PrefMap: Visualization of fuzzy pairwise preference structures

Abstract

Fuzzy pairwise preferences are an important model to specify and process expert opinions. A fuzzy pairwise preference matrix contains degrees of preference of each option over each other option. Such degrees of preference are often numerically specified by domain experts. In decision processes it is highly desirable to be able to analyze such preference structures, in order to answer questions like: Which objects are most or least preferred? Are there clusters of options with similar preference? Are the preferences consistent or partially contradictory? An important approach for such analysis is visualization. The goal is to produce good visualizations of preference matrices in order to better understand the expert opinions, to easily identify favorite or less favorite options, to discuss and address inconsistencies, or to reach consensus in group decision processes. Standard methods for visualization of preferences are matrix visualization and chord diagrams, which are not suitable for larger data sets, and which are not able to visualize clusters or inconsistencies. To overcome this drawback we propose PrefMap, a new method for visualizing preference matrices. Experiments with nine artificial and real–world preference data sets indicate that PrefMap yields good visualizations that allow to easily identify favorite and less favorite options, clusters, and inconsistencies, even for large data sets.

Keywords

fuzzy preference relations visualization multidimensional scaling

1 Introduction

Visual analytics [17] plays an important role in human (individual and group) decision making processes. Decision making processes are often based on individual ratings of options and/or on quantitative relations between pairs of options. Such pairwise relations include similarity, dissimilarity, and preference. Preferences can be specified by rank order, utility, additive or multiplicative pairwise preference. For the analysis of preference structures and for reaching consensus in group decision processes it is highly beneficial to be able to visualize preference structures. The goal of such visualization is to be able to answer questions like: Which objects are most or least preferred? Are there clusters of options with similar preference? Are the preferences consistent or partially contradictory?

Preference modeling and its application in decision making processes has been extensively studied in [10, 29]. An important application field of preference models is in group decision processes [3 , 16] with numerous practical applications including robotics [35], product recommendations [12], or human resources [36]. Important mathematical properties of pairwise preference relations, in particular consistency, have been discussed in [13 , 33]. Multidimensional scaling [7 , 41] is one of the most popular approaches to visualize pairwise dissimilarity and similarity relations with many practical applications such as visualization of text streams [31], genomics [11], music archives [24], or game play [37]. Chord diagrams [19] have been originally developed for visualizing genome similarities, but have also been applied in a variety of other domains, for example city traffic analysis [42] or network anomaly detection [15, 22], and they can also be applied to preference data. For a more general overview of fuzzy decision making we refer to [2].

In this paper we propose a novel preference visualization method called PrefMap (inspired by multidimensional scaling) that produces geometrical mappings of the considered objects taking into account the given preference structure. All three considered methods (matrix visualization, chord diagrams, and the new PrefMap method) are experimentally evaluated using nine preference data sets: three artificial data sets, four example data sets from group decision making, and two real world data sets (La Liga and YouTube Comedy Slam).

This paper is structured as follows: Section 2 introduces the mathematical concepts for the different types of preference structures and mappings between these structures. Section 3 presents the nine preference data sets used in our experimental evaluations. Section 4 discusses the application of matrix visualization and chord diagrams to preference matrices. Section 5 introduces PrefMap, a new method for the visualization of pairwise preference matrices. Finally, Section 6 summarizes our conclusions.

2 Preference models and transformations

A preference model is a way to quantitatively specify preferences over a set of objects O = {o₁, …, o_n}, for example books, movies, soccer teams, suppliers, equity funds, and so on, as a basis for (individual or group) decision making. Preferences may be specified manually (which is only feasible for small numbers of objects) or may be constructed from data (which also allows to deals with large numbers of objects) [9, 14]. We distinguish rank orders, utilities, and pairwise preferences.

A rank order assigns each object i = 1, …, n a rank r_i ∈ {1, …, n}, where an object with rank 1 has the highest preference, rank 2 the second highest, and so on. A rank order is called unique if and only if all r_i are different, i.e. r_i ≠ r_j for all i, j = 1, …, n, i ≠ j.

