Assigning Numerical Scores to Linguistic Expressions

axioms Article Assigning Numerical Scores to Linguistic Expressions María Jesús Campión 1, Edurne Falcó 2, José Luis García-Lapresta 3,* and Esteban Induráin 4 1Inarbe (Institute for Advanced Research in Business and Economics), Departamento de Matemáticas, Universidad Pública de Navarra, 31006 Pamplona, Spain; [email protected] 2VIRENA Navarra SL, Beriáin, 31191 Navarra, Spain; [email protected] 3PRESAD Research Group, BORDA Research Unit, IMUVA, Departamento de Economía Aplicada, Universidad de Valladolid, 47011 Valladolid, Spain 4 InaMat (Institute for Advanced Materials), Departamento de Matemáticas, Universidad Pública de Navarra, 31006 Pamplona, Spain; [email protected] *Correspondence: lapr[email protected]; Tel.: +34-983-184-391; Fax: +34-983-423-299 Academic Editor: Radko Mesiar Received: 7 June 2017; Accepted: 30 June 2017; Published: 6 July 2017 Abstract: In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through distances in a graph plus some suitable penalization of imprecision. The relationship between this setting and the classical problems of numerical representability of orderings, as well as extension of orderings from a set to a superset is also explored. Finally, aggregation procedures of qualitative rankings and scorings are also analyzed. Keywords: decision making; imprecision; qualitative scales; linguistic expressions; scores; orderings; aggregation MSC: 62C86 (Primary); 03E72; 06A06; 91B06; 91B14 (Secondary) 1. Introduction Let U stand for a nonempty set, called universe. It is clear that comparing the elements of U through some sort of binary relation R such that uRv is immediately understood as “the element u is not worse than the element v ” gives less information than assigning scores to each element of U . A scoring function is a real-valued function S:U→R . The interpretation is obvious: if the score S(u) of the element u∈U is not smaller than the score S(v) of the element v , then the above conclusion, namely, “the element u is not worse than the element v ” is also achieved. However, if an ordering given on U through a binary relation R is also represented by a scoring function, say S:U→R , the composition of f◦S of the map S with no matter what strictly increasing function f:R→R also represents the same ordering R on U . In other words: uRv⇔S(u)≥S(v)⇔f(S(u)) ≥f(S(v)) holds true for all u,v∈U. Despite the ordering being the same, the scores have important consequences and collateral meanings. A typical example corresponds to a firm that wants to hire people, and prepares a test or exam consisting of one hundred tasks to be done. Suppose the scores of two candidates u and v have been just S(u) = 1 and S(v) = 2 (out of 100). Obviously, we would say that v is better than u , but both of them having such poor results, the firm would reject hiring anyone among them. If, instead, the scores were S(u) = 99 and S(v) = 100, again we would say that v is better than u , but now, quite probably, the firm would be willing to hire both candidates. Axioms 2017,6, 19; doi:10.3390/axioms6030019 www.mdpi.com/journal/axioms Axioms 2017,6, 19 2 of 17 The most typical scoring procedures consist of numerical scales. Nevertheless, due to the vagueness and imprecision involved in a wide variety of practical situations in which some sort of evaluation is made, sometimes the use of finite measurement scales based on linguistic terms or expressions seems to be more suitable than just using numerical scales. In this case, if L is a set of linguistic terms (also known as labels), such as { very bad, bad, poor, fair, good, very good, excellent } , a scoring function on Utaking values on Lis a function S:U→Lthat can obviously be