Journal Pre-proof Building rankings encompassing multiple criteria to support qualitative decision-making

Decision makers are commonly challenged with comparing, and ultimately ranking, elements with regards to the degree to which they satisfy multiple criteria and in terms of their own preferences.This calls for a new decision making framework, which we formally present here. Within such a framework, we present multi-criteria lex-cel : a new method for ranking single elements. Furthermore, we formally establish that our contributions generalise recent results in the social choice literature. We also illustrate our contributions through a case study that poses an ethical decision-making problem


Introduction
Rankings establish comparisons between individual objects (or sets of objects) that are useful for many applications. Consider, for example, the widely studied problem of college admissions [49]. Ranking solutions have been proposed to solve this problem [39] and other similar problems such as committee 5 selection [21]. Thus, they have been long investigated in the literature. Without aiming for completeness, here we highlight three different bodies of work related to rankings. Firstly, the literature has countless examples of works studying voting and ranking aggregation. A representative example of recent developments in this area include the work of Aledo et al.
[2] on a highly scalable algorithm 10 to aggregate general rankings, and Miebs et al. [30] who study heuristic algorithms to aggregate incomplete partial rankings. Secondly, Barbera et al. [8] study functions that transform rankings of individual elements into rankings of sets of these elements. Maxmin and minmax [5] or leximin and leximax [38] are examples of such functions. Thirdly, Moretti and Ozturk [35] introduce the 15 social ranking as a mapping that transforms a ranking of sets of elements into a ranking of the individual elements of these sets. Social rankings have been extensively studied: Haret et al. [19] base their work on the ceteris paribus majority principle; Khani et al. [20] focus on the notion of marginal contribution; and Doignon et al. [14] study the stability of social scorings (a concept related 20 to social rankings). We can even find the usage of social rankings in ethical decision-making [43].
A particularly interesting social ranking is lex-cel introduced by Bernardi et al. [12] which focuses on lexicographical preferences and satisfies some desirable properties. Lex-cel has caught the attention of the social choice community, so 25 much so that there have recently been many works, generalising it [3,9], proving it is not manipulable [4], applying it to coalition formation [22], and defining a new social choice function based on it [27].
A common assumption in the ranking literature is the existence of preferences regarding the elements or sets of elements from which to build a ranking. 30 Indeed, this assumption is reasonable when considering a limited number of candidate elements -e.g. in a presidential election. However, when considering many candidates -e.g. in scholarship assignments -it is common practice simply to be provided with the criteria for establishing the element ranking, rather than an explicit ranking over the individual elements. Moreover, if multiple cri- 35 teria are considered, we may also be provided with preferences regarding them.
For example, when designing a diet, we may consider different criteria regarding the food to choose -such as its healthiness or tastiness -and preferences over these criteria -e.g., healthiness preferred over tastiness.
Against this background, in this paper we develop novel formal tools to help 40 decision-makers take qualitative decisions about multiple options while considering their preferences. In particular, we propose multi-criteria-based ranking (MC ranking), a method to rank individual elements based on: i) how they relate to the criteria (e.g., if the food is tasty); and ii) the preferences over these criteria. Specifically, when building MC rankings, we consider that candidate 45 elements may relate to a given criterion to different degrees. Thus, following the example of a diet, broccoli cheddar soup can be considered to be not particularly healthy, a Caesar salad to be healthy, and steamed vegetables to be very healthy. Moreover, these qualitative relationships may even be negative -with sausages considered unhealthy and chips very unhealthy. Overall, our multi-50 criteria-based ranking encompasses rich qualitative element-criterion relations that produce a comprehensive ranking over the individual candidate elements.
Briefly, the contributions of this paper are:    [35], while multi-criteria lex-cel generalises the lex-cel ranking function introduced in [12]. 5. A case study posing an ethical decision-making problem that illustrates to order theory in Section 2, while Section 3 formalises labels and label systems.
Next, Section 4 formalises MC rankings, as well as the property of dominance, while Section 5 introduces MC lex-cel. Subsequently, Section 6 studies the 70 relation of our MC rankings and MC lex-cel with the literature on social choice.
Finally, Section 7 analyses a case study in ethical decision-making, while Section 8 shows an application in participatory budgets, and Section 9 discusses our conclusions.

Background: Order theory 75
Let X be a set of objects. A binary relation on X is said to be: reflexive, if for each x ∈ X, x x; transitive, if for each x, y, z ∈ X, (x y and y z) ⇒ x z; total, if for each x, y ∈ X, x y or y x; antisymmetric, if for each x, y ∈ X, x y and y x ⇒ x = y. We can define preferences among the elements of X by means of binary relations. Moreover, we can categorise the 80 type of preferences depending on the properties they hold as follows.
Def. 1 (Preorder, ranking, linear order and partial order). A preorder (or quasi-ordering) is a binary relation that is reflexive and transitive. A preorder that is also total is called a total preorder or ranking. A total preorder that is also antisymmetric is called a linear order. A preorder that is antisymmetric 85 but not total is called partial order.
We build a lexicographical order for two tuples by comparing them elementwise from left to right. While the elements in both tuples are the same, we move to the next position on the tuples. We traverse the tuples until two elements differ (one is preferred over the other). The more preferred tuple is the one 90 containing the more preferred element. If all elements are the same, the tuples are deemed equal. Formally: Def. 2. Given two tuples t, t , with t = (t 1 , . . . t q ) and t = (t 1 , . . . t q ), we define the lexicographical order of tuples ≥ lex as: t ≥ lex t ⇔ if either t = t or ∃i ∈ {1, . . . q} s.t. t i > t i and ∀j < i, t j = t j (note that t = lex t ⇔ t = t ).