A utility vector u = (u₁, …, u_n) contains the degrees of utility u_i of each object o_i. Often utilities are normalized to u_i ∈ [0, 1]. A utility vector is called unique if and only if all u_i are different, u_i ≠ u_j for all i, j = 1, …, n, i ≠ j. Each utility vector u induces a rank order r, which is obtained by sorting the elements in u in decreasing order, and then setting each r_i to the index of u_i in the sorted list u^*, so r_i = j if and only if $u_{j}^{*} = u_{i}$ . We call this mapping the U2R transformation. Each unique utility vector induces a unique rank order. We define a canonical utility vector u induced by a unique rank order r by u_i = (n - r_i)/(n - 1). We call this mapping the R2U transformation. Different (non–canonical) utility vectors may yield the same rank order, so a utility vector contains more information than a rank order.

A pairwise preference matrix P contains the elements p_ij which quantify the degree of preference of object o_i over o_j. P is called a normalized additive preference matrix [10] if and only if p_ij ∈ [0, 1] and p_ii = 0.5. P is called a normalized multiplicative preference matrix [29] if and only if p_ij > 0 and p_ii = 1. Normalized additive preferences p_a ∈ [0, 1] may be mapped to normalized multiplicative preferences $p_{m} \in [1 / \hat{p}, \hat{p}]$ by $p_{m} = {\hat{p}}^{2 p_{a} - 1}$ (1) with a suitable parameter $\hat{p} > 0$ , and normalized multiplicative preferences $p_{m} \in [1 / \hat{p}, \hat{p}]$ may be mapped to additive preferences p_a ∈ [0, 1] by $p_{a} = \frac{1}{2} + \frac{log p_{m}}{2 log \hat{p}}$ (2) We call these mappings the PA2PM and PM2PA transformations, respectively. An additive preference matrix P is called reciprocal if and only if p_ij + p_ji = 1, and a multiplicative preference matrix P is called reciprocal if and only if p_ij · p_ji = 1. An additive preference matrix P is called consistent [32, 33] if and only if $(p_{ij} - 0.5) + (p_{jk} - 0.5) = (p_{ik} - 0.5)$ (3) for all i, j, k = 1, …, n. A multiplicative preference matrix P is called consistent [32, 33] if and only if $\frac{p_{ji}}{p_{ij}} \cdot \frac{p_{kj}}{p_{jk}} = \frac{p_{ki}}{p_{ik}}$ (4) PA2PM transformation (1) of any consistent additive preference matrix will yield a consistent multiplicative preference matrix, and PM2PA transformation (2) of any consistent multiplicative preference matrix will yield a consistent additive preference matrix.

Also utility vectors may be transformed to preference matrices [25]. Any utility vector u may be transformed to a consistent reciprocal additive preference matrix P using the U2PA transformation [26] $p_{ij} = \frac{1}{2} (u_{i} - u_{j} + 1)$ (5) Any consistent reciprocal additive preference matrix P may be transformed to a utility vector u by choosing the value of one element of u and then computing all other elements using the P2UA transformation [26] $u_{i} = 2 p_{ij} + u_{j} - 1$ (6) For normalized utility vectors U2PA will yield normalized preferences, and for normalized preferences P2UA will yield normalized utility vectors.

Any utility vector u may be transformed to a consistent reciprocal multiplicative preference matrix P using the U2PM transformation [26] $p_{ij} = \frac{u_{i} (1 - u_{j})}{u_{i} (1 - u_{j}) + u_{j} (1 - u_{i})}$ (7) Any consistent reciprocal multiplicative preference matrix P may be transformed to a utility vector u by choosing the value of one element of u and then computing all other elements using the P2UM transformation [26] $u_{i} = \frac{u_{j} p_{ij}}{1 - u_{j} - p_{ij} + 2 u_{j} p_{ij}}$ (8)

