Yiru Zhang

Tassadit Bouadi
email: tassadit.bouadi@irisa.fr

Yewan Wang
email: yewan.wang@gmail.com

Arnaud Martin
email: arnaud.martin@irisa.fr

A distance for evidential preferences with application to group decision making

Keywords: Preference modelling, Theory of belief functions, Decision making, Distance measure

In this paper, we focus on measuring the dissimilarity between preferences with uncertainty and imprecision, modelled by evidential preferences based on the theory of belief functions. Two issues are targeted: The first concerns the conflicting interpretations of incomparability, leading to a lack of consensus within the preference modelling community. This discord affects the value settings of dissimilarity measures between preference relations. After reviewing the state of the art, we propose to distinguish between two cases: indecisive and undecided, respectively modelled by a binary relation and union of all relations. The second concerns a flaw that becomes apparent when measuring the dissimilarity in the theory of belief functions. Existing dissimilarity functions in the theory of belief functions are not suitable for evidential preferences, because they measure the dissimilarity between preference relations as being identical. This is counter-intuitive and conflicting with almost all the related works. We propose a novel distance named Unequal Singleton Pair (USP) distance, able to discriminate specific singletons from others when measuring the dissimilarity. The advantages of USP distances are illustrated by the evidential preference aggregation and group decision-making applications. The experiments show that USP distance effectively improves the quality of decision results.

Introduction

Preference is a traditional topic in human history and its corresponding study is of great interest in various domains, such as sociology [START_REF] Clark | Community structure, decision-making, budget expenditures, and urban renewal in 51 american communities[END_REF], economy [START_REF] Simon | Rational decision making in business organizations[END_REF], and more specifically group decision making [START_REF] Meng | Group decision making with heterogeneous intuitionistic fuzzy preference relations[END_REF]. As new applications emerge, preference modelling continues to attract the attention of recent research communities in computer science, such as social networks [START_REF] Biswas | An mcdm integrated adaptive simulated annealing approach for influence maximization in social networks[END_REF], and deep learning communities [START_REF] Ni | An effective recommendation model based on deep representation learning[END_REF], etc.

In reality, preferences are not always expressed in a firm or consistent manner, uncertainty and imprecision may exist in preference information, facing unknown situations, or multiple conflicting information sources. In this paper, we accept terms uncertainty and imprecision for two different concepts of imperfectness. Term uncertainty refers to a degree of conformity to reality, while term imprecision refers to quantitative default of knowledge on the information contained. A simple example is given here for a better understanding of the two terms, while other scenarios of imperfectness are also possible:

Example 1. In the US presidential election of 2020 between Joe Biden and Donald Trump, two pieces of information are given: the first is a statement, saying Joe Biden has 50% chances to win. The second is a data missing case, interpreted as Biden or Trump will win.

In this example, the first case expresses a piece of uncertain preference information, while the certainty is 50%. The second expresses a piece of preference information with imprecision. Indeed, the second is an extreme case of imprecision, also known as total ignorance. Some previous works have already investigated imperfect preferences. For example, the authors of [START_REF] Orlovsky | Decision-making with a fuzzy preference relation[END_REF] proposed a fuzzy model to cope with imprecise preferences. Recently, evidential preference models based on the Theory of Belief Functions (TBF) have been proposed and attracted much attention in preference modelling venues [START_REF] Denoeux | Evidential reasoning in large partially ordered sets[END_REF][START_REF] Masson | Modelling and predicting partial orders from pairwise belief functions[END_REF][START_REF] Zhang | Preference fusion and condorcet's paradox under uncertainty[END_REF]. TBF (also referred to as Dempster-Shafer Theory [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] or Evidence Theory) is a powerful mathematical tool for the modelling of uncertainty and imprecision, widely applied in information fusion and decision making. In addition to the capability of characterizing uncertainty and imprecision, the evidential preference model handles as well the aforementioned ambiguity of the interpretation of "incomparability", in a unified framework.

Knowledge with uncertainty is depicted by degree values on the corresponding event, while knowledge with imprecision is depicted by the union set of possible singletons. Thus, the evidential preference model is able to express these two aspects of imperfectness in preference information faithfully. A more detailed introduction on the TBF and evidential preference model will be provided later in Section 2. Dissimilarity measure between imperfect preference relations is another issue targeted in this work in addition to preference modelling. Dissimilarity plays a vital role in preferenceaware systems concerning various applications. For example, distance-based consensus methods is an important category in preference aggregation. Indeed, some conventional voting rules are also considered distance-based consensus methods using different distance functions, as explained in [START_REF] Viappiani | Characterization of scoring rules with distances: application to the clustering of rankings[END_REF]. In social network analysis, distances between preferences are applied for community detection and neighborhood-based recommendation systems [START_REF] Liu | A new user similarity model to improve the accuracy of collaborative filtering[END_REF][START_REF] Cacheda | Comparison of collaborative filtering algorithms: Limitations of current techniques and proposals for scalable, high-performance recommender systems[END_REF].

Dissimilarity functions depend predominantly on data representation. For example, for preferences represented as orders or list-wise, correlation coefficients are widely applied, such as Kendall's τ , Spearman's ρ, Pearson's r or their extended versions [START_REF] Fagin | Comparing top k lists[END_REF]. In the case of a pair-wise representation of the preferences, dissimilarities are measured between binary relations [START_REF] Kemeny | Mathematical models in the social sciences, chapter preference rankings: An axiomatic approach[END_REF][START_REF] Roy | Criterion of distance between technical programming and socio-economic priority[END_REF][START_REF] Ben Khelifa | A distance-based collective weak ordering[END_REF][START_REF] Jabeur | A distance-based collective preorder integrating the relative importance of the group's members[END_REF]. Bouyssou et al. [START_REF] Bouyssou | Binary relations and preference modelling[END_REF] mentioned that binary relation is the central tool in most models of preferences. Thus, measuring the dissimilarity between pairwise preferences is a fundamental issue. Usually, between any two alternatives a i and a j , four exclusive binary relations are possible: "strict preference (a i a j )", "inverse strict preference (a i ≺ a j )", [START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF] "indifference (a i ≈ a j )", and "incomparability (a i ∼ a j )". The dissimilarity between preference relations are mainly measured through these four relations. However, the definition of "incomparability" may correspond to different interpretations, resulting in different dissimilarity value settings. Two major issues are encountered when measuring the dissimilarity between evidential preferences. The first is caused by the ambiguous interpretations of incomparability while the second exists in the context of the theory of belief functions.

First issue: different interpretations of "incomparability" Relation incomparability between two alternatives is defined as neither prefer, nor indifferent (with the formal definition given in Section 2.1). However, this definition is ambiguous 1 . In the literature, we have identified two interpretations of incomparability, considered either as an indecisive or an undecided situation.

Example 2. Indecisive case: Which job is better? [START_REF] Deparis | When conflict induces the expression of incomplete preferences[END_REF] Alice has two available job offers. The first job is poorly paid but corresponds well to Alice's interests while the second is well paid but does not match Alice's interests. As a consequence, Alice finds herself in a dilemma when she has to choose between the two jobs.

In Example 2, incomparability is caused by conflicting preferences associated with different criteria, making the decision indeterminable. We interpret this case as indecisive.

Example 3. Undecided case: Missing opinions in Sushi survey. [START_REF] Kamishima | Nantonac collaborative filtering: recommendation based on order responses[END_REF] In the Sushi preference data set, the respondent (agent) is asked to give her/his preferences on 10 types of sushi randomly selected among 100 sushi types. The preferences are expressed in the forms of both score and order. However, the preferences on the other 90 sushi of each agent are missing, making them incomparable.

The second example shows a case where incomparability is caused by missing information, which relates to the ignorance of knowledge. We interpret this case as undecided.

To avoid misleading in engineering aspects, we need to mention that missing information is not always interpreted as ignorance. In some scenarios, indifference or not preferred could be better interpretations. For example, in democratic voting systems, blank votes are usually considered as abstention; In a top-k ranking system, alternatives without preference information are considered as lower-ranked .