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The lexicographical order for tuples is used in the definition of the lex-cel ranking [12]. Let X be a set of elements, and S a ranking over the power set P(X), then lex-cel builds an element ranking e by means of assigning a tuple to each element (noted θ(x)). To build this tuple, consider the quotient set P(X)/∼ S with quotient order Σ 1 S Σ 2 S · · · S Σ q . Then, θ(x) is defined 100 as: Lex-cel ranks elements in X by comparing lexicographically their corresponding θ tuples: x e y ⇔ θ(x) ≥ lex θ(y).

Relating elements to criteria
As previously introduced, when ranking candidate elements according to 105 given criteria, we consider those candidate elements that are related to the given criteria. Specifically, we enrich the expressivity of these relations by means of graded labels. As we assume humans will assign semantics to the labels and will specify these relations, we take inspiration from the widely-used Likert scale [36] 1 , and specify that graduation ranges from negative labels, which signal that an 110 element is detrimental to a criterion, through a neutral label, to positive labels, which indicate that an element aligns with a criterion. Next, we introduce the notions of label system, the object that defines labels for relating elements and criteria and their semantics, and labelling, a function to relate elements to criteria through labels.

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A label system contains a set of labels and an order over them to establish their grading. This set of labels must contain a neutral label, lying between positive and negative labels. Positive labels are those that are more preferred than the neutral label, whereas negative labels are those that are less preferred than the neutral label. In terms of label grading, the more preferred a positive 120 label, the higher the degree of alignment between an element and the criterion it is meant to represent. Conversely, the less preferred a negative label, the higher the detrimental degree.
Def. 3 (Label system). A label system is a pair L, > L , where L is a set of labels, and > L is a linear order over L. A label system includes a neutral 125 label 2 l 0 ∈ L. Labels more preferred than l 0 are positive labels, whereas those less preferred than l 0 are negative labels.
Note that a label system does not need to have a negative label for each positive label. In fact, it might only have positive labels. However, a label system with more labels of one type than of another one hinders the task of 130 comparing labels. For example, given l 2 > L l 1 > L l 0 > L l −1 , it is unclear whether the positive counterpart of l −1 is l 2 because both labels are the most extreme ones, or if it is l 1 , because they are equally separated from l 0 . To avoid these uncertainties, we focus on a particular type of label systems: the so-called symmetric label systems, for which each positive label has a negative 135 counterpart. To ease their definition, we first introduce two auxiliary functions, namely the sign and strength of a label, which also provide a useful notation for the forthcoming sections. Given a label system, the sign function signals whether a label is positive (1), negative (-1), or the neutral label (0).
The strength function characterises the label's degree of preference in the label system order. In particular, we consider that, given a label l, the more labels between l and l 0 in the label order, the greater its strength. Formally: Def. 4 (Symmetric label system). A label system L, > L is symmetric if ∀l ∈ L, ∃l ∈ L, such that sgn(l) = −sgn(l ) and stg(l) = stg(l ).

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Symmetric label systems have the same number of positive and negative labels. Note that, without loss of generality, any label system can be transformed into a symmetric label system by simply adding superfluous labels. Hereafter, we only consider symmetric label systems. Also, we can uniquely note each label in the label system as l sgn(l)·stg(l) (for example, we note as l −2 the label 150 of sign -1 and strength 2).
Example 1. Consider L = {l 1 , l 0 , l −1 , l −2 } to be a label system with order Note, for example, that l −2 is the label of sign sgn(l −2 ) = −1 and strength str(l −2 ) = 2. Moreover, as we require symmetry, we can add an additional superfluous label l 2 to transform the system into a 155 symmetric label system L = {l 2 , l 1 , l 0 , l −1 , l −2 } with order Using a label system, a decision-maker can relate an element with a criterion by means of a labelling function.
Def. 5. Given a set of elements X, a set of criteria C, and a label system 160 L, > L , a labelling is a function λ : X × C → L that assigns a label in L to each pair of elements in X and criterion in C, thereby establishing the relation between the element and the criterion. We note as L(X, C) the set of all possible labellings over X and C.
If λ(x, c) = l, we say that element x is related to criterion c with degree l.

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From equation 3, we also say that the strength of the relation is stg(l). For example, recalling the diet example, a labelling would relate steamed vegetables

Multi-criteria-based rankings
As mentioned above, we assume that the decision maker establishes a set of criteria and knows their preferences over them. We have learnt in Section 3 175 how to relate elements to criteria. Our goal is to build a ranking of the single elements in X from: (i) the relationships between elements and criteria; and (ii) preferences regarding criteria. We will call such ranking a multi-criteria-based ranking (MC ranking). In this section, we formally define this ranking, as well as the fundamental notion of dominance for MC rankings.