Any rank order r may be transformed to a consistent reciprocal additive preference matrix P by first applying R2U and then U2PA transformation, which yields a so–called canonical reciprocal additive preference matrix [13, 27] whose elements can be computed as $p_{ij} = \frac{1}{2} + \frac{r_{j} - r_{i}}{2 (n - 1)}$ (9) Any reciprocal additive preference matrix P may be transformed to a rank order r by first applying P2UA and then U2R transformation. Any rank order r may be transformed to a consistent reciprocal multiplicative preference matrix P by first applying R2U and then U2PM transformation, which yields a so–called canonical reciprocal multiplicative preference matrix [13, 27] with $p_{ij} = \frac{(n - r_{i}) (r_{j} - 1)}{(n - r_{i}) (r_{j} - 1) + (n - r_{j}) (r_{i} - 1)}$ (10) Any reciprocal multiplicative preference matrix P may be transformed to a rank order r by first applying P2UM and then U2R transformation.

Fig. 1 illustrates the transformations between the different preference models discussed in this section.

Fig. 1

Preference models and transformations between these models.

The visualization techniques discussed in this paper will focus on additive preferences P_a. To visualize other types of preferences, for example multiplicative preferences P_m, we first transform these to additive preferences P_a and then apply any of the presented visualization techniques.

3 Preference data sets

This section presents nine (artificial and real world) normalized additive preference data sets with different properties that will be used to illustrate and evaluate the visualization techniques discussed in this paper.

3.1 Artificial data sets

To construct a first simple normalized artificial additive preference data set we require reciprocity and consistency. We use the U2PA transformation (5) to construct a reciprocal consistent preference data set. Consistency (3) requires to consider at least n = 3 objects. We want a symmetric case and we want to maximize utility spread, so we choose the utility vector u = (1, 0.5, 0). For this utility vector, U2PA (5) yields the symmetric reciprocal consistent additive preference matrix $P_{1} = (\begin{matrix} 0.5 & 0.75 & 1 \\ 0.25 & 0.5 & 0.75 \\ 0 & 0.25 & 0.5 \end{matrix})$ (11)

Next we keep the same structure but abandon consistency and choose p₁₂ = p₂₃ = 0.6, and for reciprocity p₂₁ = p₃₂ = 0.4, which yields $P_{2} = (\begin{matrix} 0.5 & 0.6 & 1 \\ 0.4 & 0.5 & 0.6 \\ 0 & 0.4 & 0.5 \end{matrix})$ (12) The reader may easily verify that this violates (3), so P₂ is symmetric and reciprocal, but not consistent.

Third we consider the case of cyclic preferences: n = 5 objects which are considered pairwise equivalent, except that object k is preferred over object k - 1, k = 1, …, n, and object 1 is preferred over object n, with preferences equal to one, which yields $P_{3} = (\begin{matrix} 0.5 & 1 & 0.5 & 0.5 & 0 \\ 0 & 0.5 & 1 & 0.5 & 0.5 \\ 0.5 & 0 & 0.5 & 1 & 0.5 \\ 0.5 & 0.5 & 0 & 0.5 & 1 \\ 1 & 0.5 & 0.5 & 0 & 0.5 \end{matrix})$ (13) Obviously, P₃ is symmetric and reciprocal, but not consistent.

3.2 Group decision making

An important application of preference models is group decision making [10, 16]. For example, Chiclana et al. [5] discussed an experiment where three experts the provide the following preference matrices P₄, P₅, P₆ for three objects: $P_{4} = (\begin{matrix} 0.5 & 0.75 & 0.87 \\ 0.25 & 0.5 & 0.66 \\ 0.13 & 0.34 & 0.5 \end{matrix})$ (14) $P_{5} = (\begin{matrix} 0.5 & 0.66 & 0.94 \\ 0.34 & 0.5 & 0.87 \\ 0.06 & 0.13 & 0.5 \end{matrix})$ (15) $P_{6} = (\begin{matrix} 0.5 & 0.66 & 0.75 \\ 0.34 & 0.5 & 0.66 \\ 0.25 & 0.34 & 0.5 \end{matrix})$ (16) For these expert preferences a collective preference relation P₇ [4] is constructed using ordered weighted averaging aggregation operators [40], which aims to represent a compromise between the three individual preferences. $P_{7} = (\begin{matrix} 0.5 & 0.71 & 0.89 \\ 0.32 & 0.5 & 0.78 \\ 0.19 & 0.3 & 0.5 \end{matrix})$ (17) The expert preference matrices P₄, P₅, P₆ are reciprocal, but the collective preference matrix P₇ is not reciprocal. None of the preference matrices P₄, …, P₇ is consistent.