Incomparability can be set without controversy when only one of the two situations mentioned before exist in the preference-aware systems. However, for the cases that merge with both situations, it is necessary to distinguish between the two interpretations. Dissimilarity measures between pair-wise preference relations may encounter such issues. Some works (e.g. [START_REF] Roy | Aide multicritère à la décision: méthodes et cas[END_REF][START_REF] Schärlig | Pratiquer Electre et Prométhée: un complément à décider sur plusieurs critères[END_REF]) differentiate between the two relations as indifference and incomparability, some works interpret incomparability as an absence of knowledge (e.g. [START_REF] Ben Khelifa | A distance-based collective weak ordering[END_REF][START_REF] Jabeur | A distance-based collective preorder integrating the relative importance of the group's members[END_REF][START_REF] Blin | A linear assignment formulation of the multiattribute decision problem[END_REF]), and some works interpret as a specific case (e.g. [START_REF] Roy | Criterion of distance between technical programming and socio-economic priority[END_REF]). Indeed, this issue exists in different works but has rarely been systematically discussed in the research community.

The flaws of dissimilarity measure are not limited to preferences. In the Theory of Belief Functions, some problems in dissimilarity measures rise to be important in the evidential preference model.

Second issue: a flaw in dissimilarity measures in the Theory of Belief Functions

In TBF, each precise event is modelled by a singleton. Dissimilarity values between different singletons are always considered equal in existing measures. This is because two singletons are considered as either different or identical, respecting a binary relation. However, with this property applied by evidential preferences, dissimilarities between any two preference relations are equal. A straightforward example is that the dissimilarity between strict preference and inverse strict preference (denoted as d( , ≺)) is identical to the dissimilarity between strict preference and indifference (denoted as d( , ≈)), formally, d( , ≺) = d( , ≈). This equality relation is counter-intuition and in conflict with all previous works in preference modelling [16-19, 25, 26].

For the smoothness of writing, related works, as well as comprehensive details about this flaw, are introduced in Section 2.3, after a preliminary introduction on the TBF.

Contributions

Facing the two issues above, three main contributions are made targeting the issues introduced above.

-The state of the art in dissimilarity measures over pairwise preferences is reviewed.

Regarding the different interpretations of "incomparability", the disagreements are clarified and characterized through the evidential preference model.

-A novel metric function for dissimilarity measure between evidential preferences is proposed by extending the existing Jousselme distance [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF], named Unequal Singleton Pair (USP) distance. USP distance takes into account the different dissimilarity values between singletons concerning the second issue and also handles well the case of imprecise information.

-The proposed USP distance is applied in group-decision making and is compared with other traditional and recent methods on both synthetic and real-world data sets. The results show that USP distance improves the quality of group decision making. Besides, a Condorcet's paradox elimination method for the evidential preference model is developed and applied as well.

The merits of the techniques proposed in this paper consist of three main parts, two of which are related to the evidential preference based on TBF.

-Evidential preferences are able to express imprecise cases, especially missing data cases faithfully;

-The ambiguous definition of incomparability is clarified by undecided and indecisive in a unified framework in evidential preference;

-USP distance is capable to discriminate singleton elements for decision making with the imprecise information taken into account. The advantage of this distance is demonstrated in group decision-making applications based on the evidential preference model.

This paper is organised as follows: preliminary notions on preference modelling, Theory of Belief Functions, and dissimilarity measures are introduced in Section 2. Afterwards, the first contribution is presented in Section 3, where dissimilarity measures over pairwise preferences are reviewed. In Section 4, the proposed novel distance for evidential objects is introduced, named Unequal Singleton Pair (USP) distance, along with a value-setting example and an application of group decision making. In Section 5, the advantages of USP distance over the other preference aggregation methods are illustrated with experiments on the sushi preference dataset in the real-world. Finally, conclusions, as well as important perspectives are, given in Section 6.

Preliminaries

In this section, preliminary knowledge on preferences and the theory of belief functions, as well as some properties of dissimilarity measures are introduced.

Preference modelling: background

Preference modelling is usually based on order theory. In this paper, we use the widely accepted notions in studies of preferences from [START_REF] Öztürke | Preference Modelling[END_REF].

Definition 1. (Binary Relation) Given a finite set of N alternatives A = {a 1 , a 2 , a 3 , . . . , a N }, a binary relation R on the alternative set A is a subset of the Cartesian product A × A, that is, a set of ordered pairs (a i , a j ) such that a i and a j are in A : R ⊆ A × A [START_REF] Öztürke | Preference Modelling[END_REF]. Definition 2. (Preference relation) Between any two alternatives a i , a j , only four relations possibly exist { , , ≈, ∼}, defined from binary relation R as:

Strict preference: a i a j ⇔ a i Ra j and a j ¬Ra i ;

Indifference: a i ≈ a j ⇔ a i Ra j and a j Ra i ; Weak preference: a i a j ⇔ a i a j or a i ≈ a j ; Incomparability: a i ∼ a j ⇔ a i ¬ a j and a i ¬ ≺ a j and a i ¬ ≈ a j .

Relation "weak preference" is the union of strict preference and indifference. Thus, between two alternatives a i and a j , the four relations { , ≺, ≈, ∼} are exclusive and complete2 , illustrated in Figure 1. Let R be a binary relation on the alternative set A, we have the definitions of different orders based on preference structure with relations "strict preference", "indifference" and "incomparability" : Definition 3. (Preference Structure) A preference structure is a collection of binary relations defined on the set A and such that:

-for each couple (a i , a j ), a i , a j ∈ A, at least one relation is satisfied; -for each couple (a i , a j ), a i , a j ∈ A, if one relation is satisfied, any other relation cannot be satisfied.

Definition 4. (Total/Weak/Partial/Quasi (pre)order) R is a total order iif. R ∈ { , ≺}; R is a weak order 3 iif. R ∈ { , ≺, ≈}; R is a partial order iif. R ∈ { , ≺, ∼}; R is a quasi-(pre)order iif. R ∈ { , ≺, ≈, ∼}.
In this paper, orders without incomparability are called complete orders, i.e. total and weak orders are complete orders.

Dissimilarity measure for quasi (pre)orders is compatible with all the other three orders as quasi (pre)order is the most general concept. However, most of the dissimilarity measure methods are designed for complete orders. The difficulty falls on the dissimilarity measure on relation "incomparable" , with more details given in Section 3.

Theory of belief functions (TBF): notions

Let Ω = {ω 1 , . . . , ω H } be a finite set representing all possible status of a categorical attribute, the uncertainty and imprecision of this attribute is expressed by Basic Belief Assignment (BBA).

Definition 5. (Basic Belief Assignment (BBA)) A (normalized) Basic Belief Assignmen (BBA) on Ω is a function m : 2 Ω → [0, 1] such that: m(∅) = 0 and X⊆Ω m(X) = 1. (1)
The subsets X of Ω such that m(X) > 0 are called focal elements, while the finite set Ω is called discernment framework. Ω is also considered as total ignorance since it represents all the possibilities. A BBA representing total ignorance (m(Ω) = 1) is also called a vacuous BBA. A BBA is simple supported if a non-zero value is assigned only on one singleton and Ω. In this paper, we accept the term evidential object as an object described by imperfect information with both uncertainty and imprecision. Thus, an evidential object is represented by a BBA. Definition 6. (Categorical BBA) A categorical BBA is a normalized BBA satisfying:

m(X) = 1 , ∀X ⊂ Ω and m(Y ) = 0 , ∀Y ⊆ Ω, Y = X. (2)
A categorical BBA on element X ∈ 2 Ω is denoted as X 0 . With the concept of simple support, we name a categorical BBA on one singleton as categorically simple supported.

In the process of information fusion, where multiple BBAs are combined into one, various combination rules are available. The selection of a pertinent combination rule depends on many criteria, in which an important one is cognitively independent. Definition 7. (Cognitively Independent Source) Sources are considered as cognitively independent if the values of BBA on any source has no communication with the others.

The conjunctive combination rule proposed by [START_REF] Smets | The combination of evidence in the transferable belief model[END_REF] is applied for finding the consensus among multiple reliable and cognitively independent sources, defined as:

m conj (X) = Y 1 ∩...∩Ys=X S s=1 m s (Y s ). (3)
For cognitive dependent sources, the mean value rule is commonly applied. Given a discernment framework Ω and sources S, BBA on source s ∈ S denoted as m s , for ∀X ∈ 2 Ω , mean value combination is defined as follows:

m mean (X) = 1 S S s=1 m s (X). (4)
These two combination rules are applied in the evidential preference aggregation process, with details given in Section 4.5.