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An MC ranking considers a set of elements X, a set of criteria C, a ranking C over the criteria, and a labelling λ relating elements to criteria and builds a ranking over the single elements in X. Formally: Def. 6. Given a set of elements X, a set of criteria C, and a set of labellings L(X, C), an MC ranking is a function mcr : L(X, C) × R(C) → R(X) that 185 associates any pair of labelling λ ∈ L(X, C) (relating elements with criteria) and ranking C ∈ R(C) (over the criteria) to another ranking mcr(λ, C ) ∈ R(X) over the elements of X.
MC rankings call for the introduction of a novel notion of dominance between the elements in X, as is common in the literature (e.g. [8] [35]). Such 190 notion of dominance must ensure that the ranking of elements is based strictly on the ranking over criteria. However, defining dominance for MC rankings is intricate due to the richness of our labelling approach. Informally, our notion of dominance requires that an MC ranking function ranks the elements in X taking into accountthe element-criterion relations, their associated labels, and the 195 criteria preferences. Thus, the more preferred a criterion with which an element relates positively, the more preferred the element. Conversely, the more preferred the criterion with which an element relates negatively, the less preferred the element. The higher the degree of the labels on these positive/negative relations, the more/less preferred the element will be. Furthermore, the larger 200 the number of positive relations and the lower the number of negative relations for an element, the more preferred the element in the ranking will be.
Our notion of dominance between two elements is founded on the dominance within each equivalence class of criteria resulting from the ranking C over criteria. Thus, consider the quotient set of criteria C/∼ C with equivalence 205 classes κ 1 , . . . , κ r , and quotient order C . Note that the criteria within each equivalence class κ ∈ C/∼ C are preferred equally. Given an equivalence class κ, our first aim is to establish whether an element x ∈ X is κ-dominant (dominant within the scope of the equivalence class κ) over another element y ∈ X.
An element will be κ-dominant over another if it relates more strongly (and Let s max = max l∈L stg(l) be the maximum strength of the labels in the label 220 system. Then, κ-dominance is defined as: Def. 8. Given two elements x, y ∈ X, a set of criteria C, a ranking over these criteria C , a symmetric label system L, > L and a criteria equivalence class na(x, κ, s) > na(y, κ, s) and ∀s > s, we have na(x, κ, s ) = na(y, κ, s ). If  while for the rest of the elements in X, their net alignment of strength 2 with 230 κ 1 is 0, which is greater than −1. Thus, we say that x 2 , x 3 , x 4 , and x 5 are Notice that the relation over the elements in X provided by κ-dominance corresponds to a lexicographical order over the tuples of net alignment values arranged in decreasing order of label strength, from s max to 1 (see relation (6) 235 in the following section for a formal definition). In the previous example, with s max = 2, the tuples of net alignment values (na(x, κ 2 , 2), na(x, κ 2 , 1)) on the criteria equivalence class κ 2 for each element in x ∈ X are (1, 0) for x 1 and x 5 , and its application has been studied extensively both from a mathematical and 255 a pragmatic perspective (see for instance [16,23,42]).
Using the concept of κ-dominance, we define dominance considering all equivalence classes in C/ ∼ C (and their quotient order C ). We say that x is dominant over y if for a given criteria equivalence class x is κ-dominant over y, while for more preferred equivalence classes they are κ-indifferent.

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Def. 9. Given two elements x, y ∈ X with criteria in C and a ranking over criteria C , we say that x is dominant over y if there is a criteria equivalence class κ ∈ C/∼ C , such that: (i) x is κ-dominant over y; and (ii) ∀κ ∈ C/∼ C , such that κ C κ, x and y are κ -indifferent. If neither element dominates the other (they are κ-indifferent ∀κ ∈ C/∼ C ), we say that they are indifferent.

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Dominance is a natural extension of the κ-dominance notion to a multi-criteria framework where a preference ranking C over criteria is given. As we will explain in detail in Section 5.2, the dominance relation over elements represents a lexicographical order aimed at rewarding the elements having excellent labelling degrees in the most preferred criteria equivalence classes.
x 5 are dominant over x 1 because they are κ 1 -dominant and that κ 1 is the most preferred class.

Multi-criteria lex-cel
Next, we introduce multi-criteria lex-cel (MC lex-cel), which is an MC rank-275 ing function. For each element in X, MC lex-cel builds a tuple, the so-called multi-criteria profile (MC profile), which summarises the relations between the element and the criteria. Then, MC lex-cel ranks the elements in X by comparing their MC profiles lexicographically. In Section 5.1, we describe how to build MC profiles, whereas Section 5.2 defines MC lex-cel and proves that it embodies 280 the dominance property in Definition 9.

Building MC profiles for elements
We will build the MC profile of an element x ∈ X as a tuple µ(x) that is meant to summarise the relations of that particular element with all the criteria at hand.

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In general terms, we build an MC profile for an element through a nested process: (1) we start considering criteria preferences, from more preferred to less preferred; (2) thereafter, we delve into each equivalence class to consider the strengths of the relations, from stronger to weaker.
Formally, we build an MC profile by considering the quotient set C/ ∼ C , 290 where κ 1 , . . . , κ q ∈ C/∼ C are criteria equivalence classes with quotient order We compose the MC profile µ(x) of an element x, from its equivalence class profiles µ(x, κ 1 ), . . . , µ(x, κ q ). An equivalence class profile µ(x, κ i ) summarises the relations between x and the equivalence class κ i . We want to ensure that 295 criteria preferences are satisfied according to C . Thus, we compose the MC profile µ(x) by considering that the relationships with more preferred criteria are positioned further to the left 3 of µ(x) as follows: Within an equivalence class κ, all criteria are preferred indifferently. Thus, what distinguishes the relations between x and κ here is their strength and sign. µ(x, κ) = (na(x, κ, s max ), . . . , na(x, κ, 1)), where, as for Definition 8, s max = max l∈L stg(l) is the maximum strength of 310 the label system. For the sake of understanding, we will now illustrate how to build the MC profiles for the elements in our running example.
Example 5. Following our running example, note that the criteria preferences the label system that we have considered contains labels of strength 2, 1 (and 0).
Hence, since the maximum strength is 2, µ(x, κ) = (na(x, κ, 2), na(x, κ, 1)) for 3 Recall that the MC lex-cel function in Section 5.2 applies a lexicographical order over µ(x), and thus the left indicates greater preference. 4 A strength zero relation (labelled l 0 ) represents that the element is neutral to the criterion.
In other words, the element does not affect the criterion (the element neither aligns with nor is detrimental to the criterion). Hence, we should not take into account these relations in the MC profile.
Again, by means of equation 6, we have that µ(x 1 , κ 2 ) = (1, 0). With these two equivalence class profiles, we can now apply equation 5 to build the MC profile of 325 x 1 as µ(x 1 ) = ((−1, 0), (1, 0)). By following an analogous procedure, we obtain the MC profiles for the rest of elements of X:

The multi-criteria lex-cel ranking function
Since the MC profile of an element x ∈ X encodes its alignment with the criteria in C, we propose comparing elements in X by comparing their MC profiles by means of their lexicographical order. This is precisely what our multi-criteria lex-cel function captures as follows: x y ⇔ µ(x) ≥ lex µ(y).
Def. 10. Given a set of elements X, a set of criteria C and a set of labellings L(X, C), the multi-criteria lex-cel (MC lex-cel) function mclex : L(X, C) × R(C) → R(X) associates to any labelling λ ∈ L(X, C) and any ranking C ∈ R(C), another ranking = mclex(λ, C ) ∈ R(X) such that for any two elements x, y ∈ X: where > lex the lexicographical order in Definition 2.
Example 6. After applying MC lex-cel to the MC profiles obtained in Example 5, we obtain the following element ranking: Our purpose now is to prove that MC lex-cel embodies dominance according 335 to Definition 9. Before doing that, we need an intermediary result showing that the lexicographical ordering of criteria profile captures κ-dominance within criteria equivalence classes.
With the help of Lemma 1, we are now ready to prove that multi-criteria lex-cel embodies dominance.

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Theorem 1. MC lex-cel embodies dominance, that is, if mclex( C ) = , then for x, y ∈ X, we have that x y ⇔ x is dominant over y.
Proof. Suppose that x y. Since has been obtained through MC lexcel, we know that µ(x) > lex µ(y). This means that ∃κ ∈ C/ ∼ C , such that µ(x, κ) > lex µ(y, κ) and ∀κ C κ, µ(x, κ ) = µ(y, κ ). Thanks to Lemma 1, we 360 have seen that this means that x is κ-dominant over y and ∀κ C κ, x and y are κ -indifferent, which is the definition of dominance of x over y. Similarly, if µ(x) = µ(y), then ∀κ, µ(x, κ) = µ(y, κ), and thus x and y are κ-indifferent, meaning that they are indifferent. As to the other direction of the proof, say that x is dominant over y. If µ(x) < lex µ(y), it would imply that y is dominant 365 over x, which contradicts our assumption. Similarly, if µ(x) = µ(y), x and y should be indifferent, again contradicting our assumption. Therefore, the only possibility is that µ(x) > lex µ(y). The same reasoning applies if we suppose that y is dominant over x or x and y are indifferent.

MC ranking and social ranking 370
In this section we explore the relation between our MC ranking and the social ranking introduced by Moretti et al. in [35]. We show that any social ranking can be encoded as an MC ranking, but that is not true the other way around. Therefore, the MC ranking is more general. Furthermore, we also show that our MC lex-cel generalises the lex-cel social ranking solution introduced by 375 Bernardi et al. in [12].
The social ranking [35] considers a set of elements X, and a ranking over coalitions of these elements, namely a ranking over P(X). The purpose of a social ranking is to transform or ground this power set ranking into a ranking over X. Formally:

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Def. 11. A social ranking is a function sr : R(P(X)) → R(X) which transforms a ranking over P(X) into a ranking over the elements of X.
The goal of a social ranking and of an MC ranking is the same: to obtain a ranking over X. Nonetheless, the starting points for the computation of the two rankings are different. While a social ranking considers a ranking over the 385 power set of X, an MC ranking considers criteria, a ranking over criteria and a labelling relating elements to criteria. Note though that it is possible to define a function that transforms a social ranking into an MC ranking. Since the input of sr is in R(P(X)) and the input of mcr is in R(C) × L(X, C), we propose a function t : R(P(X)) → R(C) × L(X, C) to transform the input of a social 390 ranking into an input for MC ranking. Therefore, this transformation function is such that t( P ) = (λ, C ). Let X be a set of elements and P a ranking over P(X), we build function t as follows: 1. We transform the sets in P(X) into criteria: C = {c S , ∀S ∈ P(X)}.
2. We obtain the ranking over criteria as a direct translation of the ranking 395 over sets: c S C c S ⇔ S P S .
3. Finally, to define a labelling function, note that a social ranking does not consider gradings. However, we can consider one label to indicate that an element aligns with criterion c S (the element appears in set S), and another label to indicate that the element is neutral with regard to this criterion (the element does not appear in S). To do so, we define labels l 1 and l 0 respectively (along with the unused l −1 to make the label system symmetric). Hence, we define the label system LS = L, ≥ L , with L = {l 1 , l 0 , l −1 }, and order l 1 ≥ L l 0 ≥ L l −1 . Then, we build a labelling λ that specifies whether an element x is related to c S with label l 1 if x ∈ S, or with label l 0 if x / ∈ S: The t function allows us to transform any social ranking input into an MC ranking input. In fact, in what follows we prove that MC rankings generalise social rankings. Before that, we need an auxiliary result regarding the properties of function t as shown by the following lemma.