interpreted as a qualitative scale of measurement. From a pure mathematical point of view, whenever L is given a linear order <L , a scoring procedure on U taking values on L can equivalently be understood through a numerical scoring function f:U→R , which is obviously a quantitative scale of measurement. Indeed, if L={l1,l2, . . . , lg} denotes the set of linguistic terms and we understand that lh<Llk holds true if and only if h≥k , then a scoring function S:U→L can be immediately interpreted by means of the numerical scoring function T◦S:U→R , where T:L→R is given by T(lh) = h , h∈ { 1, . . . , g} . In this situation, namely when L is given a linear order, the use of a scoring function taking values on L or an equivalent numerical scoring function is, perhaps, a matter of taste (see García-Lapresta et al. [1]) . However, sometimes, in a higher degree of vagueness or imprecision, and even in the case of using a set L={l1 , l2 , . . . , lg} of linguistic terms, it could happen that the agents that assign scores to the elements of a nonempty set U may hesitate when, given an element u∈U , the score S(u) needs to be determined. A typical example is an approximation (based on data as climate, composition of soil, amount of rain water per year, etc.) of the quality of the wine produced in a vineyard. An expert could hesitate here, and instead of saying “no doubt, it will be very good”, she/he could declare something as “from good to excellent”, so using a linguistic expression instead of just a linguistic term. Our approach concerning the imprecision is based on an adaptation of the absolute order of magnitude spaces introduced by Travé-Massuyès and Dague [ 2 ] and Travé-Massuyès and Piera [ 3 ]; more specifically in the extensions devised by Roselló et al. [ 4 – 6 ] (see also Agell et al. [ 7 ] and Falcó et al. [8,9]). At this stage, several mathematical problems appear, namely (i) If a set L={l1 , l2 , . . . , lg} of linguistic terms has been endowed with a linear order <L , try to define a linear order on the set of linguistic expressions based on L . Needless to say, this “extended” order should preserve or be compatible with <L when acting on the labels of L considered as special cases of linguistic expressions. (ii) Compare different linear orders (when available) defined on a set of linguistic expressions, trying to explain which is the most suitable one to be used in some practical concrete situation. (iii) Study problems of aggregation of a finite number of scorings based on linguistic terms and expressions. Having in mind these questions, the structure of the paper goes as follows. After the Introduction (Section 1) and a compulsory Section 2of preliminaries, previous results and necessary background, in Section 3, we study sets of linguistic expressions and define suitable orderings on these kinds of sets. In Section 4, we analyze some questions relative to the aggregation of rankings and scorings based on qualitative scales. A final Section 5of concluding remarks, suggestions for further research and open questions closes the paper. 2. Previous Concepts and Results Intuitively, a ranking on a universe U is some kind of ordering defined on U , usually defined through a binary relation R on U , such that uRv means “the element u is at least as good as the element v ”. A scoring function on U is a map S:U→R . Notice that a scoring function S immediately defines a ranking R on U , by declaring uRv⇔S(u)≥S(v) , for all u , v∈U . For finite sets, a sort of converse is also true, namely, each ranking can be represented by a scoring function (see e.g., the first three chapters in Bridges and Mehta [10] for further details). Axioms 2017,6, 19 3 of 17 Let us formalize all this through some