3.3 Real world data sets

Besides the seven artificial data data sets presented in the previous sections we also consider two real world data sets: pairwise preferences extracted from the Spanish La Liga data set [23] and the YouTube Comedy Slam preference data set [30].

The Spanish La Liga data set [23] available at http://datahub.io contains match results for several seasons of the Spanish soccer league La Liga. This data set has been used for testing statistical diagnosis [8, 21] and prediction [6] methods. For our experiments we use the data from the second half of season 2017/18, which contain the results for the matches of the 20 La Liga 2017/18 teams against each other team, so the number of matches is 20 × 19/2 =190. For each of these matches we consider the full time result which is either a home win (H), a draw (D), or an away win (A). For each match between a home team with index i and an away team with index j we set p_ij = 1 for H, p_ij = 0.5 for D, and p_ij = 0 for A, and set p_ji = 1 - p_ij for reciprocal preferences. We denote the resulting 20 × 20 preference matrix as P₈, which is reciprocal but not consistent.

The YouTube Comedy Slam preference data set [1, 30] available at http://archive.ics.uci.edu contains about 1.7 million votes from a video rating experiment for about 20.000 videos, where random pairs of these videos were shown to the users and the users were asked to vote for the video that they found funnier. The original data set is split into training and test data; here we use the merged full data set. Each video has an ID like for example -0_iw2b9fXY and the corresponding video can be retrieved using the URL http://www.youtube.com/watch?v= -0_iw2b9fXY. For each individual vote the data set contains the IDs of the two considered videos, the vote result (left or right is funnier), but no other information about the vote like the ID of the user. For each pair of video IDs, say i and j, we count how often i is preferred over j (c_ij) and how often j is preferred over i (c_ji). Despite the large number of votes many pairs of video IDs do not occur in the data set, c_ij = c_ji = 0. To construct a complete preference data set we try to find the largest index set I ⊂ {1, …, n} so that c_ij > 0 and c_ji > 0 for all i, j ∈ I, which is an NP complete problem. Random search finds an index set with 41 elements. For these we set p_ij = c_ij/c_ji, p_ji = c_ji/c_ij, which yields a 41 × 41 reciprocal multiplicative preference matrix. We apply the PM2PA transformation (2) and finally obtain a 41 × 41 reciprocal additive preference matrix which we denote as P₉, and which is reciprocal but not consistent.

4 Standard visualization methods

In this section we discuss two popular standard methods for visualizing pairwise relations: matrix visualization and chord diagrams.

4.1 Matrix visualization

A straightforward way of visualizing preference matrices is to apply standard matrix visualization techniques [38]. An n × n matrix is visualized by an n × n grid of squares whose grey values correspond to the values of the corresponding matrix elements. For preference matrices we can use a linear grey scale from black corresponding to p = 0 to white corresponding to p = 1. Diagonals with neutral preference p = 0.5 will appear in grey. If the number of elements n is low, then we may additionally display the numerical value of each matrix element in the corresponding square, as white text for p < 0.5 and black text for p ≥ 0.5.

Fig. 2 shows the matrix visualizations of the artificial data sets P₁, P₂, P₃.

Fig. 2

Matrix visualizations of the artificial data sets.