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Lemma 2. The t function is injective, but not exhaustive.
Proof. Suppose that t is not injective. Thus, for a given power set P(X), there are two different rankings , ∈ R(P(X)), such that t( ) = t( ). Since , are different rankings, ∃Y, Z ∈ P(X), such that Z Y , while Z Y .
Note though that in these cases when applying t, we would have that c Z C c Y injective. In terms of exhaustivity, t is not exhaustive because labellings using labels other than l 1 and l 0 can never be the image of a social ranking.
Thanks to lemma 2, we prove our first general result.
Theorem 2. MC rankings generalise social rankings. That is, given a set of 410 elements X, a power set P(X), a ranking over the power set P , and a social ranking sr : R(P(X)) → R(X), there exists an MC ranking mcr, such that sr( P ) = mcr(t( P )), but the reverse does not hold in general.
Proof. To prove the theorem we have to find a mcr function such that sr( P ) = mcr(t( P )). Consider mcr = sr • t −1 . In this case, we would have that 415 mcr(t( P )) = sr(t −1 (t( P ))) = sr( P ). In the previous lemma we have seen that t is injective but not exhaustive in general, meaning that in general it is not invertible. Note though that t is invertible when restricted to the domain t(R(X)). In this case, since we start in this domain, t −1 exists, meaning that mcr = sr • t −1 is a valid function which proves the theorem.

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This last theorem proves that all social rankings can be cast as an equivalent MC ranking. Also, since t is not exhaustive there are many MC rankings that cannot be cast as social rankings, meaning that the MC ranking is more general.
Regarding this last result, an interesting question we have to address is the relation between MC lex-cel and lex-cel (see Section 2). The next theorem 425 shows that MC lex-cel generalises lex-cel.
Proof. Suppose that lex( P ) = e and mclex(t( P )) = e . We will see that given x, y ∈ X, x e y ⇔ x e y. We start with x e y ⇒ x e y. First,

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Recall that the labelling obtained by t is built following λ(x, c S ) = l 1 ⇔ x ∈ S.

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When it comes to the reverse implication, x e y ⇒ x e y, suppose that x e y. In this case, x e y cannot happen because we have seen above that it would imply that x e y, which is not true. Then, the only possibility is that x e y. We can follow the same reasoning to prove that x ≺ e y ⇒ x ≺ e y and x ∼ e y ⇒ x ∼ e y. 460

Case study: a value alignment problem
The purpose of this section is to illustrate how MC-lexcel can be used to solve a value alignment problem, that is, a decision-making problem where elements have to be chosen by considering their alignment with multiple moral values.
In particular, we focus on the problem tackled by a decision maker (e.g. a 465 policy maker) when tasked with selecting the collection of regulatory norms that are most closely aligned with the moral values of a society. In Section 7.1, we introduce the decision-making problem. In Section 7.2 we discuss how to exploit MC-lexcel to solve the decision-making problem computationally.
Finally, in Section 7.3 we discuss a case study in a healthcare context, concerned 470 with selecting norms related to hospital admission. Furthermore, we compare the qualitative solving method detailed in Section 7.2 with existing methods in the literature.

Defining the value alignment problem
Within societies, norms have long been used as a coordination mechanism 475 [6]. On the one hand, the literature on Normative multi-agent systems has traditionally focused on establishing norms to regulate agents' behaviour by means of: emergence [40,47] , an empirical bottom-up approach; off-line norm synthesis [1], a formal top-down approach; and on-line norm synthesis [32,33], which is empirical and top-down. We refer to the set of norms enacted in a 480 society as a norm system. On the other hand, norms have also been related to moral values 5 [17], which are used as a guiding criteria for the selection [46,45] or synthesis [31] of the norm system to be enacted. Indeed, composing a set of norms that promote ethical behaviour (i.e., moral values) naturally induces this ethical behaviour in the society. Moreover, if different moral values can 485 be promoted, then it seems reasonable to prioritise the most preferred ones.
Consider, for example, a government that enacts norms limiting pollution. In this case, we can confidently infer that this government prioritises sustainability over other values such as development.
However, the problem of selecting the regulatory norms that align best with the ethical principles of a society (or, in other words, the most value-aligned norm system) is not straightforward. In addition to the different values and preferences over them that a society may have, we must also consider whether norms actually promote or demote those values, as well as the degree of promotion/demotion. Some of the literature in Philosophy discusses a number of 495 these aspects [18]. Nonetheless, in the Artificial Intelligence literature, while value promotion and demotion are commonly considered, the degrees of such relations are not typically considered (e.g. [28], [11], [43]). In fact, to the best of our knowledge such aspects have only been considered in legal cases [10].
Against this background, we introduce our value alignment problem while 500 considering promotion and demotion relationships between norms and values as first-class citizens. Thus, we first introduce the formal objects required for the problem, namely: norms, value system, and the relationships between norms and values.
We define the core notion of our problem, the norm, as a simplification of Furthermore, we consider a propositional language PL (with propositions in P and the logical operator "and"), a set of states S, and a state transition function that changes the state of the world when agents perform actions (following the 510 multi-agent system model introduced by Morales et al. in [32,34]). Then, a norm is composed of a precondition ϕ ⊆ P (with an "and" semantic between propositions), an action in A, and a deontic operator θ to establish Obligations (Obl), Permissions (P er), and Prohibitions (P rh). With these definitions in place, we define a norm as: 515 Def. 12 (Norm). A norm is a pair ϕ, θ(a) , where ϕ is a precondition in the language PL; a ∈ A is the regulated action, and θ ∈ {Obl, P er, P rh} is a deontic operator.
Example 7. Within a healthcare context, we may have a norm permitting the hospital admission of incoming patients: patient in, Per(admit) .

