Aggregating opinions in non-uniform ordered qualitative scales

Aggregating opinions in non-uniform ordered qualitative scales Jos´e Luis Garc´ıa-Lapresta PRESAD Research Group, BORDA Research Unit, IMUVA, Departamento de Econom´ıa Aplicada, Universidad de Valladolid, Spain David P´erez-Rom´an PRESAD Research Group, BORDA Research Unit, Departamento de Organizaci´on de Empresas y Comercializaci´on e Investigaci´on de Mercados, Universidad de Valladolid, Spain Abstract This paper introduces a new voting system in the setting of ordered qualitative scales. The process is conducted in a purely ordinal way by considering an ordinal proximity measure that assigns an ordinal degree of proximity to each pair of linguistic terms of the qualitative scale. Once the agents assess the alternatives through the qualitative scale, the alternatives are ranked according to the medians of the ordinal degrees of proximity between the obtained individual assessments and the highest linguistic term of the scale. Since some alternatives may share the same median, an appropriate tie-breaking procedure is introduced. Some properties of the proposed voting system have been provided. Keywords: group decision making; qualitative scales; ordinal proximity measures. 1. Introduction Ordered qualitative scales are common in social sciences, engineering, computer sciences and other fields, because they are more appropriate than numerical scales for dealing with the vagueness and imprecision of human beings when evaluating different alternatives. Some ordered qualitative scales are uniform: the psychological proximity between each pair of consecutive terms of the scale Email addresses: [email protected] (Jos´e Luis Garc´ıa-Lapresta), [email protected] (David P´erez-Rom´an) Preprint submitted to Applied Soft Computing May 21, 2017 Applied Soft Computing, 2018, Volume 67, Pages 652-657. https://doi.org/10.1016/j.asoc.2017.05.064 is the same, e.g. the scale {‘very bad’, ‘bad’, ‘regular’, ‘good’, ‘very good’}. Usually, this is the case of Likert-type scales [20]. However, not all ordered qualitative scales are uniform. For instance, the scale {‘reject’, ‘major revision’, ‘minor revision’, ‘accept’}, that some scientific journals use for evaluating papers, may be considered as non-uniform (see Garc´ıa-Lapresta and P´erez-Rom´an [14] for empirical evidence). Although ordered qualitative scales consist of vague linguistic terms, sometimes these terms are represented by exact numerical values. For instance, the International Association of Oenologists considers that each attribute of a wine is evaluated in an ordered qualitative scale of seven linguistic terms: {‘bad’, ‘mediocre’, ‘inadequate’, ‘passable’, ‘good’, ‘very good’, ‘excellent’}and each term is associated with an integer number (see Balinski and Laraki [4]). In spite of the fact of this practice has been widely used in the literature (see, for instance, Franceschini and Romano [10] and Averkin et al. [1]), it is meaningless because different codifications of the same ordered qualitative scale could generate different outcomes when aggregating individual assessments (see Roberts [23] and Franceschini et al. [9], among others). In order to capture the vagueness of ordered qualitative scales, some authors assign other cardinal objects, such as intervals of real numbers or fuzzy numbers, to the linguistic terms of the scale (see, for instance, Zadeh [24], Bass and Kwakernaak [6] and Chen and Hwang [7]). Again, these cardinal representations may be considered as meaningless. Herrera and Mart´ınez [17, 18] introduce the 2-tuple linguistic model for aggregating linguistic information in the setting of uniform ordered qualitative scales. The authors identify each linguistic term of the scale with its position in the scale; after an aggregation process, the outcome is represented by a pair (2-tuple) consisting of a linguistic term and a numerical value that measures the deviation with respect to the linguistic term. Thus, in practice, 2-tuples and real numbers are identical. The procedure is completed with a linear order on the set of 2-tuples that permits rank order the outcomes generated by the aggregation process. Although the procedure manages linguistic information, it is mathematically equivalent to work with numerical values (see Garc´ıa-Lapresta [11]). Herrera et al. [15, 16] extend the 2-tuple linguistic model to the case of unbalanced qualitative scales by considering additional linguistic terms and under a high computational cost (see also