Properties of dissimilarity measures over evidential objects

Metrics, or distance functions are usually applied to measure the dissimilarity between two objects. Definition 8. (Metric (distance)) A metric (distance) function d on a set T is defined as:

d : T × T → R ≥0 ,
where R ≥0 denotes the set of non-negative real numbers. Given any three elements t i , t j , t k ∈ T , the following conditions are satisfied:

1. d(t i , t j ) ≥ 0 : non-negativity; 2. d(t i , t j ) = 0 ⇔ t i = t j : identity of indiscernible; 3. d(t i , t j ) = d(t j , t i ) : symmetry; 4. d(t i , t j ) ≤ d(t i , t k ) + d(t k , t j ) :
triangle inequality.

(

) 5
Usually, the term dissimilarity refers to semi-metric, in which the property of triangle inequality does not hold.

In TBF, different dissimilarity measures are applicable to BBAs. From geometrical and statistical views, these measures are categorized into three parts: distance [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF][START_REF] Mahler | Combining ambiguous evidence with respect to ambiguous a priori knowledge. i. boolean logic[END_REF], conflicts 4 [31], and divergence [START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF]. Comprehensive surveys are respectively available in [START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF] and [START_REF] Jousselme | Distances in evidence theory: Comprehensive survey and generalizations[END_REF]. We only introduce the properties required in this paper.

As BBAs are defined in space of 2 Ω , in addition to the properties for metrics in Equation set [START_REF] Ni | An effective recommendation model based on deep representation learning[END_REF], properties considering the structure of discernment are specifically required in TBF. The authors in [START_REF] Jousselme | Distances in evidence theory: Comprehensive survey and generalizations[END_REF] resumed three structural properties: Strong structural property (with consideration of the interaction between focal elements), weak structural property (with consideration of the cardinality of focal elements), and structural dissimilarity (with consideration of the interaction between sets of focal elements). We accentuate the strong structural property hereby, which is the strictest one.

Property 1. Strong structural property: A distance measure function d between two

BBAs m 1 and m 2 is strongly structural if its definition accounts for the interaction between the focal elements of m 1 and m 2 .

According to [START_REF] Jousselme | Distances in evidence theory: Comprehensive survey and generalizations[END_REF], the structural property is usually considered as the axiomatic metric property for distance measure between evidence objects. A popular distance that is strongly structural is Jousselme distance [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF], defined as:

d Jousselme (m 1 , m 2 ) = (m 1 -m 2 ) T J acc(m 1 -m 2 ), (6)
where J acc is the matrix whose elements are Jaccard indices:

Jacc(X 1 , X 2 ) = |X 1 ∩ X 2 | |X 1 ∪ X 2 | , for X 1 , X 2 ∈ 2 Ω \ ∅. (7)
Matrix J acc is positive definite and the properties of metric (Equation set (5)) hold in Jousselme distance, proved in [START_REF] Bouchard | A proof for the positive definiteness of the jaccard index matrix[END_REF].

Contribution I: A synthetic review of dissimilarity measures over pairwise preferences

Dissimilarity measures over preferences can be grouped into two categories: between pairwise preferences (binary relations) and between list-wise preferences (preference orders). In this section, the former one is focused, where axiomatic distances and Hamming distance are mostly applied. Among these distances, different value settings concerning incomparability are accepted, caused by the ambiguous interpretations aforementioned. In this section, starting with a synthetic review of dissimilarity over pair-wise preferences, we introduce how the evidential preference model distinguish these interpretations in a unified framework.

Axiomatic distances

In axiomatic distances, axioms are used to help to define values of dissimilarity between preference relations. These axioms are extracted either from the definitions or intrinsic properties of preference relations. Axiomatic distances were originally proposed for preference aggregation in distance-based consensus models.

Early research was initiated by Kemeny and Snell [START_REF] Kemeny | Mathematical models in the social sciences, chapter preference rankings: An axiomatic approach[END_REF], where a model for combining preferences (in weak orders) into a group consensus was proposed (a.k.a. Kemeny-Snell model, marked as KS model in this paper). However, the definition given in [START_REF] Kemeny | Mathematical models in the social sciences, chapter preference rankings: An axiomatic approach[END_REF] to the partial order does not include relation indifference since it supposes that an alternative a i is either preferred or incomparable to another alternative a j . J.M. Blin [START_REF] Blin | A linear assignment formulation of the multiattribute decision problem[END_REF] adopted the KS model and proposed an aggregation method by distance minimization. Cook et al. [START_REF] Cook | Information and preference in partial orders: a bimatrix representation[END_REF] (marked as CS model) consider incomparability by introducing a comparison matrix . In the CS model, the incomparability is defined as not compared while the axioms respect definitions in Figure 1. Roy and Slowinski [START_REF] Roy | Criterion of distance between technical programming and socio-economic priority[END_REF] (marked as RS model) proposed logical conditions to build a distance measure between pairs of binary relations. The authors in [START_REF] Ben Khelifa | A distance-based collective weak ordering[END_REF] (marked as KM model) respected the RS model and suggested a dissimilarity measure with supplementary conditions. Then, Jabeur et al. [START_REF] Jabeur | A distance-based collective preorder integrating the relative importance of the group's members[END_REF] proposed numeric values for dissimilarity measure between different relations based on [START_REF] Ben Khelifa | A distance-based collective weak ordering[END_REF] (marked as JMK model) by taking the centroid value in a restrained space created by the axioms. In this paper, we refer to the distances applied in different models by corresponding model names.

Several contradictions exist within these models, as depicted in Figure 2, where Ω denotes the ignorance of knowledge. In the RS model, incomparability is interpreted as:

Incomparability relation is the affirmation of the incapacity to establish the relation type: there is no indifference, no weak preference and no strict preference between the two alternatives [START_REF] Roy | Criterion of distance between technical programming and socio-economic priority[END_REF]. This interpretation matches the Venn diagram in Figure 2a. However, in other models, such as CS, KM, and JMK models, incomparability is interpreted as the two alternatives are not compared (i.e. undecided ). This interpretation corresponds to the illustration in Figure 2b. Axioms accepted in each model are listed in Table 1. These axioms concern the preference relations , ≺, ≈, ∼, ? where the symbol ? represent ignorance induced from the undecided case, (i.e. ? = { ∪ ≺ ∪ ≈ ∪ ∼}). Distance function between preference relations is denoted by d ∆ . In order to distinguish the two interpretations of incomparability, we remark axioms on the undecided version as BIS.

Axiom Content Model reference
Axiom 1: properties of metric (Equations ( 5)) all models Axiom 2:

d ∆ ( , ≈) = d ∆ (≺, ≈) and d ∆ ( , ∼) = d ∆ (≺, ∼) RS model BIS: d ∆ ( , ?) = d ∆ (≺, ?) CS and KM models Axiom 3: d ∆ ( , ≈) + d ∆ (≈, ≺) = d ∆ ( , ≺) all models Axiom 4: d ∆ ( , ∼) ≤ d ∆ (≈, ∼) RS model BIS 1: d ∆ ( , ?) ≤ d ∆ (≈, ?) CS model BIS 2: d ∆ ( , ?) = d ∆ (≈, ?) KM model Axiom 5: d ∆ (≈, ∼) ≤ d ∆ (≈, )
CS model BIS:

d ∆ (≈, ?) ≤ d ∆ (≈, ?) CS and KM models Axiom 6: d ∆ ( , ≺) = max({d ∆ (R, R ) : R, R ∈ { , ≺, ≈, ∼}}) all models Axiom 2 implies that and ≺ are opposite relations, and symmetric to indifference ≈. Axiom 6 indicates that strict preference and inverse preference relations are most distinguished. CS and KM models accept the interpretation of undecided for incomparability, referring to the knowledge of ignorance. CS and KM models confuse the definition of "incomparability" with "ignorance", where "ignorance" is caused by having no knowledge of the preference relation.

Apart from axiomatic preference distances, Minkowski distances based on the encoding of different preference relations are often applied. The most popular one is the Hamming distance.

