A Measurement for Essential Conﬂict in Dempster-Shafer Theory

Wenjun Ma, Jieyu Zhan

∗

and Yuncheng Jiang

Guangzhou Key Laboratory of Big Data and Intelligent Education, School of Computer Science,

South China Normal University, Guangzhou, China

Keywords:

D-S Theory of Evidence, Conﬂict Management, Essential Conﬂict, Information Fusion.

Abstract:

Dempster’s combination rule in Dempster-Shafer (D-S) theory is widely used in data mining, machine learn-

ing, clustering and database systems. In these applications, the counter-intuitive result is often obtained with

this rule when the combination of mass function is performed without checking whether original beliefs are

in conﬂict. In this paper, a new type of conﬂict called essential conﬂict has been revealed with two character-

istics: (i) it is an essential factor that leading to the counter-intuitive result by turning a possible state into a

necessary true state or an impossible state; (ii) it cannot be corrected by the combination process of any new

mass functions. After showing that the existing conﬂict measurements in D-S theory have the limitations to

address the essential conﬂict and presenting a formalism about the concept of essential conﬂict, we propose

a measurement of essential conﬂict between two mass functions based on the mass value and the intersection

relation of their focal elements. We argue that if there exists a focal element of one mass function, such that

the intersection of it and any focal element of another mass function is an empty set, then the essential conﬂict

is caused and Dempster’s combination rule is not applicable.

1 INTRODUCTION

Dempster-Shafer (D-S) theory (Dempster, 2008;

Shafer, 1976) is a powerful tool for modelling and

reasoning with ambiguous information in applications

(Ma et al., 2013), such as information fusion (Hong

et al., 2016), pattern recognition (Jiang et al., 2016),

and decision making (Ma et al., 2017). In this theory,

when multiple pieces of evidence for a proposition

are accumulated from multiple distinct sources, they

need to be combined to see how strongly they support

the proposition together (Jiang and Zhan, 2017). And

Dempster’s combination rule is widely employed to

do this. Nevertheless, many researchers challenge its

validity and consistency when it is used to combine

highly conﬂicting evidence (Destercke and Burger,

2012; Liu, 2006), which makes the use of Dempster’s

combination rule questionable.

To remedy this weakness, two major viewpoints

have been proposed. The ﬁrst viewpoint suggests that

we should develop a new combination rule to replace

Dempster’s combination rule and redistribute the con-

ﬂict, while the researches holding with the second

viewpoint suggest that we should consider the con-

ditions in which Dempster’s combination rule is safe

∗

Corresponding author

to be used and modify the belief function if the con-

ditions are unsatisﬁed. Nevertheless, in the current

methods, the alternative rules do not get wide ac-

ceptance in real-world application, and the proposed

Dempster’s rule combination conditions is controver-

sial. (More details discuss in Section 3)

Since for both viewpoints, the fundamental ques-

tion that what does conﬂict mean among evidence is

important and still an open issue, in this paper, we fo-

cus on this issue and propose to study the notion of

conﬂict from a different perspective based on the in-

tersection of the focal elements of two original mass

functions. More speciﬁcally, ﬁrstly in this paper we

reveal a new type of conﬂict: essential conﬂict. Two

characteristics of it are analysed. (1) Belief Absolu-

tization: such conﬂict can turn a possible state of an

original mass function into a necessary true state or an

impossible state in the combination result with Demp-

ster’s combination rule; (2) Uncorrectable Assertion:

such conﬂict cannot be corrected by the combination

of new mass function. Then after presenting a for-

mal deﬁnition of essential conﬂict and revealing the

properties of conﬂict in the combination process, we

argue essential conﬂict is a more importance factor

to show the essence and uncorrectable disagreement

between sources. Hence, a measurement of essential

conﬂict is given between two mass functions in D-S

1282

Ma, W., Zhan, J. and Jiang, Y.

A Measurement for Essential Conﬂict in Dempster-Shafer Theory.

DOI: 10.5220/0010394512821289

In Proceedings of the 13th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2021) - Volume 2, pages 1282-1289

ISBN: 978-989-758-484-8

theory based on their focal element set and the origi-

nal mass values. Also, we will analyse the properties

of such new conﬂict measurement. Finally, examples

are given to illustrate the advantages of our method.

This paper advances the state of the art in the

area of D-S theory in the following aspects: (i) re-

veal a new type of conﬂict in the combination process:

essential conﬂict that highly related to the counter-

intuitive results with Dempster’s combination rule;

(ii) give a formal deﬁnition to represent the essential

conﬂict; (iii) design a measurement for identifying

conﬂict between two mass functions more effectively

and less conservatively.

The rest of this paper is organised as follows. Sec-

tion 2 recaps background knowledge. Section 3 dis-

cusses related work. Section 4 gives a formal deﬁni-

tion of the essential conﬂict. Section 5 analyses the

properties of the essential conﬂict. Section 6 gives

a novel essential conﬂict measurement with desirable

properties and illustration examples. Finally, Section

7 concludes the paper with future work.

2 PRELIMINARIES

This section recaps some base concepts in D-S theory.

Deﬁnition 1. (Shafer, 1976) Let Θ = {ω

, . . . , ω

} be

a set of exhaustive and mutually exclusive elements

(i.e., states of the world), called a frame of discern-

ment (or simple a frame). Function m :2

→[0, 1] is a

mass function if m(

0) = 0 and

∑

A⊆Θ

m(A) = 1.