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Let N be a set of candidate norms; the norms in N might have relationships between them [46]. We consider two types of such norm relations, namely norm exclusivity and norm generalisation and note them as R x , and R g respectively (we assume the decision-maker has sufficient knowledge of the domain to detect and provide these norm relations). On the one hand, we say n, n are exclusive 525 norms, noted as (n, n ) ∈ R x , when we cannot enact both of them simultaneously. On the other hand, we say they have a direct generalisation relation, noted (n, n ) ∈ R g , meaning n is more general than n . With regards to generalisation relations, we note as S(n) and A(n), the successors and ancestors of n respectively. Formally: 530 Def. 13. Given a norm n ∈ N , its ancestors are the norms that (directly or indirectly) generalise it: A(n) = {n ∈ N : ∃n 1 , . . . , n k , and (n , n 1 ), . . . , (n k , n) ∈ R g }. Conversely, successors are the norms that are (directly or indirectly) generalised by n: S(n) = {n ∈ N : ∃n 1 , . . . , n k , and (n, n 1 ), . . . , (n k , n ) ∈ R g }.
Norms and their relations form a structure called a norm net. Def. 16. Let N, R be a norm net, then we consider a norm system Ω ⊆ N to be sound iff it is: • Conflict-free: ∀n i , n j ∈ Ω, (n i , n j ) / ∈ R x 545 • Non-redundant: ∀n, with |S(n)| > 1, thenS(n) Ω.
WhereS(n) = {n ∈ N, (n, n ) ∈ R g } stands for the set of direct successors.

Moral values are principles deemed valuable by a society [48]. Ethical choice
typically implies a set of moral values [13] and preferences regarding them.
Indeed, some values are preferred over others [11], and these preferences must 550 impact the decision-making process. For that reason, we consider the value system to be a structure formed by moral values and their preferences [11,28,46]. Thus, we say that the value system guides ethical reasoning. While several types of preferences have been used to formalise this structure, we favour rankings because they are the least restrictive preference structures satisfying 555 totality. In this manner, given any pair of values, we can assert a preference between them (which may not be possible with non-total preferences such as partial orders). Therefore, we define the value system as follows.
Def. 17. Let V be a non-empty set of moral values, and v a ranking over V , we call value system the tuple V, v . the degree of promotion/demotion of labels. Thus, given a label l ∈ L, the more labels between l and l 0 , the larger its promotion/demotion degree (i.e., the stronger l is).
Thanks to the objects formally introduced so far, we are ready to introduce our decision-making problem, the so-called generalised value-aligned norm se-575 lection problem (GVANS) 6 . The input of the GVANS problem is: (i) a norm net N, R ; (ii) a value system V, v ; and (iii) a symmetric label system L, > l , λ that sets the relation between norms and values. Solving a GVANS problem consists in composing the sound norm system which best aligns with the value system, taking into account the degree of promotion/demotion of norm-value 580 relations as expressed by the label system.

Solving the value alignment problem
When deciding on the most value-aligned norm system, we follow the following proposition: the more preferred the values promoted by a norm system, the more preferred the norm system, or, in other words, the more value-aligned. To 585 obtain the most value-aligned norm system (i.e., to solve a GVANS problem) we will proceed in two steps.
First, we exploit MC-lexcel to obtain a ranking over individual norms from a ranking over values in a value system. This is straightforward if we consider that the values in V act like criteria (i.e. C = V ), and value preferences are 590 cast over the elements of the decision (i.e. the norms in N ). Importantly, our aim is to use the norm ranking to later select the set of norms that best aligns with the value system. Since norms can both promote and demote values, there might be norms which, overall, demote more preferred values than those they promote. We call these norms non-beneficial norms. In contrast, beneficial 595 norms are those that promote more preferred values than those they demote. A simple informal way to differentiate between beneficial and non-beneficial norms is to compare them to a neutral norm n 0 . We define n 0 as an artificial norm that is neutral with regards to all the moral values in the value system. Thus, informally: 600 Def. 18. A beneficial norm is a norm that is more preferred than n 0 . Norms less preferred than or indifferently preferred to n 0 are non-beneficial norms. We note as N ben ⊆ N the subset of beneficial norms in N .
When selecting a set of norms, we want to select only beneficial norms and avoid non-beneficial norms. In other words, the solution to the GVANS problem is a 605 set of norms in N ben . With MC-lexcel we can obtain a ranking that allows us to compare norms, but we must also know which norms are beneficial and which are not. In line with Definition 18 above, we exploit MC profiles to differentiate between them. Thus, in the case of MC profiles: Def. 19. We say that a norm n ∈ N is beneficial if µ(n) > lex µ(n 0 ). On the 610 other hand, a norm is non-beneficial if µ(n 0 ) ≥ lex µ(n). Thus, in this case, Indeed, since we build the ranking from the MC profiles of norms, a norm that is less preferred than n 0 will be a norm whose MC-profile is worse than that of a totally neutral norm. This is the case when an MC profile contains more 615 demotion labels than promotion labels, or contains demotion labels associated to more preferred values. Thus, by applying MC-lexcel considering N ∪{n 0 }, we obtain a ranking mclex( v ) = n in which not only we can compare norms, but also n 0 partitions norms between beneficial (when n n n 0 ) and non-beneficial (when n 0 n n) norms.