Hamming distance

Hamming distance for preference relation is based on the encoding of preference. Given two alternatives a i and a j , preference relations are encoded in binary by flattening preference matrix, with corresponding Hamming distance values resumed in Table 2. R a i a j a i a j a i 0 1 a j 0 0 Code 0100 a i ≺ a j a i a j 0 0 1 0 0010

a i ≈ a j a i a j 0 1 1 0 0110 a i ∼ a j a i a j 0 0 0 0 0000 d ∆ H ≺ ≈ ∼ 0 2 1 1 ≺ 2 0 1 1 ≈ 1 1 0 2 ∼ 1 1 2 0

Summary of distances between pair-wise relations

The inequality relations applied in each distance are summarized in Figure 3. An impor- tant advantage of axiomatic methods is that these dissimilarity measures are metrics with the properties guaranteed. However, the above research works show that the dissimilarity measures vary with different axioms accepted, these methods have a strong dependency on the interpretation of different preference relations.

Clarification of "incomparability" in evidential preference model

The different interpretations of incomparability can be simultaneously characterized by the evidential preference model in a unified framework.

The evidential preference model is originally proposed by [START_REF] Masson | Modelling and predicting partial orders from pairwise belief functions[END_REF] on weak orders (corresponding to Figure 2b) and extended to quasi orders (corresponding to Figure 2a) with the consideration of incomparability by [START_REF] Zhang | Preference fusion and condorcet's paradox under uncertainty[END_REF]. Consider the case that a set of agents U expressing their preferences between every pair over alternative set A. Each agent is a source, denoted [START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF] as u, u ∈ U. For any alternative pair a i , a j ∈ A, four relations are possible. Therefore, the discernment framework Ω pref ij is defined as:

Ω pref ij = {ω R ij |R ∈ { , ≺, ≈, ∼}}. (8)
In evidential preference, the case of undecided is characterized by total ignorance Ω pref ij , while the case of indecisive by a specific binary relation incomparability ω ∼ ij . Therefore, different interpretations are distinguished in a unified framework. The degree of uncertainty on preference relation is represented by values on singletons, which can be regarded as a fuzzy value 5 . The imprecision is characterized by values on union sets.

With the combination rules in the framework of the theory of belief functions, the evidential preference model is effective in group decision-making with imperfect preference information sources. A more in-depth discussion on this topic is given in Section 4.5.

Contribution II: USP distance -A novel measure suitable for evidential preferences

Despite that evidential preference can faithfully express all types of preference, none of the existing distances in the context of the theory of belief functions is able to properly measure the dissimilarity between evidential preferences, with aforementioned inequality relations respected. In this section, we analyze the properties that disable a relevant dissimilarity measure and propose a new distance adapted to evidential preferences, namely Unequal Singleton Pair (USP) distance. The analysis starts with a detailed introduction to this flaw.

A flaw in the dissimilarity measure over evidential preferences

The fact that existing distances in the theory of belief functions are not relevant for evidential preference is caused by a contradiction between their required properties. In Section 3, it is concluded that Equation ( 9) is always applied in measuring the dissimilarity between preference types.

d ∆ ( , ≺) > d ∆ ( , ≈). (9)
This inequality relation is rewritten as Equation ( 10) adapted to evidential preferences.

d({ω } 0 , {ω ≺ } 0 ) > d({ω } 0 , {ω ≈ } 0 ), (10)
where {ω R } 0 represents the categorical BBA on singleton element ω R (see Definition 6). However, the existing dissimilarity measures do not distinguish values between singletons in TBF, thus, none of them satisfies Equation [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF]. In TBF, a discernment framework (Ω = {ω 1 , ω 2 , . . . , ω H }) requires that all the singleton elements are mutually exclusive (see Section 2.2). With the property of exclusivity, any two elements ω i , ω j ∈ Ω are either identical or distinguished, with respect to a binary relation. Existing distances apply this binary relation by an assumption that dissimilarity values between singleton elements are equal, precisely: Assumption 1. In a discernment framework Ω = {ω 1 , ω 2 , . . . , ω H }, the dissimilarity between any two different singletons is a constant (normalized as 1), formally,

∀ω m , ω n ∈ Ω, ω m = ω n , d({ω m } 0 , {ω n } 0 ) ≡ 1. (11)
Obviously, Equation [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF] and Equation ( 11) are contradictory. As a consequence, Assumption 1 is not suitable for measuring the dissimilarity between evidential preferences. To propose a theoretically correct solution, an analysis of necessary properties for evidential preferences is conducted.

Dissimilarity measure properties for evidential preferences

Three essential properties are required for dissimilarity measures on evidential preferences.

Property 2. The dissimilarity is measured by a metric, i.e. the properties of metric (Definition 8) hold.

Property 3. The dissimilarity measure takes the difference of dissimilarity between singletons into account.

Property 3 abandons Assumption 1. With the account that dissimilarities are measured between BBAs in the space of 2 Ω , the following property must hold as well, matching the property of strongly structural (see Property 1 in Section 2.3), Specifically described as:

∀X, Y ∈ 2 Ω , d(X 0 , Y 0 ) ≤ d({X \ (X ∩ Y )} 0 , Y 0 ). (12)
By applying these properties on weak orders, represented by Ω = {ω , ω ≺ , ω ≈ }, the dissimilarity values are set as:

d({ω } 0 , {ω ≺ } 0 ) = 1; (13a) d({ω } 0 , {ω ≈ } 0 ) = d ∆ (≺, ≈) =p. ( 13b
)
where p is a dissimilarity value between 0 and 1 (strictly smaller than 1). With respect to Equation ( 12), the three relations below yield:

1. Between strict preference and union with indifference ≈:

d({ω } 0 , {ω , ω ≈ } 0 ) < d({ω } 0 , {ω ≈ } 0 ) ⇒ d({ω } 0 , {ω , ω ≈ } 0 ) ∈ (0, p). (14a)
Symmetrically,

d({ω ≺ } 0 , {ω ≺ , ω ≈ } 0 ) < d({ω ≺ } 0 , {ω ≈ } 0 ) ⇒ d({ω ≺ } 0 , {ω ≺ , ω ≈ } 0 ) ∈ (0, p). (14b)
2. Between strict preference and total ignorance (missing data):

d({ω } 0 , Ω 0 ) > d({ω } 0 , {ω , ω ≈ } 0 ). (15a)
Symmetrically,

d({ω ≺ } 0 , Ω 0 ) > d({ω ≺ } 0 , {ω ≺ , ω ≈ } 0 ). (15b)
3. Between strict preference and union of other two preference types (≺ and ≈):

d({ω } 0 , {ω ≈ } 0 ) < d({ω } 0 , {ω ≺ , ω ≈ } 0 ) < d({ω } 0 , {ω ≺ } 0 ). (16a)
Symmetrically,

d({ω ≺ } 0 , {ω ≈ } 0 ) < d({ω ≺ } 0 , {ω , ω ≈ } 0 ) < d({ω } 0 , {ω ≺ } 0 ). ( 16b
)
To apply Properties 2, 3, and 4, we propose Unequal Singleton Pair (USP) distance as a solution.

USP distance-a distance for unequal singleton pairs over BBAs

In TBF, Jousselme distance [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF] is widely applied because this distance is strongly structural by taking into account the imprecision in the structure with respect to Property 2 and Property 4. Inspired by the methodology in the conception of Jousselme distance, we propose a novel distance named Unequal Singleton Pair (USP) distance, with the Property 3 imposed.