definitions. Definition 1. Let U denote a universe. A preorder < on U is a binary relation on U which is reflexive and transitive. An antisymmetric preorder is said to be a partial order. A total preorder < on U is a preorder such that u<v or v <u holds true for all u,v∈U. A total order is also called a linear order. If < is a preorder on U , then the associated asymmetric relation is denoted by  , whereas ∼ stands for the associated equivalence relation. These relations are respectively defined by uv⇔(u<v) and not (v<u) , and u∼v⇔(u<v) and (v<u) , for all u , v∈U . Notice that if < is a linear order, then uv⇔(u<v)and (u6=v). Definition 2. Let U denote a universe. Let < stand for a total preorder defined on U . We say that < is representable if there exists a real-valued function S:U→R such that u<v⇔S(u)≥S(v) holds true for all u,v∈U. As commented before, it happens that, provided that U is finite, any total preorder defined on U is representable (see Bridges and Mehta [ 10 ]). A function S that represents a total preorder < defined on Uis usually said to be a utility function for <. Remark 1. Notice that a ranking (e.g., a total preorder) on a universe U is a qualitative scale of measurement, whereas a scoring function is a quantitative scale. Other typical sorts of qualitative scales are those based on labels and linguistic expressions (see next Definition 3). If S is a utility function that represents a total preorder < on a universe U , and T:R→R is a strictly increasing function, it is straightforward to see that the composition T◦S:U→R is also a utility function that represents < . Consequently, there are infinitely many utility functions that represent a given total preorder <on a finite universe U. Definition 3. Let L={l1 , l2 , . . . , lg} denote a finite nonempty set of linguistic terms (labels), with g ≥3. The set of linguistic expressions associated with L is defined as L={[lh,lk]|lh,lk∈L, 1 ≤h≤k≤g}, where [lh , lk] = {lh , lh+1 , . . . , lk} . Since the linguistic expression [lh , lh] is the singleton {lh} , by convention, it can be replaced by the label lh. In this way, L ⊂L. Remark 2. The set of linguistic expressions can be represented by a graph GL . In the graph, the lowest layer represents the linguistic terms lh∈L⊂L , the second layer represents the linguistic expressions created by two consecutive labels [lh , lh+1] , the third layer represents the linguistic expressions generated by three consecutive labels [lh , lh+2] , and so on up to last layer, where we represent the linguistic expression [l1 , lg] . As a result, the higher an element of the graph GLis, the more imprecise it becomes. The vertices in GL are the elements of L and the edges E − F , where E= [lh , lk] and F= [lh,lk+1], or E= [lh , lk] and F= [lh+1 , lk] . Figure 1shows the graph representation for g=5. Axioms 2017,6, 19 4 of 17 [l1,l5] [l2,l5][l1,l4] [l3,l5][l2,l4][l1,l3] [l4,l5][l3,l4][l2,l3][l1,l2] l5 l4 l3 l2 l1 Figure 1. Graph representation of the linguistic expressions for g=5. 3. Defining Orderings on Sets of Linguistic Expressions Suppose that the set L={l1 , l2 , . . . , lg} of linguistic terms is endowed with the linear order < given by lh<lk⇔h≥k, for all h,k∈ {1, . . . , g}. We want now to extend the linear order < to a new linear order defined on the whole set L of linguistic expressions on L. This problem has several positive solutions. Remark 3. To extend the linear order < on L to a new linear order defined on L , first, we may consider a partial order <1 defined on L as follows: [lh , lk]<1[lh , lk] (reflexivity), and also {lh}= [lh,lh]<1[lk,lk] = {lk} ⇔ h≥k, for all h,k∈ {1, . . . , g}(compatibility with <on L). Then, we may apply the well-known Szpilrajn’s theorem ([ 11 ]) that