Mart´ınez and Herrera [22]). Bartczuk et. al. [5] modify the previous model by introducing numerical correction factors in 2 the extended linguistic terms. This new model is computationally less expensive than the previous one and provides a simpler semantics. In the mentioned approaches, the linguistic information and its aggregation are managed through cardinal objects and techniques. In this paper, we do not represent linguistic terms of ordered qualitative scales by any mathematical object. Instead, we consider psychological proximities among linguistic terms in a purely ordinal way, without using numerical distances, but ordinal degrees. In real life, it is usual to make comparisons between proximities of different pairs of objects in a vague and ordinal fashion. For instance, we say “Rome is closer to Naples than to Milan”, “Budapest is closer to Vienna than Paris is to Athens”, etc. An excellent example of the use of ordinal proximities can be found in the following sentence of the Amos Oz’ novel Suddenly in the Depth of the Forest: “... fairly close to Maya’s back but not as close as she was to the stranger, and slightly closer than she was to the opening of the cave”. These kinds of ordinal comparisons will be taken into account in the setting of ordered qualitative scales through the notion of ordinal proximity measure, introduced by Garc´ıa-Lapresta and P´erez-Rom´an [14] to deal with the psychological proximities among linguistic terms of ordered qualitative scales. In the Majority Judgment (MJ) voting system, introduced by Balinski and Laraki [2, 3]), agents evaluate the alternatives through the linguistic terms of an ordered qualitative scale. In MJ, the alternatives are ranked according to the medians of the obtained assessments. The authors also propose two different tie-breaking processes for obtaining the final ranking. Despite the fact that the qualitative scales considered by the authors are not necessarily uniform, the authors did not take this aspect into account. In this paper, we use the new approach of ordinal proximity measures for designing a voting system that ranks the alternatives evaluated by the agents by means of an ordered qualitative scale. The proposed voting system is related to MJ, but we pay special attention to the ordinal proximities among the terms of the corresponding ordered qualitative scale. Concretely, alternatives are ranked according to the medians of the ordinal proximities between the individual assessments and the highest term of the scale. A tie-breaking procedure that takes into account the ordinal proximities among linguistic terms is also proposed. We also briefly show some properties of the devised voting system. It should be noted that our approach shares with some soft computing methodologies the tolerance for imprecision, uncertainty and subjectivity, under a mathematical foundation (see Zadeh [25], Karray and De Silva [19] and 3 Magdalena [21], among others). The rest of the paper is organized as follows. Section 2 is devoted to ordinal proximity measures. In Section 3 we introduce and analyze the proposed voting system. Section 4 includes an example that illustrates how the voting system works. In Section 5 we extend the voting system to the case of multiple criteria. Finally, Section 6 concludes the paper with some remarks. 2. Ordinal proximity measures We consider that each individual of a group of agents assigns a linguistic term to every feasible alternative. These linguistic terms belong to an ordered qualitative scale L={l1, . . . , lg}, arranged from worst to best, l1<· · · < lg, where the granularity of Lis at least 3, i.e., g≥3. We now recall the notion of ordinal proximity measure, introduced by Garc´ıaLapresta and P´erez-Rom´an [14]. It is a mapping that assigns an ordinal degree of proximity to each pair of linguistic terms of an ordered qualitative scale L. These ordinal degrees of proximity belong to a linear order ∆ = {δ1, . . . , δh}, with δ1 · · ·  δh, being δ1and δhthe maximum and minimum degrees of proximity, respectively. It is important noticing that the elements of ∆ are not numbers. In fact, they are only abstract objects, without meaning, representing different degrees of proximity. As usual in the setting of linear orders, δrδsmeans δrδsor δr=δs; and δr≺δsmeans δsδr. Definition 1. ([14]) An ordinal proximity measure on Lwith values in ∆ is a mapping π:L2−→ ∆, where π(lr, ls) = πrs means the degree of proximity between lrand ls, satisfying the following conditions: 1. Exhaustiveness: For every δ∈∆, there exist lr, ls∈ L such that δ=πrs. 2. Symmetry:πsr =πrs, for all r, s ∈ {1, . . . , g}. 