For P₁ (left) and P₂ (middle) the preference p₁₃ = 1 is highlighted (white box), and symmetry along the main axis is clearly visible (p₁₂ = p₂₃, p₂₁ = p₃₂, but it cannot be recognized whether the preferences are consistent (left: yes; middle: no). For P₃ (right) the preferences above the main diagonal p₁₂ = p₂₃ = p₃₄ = p₄₅ = 1 and in the bottom left corner p₅₁ = 1 are highlighted (white box), but the circular preference structure is not clearly apparent.

Fig. 3 shows the matrix visualizations of the group decision making data sets: the individual preference matrices P₄, P₅, P₆ left of the vertical line, and the collective preference matrix P₇ right of the vertical line.

Fig. 3

Matrix visualizations of the group decision making data sets.

All four matrices show a similar preference structure with similar grey values with a clear maximum preference at p₁₃ (top right), and variations of the grey values indicate small differences between the preferences. The grey values of the collective preference matrix (right) are more or less averages of the grey values of the individual preference matrices (left), but this is not easy to see.

Fig. 4 shows the matrix visualization of the La Liga data set P₈.

Fig. 4

Matrix visualization of the La Liga data set P₈.

Each box off the main diagonal represents one match. The boxes in each row indicate how the corresponding team has played against each other team (black 0.0: lost, grey 0.5: tie, white 1.0: won). Teams with many losses (dark rows) such as Malaga, and teams with many wins (light rows) such as Real Madrid can be identified, but not easily.

Fig. 5 shows the matrix visualization of the YouTube Comedy Slam preference data set P₉.

Fig. 5

Matrix visualization of the YouTube Comedy Slam data set P₉.

Here the number of objects (41) does not allow to display the numerical values of the preferences. Some high and low preferences (light and dark boxes) are easily recognized. The row -3P57xi7jMA is relatively dark, and the column -3P57xi7jMA is relatively light, respectively. More information about the preference structure can not be easily recognized from this diagram.

4.2 Chord diagrams

A chord diagram [19] is a circular diagram visualizing the similarity structure between pairs of objects. Chord diagrams have been primarily developed for genome visualization, but have also been applied in a variety of other domains, for example city traffic analysis [42] or network anomaly detection [15, 22]. Here, we use chord diagrams to visualize pairwise preferences instead of similarities.

In a chord diagram for visualizing an n × n similarity matrix S each of the n objects is represented by an arc segment of a full circle whose length Δφ_i is proportional to the the sum of the similarities to the other objects. $Δ φ_{i} \sim \frac{\sum_{k = 1}^{n} s_{ik}}{\sum_{j = 1}^{n} \sum_{k = 1}^{n} s_{jk}}$ (18) Each pair of circle segments is connected by a curved chord whose width is proportional to the similarity s_ij between the corresponding pair. To visualize an n × n preference matrix P we may choose the arc segment lengths proportional to the sum of outgoing and incoming preference values. $Δ φ_{i} \sim \frac{\sum_{k = 1}^{n} (p_{ik} + p_{ki})}{\sum_{j = 1}^{n} \sum_{k = 1}^{n} (p_{jk} + p_{kj})}$ (19) The chords are connected to the circle segments at the higher preference ends. At the lower preference ends we keep a small gap between the chords and the circle segments. So if p_ij > p_ji, then the corresponding chord will be connected to segment i but not to segment j (where for ties p_ij = p_ji the direction is arbitrary). The width of each chord is proportional to p_ij at segment i and 1 - p_ji at segment j, and is interpolated along the chord. If P is reciprocal, then p_ij = 1 - p_ji, so each chord has the same width at each end. We have used the Circos software to generate the chord diagrams presented in this paper.