ensures that any partial order on a given set can be extended to a linear order on that set. 3.1. Constructive Extensions: The Lexicographic Order and the Canonical Order Instead of using a powerful result as Szpilrajn’s theorem as commented in Remark 3, when L={l1,l2, . . . , lg} , we can directly furnish some extensions in a constructive way, as stated in Proposition 1and Proposition 2, whose straightforward proofs are omitted for the sake of brevity. Proposition 1. The binary relation <`on Ldefined as [lh,lk]<`[lh0,lk0]⇔     h>h0 or h=h0and k ≥k0 is a linear order that preserves the given order <on L ⊂L. It is called the lexicographic order on L. Notice that the real valued function S:L→R given by S([lh , lk]) = (g·h) + k is a utility representation for <`. Proposition 2. ([8]) The binary relation <con Ldefined as [lh,lk]<c[lh0,lk0]⇔     h+k>h0+k0 or h+k=h0+k0and k −h≤k0−h0 Axioms 2017,6, 19 5 of 17 is a linear order, and it is called the canonical order on L. It also preserves the given order <on L ⊂L. Example 1. For g =5, the elements of Lare ordered as follows with respect to the lexicographic order: l5`[l4,l5]`l4`[l3,l5]`[l3,l4]`l3`[l2,l5]`[l2,l4]` `[l2,l3]`l2`[l1,l5]`[l1,l4]`[l1,l3]`[l1,l2]`l1. Moreover, through the canonical order (see also Figure 2), we get: l5c[l4,l5]cl4c[l3,l5]c[l3,l4]c[l2,l5]cl3c[l2,l4]c c[l1,l5]c[l2,l3]c[l1,l4]cl2c[l1,l3]c[l1,l2]cl1. [l1,l5] [l2,l5][l1,l4] [l3,l5][l2,l4][l1,l3] [l4,l5][l3,l4][l2,l3][l1,l2] l5 l4 l3 l2 l1 Figure 2. Canonical order in Lfor g=5. 3.2. Numerical Representation of the Canonical Order Based on the Geodesic Metric The canonical order on L can be represented by utility functions u:L−→ R in such a way that E cF ⇔ u(E)>u(F), for all E,F ∈ L. Among the infinite possible utility representations of the canonical order on L , we are interested in those based on distances. In addition, concerning metrics on a graph, perhaps the most typical one is the so-called geodesic metric. Definition 4. The geodesic metric dG:L2→Ris defined as dG[lh,lk],[lh0,lk0]=|h−h0|+|k−k0|(1) for all E= [lh,lk],F= [l0 h,l0 k]∈L. Remark 4. Notice that dG(lh , lh+1) = 2for every h∈ { 1, . . . , g} , so that L becomes uniform, in the sense that adjacent linguistic terms are equidistant (see e.g., Herrera et al. [ 12 ] and García-Lapresta and Pérez-Román [13,14] for dealing with non uniform qualitative scales). Given E= [lh , lk]∈L , with # E , we denote the cardinality of E , i.e., the number of labels in the interval [lh,lk]: #E=k+1−h. Axioms 2017,6, 19 6 of 17 In the following result, we establish that the canonical order on L introduced in Proposition 2can be defined through the geodesic distances to the highest possible assessment and the cardinality of the linguistic expressions. Proposition 3. For all E,F ∈ L, it holds E cF ⇔      dG(E,lg)<dG(F,lg), or dG(E,lg) = dG(F,lg)and #E<#F. Proof. It follows from Proposition 2and Equation (1). Remark 5. To distinguish between elements E , F ∈ L , the second condition that appears in the statement of Proposition 3, namely the one that reads “or dG(E , lg) = dG(F , lg) and # E< # F ” is crucial. In fact, suppose that on L we use only a scoring function S based on the geodesic metric, and given by S(E) = dG(E , lg) . Then, it is straightforward to see that two nodes E , F that appear in the same vertical in the graph GL are assigned the same score, which is S(E) = dG(E , lg) = dG(F , lg) = S(F) . This scoring function S defines an ordering <∗ on Lby declaring E<∗F ⇔ S(E)≥S(F), for all E,F ∈ L. However, since g ≥3, it happens that <∗is a total preorder, but not a linear order. In other words, the geodesic distance is, so-to-say, miopic since it cannot detect the difference between two elements in the same vertical in GL . For instance, if L={ very bad, bad, poor, fair, good, very good, excellent } , the linguistic expressions “very good” and “from good to excellent” would be assigned the same score through the geodesic metric. 