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The next step is to use the norm ranking to compose the desired set of valuealigned norms. Since only beneficial norms should be taken into account when composing the norm set, we discard the non-beneficial norms hereafter. Hence, we now consider the ranking only over beneficial norms ben n obtained from the MC ranking over all norms. We formalise this using the following restriction:
Our final step consists in transforming the ranking over beneficial norms into a ranking over norm systems. For that, we utilise the anti-lexcel operator 630 introduced in [43]. Let N ben be a set of beneficial norms, and ben n a ranking over these norms, the anti-lex-cel function ale : R(N ben ) → R(P(N ben )) is a lifting function which generates a ranking over subsets of beneficial norms, namely over the norm systems in P(N ben ). Therefore, the composition of MC-lexcel, the restriction to beneficial norms, and anti-lex-cel, transforms preferences over 635 values into a value system for preferences over beneficial norm systems. We formally define this composition as follows: Def. 21. We call nsr : R(V ) → R(P(N ben )) (nsr for norm system ranking) the function nsr = ale • ben • mclex. Thus, for a value ranking v ∈ R(V ), nsr( v ) = ale(ben(mclex( v ))) = is a ranking over norm systems (introduced 640 in Definition 15) composed of beneficial norms.
The solution to the GVANS problem at hand will be the most preferred sound norm system produced by a norm system ranking. Unfortunately, although a norm systems ranking helps us obtain the solution to a GVANS problem, the cost of building a whole ranking over norm systems (elements 645 in P(N ben )) turns out to be rather costly. As discussed in This BIP only employs |N | decision variables and can be solved with the aid of standard BIP solvers (e.g. CPLEX 7 or Gurobi 8 ). We propose following the same approach here. Figure 1 shows the steps of our computational approach to compute the most value-aligned norm system. First, given a set of norms and 655 a value system as input, we apply MC-lexcel to obtain a norm ranking. After that, we restrict the norm ranking to beneficial norms, and then we use this beneficial norm ranking to do the encoding of the GVANS problem as a BIP.
Finally, we solve the BIP with the aid of standard BIP solvers. Appendix A provides more details on this approach. We refer the reader to that appendix 660 for details on encoding a GVANS problem.
Henceforth, we refer to the method outlined in Figure 1 as the qualitative approach with graded value promotion and demotion. The next section illustrates and compares it to previous approaches.

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Following Example 7 on healthcare, here we introduce a simple example that illustrates the qualitative approach with graded value promotion and demotion described in Section 7.2. Furthermore, we use it to compare our approach to those in the literature. On the one hand, [46] proposes a numerical approach that first assigns a utility to each norm -which represents 670 their value alignment-and then selects norms by maximising their cumulative utility. Here, we show that asserting norm utilities may introduce biases that our qualitative approach avoids. On the other hand, although the work in [43] is also qualitative, it has limited expressiveness, since it does not allow for demotion, nor for different degrees of promotion/demotion. Overall, we show 675 that, for specific cases, these other methods in the literature fail to produce a norm system that is the most closely aligned with the given moral values.
As previously mentioned, our case study focuses on selecting norms related to hospital admission. In particular, as Figure 2 shows, we consider four norms: • AE = patient in, Per(admit) : Allow admission to Everybody;  Table 1 detail the λ function that completes our label system. Overall, general norms that apply to everybody are strongly related to the values. This is also the case for elders, since they are most likely to require admission. Alternatively, 700 we consider the relationship with young people to be less strong, since they are less likely to require admission.
From here, we apply our qualitative approach with graded value promotion and demotion to compute the norm ranking as AE n AY n n 0 n DL n DE.
This is because using Equation  Alternatively, when considering the quantitative utilitarian approach in

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[46], the task of assigning (and justifying) numerical degrees of promotion/demotion turns out to be more difficult. U columns in Table 1  Subsequently, the procedure in [46] computes norm utilities -for simplicity, we take the random ∈ (0, 1] to be 0.5 -as: u(AE) = 0.5, u(DE) = −1.5, 725 u(AY) = 0.6, u(DL) = −1, 2 so that the sound norm system with the highest utility is {AY}. This means that the quantitative utilitarian method selects a norm system that fails to regulate admissions of elder people. This is because AE strongly demotes the AUS value, and this diminishes its utility.
If we now consider the binary qualitative approach used by [43], the B 730 columns in Table 1 are limited to representing promotion (1) and no-promotion (0). This method produces a norm ranking of AE ∼ n AY n DE ∼ n DL. Indeed, although most preferred values should prevail, the quantitative method also fails to capture the absolute preferences of the value system.
The advantages of our method are two-fold. First, its graded qualitative labels for promotion and demotion are much simpler to define -and less prone to biases -than numerical degrees, and it also provides far more expressive-755 ness than binary promotion alone. Second, its ranking method captures the preferences of the value system in the selection of the norm system to enact.