Fig. 6 shows the chord diagrams of the artificial data sets P₁, P₂, P₃. The objects are arranged along the circle counter–clockwise from the top. For P₁ (Fig. 6 left) object 1 has higher preference than objects 2 and 3, and object 2 has higher preference than object 3, so segment 1 does dot have a gap, segment 2 has a gap at the chord to 1, and both chords at segment 3 have gaps. The preferences of 1 over 2 and 2 over 3 are the same, so both chords at the bottom have the same width. The preference of 1 over 3 is twice as large, so the width of the top chord is twice the width of the bottom chords. For P₂ (Fig. 6 middle) the pattern is similar but the top chord is wider and the bottom chord is narrower. From the chord diagrams of P₁ and P₂ it is not clear which of both is consistent (P₁). For P₃ (Fig. 6 right) the chord diagram clearly exhibits the circular preference structure from 1 to 2, 2 to 3, and so on. However, such circular structures or chains can only be detected if the objects are arranged in a suitable order. If the order of the objects is permuted, then such structures may not be recognizable any more.

Fig. 6

Chord diagrams of the artificial data sets.

Fig. 7 shows the chord diagrams of the group decision making data sets: the individual preferences P₄, P₅, P₆ and the collective preferences P₇.

Fig. 7

Chord diagrams of the group decision making data sets.

The chord diagrams of P₄, …, P₇ look similar to those of P₁ and P₂ (Fig. 6). The different preference structures of the individual preferences P₄, P₅, P₆ can be clearly distinguished, and the chord widths for the group decision P₇ visually appear as approximate averages of the individual preferences P₄, P₅, P₆.

Fig. 8 shows the chord diagram of the La Liga data set P₈.

Fig. 8

Chord diagram of the La Liga data set P₈.

Objects with higher preferences such as “Barcelona” can be easily identified by their shorter gap, and objects with lower preferences such as “Malaga” can be easily identified by their longer gap. Due to the large number of chords, however, the actual pairwise preference structure does not become clear in the chord diagram.

Fig. 9 shows the chord diagram of the YouTube Comedy Slam preference data set P₉.

Fig. 9

Chord diagram of the YouTube Comedy Slam data set P₉.

Here, the number of objects is too large to include the object labels in readable fonts. Again, objects with higher and lower preferences can be identified by their gap lengths. Some wider chords (higher preferences) can be identified, but still the overall preference structure is quite unclear.

5 Preference mapping

In this section we introduce a new method for visual analytics of pairwise preference relations called preference mapping or short PrefMap that is inspired by multidimensional scaling [7 , 41]. We denote preference over ambiguity as $x_{ij} = p_{ij} - 0.5$ (20) and define the positive definite matrix $X \cdot X^{T} = (P - 0.5 \cdot 1^{T} \cdot 1) \cdot (P - 0.5 \cdot 1^{T} \cdot 1)^{T}$ (21) where 1 denotes a row vector of n ones, so 1^T1 is an n × n matrix of ones. In a similar way as in multidimensional scaling we apply spectral decomposition $X \cdot X^{T} = V \cdot Λ \cdot V^{'}$ (22) where Λ is the diagonal matrix of the eigenvalues of X · X^T and V is the matrix of the corresponding eigenvectors. This yields the mapping $X_{q} = V_{q} \cdot \sqrt{Λ_{q}}$ (23) for q ∈ {1, …, n}, where $\sqrt{Λ_{q}}$ is the diagonal matrix with the square roots of the q largest eigenvalues of X and V_q is the matrix of the corresponding eigenvectors. In the following we will focus on two–dimensional diagrams and therefore set q = 2.

PrefMap has the interesting and useful property to be able to easily identify if a preference matrix is consistent or not. To see this, imagine that P is a consistent additive preference matrix. Then with the U2PA transformation (5) we can write (20) as $x_{ij} = \frac{1}{2} (u_{i} - u_{j})$ (24) and further $X = \frac{1}{2} (u^{T} \cdot 1 - 1^{T} \cdot u)$ (25) where $X^{T} = \frac{1}{2} (1^{T} \cdot u - u^{T} \cdot 1) = - X$ (26) so $X \cdot X^{T} = - X^{2}$ (27) Hence, the eigenvalues Λ of X · X^T can be computed from the eigenvalues Λ_X of X as $Λ = - Λ_{X}^{2}$ (28) We define $Y = \frac{1}{2} \cdot u^{T} \cdot 1$ (29) and can then write $X = Y - Y^{T}$ (30) Due to linear dependency the matrix Y (and Y^T) has only one non–zero eigenvalue $λ_{Y} = \frac{1}{2} \sum_{k = 1}^{n} u_{k}$ (31) Further, because of (30), X is a skew–symmetric matrix and has two conjugated imaginary eigenvalues, and with (28), X · X^T has two equal nonzero eigenvalues. This leads to a very interesting property of preference mapping: For any consistent additive preference matrix, P preference mapping will yield n collinear points. If the points are not collinear, then P contains inconsistencies, where points off the main axis correspond to the objects causing inconsistency.