3.3. GeodesIc Metric Matching Penalization of Imprecision Another important feature that carries the second condition in the statement of Proposition 3and says “or dG(E , lg) = dG(F , lg) and # E< # F ” is warning us about the role played by imprecision. If a linguistic expression F consists of more labels than another one E , we understand that “ F is more imprecise than E ” or, equivalently, “ E is more accurate than F ”. Taking this fact into account, it seems interesting to search for measurements or scoring methods that, someway, penalize the imprecision when comparing terms in L, as next Proposition 4does. Proposition 4. For every α≥0, the function dα:L2−→ R, defined as dα(E,F) = dG(E,F) + α|#E − #F|, i.e., dα[lh,lk],[lh0,lk0]=|h−h0|+|k−k0|+α|h0−h+k−k0|(2) for all E= [lh,lk],F= [l0 h,l0 k]∈L, is a metric, and it is called the metric associated with α. Proof. Since dα is a positive linear combination of the geodesic metric dG and the pseudometric d:L2−→ Rdefined as d(E,F) = |#E − #F|, then dαis a metric. Proposition 5. Given α≥0, the following statements are equivalent: 1. ∀E,F ∈ LE cF ⇔ dα(E,lg)<dα(F,lg). 2. α∈Ig, where Ig=0, 1 g−2, if g is odd, and Ig=0, 1 g−1, if g is even. Proof. 1⇒2) Consider gis odd. By Proposition 2, we have [l1,lg]chlg−1 2,lg+1 2i. Axioms 2017,6, 19 7 of 17 Then, by hypothesis, we have dα[l1,lg],lg<dαhlg−1 2,lg+1 2i,lg. (3) Taking into account Equation (2), dα[l1,lg],lg=|1−g|+|g−g|+α|g−1+g−g|=g−1+α(g−1) and dαhlg−1 2,lg+1 2i,lg=  g−1 2−g + g+1 2−g +α g−g−1 2+g+1 2−g =g+α. Then, from Inequality (3), we obtain g−1+α(g−1)<g+α, i.e., α<1 g−2. In order to prove that α> 0, suppose by way of contradiction that α= 0. By Proposition 2, we have lg+1 2c[l1,lg]. Then, by hypothesis, we have dαlg+1 2,lg<dα([l1,lg],lg). Taking into account Equation (2), dαlg+1 2,lg=g−1=dα([l1,lg],lg), which is a contradiction. Consequently, α∈Ig=0, 1 g−2. We now consider that gis even. By Proposition 2, we have hlg 2,lg 2+1ic[l1,lg]. Then, by hypothesis, we have dαhlg 2,lg 2+1i,lg<dα[l1,lg],lg. (4) Taking into account Equation (2), dαhlg 2,lg 2+1i,lg= g 2−g+ g 2+1−g+αg−g 2+g 2+1−g=g. Then, from Inequality (4), we obtain g−1+α(g−1)<g, i.e., α<1 g−1. In order to prove that α> 0, suppose by way of contradiction that α= 0. As mentioned above, we have dαhlg 2,lg 2+1i,lg<dα([l1,lg],lg). Axioms 2017,6, 19 8 of 17 However, taking into account Equation (2), dαhlg 2,lg 2+1i,lg=g−1=dα([l1,lg],lg), we obtain a contradiction. Consequently, α∈Ig=0, 1 g−1. 2⇒1) It is a routine. Definition 5. Given α∈Ig, the scoring function Sα:L−→ Ris defined as Sα(E) = dα(l1,lg)−dα(E,lg), for every E ∈ L. It is easy to see that, for every [lh,lk]∈L, it holds Sα[lh,lk]=h+k−2−α(k−h). (5) Proposition 6. Given α∈Ig, for all E,F ∈ L, it holds E cF ⇔ Sα(E)>Sα(F). Proof. By Proposition 5. Example 2. Taking into account the condition appearing in Proposition 5for g= 5, i.e., 0 <α<1 3 , and Equation (5), we have Sα(l5) = 8>Sα([l4,l5]) = 7−α>Sα(l4) = 6>Sα([l3,l5]) = 6−2α> Sα([l3,l4]) = 5−α>Sα([l2,l5]) = 5−3α>Sα(l3) = 4> Sα([l2,l4]) = 4−2α>Sα([l1,l5]) = 4−4α>Sα([l2,l3]) = 3−α> Sα([l1,l4]) = 3−3α>Sα(l2) = 2>Sα([l1,l3]) = 2−2α> Sα([l1,l2]) = 1−α>Sα(l1) = 0. 