$ is given by
${g}_{\mathrm{s}}\left(x\right){g}_{\mathrm{s}}\left(y\right)$, where
${g}_{\mathrm{s}}(\cdot )={{\displaystyle \sum}}_{i=1}^{N}\mathrm{max}\left({u}_{i}(\cdot ),C\right)$ is the secondary outcome score. The tie is resolved in favor of
$x$ if and only if
${g}_{\mathrm{s}}\left(x\right)>{g}_{\mathrm{s}}\left(y\right)$, i.e., if
$x$ delivers a higher total utility than
$y$ to agents with utilities above the compassion threshold. To summarize, the sufficientist ranking of outcomes is generated not by a single social utility function but by a pair of separable outcome scores: the primary outcome score
${g}_{\mathrm{p}}(\cdot )$, which is a type of prioritarian aggregating function (discussed below), applied to utility profiles truncated above the compassion threshold and the secondary outcome score
${g}_{\mathrm{s}}(\cdot )$, which is the utilitarian aggregating function, applied to utility profiles truncated below the compassion threshold. Note that the primary outcome score
${g}_{\mathrm{p}}(\cdot )$ alone may not agree with the Pareto ranking for outcomes in which some agents attain utilities higher than the compassion threshold, but that in such cases the secondary outcome score
${g}_{\mathrm{s}}(\cdot )$ ensures the Pareto inclusiveness of the sufficientist ranking. Since the sufficientist ranking of outcomes cannot be operationalized with a single aggregating function (it requires two functions
${g}_{\mathrm{p}}$ and
${g}_{\mathrm{s}}$ to be applied in a lexicographic order), this approach is omitted from
Figure 2 and
Table 5.
The core premise of the prioritarian ranking of outcomes is that improving the utilities of agents with low utilities is ethically more important than improving the utilities of agents with high utilities. Accordingly, in the prioritarian approach, the strength of the claim of agent $i$ for an outcome $x$ over an alternative outcome $y$ is given by the difference $h\left({u}_{i}\left(x\right)\right)h\left({u}_{i}\left(y\right)\right)$, where $h$ is a strictly increasing and strictly concave function. Then, the total strength of claims among all agents $i=1,\dots ,N$ in favor of $x$ over $y$ can be expressed as ${{\displaystyle \sum}}_{i=1}^{N}\left(h\left({u}_{i}\left(x\right)\right)h\left({u}_{i}\left(y\right)\right)\right)=g\left(u\left(x\right)\right)g\left(u\left(y\right)\right)$, where $g\left(u(\cdot )\right)={{\displaystyle \sum}}_{i=1}^{N}h\left({u}_{i}(\cdot )\right)$ and negative values quantify the strength of claims in the opposite direction. Outcome $x$ is preferred in the prioritarian ranking over outcome $y$ if the total strength of claims for $x$ outweigh the total strength of claims for $y$, i.e., if $g\left(u\left(x\right)\right)>g\left(u\left(y\right)\right)$. Thus, we can associate a separable aggregating function $g\left(u\right)={{\displaystyle \sum}}_{i=1}^{N}h\left({u}_{i}\right)$ with the prioritarian approach because it defines a social utility function $g\left(u(\cdot )\right)={{\displaystyle \sum}}_{i=1}^{N}h\left({u}_{i}(\cdot )\right)$ that generates a prioritarian ranking of outcomes.
The prioritarian aggregating function
$g\left(u\right)={{\displaystyle \sum}}_{i=1}^{N}h\left({u}_{i}\right)$ has several convenient properties that have important interpretations. As
$h(\cdot )$ is strictly increasing,
$g(\cdot )$ has the SI property, thus ensuring that the social utility function
$g\left(u(\cdot )\right)$ generates a Paretoinclusive ranking of outcomes. Moreover, the permutation invariance of
$g(\cdot )$ and the strict concavity of
$h(\cdot )$ imply that
$g(\cdot )$ is Schurconcave and thus has the PD property. Consequently, whenever it is possible to reduce the utility of a betteroff agent by a given amount and transfer it to a worseoff agent without losses and without reversing the utility ranking of the agents involved in the transfer, the resultant utility profile is always preferred in the prioritarian ranking over the original utility profile. Such lossfree transfers are not always possible, e.g., due to the shape of the set of attainable utility profiles; thus, making the distribution of utility among agents more equal may come at the price of reducing total utility. The prioritarian ranking generated by the social utility function
$g\left(u(\cdot )\right)={{\displaystyle \sum}}_{i=1}^{N}h\left({u}_{i}(\cdot )\right)$ allows specifying how much of total utility can be forfeited for attaining a higher equality of the utility distribution, or, conversely, how high an increase in inequality can be accepted for improving efficiency, i.e., total utility. Indeed, it can be shown that for any prioritarian aggregating function
$g$ there exists an inequality measure
${I}_{g}$ such that the ranking based on the following rule—outcome
$x$ is at least as socially preferred as outcome
$y$ if
$\left(1{I}_{g}\left(u\left(x\right)\right)\right){{\displaystyle \sum}}_{i=1}^{N}{u}_{i}\left(x\right)\ge \left(1{I}_{g}\left(u\left(y\right)\right)\right){{\displaystyle \sum}}_{i=1}^{N}{u}_{i}\left(y\right)$—is equivalent to prioritarian ranking, which based on the following rule—
$x$ is socially at least as good as
$y$ if
$g\left(u\left(x\right)\right)\ge g\left(u\left(y\right)\right)$ ([
19], p. 120).
If, in addition to the properties discussed above, the prioritarian aggregating function
$g\left(u\right)={{\displaystyle \sum}}_{i=1}^{N}h\left({u}_{i}\right)$ also has the RRI property, it can be shown that it then must belong to the class of socalled Atkinson functions, which are presented in
Table 5 ([
11], p. 68). The shape parameter
$\gamma >0$ of the Atkinson function
${g}_{\gamma}$ can be interpreted as the “marginal rate of moral substitution”, which quantifies the critical tolerable loss in transfers of utility from betteroff to worseoff agents ([
19], p. 385). The parameter
$\gamma $ can thus also be understood to quantify the degree of “inequality aversion.” Indeed, it can be shown that the ranking of outcomes generated by the social utility function
${g}_{\gamma}\left(u(\cdot )\right)$ is equivalent to the ranking generated by the social utility function
$\left(1+{I}_{\gamma}\left(u(\cdot )\right)\right){{\displaystyle \sum}}_{i=1}^{N}{u}_{i}(\cdot )$, where
${I}_{\gamma}$ is the negative Atkinson inequality index (
Table 1) with the inequality aversion parameter
$\gamma $ ([
19], p. 121).