Assumption 1 is guaranteed in by Jaccard index (Equation ( 6)), with Property 4, recalled here:

Jacc(X 1 , X 2 ) = |X 1 ∩ X 2 | |X 1 ∪ X 2 | , for X 1 , X 2 ∈ 2 Ω \ ∅.
To drop Assumption 3, we define USP distance by modifying Jaccard index matrix as follows.

d U SP (m 1 , m 2 ) = (m 1 -m 2 ) T Sim(m 1 -m 2 ), (17)
where Sim differs from J acc, also referring to the similarity matrix between different singleton elements. In Sim, the similarity value between two elements X 1 and X 2 , (X 1 , X 2 ∈ 2 Ω ) is denoted as sim(X 1 , X 2 ). In the first place, with a similar form [START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF] as Jaccard index, sim(X 1 , X 2 ) is defined as the division between two virtual functions: resemb(X 1 , X 2 , . . . , X M ) and entire(X 1 , X 2 , . . . , X M ):

sim(X 1 , X 2 ) = resemb(X 1 , X 2 ) entire(X 1 , X 2 ) , (18)
where resemb(X 1 , X 2 ) describes the cause of the similarity between X 1 and X 2 (say resemblance), and entire(X 1 , X 2 ) the entire part concerned by X 1 and X 2 . resemb(•) and entire(•) are defined as virtual functions. With Assumption 1, the functions resemb ass1 (X 1 , X 2 , . . . , X M ) and entire ass1 (X 1 , X 2 , . . . , X M ) are overrode as:

resemb ass1 (X 1 , X 2 , . . . , X M ) =| M i=1 X i |, entire ass1 (X 1 , X 2 , . . . , X M ) =| M i=1 X i | . (19) (20)
Obviously, the similarity matrix Sim is identical to Jaccard matrix J acc.

Here, we propose a new method to drop Assumption 1. Define a set of elements in 2 Ω , W = {X 1 , X 2 , . . . , X M }, therefore W ⊆ 2 Ω . Denote resemb(W ) for resemb(X 1 , X 2 , . . . , X M ) and entire(W ) for entire(X 1 , X 2 , . . . , X M ) to simplify the expression. The size of W is defined by the number of elements X ∈ 2 Ω , denoted by |W |. Singletons in W is defined by the union of all elements in W , formally:

∪W = X i ∈W X i . (21)
Denote the subset of W by W sub , by dropping Assumption 1, entire(W ) is defined as a generalized version of cardinal function on the union sets:

entire(W ) = ω∈∪W entire(ω) - W sub ⊆W,|W sub |=2 resemb(W sub ) + W sub ⊆W,|W sub |=3 resemb(W sub ) - W sub ⊆W,|W sub |=4 resemb(W sub ) + . . . + W sub ⊆W,|W sub |=|2 Ω | resemb(W sub ) × (-1) |2 Ω | = ω∈∪W entire(ω) + |2 Ω | t=1 W sub ⊆W,|W sub |=t resemb(W sub ) × (-1) t . (22)
In practice, with only the similarity between two singleton elements given, a unique solution for entire(W ) may not be reached by Equation [START_REF] Kamishima | Nantonac collaborative filtering: recommendation based on order responses[END_REF]. To enable and simplify the calculation, two more assumptions are imposed, defined as follows.

March 17, 2021

Assumption 2. In a set of exclusive elements Ω = {ω 1 , ω 2 , . . . , ω H }, any resemblance is only shared by maximal two elements, formally:

resemb(W ) = 0, ∀W ⊆ 2 Ω , |W | ≥ 3. (23)
Assumption 2 guarantees that a unique solution exists for resemb(ω m , ω n ) with only pairwise similarity sim(ω m , ω n ) given.

Assumption 3. To normalize the element values in the calculation, entire part concerned by one singleton is assigned to 1, formally:

entire(ω) = 1, ∀ω ∈ Ω. ( 24
)
With Assumptions 2 and 3, given similarity sim(ω m , ω n ), the resemblance is calculable and a unique solution exists. Deduced from Equation ( 22), we have:

entire(X, Y ) = ω∈X∪Y entire(ω) - ωm∈X ωn∈Y m =n resemb(ω m , ω n ). (25)
Hence, Equation ( 18) is overrode as:

sim(X 1 , X 2 ) = ωm∈X 1 ωn∈X 2 m =n resemb(ω m , ω n ) ω∈X 1 ∪X 2 entire(ω) - ωm∈X 1 ωn∈X 2 m =n resemb(ω m , ω n ) . ( 26
)
To guarantee the Assumption 2 given by Equation ( 23), the following condition holds:

ωm,ωn∈Ω ωm =ωn sim(ω m , ω n ) ≤ 1. (27)
Property 2 (property of the metric system) is guaranteed because the USP distance is in the format of the Mahalanobis distance. Moreover, extended from the Jousselme distance, Property 4 is kept by different values in the similarity matrix Sim.

For a better understanding, we demonstrate the inference and calculation of the USP distance in the evidence preference by a graphical illustration, followed by an application of decision-making thereafter.

Illustrative example-USP for evidential preferences

The calculation of the similarity matrix can be aided by graphics as depicted in Figure 4. Evidential preference is taken as an example, with four singletons elements: ω , ω ≺ , ω ≈ and ω ∼ . In accordance with the exclusivity of the discernment framework, Figure 4a shows [START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF] that singletons do not share any overlapped parts. Obviously, Figure 4a is isomorphic with Figure 1. In the conception of USP, we assume that the pairwise dissimilarity difference between singletons can be described by an overlapped zone in another space than the one in which they are defined. More precisely, outside the original S space, the singletons can have an overlapping part in a different S * space, represented by the overlapping space between singletons in figure 4b. In this way, the overlapped part could be used to show the difference in dissimilarities between the singletons elements. Thus, the overlap area in space S * allows calculating the similarity between simple elements. In addition, the constraints set by the functions Assumption 2 and Assumption 3 are respectively interpreted as the overlapped zones are shared by 2 elements, and the area of an element is valued as 1.

Table 3 gives the similarity values between ω R , R ∈ { , ≺, ≈, ∼}. The function resemb(•)

sim(•) ω ω ≺ ω ≈ ω ∼ ω 1 0 x y ω ≺ 0 1 x y ω ≈ x x 1 z ω ∼ y y z 1
valuing the overlapped areas are assigned with p 1 , p 2 , p 3 corresponding to Figure 4 as follows:

resemb(ω , ω ≈ ) = resemb(ω ≺ , ω ≈ ) = p 1 ; resemb(ω , ω ∼ ) = resemb(ω ≺ , ω ∼ ) = p 2 ; resemb(ω ≈ , ω ∼ ) = p 3 . ( 28
)
Plugging Equation [START_REF] Öztürke | Preference Modelling[END_REF] in Equations ( 24) and ( 25), the following relations are established:

               x = p 1 2 -p 1 ⇒p 1 = 2x 1 + x ; y = p 2 2 -p 2 ⇒p 2 = 2y 1 + y ; z = p 3 2 -p 3 ⇒p 3 = 2z 1 + z . (29)
Thus, similarities in 2 Ω ×2 Ω are calculated by applying Equation ( 26), and Assumption 2 yields:

2p 1 + p 3 ≤ 1 ⇒ 4x 1 + x + 2z 1 + z ≤ 1 ⇒ 2 x + 1 + 1 z + 1 ≤ 5 2 ; 2p 2 + p 3 ≤ 1 ⇒ 4y 1 + y + 2z 1 + z ≤ 1 ⇒ 2 y + 1 + 1 z + 1 ≤ 5 2
;

p 1 + p 2 ≤ 1 ⇒ 2x 1 + x + 2y 1 + y ≤ 1 ⇒ 1 x + 1 + 1 y + 1 ≤ 3 2 . ( 30
)
Equation set (30) defines the limit of value setting, guarantee axiomatic properties, and metric properties. With this dissimilarity value calculation method, we apply USP distance in group decision-making applications.

Application: Group decision making

The USP distance finds its application in group decision making between pairwise alternatives, represented by a i , a j ∈ A. Respectively, the discernment framework is designated by Ω ij (see Equation ( 8)) and the decided relation by R ij . The procedure consists of two steps: combination and decision, as shown in Figure 5. As mentioned above, in CS and KM models, incomparability interprets the absence of information, (referred to as undecided ), however, the final consensus decision strategies are possible to converge to the absence of information (undecided ) as a consequence, which is meaningless in decision making. To solve this problem, the conjunctive combination rule for BBAs is more adaptable. More specifically, LNS-CR (Conjunctive Rule for Large Number of Sources) [START_REF] Zhou | A belief combination rule for a large number of sources[END_REF] is applied as it is suitable for the aggregation of preferences without uncertainty, represented by categorically simple supported BBAs (see Definition 6).

m Ω ij s m Ω ij 1 . . . Conjunctive combination m Ω ij Decision R ij ∈ { , ≺, ≈, ∼}
In the decision step, a distance-based strategy is applied, which is originally introduced in [START_REF] Essaid | A distance-based decision in the credal level[END_REF], defined by:

R ij = argmin R∈{ ,≺,≈,∼} (d(m Ω ij , {ω R } 0 )), (31)
where m Ω ij denotes a BBA for the evidential preference between the alternatives a i and a j , and {ω R } 0 the categorical BBAs respectively representing the relations , ≺, ≈, and ∼. With USP distance applied in the decision rule above, the dissimilarities in singleton pairs are significantly discriminated against. The advantages are further discussed with experiments in Section 5.