3.4. Extensions Based on Different Criteria As we have already seen, both the lexicographic and the canonical linear orders on L extend the given order on L . Although it is always possible to extend a linear order from a given finite set U to its power set, a typical question that, as a matter of fact, gave rise to a battery of classical papers from the 1970s (see e.g., Gärdenfors [ 15 ], Kannai and Peleg [ 16 ], Barberà and Pattanaik [ 17 ], Fishburn [ 18 ], Bossert [ 19 ] and Bossert et al. [ 20 ]) is whether or not it is possible to perform an extension that follows a list of criteria imposed a priori. Sometimes, the extension is not possible because the criteria used are, so-to-say, contradictory. However, perhaps surprisingly, there are other situations in which the extension is not possible because of a combinatorial explosion, which, due to the much bigger cardinality 2 #U of the power set of U , does not leave room to accommodate all terms of the power set, in an extended linear order, accomplishing all the criteria. Perhaps the most famous result in this direction is the so-called theorem of impossibility of extension due to Kannai and Peleg [16] (for a detailed account, see Barberà et al. [21]). Consider the following example. Example 3. Let U={u1 , u2 , . . . , up} be a finite universe. Assume that U is endowed with a linear order < such that ui<uj⇔i≥j . Consider the following two criteria for an extension, say <ext of < to the power set of U. Axioms 2017,6, 19 9 of 17 (i) A criterion [M] based on the idea of a mean value, so that if uiuj in U , then {ui} ext {ui,uj} ext {uj}holds true. (ii) A criterion [C] based on the idea of cardinality, in the sense “the bigger, the better”, so that given ui,uj∈Usuch that ui6=uj, then {ui,uj} ext {ui}and also {ui,uj} ext {uj}hold true. In this example, it is clear that these two criteria are contradictory. Unless the set U is a singleton, no extension <ext can at the same time satisfy [M] and [C]. By the way, suppose that we use the criterion [C], jointly with a rule [B] that pays attention to the best elements of the corresponding subsets, as follows: when two subsets A , B⊂U have the same cardinality, pay attention to the best elements max A , max B as regards < . If max A=max B=ui , then pay attention to max(A\ {ui}) and max(B\ {ui}) , and so on. We proceed this way until we declare ABor BA. Starting with the set of labels L={l1 , l2 , . . . , lg} , we extend the usual ordering not to the whole power set of L , but to the set of linguistic expressions L , and accomplishing the criteria [C] and [B], and it is well understood that [C] is accomplished in this context if, for every [lh , lk]∈L such that k<g it holds [lh , lk+1]ext [lh , lk] , and for every [lh , lk]∈L such that h> 1, it holds that [lh−1 , lk]ext [lh , lk] . Calling <cm to the extended linear order on L, for g=5, we get: [l1,l5]cm [l2,l5]cm [l1,l4]cm [l3,l5]cm [l2,l4]cm [l1,l3]cm [l4,l5]cm [l3,l4]cm [l2,l3]cm [l1,l2]cm l5cm l4cm l3cm l2cm l1. The linear order <cm is said to be the cardinality-maximality extension of <. Remark 6. The canonical order <c on L only satisfies the criterion [M], well understood that [M] is accomplished in this context if, for every lh∈L such that h<g , it holds that {lh+1} ext [lh , lh+1]ext {lh} . The lexicographic order <`fails to satisfy both [M] and [C]. The cardinality-maximality extension <cm satisfies [C], but not [M]. Taking into account the last ideas in Example 3, in which an extended order was defined on L but not necessarily on the whole power set of L , we may realize that, at this stage, an interesting question appears. Suppose that we are given a set of criteria for extension of a linear order from a set U to its power set. Assume also that, as it is the case of Kannai–Peleg’s theorem ([ 16 ]), such set of criteria provokes an impossibility result. Even if this happens, when we try to extend the ordering < defined on the set L={l1 , l2 , . . . , lg} of labels, to its corresponding set L of linguistic expressions, it may still happen that an extension that accomplishes the criteria is possible. The reason