For q = 2 dimensional visualizations we rotate X₂ so that objects with high preferences are located on the top, and objects with low preferences are located on the bottom. To do so, we compute the main direction of preferences as the sum of all difference vectors weighted by preference minus 0.5, $y = \sum_{i = 1}^{n} \sum_{j = 1}^{n} (p_{ij} - 0.5) \cdot (x_{i} - x_{j})$ (32) $= 1 \cdot (X^{T} - X) \cdot X_{2}$ (33) and rotate X₂, so that the direction of y is turned to the vertical unit vector (0, 1). $X_{2}^{'} = X_{2} \cdot (\begin{matrix} y_{2} & y_{1} \\ - y_{1} & y_{2} \end{matrix})$ (34) Notice that we ignore scaling here. We further flip the horizontal axis, if the horizontal component of the main direction of preferences $y^{'} = 1 \cdot (X^{T} - X) \cdot X_{2}^{'} \cdot (\begin{matrix} 1 \\ 0 \end{matrix})$ (35) is negative, so $X_{2}^{″} = {\begin{matrix} X_{2}^{'} & if y^{'} \geq 0 \\ X_{2}^{'} \cdot (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}) & if y^{'} < 0 \end{matrix}$ (36)

Fig. 10 shows a comparison between the two–dimensional multidimensional scaling and preference mapping algorithms.

Fig. 10

Comparison between the two–dimensional multidimensional scaling and preference mapping (PrefMap) algorithms.

Besides visualization, eigenvector methods have also been considered in improving consistency for pairwise preference relations [39].

In a PrefMap diagram the n objects are represented by nodes (grey boxes) at the locations $X_{2}^{″}$ , and each pair of nodes is connected by an edge. The grey value of each box corresponds to the average preference of the corresponding object, light grey for average preference 1 and black for average preference 0. The index or name of each object is displayed in each box, as white text for average preference <0.5 and black text otherwise. Each edge between node i and j, i, j = 1, …, n is displayed in grey if p_ij = p_ji = 0.5. Otherwise, it is light grey at the node with higher preference, say i, and black at the node with lower preference, say j. The relative length of the light grey segment (excluding the edge tips) corresponds to p_ij/(p_ij + p_ji), and the relative length of the black segment corresponds to p_ji/(p_ij + p_ji).

To illustrate the properties of PrefMap we first look at two important aspects: How does PrefMap handle most or least preferred objects, and what happens if the preference matrix is inconsistent? Fig. 11 shows the PrefMaps of the artificial data sets P₁, P₂, P₃.

Fig. 11

PrefMaps of the artificial data sets P₁, P₂, P₃.

For P₁ we have $p_{1 k} \geq p_{ik}$ (37) for all i, k = 1, …, 3, so the first object is most preferred, and we have $p_{3 k} \leq p_{ik}$ (38) for all i, k = 1, …, 3, so the third object is least preferred, and, as pointed out above, the preference structure is consistent. Accordingly, the PrefMap for P₁ (Fig. 11 left) shows the most preferred object 1 on top and the least preferred object 3 on the bottom, and due to consistency all three points are collinear. Moreover, the distance between objects 1 and 2 is equal to the distance between objects 2 and 3, because p₁₂ = p₂₃.

Also for P₂, the first object is most preferred (37), and the third object is least preferred (38), but here, as pointed out above, the preference structure is not consistent. Accordingly, the PrefMap for P₂ (Fig. 11 middle) shows the most preferred object 1 on top and the least preferred object 3 on the bottom, but due to inconsistency the three points are not collinear.