Illustrative examples and experiments

In this section, three experiments are conducted. In the first experiment, we compare decision-making methods by examples on synthetic data with traditional methods, the impact of missing data is particularly studied. In the second experiment, we compare the differences between USP distance and Jousselme distance in decision making, particularly confronting a Condorcet Paradox. Afterwards, in the third experiment, group decision making is evaluated based on real-world Sushi preference data.

On synthetic data

In the first and second experiments, preferences are generated in quasi order with missing data. That is to say, between alternatives a i and a j , in addition to the four preference relations , ≺, ≈, and ∼ (indecisive), the case of missing information (undecided ) is also considered, denoted as ?.

Exp. 1: Impact of missing data in preference aggregation

Firstly, preference aggregation results with different numbers of each preference relation are compared. Following aggregation rules compatible with quasi order are considered: Borda rule, distance-based consensus rule with KM distance and CS distance, evidential preference with Jousselme distance (denoted as EP J ), and with USP distance (denoted as EP U SP ). In Section 3.1, we mentioned that KM and CS models are not applicable to measuring incomparability as indecisive. To make a fair comparison, preferences with incomparability interpreted as undecided (denoted with the symbol ?) rather than indecisive (∼) are generated. Since traditional preference models without uncertainty are compared, BBAs are categorical in the evidential preference model.

To calculate USP distance, similarity values corresponding to Figure 6 are applied, where extreme values are taken, i.e. the equality conditions in Formulas (30) are accepted. Table 4 illustrates the agent number settings, with # , # ≺ , # ≈ , # ∼ . # ? denoting the number of agents giving each preference relation. These numbers are particularly set to highlight the differences among aggregation strategies.

Aggregation results are illustrated in Table 5. EP X Sand EP X Ω respectively represent the decision on a singleton (in space of Ω pref ) and the power set (in space of 2 Ω pref ) with the evidential preference model. X is written as J and U SP to respectively denote Jousselme distance and USP distance. Only non-zero values in the combined BBAs are illustrated in [START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF] Table 4: Experiment settings of Exp. 1: column "combined BBA" (without the value on ∅). From the results, we draw the following conclusions:

SettingN • # # ≺ # ≈ # ∼ # ? 1 5
N • KM CS Borda EP J S EP J Ω EP U SP S EP U SP Ω combined BBA singletons m ω ω ≺ ω ≈ Ω 1 ? ≈ Ω ≈ ∪ ≺ ∪ ≈ or Ω 0.
sim ≺ ≈ ∼ 1 0 1/3 0 ≺ 0 1 1/3 0 ≈ 1/3 1/3 1 0 ∼ 0 0 0 1
1. In N • 1, 2, 3, 5, 8, 9, KM and CS models return undecided case (denoted by ?). This indicates that consensus rules based on axiomatic distances are not appropriate when a big part of data is missing. 2. From N • 1˜6, we observe that Ω (equivalent to the undecided case ?) is one of the possible decision results in EP J Ω and EP U SP Ω. This issue is similar to the above one and reflects that decisions in space of 2 Ω are easily influenced by missing data. 3. The decision result from the Borda rule is always precise. However, relations ≈, ∼ and ? are not distinguished. Indeed, the Borda rule merely compares the number of and ≺ with other types of preferences neglected. 4. In strategies for evidential preference model, between two distances in settings N • 3 and 4, EP U SP S gives a more precise result than EP J S. This is because those decisions on singletons are discriminated against with USP distance.

Exp. 2: Condorcet's paradox avoidance in preference aggregation

Condorcet's paradox is a traditional issue in preference aggregation problems, reflecting a phenomenon that collective preferences are cyclic while all agents' preferences are not. An initial discussion and with an avoidance solution evidential preference model is introduced in [START_REF] Zhang | Preference fusion and condorcet's paradox under uncertainty[END_REF]. In this experiment, we demonstrate different results between EP U SP and EP J strategies concerning Condorcet's paradox.

In Figure 7, we generate BBA of three agents U = {u 1 , u 2 , u 3 } expressing their uncertain preferences on three alternatives A = {a 1 , a 2 , a 3 }. The aggregation results decided by rules EP J and EP U SP are illustrated in Figure 8. It can be observed that EP J decision strategy returns a cyclic preference order, indicating Condorcet's paradox, while EP U SP decision strategy returns indifference among the three alternatives. This shows the advantage of USP distance. Still, we need to point that EP U SP does not definitively eliminate Condorcet's paradox. The example here demonstrates only a specific case. Condorcet's paradox is not definitively avoided by EP U SP . To avoid Condorcet's paradox, we develop an elimination method for the evidential preference model.

1 2 3 u 1 1 2 3 u 2 1 2 3 u 3 m(ω ) m(ω ≺ ) m(ω ≈ ) BBA of agent u 1 (a 1 , a 2 ) 0.7 0 0.3 (a 1 , a 3 ) 0.9 0 0.1 (a 2 , a 3 ) 0.7 0 0.3 BBA of agent u 2 (a 1 , a 2 ) 0 0.9 0.1 (a 1 , a 3 ) 0 0.7 0.3 (a 2 , a 3 ) 0.7 0 0.3 BBA of agent u 3 (a 1 , a 2 ) 0.7 0 0.3 (a 1 , a 3 ) 0 0.7 0.3 (a 2 , a 3 ) 0 0.9 0.1
Figure 7: BBA value settings of evidential preferences from 3 agents. We denote the set of edges (preference relations) as:

1 2 3 EP J 1 2 3 EP U SP m(∅) m(ω ) m(ω ≺ ) m(ω ≈ ) m(Ω) (a 1 , a 2 ) 0.
R Condorcet = {R ij }, R ij ∈ { , ≺},
where i, j are the index of alternative pair (a i , a j ) in the cyclic graph representing Condorcet's paradox. Among all edges in the cyclic graph, the one whose corresponding BBA is closest to the categorical BBA of its opposite relation {ω R oppo ij } 0 is searched, where R oppo ij represent the opposite relation to R ij .Then, this edge is replaced by its opposite relation. Formally, the edge to be replaced, denoted by R replace , is determined by:

R replace = argmin R ij ∈R Condorcet (d(m Ω ij , {ω R oppo ij } 0 )). (32)
This method takes the idea from Dodgson's voting method [START_REF] Dodgson | A method of taking votes on more than two issues[END_REF], and it degrades to Dodgson's method in traditional cases when preferences are certain. This method for Condorcet's paradox elimination is applied in the following experiment on data from the real world.

Exp. 3: Aggregation of conflicting preferences over Sushi preference dataset

In this part, we demonstrate the group decision-making with USP distance in the evidential preference model on the preference dataset [START_REF] Kamishima | Nantonac collaborative filtering: recommendation based on order responses[END_REF] (referred to as SUSHI in the following text). In this dataset, only three preference relations exist between any two alternatives:

, ≺ and ≈. Missing data are considered as total ignorance in the evidential preference model as ∪ ≺ ∪ ≈.