is that L is much smaller than the whole power set of L . Sometimes, due to this restriction of domain, the extension could still be obtained. Similar questions, namely impossibility theorems in which the intrinsic impossibility actually disappears when a suitable restriction of domain is done, have been analyzed in depth in the contexts of Social Choice (see e.g., Gaertner [22]). Let us explore this fact in more detail. Given a finite set U={u1 , u2 , . . . , up} endowed with a linear order < such that ui<uj⇔i≥j , suppose that we want to define an extension <ext on the power set of U. We say that <ext satisfies: (i) The Gärdenfors principle [G] (see Gärdenfors [ 15 ]) if, for all A⊆U and ui∈U , it holds that uimax(A)⇒A∪ {ui} ext Aand min(A)ui⇒Aext A∪ {ui}. (ii) The weak monotonicity principle [W] if, for all A , B⊆U and ui/∈A∪B , it holds that Aext B⇒ A∪ {ui}<ext B∪ {ui}. A version of the key Kannai–Peleg impossibility theorem (see Kannai and Peleg [ 16 ], Fishburn [ 18 ], Barberà et al. [17] and Bossert [19]) reads as follows. Axioms 2017,6, 19 16 of 17 However, many open problems remain. To provide only a few examples, we could say at this point that: (i) A systematic study of criteria to extend a linear order from a set to its power set, trying to classify the criteria in, so-to-say, contradictory families, so that any two criteria coming from different families give raise to an impossibility result. (ii) The systematic development of a standard procedure to determine which rating, scoring and/or aggregation procedure is the most suitable, maybe taking into account some previous criteria. Needless to say, all of this leaves enough room for further research to be developed in the future. Acknowledgments: This work has been partially supported by the Spanish Ministerio de Economía y Competitividad (projects ECO2015-65031-R, MTM2015-63608-P, ECO2016-77900-P and TIN2016-77356-P), ERDF and the Research Services of the Universidad Pública de Navarra (Spain). Thanks are also given to Remedios Marín and Íñigo Arozarena (Departamento de Tecnología de Alimentos, Universidad Pública de Navarra, Pamplona, Spain) for their valuable suggestions and comments. Author Contributions: The authors contributed equally to this work. Conflicts of Interest: The authors declare no conflict of interest. References 1. García-Lapresta, J.L.; Aldavero, C.; de Castro, S. A linguistic approach to multi-criteria and multi-expert sensory analysis. In Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Montpellier, France, 15–19 July 2014; Volume 443, pp. 586–595. 2. Travé-Massuyès, L.; Dague, P. (Eds.) Modèles et Raisonnements Qualitatifs; Hermes Science: Paris, France, 2003. 3. Travé-Massuyès, L.; Piera, N. The orders of magnitude models as qualitative algebras. In Proceedings of the 11th International Joint Conference on Artificial Intelligence, Detroit, Michigan, 20–25 August 1989; pp. 1261–1266. 4. Roselló, L.; Prats, F.; Agell, N.; Sánchez, M. Measuring consensus in group decisions by means of qualitative reasoning. Int. J. Approx. Reason. 2010,51, 441–452. 5. Roselló, L.; Prats, F.; Agell, N.; Sánchez, M. A qualitative reasoning approach to measure consensus. In Consensual Processes, STUDFUZZ; Herrera-Viedma, E., García-Lapresta, J.L., Kacprzyk, J., Nurmi, H., Fedrizzi, M., Zadr˙ ozny, S., Eds.; Springer: Berlin, Germany, 2001; Volume 267, pp. 235–261. 6. Roselló, L.; Sánchez, M.; Agell, N.; Prats, F.; Mazaira, F.A. Using consensus and distances between generalized multi-attribute linguistic assessments for group decision-making. Inf. Fusion 2014,17, 83–92. 7. Agell, N.; Sánchez, M.; Prats, F.; Roselló, L. Ranking multi-attribute alternatives on the basis of linguistic labels in group decisions. Inf. Sci. 2012,209, 49–60. 