P₃ does not have any most preferred or least preferred object but the objects have a circular preference structure …→1→ 2 →3 → 4 →5 → 1 → …. Accordingly, the PrefMap for P₃ (Fig. 11 right) shows the five objects as equally spaced points on a circle, here as edges of an equilateral pentagon, with nodes arranged according to the circular preference structure. The interior edges appear in grey which corresponds to preference 0.5. The orientation of this PrefMap may be rotated arbitrarily because in P₃ all circular preferences are equal $p_{12} = p_{23} = p_{34} = p_{45} = p_{51} = 1$ (39)

Fig. 12 shows the PrefMaps of the group decision making data sets P₄, …, P₇.

Fig. 12

PrefMaps of the group decision making data sets P₄, …, P₇.

For better comparison all four PrefMaps are shown in one plot. The object indices are only displayed in the PrefMap for the aggregated preference structure P₇. The nodes appear almost collinear, so the preferences are almost consistent. In all four cases object 1 is on top, then 2, then 3. Each of the indexed nodes of the aggregated preferences P₇ is located close to the center of the corresponding non–indexed nodes of the individual preferences P₄, P₅, P₆.

Fig. 13 shows the PrefMap of the La Liga data set P₈.

Fig. 13

PrefMap of the La Liga data set P₈.

This PrefMap is not at all collinear but shows large variation along the horizontal axis, so the preference structure is quite inconsistent. The three top nodes (Barcelona, Real Madrid, Ath Madrid) and the three bottom nodes (Malaga, Las Palmas, La Coruna) have a large horizontal spread, so there is no clear winner or loser node. In the middle, two nodes are very far from each other but with very similar overall preference (Espanol and Alaves), which indicates inconsistent preferences for these two nodes over the other nodes (i.e. Espanol often won against teams that won against Alaves, and vice versa).

Fig. 14 shows the PrefMap of the YouTube Comedy Slam preference data set P₉.

Fig. 14

PrefMap of the YouTube Comedy Slam data set P₉.

This preference structure appears mostly consistent, except for some outliers, most prominently one outlier at bottom left (-3P57xi7jMA) which is the least preferred but which is also quite inconsistent with the other nodes. The node at the top (-0_iw2b9fXY) is very clearly preferred.

6 Conclusion

We have considered the problem of visualizing pairwise preference structures. Standard methods for visualizing preference matrices are matrix visualization and chord diagrams. In this paper we have introduced PrefMap, a new method for visualizing pairwise preference structures, that is inspired by multidimensional scaling. All three methods have been experimentaly evaluated with three artificial data sets, four data sets from group decision making, and two real world data sets (La Liga and YouTube Comedy Slam). Fig. 15 summarizes the features of the three considered visualization methods.

Fig. 15

Features of the three considered visualization methods.

Matrix visualization and chord diagrams work well for small data sets but become quite confusing for large data sets, as shown for example in Figs. 5 and 9. In contrast, PrefMap produces good visualizations also for large data sets, as shown for example in Fig. 14. In matrix visualization preferred options may be identified as light rows or dark columns, but these are hard to see. In chord diagrams preferred options can be identified by dark arcs, but these are difficult to identify for large data sets. In PrefMaps, preferred options are clearly mapped to the top, and less preferred options are on the bottom. In contrast to matrix and chord diagrams, PrefMaps are also able to indicate whether the preference structure contains clusters (options with similar preferences, which appear close to each other, like for example Getafe, Levante, Eibar, Celta, Ath Bilbao, and Leganes in Fig. 13) of whether the preference structure contains inconsistencies (options whose preference structures do not match, which appear horizontally spread, like for example Espanol and Alaves in Fig. 13). So, PrefMaps are a very useful tool for visualizing preference matrices, especially for large numbers of options, which allows to easily identify important properties of the preference structures such as preferred options, clusters, and inconsistencies.

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