Several most recent Multi-Criteria Decision-Making (MCDM) methods are compared with our method. [START_REF] Ye | A novel decision-making approach based on three-way decisions in fuzzy information systems[END_REF][START_REF] Zhan | Covering-based variable precision fuzzy rough sets with PROMETHEE-EDAS methods[END_REF][START_REF] Zhang | TOPSIS-WAA method based on a covering-based fuzzy rough set: an application to rating problem[END_REF][START_REF] Zhan | Three-way multi-attribute decision-making based on outranking relations[END_REF]. We have chosen PROMETHEE-EDAS [START_REF] Zhan | Covering-based variable precision fuzzy rough sets with PROMETHEE-EDAS methods[END_REF], TOPSIS-WAA [START_REF] Zhang | TOPSIS-WAA method based on a covering-based fuzzy rough set: an application to rating problem[END_REF], and 3WD-MADM method [START_REF] Zhan | Three-way multi-attribute decision-making based on outranking relations[END_REF]. These three methods are fuzzy extensions of traditional PROMETHEE [START_REF] Brans | Note-a preference ranking organisation method: (the PROMETHEE method for multiple criteria decision-making)[END_REF], TOPSIS [START_REF] Tzeng | Multiple attribute decision making: methods and applications[END_REF] and ELECTRE I [START_REF] Roy | Classement et choix en présence de points de vue multiples[END_REF] respectively, which are classic MCDM solutions. Besides, these methods are recent enough to demonstrate the effectiveness of the proposed method. It should be pointed out that the group decision-making method based on evidential preference is only adaptable for mono-criteria, where each alternative is represented by a single value. To adjust the MCDM method for group decision making, the score of each agent is considered as a criterion of alternatives. For example, the scores given by agents u 1 , u 2 , u 3 alternatives a 1 , a 2 , a 3 are represented by matrix H, as shown in Table 6. The weight of each criterion (agent) is identical and all agents play a beneficial role in MCDM. The order data are converted into a score from 9 to 0, with the most preferred given 9 and least given 0. To apply the fuzzy MCDM method and normalize the score, both score and order data are fuzzified by:

F uzzy(a i , u j ) = Score(a i , u j ) max(Score(a i , u)|u ∈ U)) , (33)
where H ij represents score on alternative a i from agent u j . These methods do not consider incomplete cases. Therefore, they are not directly applicable for Sushi preference data. For comparison, missing data are completed by the average value of all existing data from agents in the corresponding region. In the 3WD-MADM method, only the average strategy is compared. Due to the consequence of the data completion strategy that alternatives are close, in parameter selection of ELECTRE I for outranking calculation, we have chosen a relatively small concordance threshold p ELECT RE = 0.55 and large discordance threshold q ELECT RE = 0.45. For a similar reason, the threshold in PROMETHEE-EDAS linear preference function p P ROM ET HEE should be small. We fix p P ROM ET HEE = 0.05 in the preference function integrated within PROMETHEE-EDAS. Without specific description, parameters are selected by default values as in [START_REF] Zhan | Covering-based variable precision fuzzy rough sets with PROMETHEE-EDAS methods[END_REF][START_REF] Zhang | TOPSIS-WAA method based on a covering-based fuzzy rough set: an application to rating problem[END_REF][START_REF] Zhan | Three-way multi-attribute decision-making based on outranking relations[END_REF].

Introduction to Sushi preference dataset

SUSHI gathers the preferences of voluntary agents located geographically in the eastern region (3257 agents) and western region (1742 agents) across Japan (respectively referred to as East and West Japan). Agents are required to express their preference over 10 sushis randomly selected from 100 sushis. The preferences are expressed in two forms: score and order. Scores range from 0 to 4 (5 levels in total). The sparsity of data is therefore 90%. In this case (preferences are given in two forms), the contradiction is inevitable. For example, between two sushis magoru and ebi, an agent may give magoru ebi in order but with identical scores. Details of the data set are depicted in Table 8. The study of Japanese food sociology tells some ground truth knowledge. People from East and West Japan have different taste preferences on Sushis. Generally, Japanese from East Japan prefer more oily and saltier food than from West Japan. The oily level denoted by oil ∈ [0, 4] is also provided for each sushi type. Originally, level 0 refers to the oiliest one and 4 the least. For the consistency with literal significance, we inverse the order, i.e. 0 refers to the least oily and 4 the most.

Aggregation and evaluation

In SUSHI, the responses in score and order from an identical agent are cognitively dependent. Therefore, the mean rule (Equation ( 4)) is suitable for the aggregation of personal preference. The preference aggregation procedure is illustrated in Figure 9, as an extended version to Figure 5 The evaluation of preference aggregation results is tricky. If an evaluation criterion on aggregated preference is fixed, it is possible to find a method based on the optimization [START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF] of this criterion. Therefore, the aggregation results are usually evaluated by the effects or consequences, such as feedback of services [START_REF] Alonso | A web based consensus support system for group decision making problems and incomplete preferences[END_REF][START_REF] Bharati | An empirical investigation of decision-making satisfaction in web-based decision support systems[END_REF] or knowledge of ground truths. These criteria require additional information.

In our experiments, we evaluate the preference aggregation results in two aspects. Firstly, we focus on the consensus degree between the final result and each expert. Secondly, we compare the result with the ground truth knowledge in terms of food habits in Japan. In the first evaluation, the Average Deviation Index (ADI) is applied, defined in Equation ( 34) [START_REF] Chen | The criteria for evaluating the effectiveness of preference aggregation methods[END_REF]:

ADI = 1 |S| s∈S τ (σ agg , σ s ), (34)
where, σ agg refers to the aggregated preference order, σ s the preference order of agent s, and τ the normalized Fagin distance [START_REF] Fagin | Comparing top k lists[END_REF]. Table 9 summarizes the ADI of all methods in comparison, including four rules of evidential preferences on orders and scores. We observe that the decision rule of EP U SP provides aggregation result with more consensus than EP J , on both scores and orders. Besides, EP U SP returns a similar result as TOPSIS-WAA, with a little improvement. The results of 3WD-MADM and PROMETHEE-EDAS are relatively unsatisfactory. The reason is similar: 3WD-MADM is calculated based on pairwise rankings with integrated ELECTRE I. PROMETHEE-EDAS is calculated based on in/out flow on pairwise alternatives. Since the data set is highly sparse and missing values are filled in with the mean value of observed data, most criteria are identical between two alternatives, and the distances between alternatives are relatively small. In the result of ELECTRE I, almost half of the alternatives in our experiments are indifferently ranked as least preferred. In the result of PROMETHEE, most preferences are weak, which the ranking result unreliable. However, ADI is not always a convincing and reliable homogeneous evaluation criterion. Lower ADI can be reached by applying a Kendall distance τ minimization strategy. This also explains the satisfactory performance of EP U SP . Indeed, in cases without uncertainty nor imprecision, the average value of USP distance between two preference orders is identical to Kendall distance with Fagin's extension [START_REF] Fagin | Efficient similarity search and classification via rank aggregation[END_REF].

To give a more concrete conclusion, in the second place, the preference aggregation results are evaluated with a piece of sociology knowledge in Japan about eating habits: people from east Japan usually prefer more oily sushi than people from west Japan. If the aggregated preferences can successfully discriminate the oiliness choice largely from the other region, the aggregation rule is believed to be appropriate. Thus, for verification, the average oiliness index (AOI) is implemented to the whole aggregated preference order. With the oiliness degree of sushi a i denoted by oil(a i ), we define AOI on the set of sushi A sushi as:

AOI(A sushi ) = 1 |A sushi | a i ∈A sushi oil(a i ). (35)
The comparison of AOI on top-k sushis between East and West Japan is calculated respectively upon the difference:

AOI dif f,k = AOI(A East k ) -AOI(A W est k ), (36)
where, AOI(A East 36), positive AOI dif f means oily sushi is more preferred in east Japan than in west Japan, and vice versa for negative AOI dif f values. It is obvious that AOI dif f has positive values globally across different k values for decision rules based on both Jousselme and USP distance, especially for top-10 sushis. The results of EP U SP and EP J correspond generally the sociological knowledge mentioned before. Nevertheless, around k = 20, AOI dif f values drop to negative, for the reason that, the 19th and 20th favorite sushis in the order are much less oily in the east than in the west Japan. The result of TOPSIS-WAA has a similar trend with EP methods but corresponds less with the sociology knowledge. 3WD-MADM and PROMETHEE-EDAS return rather random results. This does not imply that these two methods are ineffective in decision-making. Indeed, the unsatisfactory results are mainly caused by the data completion strategy. With most of the missing data completed by the mean value, the difference between alternatives is relatively little. Therefore, MCDM methods based on preference intensity encounter problems in distinguishing the alternatives.

Still, we need to note that people in east Japan prefer more oily sushi is a rough piece of knowledge rather than a strict rule. This statement does not imply that all oily sushi should be preferred in east Japan than west, and it is common that some oily sushis are preferred in both regions, as the existence of several oily sushi more welcomed in east Japan is always reasonable.