8. Falcó, E.; García-Lapresta, J.L.; Roselló, L. Aggregating imprecise linguistic expressions. In Human-Centric Decision-Making Models for Social Sciences; Guo, P., Pedrycz, W., Eds.; Springer: Berlin, Germany, 2014; pp. 97–113. 9. Falcó, E.; García-Lapresta, J.L.; Roselló, L. Allowing agents to be imprecise: A proposal using multiple linguistic terms. Inf. Sci. 2014,258, 249–265. 10. Bridges, D.S.; Mehta, G.B. Representations of Preference Orderings; Springer: Berlin/Heidelberg, Germany, 1995. 11. Szpilrajn, E. Sur l’extension de l’ordre partiel. Fundam. Math. 1930,16, 386–389. 12. Herrera, F.; Herrera-Viedma, E.; Martínez, L. A fuzzy linguistic methodology to deal with unbalanced linguistic term sets. IEEE Trans. Fuzzy Syst. 2008,16, 354–370. 13. García-Lapresta, J.L.; Pérez-Román, D. Ordinal proximity measures in the context of unbalanced qualitative scales and some applications to consensus and clustering. Appl. Soft Comput. 2015,35, 864–872. 14. García-Lapresta, J.L.; Pérez-Román, D. Aggregating opinions in non-uniform ordered qualitative scales. Appl. Soft Comput. 2017, in press, doi:10.1016/j.asoc.2017.05.064. 15. Gärdenfors, P. Manipulation of social choice functions. J. Econ. Theory 1976,13, 227–228. 16. Kannai, Y.; Peleg, B. A note on the extension of an order on a set to the power set. J. Econ. Theory 1984 ,32, 172–175. Axioms 2017,6, 19 17 of 17 17. Barberà, S.; Pattanaik, P.K. Extending an order on a set to the power set: some remarks on Kannai and Peleg’s approach. J. Econ. Theory 1984,32, 185–191. 18. Fishburn, P.C. Comment on the Kannai–Peleg impossibility theorem for extending orders. J. Econ. Theory 1984,32, 176–179. 19. Bossert, W. On the extension of preferences over a set to the power set: An axiomatic characterization of a quasi-ordering. J. Econ. Theory 1989,49, 84–92. 20. Bossert, W.; Pattanaik, P.; Xu, Y. Ranking opportunity sets: an axiomatic approach. J. Econ. Theory 1990 ,63, 326–345. 21. Barberà, S.; Bossert, W.; Pattanaik, P.K. Ranking sets of objects. In Handbook of Utility Theory, Volume II: Extensions; Barberà, S., Hammond, P.J., Seidl, C., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004; pp. 893–977. 22. Gaertner, W. Domain Conditions in Social Choice; Cambridge University Press: Cambridge, UK, 2001. 23. Pattanaik, P.K.; Xu, Y. On ranking opportunity sets in terms of freedom of choice. Rech. Écon. Louvain 1990 , 56, 383–390. 24. Arrow, K.J. Social Choice and Individual Values, 2nd ed.; Wiley: New York, NY, USA, 1963. 25. Kelly, J.S. Social Choice Theory; Springer: New York, NY, USA, 1988. 26. Campbell, D.E.; Kelly, J.S. Impossibility theorems in the Arrovian framework. In Handbook of Social Choice and Welfare; Arrow, K.J., Sen, A.K., Suzumura, K., Eds.; North Holland: Amsterdam, The Netherlands, 2002; Volume 1, pp. 35–44. 27. Campión, M.J.; Candeal, J.C.; Catalán, R.G.; de Miguel, J.R.; Induráin, E.; Molina, J.A. Aggregation of preferences in crisp and fuzzy settings: Functional equations leading to possibility results. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 2011,19, 89–114. 28. García-Lapresta, J.L.; González del Pozo, R. An ordinal multi-criteria decision-making procedure in the context of uniform qualitative scales. In Soft Computing Applications for Group Decision-Making and Consensus Modeling; Collan, M., Kacprzyk, J., Eds.; Studies in Fuzziness and Soft Computing; Springer: Berlin/Heidelberg, Germany, 2017; in press. 29. Agell, N.; Sánchez, G.; Sánchez, M.; Ruiz, F.J. Selecting the best taste: A group decision-making application to chocolates design. In Proceedings of the 2013 IFSA-NAFIPS Joint Congress, Edmonton, AB, Canada, 24–28 June 2013; pp. 939–943. c 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).