3. Maximum proximity:πrs =δ1⇔r=s, for all r, s ∈ {1, . . . , g}. 4. Monotonicity:πrs πrt and πst πrt, for all r, s, t ∈ {1, . . . , g}such that r < s < t. We note that the previous conditions are independent (see Garc´ıa-Lapresta and P´erez-Rom´an [14, Prop. 1]). We say that an ordinal proximity measure π:L2−→ ∆ is uniform if πr(r+1) =πs(s+1) for all r, s ∈ {1, . . . , g −1}, and totally uniform if πr(r+t)= πs(s+t)for all r, s, t ∈ {1, . . . , g −1}such that r+t, s +t≤g. 4 Each ordinal proximity measure π:L2−→ ∆ will be represented by a g×gsymmetric matrix with coefficients in ∆, being the elements in the main diagonal πrr =δ1,r= 1, . . . , g:         π11 · · · π1s· · · π1g · · · · · · · · · · · · · · · πr1· · · πrs · · · πrg · · · · · · · · · · · · · · · πg1· · · πgs · · · πgg         . This matrix will be called proximity matrix associated with π. If we consider the conditions appearing in Definition 1, we would only need to show the upper half proximity matrix           δ1π12 π13 · · · π1(g−1) π1g δ1π23 · · · π2(g−1) π2g · · · · · · · · · δ1π(g−1)g δ1           . It is important noticing that the minimum proximity between linguistic terms is only reached when comparing the extreme linguistic terms: πrs = δh⇔(r, s)∈ {(1, g),(g, 1)}(see Garc´ıa-Lapresta and P´erez-Rom´an [14, Prop. 2]). In the following example we illustrate an ordered qualitative scale of four linguistic terms with two extreme ordinal proximity measures. Example 1. Consider g= 4, where five ordinal degrees, not necessarily different, have to be assigned (see Fig. 1) and h∈ {4,5,6,7}. l1· · · π12 l2· · · π23 l3 π13 · · · π34 l4 π24 Figure 1: Ordinal degrees for g= 4. 5 It is worth mentioning that for g= 4 there are 51 different ordinal proximity measures (Garc´ıa-Lapresta et al. [13]). 1. The simplest case corresponds to ∆ = {δ1, δ2, δ3, δ4}, with πrr =δ1, π12 =π23 =π34 =δ2,π13 =π24 =δ3and π14 =δ4, i.e., the totally uniform ordinal proximity measure, with associated matrix1 A222 =     δ1δ2δ3δ4 δ1δ2δ3 δ1δ2 δ1     that can be visualized in Fig. 2. l1l2l3l4 Figure 2: Ordinal proximity measure with associated matrix A222. 2. We now consider ∆ = {δ1, δ2, δ3, δ4, δ5, δ6, δ7}, with πrr =δ1,π34 =δ2, π23 =δ3,π12 =δ4,π24 =δ5,π13 =δ6and π14 =δ7. In this case, the ordinal proximity measure has the following associated matrix A432 =     δ1δ4δ6δ7 δ1δ3δ5 δ1δ2 δ1     that can be visualized in Fig. 3. l1l2l3l4 Figure 3: Ordinal proximity measure with associated matrix A432. 3. The voting system Consider a set of agents A={1, . . . , m}, with m≥2, that have to evaluate a set of alternatives X={x1, . . . , xn}, with n≥2, through an ordered qualitative 1The subindices 222 of the matrix A222 correspond to the subindices of the δ’s appearing in the coefficients just over the main diagonal. We follow the same pattern in subsequent matrices. 6 scale L={l1, . . . , lg},l1<· · · < lg, with g≥3, and an ordinal proximity measure π:L2−→ ∆. The assessments provided by the agents to the alternatives are collected in aprofile, that is a matrix V=         v1 1· · · v1 i· · · v1 n · · · · · · · · · · · · · · · va 1· · · va i· · · va n · · · · · · · · · · · · · · · vm 1· · · vm i· · · vm n         that consists of mrows and ncolumns of linguistic terms, where the element va i∈ L is the linguistic assessment given by the agent a∈Ato the alternative xi∈X. 3.1. Ranking the alternatives To rank the alternatives, the procedure is divided in the following steps. 1. For each alternative xi∈X, consider the assessments obtained by xifor all the agents: v1 i, . . . , vm i∈ L (column iof V). 2. For each alternative xi∈X, calculate the ordinal proximities between the assessments obtained by xiand the highest linguistic term lg: πv1 i, lg, . . . , π (vm i, lg)∈∆. In a different setting, Falc´o et al. [8] rank order linguistic assessments taking into account their distances to the highest linguistic term of the ordered qualitative scale (the less, the better). However, in the present approach, when considering ordinal proximities between linguistic assessments and the highest linguistic term of the ordered qualitative scale, the pattern is just the opposite (the more, the better), because the notions of distance and proximity are antonyms. 