Conclusion and discussion

In this paper, we originally bring the discussions on the ambiguity in the interpretation of incomparability. We find that most of the previous studies do not distinguish between interpretations of incompatibility according to different situations. In some works, incomparability is interpreted as undecided because of certain criteria, while in others it is interpreted as undecided because of a lack of observed information. This ambiguity could lead to controversy in dissimilarity values, especially for pair-wise preference relations. Moreover, reasoning the undecided case as indifference is not a faithful way and may lead to unexpected consequences. This issue has rarely been discussed in the research community. Therefore, in order to clarify the different dissimilarity measures at the definition level, we identify and categorize the two interpretations of incomparability as indecisive and undecided respectively. With the help of the evidential preference model based on the theory of belief functions, we are able to distinguish indecisive and undecided interpretations and indifference relation faithfully in a unified framework: the indifference and indecisive cases are exclusive with other preference relations by being considered as specific binary relations, while the undecided case of incomparability is considered as total ignorance represented by the union of all possible relations.

In the second place, we reveal a flaw in existing dissimilarity measures in the theory of belief functions while applied on evidential preferences after a thorough study on existing distances between preference relations. Due to the property of belief functions, dissimilarities between exclusive singletons are considered equal, which raises a contradiction with all previous works in preference modelling. Obviously, this property of belief functions is not suitable for evidential preferences. To solve this issue, we propose a novel dissimilarity measure function, named USP distance, for BBAs (evidential objects with both uncertainty and imprecision). USP distance is an extension of Jousselme distance, which takes the difference of dissimilarity between singletons into consideration. In the case that dissimilarities between different singletons are equal, USP distance degrades to Jousselme distance. USP distance is applied in group decision making with evidential preference and shows a better performance than Jousselme distance. The advantages of USP distance are also justified by experimental comparison with several MCDM methods.

In this paper, the application of USP distance is limited in evidential preference aggregation. USP distance can have more potential utilities in decision making, not only limited in the circumstances of preference modelling. For example, in a decision system where some alternatives are riskier than others, it is unfavourable to choose the risky alternatives. Traditionally, such discrimination is realised by utility functions, as introduced in [START_REF] Denoeux | Decision-making with belief functions: A review[END_REF]. USP distance brings another possible solution. By assigning the pair-wise dissimilarity with different values, singletons are possible to be discriminated against by USP distance. A comparative study between the two solutions is in the scope of our future work.

Moreover, other distance-based applications, notably learning tasks on evidential objects with USP distance, are left to be justified. Here, we propose a potential issue in this task as a discussion topic without being solved, with the example of clustering over evidential preferences. Clustering preferences is an important technique, especially in recommendation systems and social networks where users are profiled by their preference information. For orders consisting of crisp preference relations, the USP distance degrades to Kendall's τ distance in total orders and Fagin's distance [START_REF] Fagin | Efficient similarity search and classification via rank aggregation[END_REF] in weak orders. Thus, it is appropriate to apply USP distance for the classification on complete preference orders. A similar application using Jousselme distance has been justified in [START_REF] Zhang | A clustering model for uncertain preferences based on belief functions[END_REF]. Nevertheless, for incomplete and imprecise preferences, where the impact of ignorance is important, an issue becomes obvious: the missing preference information modelled by vacuous BBAs (see Definition 5) are identical and are classified into the same group based on the dissimilarity. With such results, imprecise information is brutally clustered into crisp clusters, losing the significance of clustering over preferences. A more rational classification result should be that these vacuous BBAs are not classifiable, i.e. the membership of these vacuous BBAs to any group is also a knowledge of total ignorance. Therefore, it is reasonable to doubt the appropriateness of existing distance functions in the theory of belief functions, including USP distance. Generally speaking, this issue is due to the failure of existing distances measuring over evidential objects with imprecision. To deal with this issue, a study on the properties of dissimilarity measure for learning tasks over evidential objects is necessary within the scope of our future work.

Figure 1 :

 1 Figure 1: Venn diagram of preference relations.

  (a) "Incomparability" as indecisive.(b) "Incomparability" as undecided.

Figure 2 :

 2 Figure 2: Venn diagrams of the preference relations.

Figure 3 :

 3 Figure 3: Inequality relation applied in axiomatic and Minkowski preference distances. d ∆ : distance between two preference relations; ?: incomparability as unobserved information (undecided); ∼: incomparability as a specific preference relation (indecisive).

Property 4 .

 4 Dissimilarity function d forms with such structure:

( a )

 a Element in original space S. (b) Element in space S * with different similarity values.

Figure 4 :

 4 Figure 4: Graphical representation of similarity matrix calculation.

Figure 5 :

 5 Figure 5: Group decision making procedure.

Figure 6 :

 6 Figure 6: Similarity value settings and graphical illustration in Exp. 1.

Figure 8 :

 8 Figure 8: Decision results from two BFpref consensus rules.

  in Section 4.5.

Figure 9 :

 9 Figure 9: Combination procedure for sushi preferences.

k)

  and AOI(A W est k ) refer respectively to the aggregation orders of top-k sushi for East and West Japan.

Figure 10 :

 10 Figure 10: Sushi AOI difference between East Japan and West Japan.

Figure 10

 10 Figure10depicts the AOI dif f of top-k preferred sushi decided by all methods in comparison, with k varying from 5 to 100. For 3WD-MADM, k varies from 5 to 40 as the last 60 sushis are all considered indifferently as not preferred. According to the definition of AOI in Equation[START_REF] Essaid | A distance-based decision in the credal level[END_REF], positive AOI dif f means oily sushi is more preferred in east Japan than in west Japan, and vice versa for negative AOI dif f values. It is obvious that AOI dif f has positive values globally across different k values for decision rules based on both Jousselme and USP distance, especially for top-10 sushis. The results of EP U SP and EP J correspond generally the sociological knowledge mentioned before. Nevertheless, around k = 20, AOI dif f values drop to negative, for the reason that, the 19th and 20th favorite sushis in the order are much less oily in the east than in the west Japan. The result of TOPSIS-WAA has a

Table 1 :

 1 Axioms accepted in different preference models:

Table 2 :

 2 Encoding of preference relations and Hamming distance:

Table 3 :

 3 Similarity sim S

* between singletons:

Table 5 :

 5 Results of Exp. 1:

Table 6 :

 6 Example: preference data in score:

	Score u 1 u 2 u 3
	a 1	4	3	2
	a 2	1	3	2
	a 3	3	0	4

Table 7 :

 7 Example: Scores converted from preference data in order:

	Order No. u 1	u 2	u 3		Converted score u 1 u 2 u 3
	a 1 a 2	3rd 2nd 1st 3rd 1st 2nd	⇒	a 1 a 2	0 3	1 1	2 2
	a 3	2nd 3rd 1st		a 3	2	3	1

Table 8 :

 8 Sushi Preference Dataset:

	# agents # alternatives sparsity	conflict between order and score	remark
	East Region 3257	100	90%	3.45‰	more oily sushi
	West Region 1742	100	90%	3.44‰	less oily sushi

Table 9 :

 9 Aggregated conflicting Sushi preferences with ADI:

		East Japan	West Japan
		Order	Score	Order	Score
	EP J	0.2685	0.2395	0.2661	0.2417
	EP U SP	0.1462	0.1172 0.1346 0.1103
	PROMETHEE-EDAS	0.1821	0.1761	0.2603	0.2609
	TOPSIS-WAA	0.1439 0.1960	0.1449	0.1666
	3WD-MADM(ELECTRE I) 0.4878	0.4638	0.6445	0.4356

[START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF]

As a i a j is equivalent to a j ≺ a i , to avoid repetitive comparisons between two alternatives, we assume that i < j in this article.

In some work, weak order is also referred to as "partial ranking", while "partial order" is another type of preference structure.[START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF]

Strictly, "conflict" is not a dissimilarity measure, or semi-metric, as the identity of in-discernment property does not hold.[START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF]

This fuzzy value does not represent a fuzzy preference. Conventional fuzzy preference express the relation between two alternatives by only one membership with 1 representing strict preference, 0 the reverse strict preference, and 0.5 the indifference.[START_REF] Xiao | A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion[END_REF]