Discussion regarding applicability
Although the main contribution of this work is theoretical, the previous section presented a norm selection case study with the aim of illustrating its ap-760 plication. However, this should not preclude the reader from understanding that our method can be applied in other scenarios. Indeed, we already mentioned some of these potential scenarios in the introduction section (college admissions, committee selections, scholarship assignments, or diet design). Here we aim to go a step further and discuss in some detail how an alternative domain in the 765 context of participatory democracy [37] may benefit from our approach.
As pointed out by the European Commission [41], democracy aims at the greater good of society, where good cannot only be measured in monetary terms, but also requires considering what citizens perceive to be valuable. As a con- Despite empowering citizens, current implementations of participatory budgeting processes suffer from a major practical caveat related to their limited convening capacity [29]. As a consequence, their representativeness may be compromised if, for instance, educated younger citizens are more inclined to it is worth mentioning that the work of Liscio et al. [25,26] can be used to detect context-specific values, and preferences over these values can be gathered 795 through surveys such as the European Values Study [15]and then aggregated by means of the state-of-the art method proposed by Lera-Leri et al. [24].
Here, we develop an alternative case study for our work by considering a participatory budgeting process in which citizens submit a set of project proposals to which we then apply our selection process based on citizens' values. 800 We illustrate this by taking real data from the participatory budgeting process held by the Barcelona city council in 2021 16 . This participatory budget is a relevant example because the process had an overall budget of 30 million euros, attracted the participation of more than 64000 citizens, and resulted in the selection of 76 (out of 184) projects being funded.  Typically, the decision selection process that is applied is the rank and select method [44]. This method is based solely on the number of votes cast as it 810 first ranks the projects in decreasing order of the number of votes received (as presented in Table 2) and then selects projects until the budget is spent. As an example, imagine Barcelona had only received these 10 project proposals and they had to decide which to select if the available budget was 2.5 million euros.
In this case, we would select projects p 1 and p 2 , with a cost of 2.45 million euros, 815 and since the cost of the next project is greater than the remaining budget, we would stop selecting projects (note that this means that we would not even select project p 4 , which could be funded with the remaining budget).
Unfortunately, participation in this process was around 4%, and we may therefore consider citizens' to be more appropriate for selecting which projects 820 to fund. Thus, we can consider that strategic areas represent different values and we can imagine that the city council of Barcelona carries out opinion polls to learn about citizens' preferences with regard to these strategic areas. Now, say that, in this hypothetical case, we conclude that citizens prefer Green spaces above all, followed by Pacification, Culture, Mobility, Monuments, and, finally,

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Sports. In particular, this means that project 2 has received a lot of support thanks to the bias of the participant base, but this would not be the case if the preferences of the whole population were considered.
From this, we can apply MC-lex-cel to obtain a ranking of proposals considering the preferences expressed regarding strategic areas (our criteria here).

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In this case, since Table 2 links each project to a single area, the ranking of projects obtained through MC-lex-cel would rank highly those projects that are related to more preferred areas (and equally those that are within the same area), hence p 1 ∼ p 3 ∼ p 4 ∼ p 6 p 5 ∼ p 8 p 9 p 10 p 7 p 2 .
Appendix B provides the encoding for participatory budgeting, which in 835 turn applies the norm selection encoding in Appendix A. The solution results in the selection of projects p 1 , p 3 , p 4 , p 5 , p 6 , p 7 , p 8 , and p 9 . Therefore, although p 2 has a large amount of votes, it is related to the least preferred area, and since it is the most expensive project, it is optimal to select the more closely aligned (and less expensive) projects. 840 Finally, note that Barcelona's Participatory Budget related each project to only one strategic area. Nonetheless, in reality, these projects will probably be related to more than one area to various degrees (e.g. p 1 is concerned with improving a park, therefore it is highly related to the area of Green Spaces, but it may also be related to Pacification to a lesser degree). If this information was 845 available, MC lex-cel could consider it to build the ranking of proposals.

Conclusions
In this paper, we have tried to make headway in supporting decision-makers that are challenged with comparing, and ultimately ranking, elements with re-gard to how such elements satisfy multiple criteria and how such criteria are 850 preferred. This calls for a new decision-making framework, which we have formally introduced here. Our framework is based on a novel method for ranking single elements.
Ranking functions have been widely used to transform rankings. For instance, the social ranking function transforms a ranking over sets of elements Finally, the paper also illustrates how they can be employed to help a decision-maker to tackle an ethical decision-making problem. Specifically, we 865 define the Generalised Value-Aligned Norm Selection (GVANS) problem and solve it with a qualitative approach with graded value promotion and demotion.
Overall, this method overcomes the shortcomings of previous methods, resulting in a norm system that is most closely aligned with the value system at hand.
In future work, we plan to study in more detail the properties of MC rankings To ensure that we select a sound norm system, the BIP encoding must also consider the following constraints: -Mutually exclusive (incompatible) norms cannot be selected at once: To select participatory budget projects, we apply the BIP encoding in Appendix A to consider preferences regarding projects. Consider a set of projects X and preferences over the projects as a ranking, with Ξ 1 , . . . Ξ r ∈ X/ and Ξ 1 · · · Ξ r . We want to maximise the preferences of the selected projects.

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The BIP encoding uses decision variables d i ∈ {0, 1} (0 for not selected, 1 for selected) for each x i ∈ X. The target function is the same formula as in Appendix A, which is the sum of decision variables d i each multiplied by a parameter related to the equivalence class x i is in (see equation A.2).
The BIP would maximise this target function and if there are exclusivity 910 or generalisation relationships between projects, this BIP encoding will also include the corresponding constraints. Similarly, budget constraints may also be added.