3. For each alternative xi∈X, arrange the previous ordinal degrees in a decreasing fashion and select the median(s)2,Mi: 2When the number of elements is odd, the median is unique. However, if that number 7 (a) If the number of assessments is odd, then we duplicate the median. Thus, Mi= (δr, δr) for some r∈ {1, . . . , h}. (b) If the number of assessments is even, then we take into account the two medians. Thus, Mi= (δr, δs) for some r, s ∈ {1, . . . , h}such that r≤s. Consequently, Mi∈∆2, where ∆2is the set of feasible medians: ∆2={(δr, δs)∈∆2|r≤s}. In the MJ voting system, Balinski and Laraki [2, 3] consider the lower median of the linguistic individual assessments as collective grade of each alternative when the number of assessments is odd (in MJ the individual assessments are arranged in an increasing manner). Choosing the lower median is not problematic when the number of agents is high, as happens in political elections. However, it can be considered as arbitrary when that number is low, as happens in small size committees. In order to avoid loss of information, it is convenient to take into account the two medians. This requires to rank order feasible medians in a suitable way. For example, in our setting, (δ2, δ3) is clearly better than (δ3, δ3), and (δ3, δ3) can be considered better than (δ2, δ4) because 3+3 = 2+4, but the dispersion (measured through the range of the subindices) is smaller in the first case than in the second one (3 −3 = 0 <2=4−2). In the next step we propose an appropriate linear order on the set of feasible medians. 4. To order the medians of ordinal proximities obtained by different alternatives in the previous step, consider the linear order on ∆2defined as (δr, δs)(δt, δu)⇔          r+s < t +u or r+s=t+uand s−r≤u−t, (1) for all (δr, δs),(δt, δu)∈∆2. is even, then there exist two medians. When the elements of a list are real numbers, the median of that list is usually defined as the arithmetic mean of the two medians. That it is impossible when the elements of the list are abstract objects, as happens when considering ordinal proximities. 8 It is easy to see that if r+s=t+u, then s−r≤u−t⇔r≥t⇔s≤u. Notice that (δr, δr)(δt, δt)⇔r≤t. 5. Finally, the alternatives are ranked according to the weak order <on X defined as xi<xj⇔MiMj. 3.2. Breaking ties Since some alternatives can share the same median(s), it is necessary to devise a tie-breaking process for ordering the alternatives. We propose to use a sequential procedure based on Balinski and Laraki [2] (see Balinski and Laraki [4] for practical examples). It consists of dropping the median(s) of the respective alternatives that are in a tie, and then select the new median(s) of the remaining ordinal degrees for the corresponding alternatives and applying the procedure given in (1). Formally, when Mi=Mj: •If mis odd, let M(1) i, M(1) j∈∆2be the medians obtained after dropping in πv1 i, lg, . . . , π (vm i, lg) and πv1 j, lg, . . . , π vm j, lgthe ordinal degree appearing in Mi=Mj, respectively. •If meven, let M(1) i, M(1) j∈∆2be the medians obtained after dropping in πv1 i, lg, . . . , π (vm i, lg) and πv1 j, lg, . . . , π vm j, lgthe pair of ordinal degrees appearing in Mi=Mj, respectively. Then, the procedure given in (1) is applied again. If M(1) i=M(1) j, then the process continues with the remaining ordinal degrees for the corresponding alternatives until the ties are broken3. It is important noticing that alternatives with different assessments never become in a final tie. 3.3. Properties We now enumerate some properties that the proposed voting system satisfies. 1. Anonymity: All individuals are treated in the same way. 2. Neutrality: All alternatives are treated in the same way. 3. Independence of irrelevant alternatives: The ranking between two alternatives only depends on the individual assessments obtained by these alternatives, being irrelevant the assessments obtained by other alternatives. 3Notice that in the following steps, the number of ordinal degrees is always even. 9 medians. This avoids loss of information and it is also a novelty with respect to other ordinal approaches ([2, 3]). As shown in Section 4, given an ordered qualitative scale, the outcome of the voting system could depend on the ordinal proximity measure fixed for describing the proximities among the linguistic terms of the scale. Then, a relevant problem is how to determine what is the most appropriate ordinal proximity measure in that scale. It is not a trivial problem and the solution may depend on the society where the voting system is applied. If several experts provide their opinions about the mentioned proximities, then an aggregation procedure is needed. 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