Here a mass function is completely ignorable if

and only if m(Θ) = 1, and F

is the focal element set

of the mass function m if for any B ∈ F

, m(B) > 0.

Deﬁnition 2. (Shafer, 1976) Let m

and m

be two

mass functions from independent and fully reliable

sources over a frame of discernment Θ. Then the com-

bined mass function from m

and m

by Dempster’s

combination rule, denoted as m

1,2

, is deﬁned as:

1,2

(x) =

(

0 if x =

1−k

(

∑

B=x

(A)m

(B)) if x 6=

(1)

where normalization constant

∑

(A)m

(B) < 1 (2)

is a classical conﬂict coefﬁcient to measure the con-

ﬂict between the pieces of evidence.

Deﬁnition 3. (Smets, 2005) Let m be a mass function

over Θ. Its associated pignistic probability function

BetP

: Θ → [0, 1] is deﬁned as:

BetP

(ω) =

∑

A⊆Θ,ω∈A

|A|

m(A)

1 − m(

, m(

0) < 1, (3)

where |A| is the cardinality of A.

Here, when an initial mass function gives m(

0) =

m(ω)

1−m(

is reduced to m(ω). BetP

(ω) is a probabil-

ity measure. It tells what is the total mass value that

a state ω can carry for decision making based on the

corresponding evidence referred by mass function m.

Finally, although Dempster’s combination rule

has been used in many real world applications, it

has been criticised upon some of its counter-intuitive

combination results (Liu, 2006; Zadeh, 1986). Per-

haps the most famous one is as follows:

Example 1 (Zadeh’s counter-example (Zadeh,

1986)). Let m

and m

be two mass functions deﬁned

on a frame of discernment Θ = {a, b, c} with:

({a}) = 0.9, m

({b}) = 0.1, m

({c}) = 0;

({a}) = 0, m

({b}) = 0.1, m

({c}) = 0.9.

They mean that option a is strongly supported by the

ﬁrst piece of evidence but absolutely denied by the

second one, option b is weekly supported by both,

and option c is strongly supported by the second but

absolutely denied by the ﬁrst. By Deﬁnition 2, if we

use Dempster’s combination rule to combine the two

mass functions, we have m

({b}) = 1.

That is, option b, hardly supported by each piece

of evidence, turns out to be fully supported after the

combination of the two mass functions. Therefore,

Zadeh argued that such a result is highly violated our

intuition about the evidence combination.

3 RELATED WORK

In general, there are two major viewpoints to im-

prove Dempster’s combination rule to resolve Zadeh’s

counter-example.

From the ﬁrst viewpoint, the counter-intuitive re-

sults were caused by Dempster’s combination rule, so

they modiﬁed combination rule and proposed a num-

ber of new evidential combination rules to remove the

defect (e.g., (Chebbah et al., 2015; Deng et al., 2014;

Dubois and Prade, 1988a; Elouedi and Mercier, 2011;

Smarandache and Dezert, 2006; Yager, 1987)). How-

ever, without a general mechanism to accurately mea-

sure the degree of conﬂict other than using the con-

ﬂict coefﬁcient k (which cannot measure the conﬂict

in the desired way), such rules are actually ad hoc be-

cause of lacking a theoretical justiﬁcation. As Smets

(Smets, 2007) pointed out, the pragmatic fact “our

rule works ﬁne” in some application cases is of course

not a proper justiﬁcation (at most a necessary con-

dition). Moreover, since the alternative combination

rules always cause higher computation complexity,

A Measurement for Essential Conﬂict in Dempster-Shafer Theory

1283

give up some desirable properties (i.e., associative and

commutative) in combination process, and encounter

new counter-intuitive behaviours in applications, such

rules do not get wide acceptance in the real-world ap-

plications. (Deng, 2015; Jiang and Zhan, 2017)

From the second viewpoint, the counter-intuitive

results were caused by abusing Dempster’s combina-

tion rule inappropriately, so they limited the condi-

tion that the Dempster’s combination rule can be used

by reconstructing the mass function (i.e., discounting

mass function and weighted averaging mass function)

(Deng et al., 2004; Dubois and Prade, 1988b; Jiang

et al., 2016; Murphy, 2000; Shafer, 1976; Smets,

2000; Wang et al., 2016; Zhao et al., 2016), or in-

troducing the open-world assumption (Deng, 2015;

Jiang and Zhan, 2017; Smets, 2000), or managing

conﬂict with conﬂict measurement (Daniel, 2014;

Jousselme et al., 2001; Jiang, 2018; Liu, 2006; Zhao

et al., 2016). For the cases of discounting mass func-

tion (Dubois and Prade, 1988b; Shafer, 1976; Smets,

2000; Zhao et al., 2016) and introducing the open-

world assumption, they just evade the criticism of

Dempster’s combination rule by making an additional

assumption, such as the assumption that the evidence

cannot be all fully reliable or the frame of discern-

ment cannot be exhaustive. Therefore, in some cases,

such ideas are too conservative since they give a too

strong limitation for the Dempster’s combination rule.

And for the cases of using weighted averaging mass

function (Deng et al., 2004; Jiang et al., 2016; Mur-

phy, 2000; Wang et al., 2016), they will cause an-

other counter-intuitive behaviour, since the original

mass function can be changed after combining with a

completely ignorable mass function (Ma et al., 2019).

Thus, the conﬂict management with conﬂict measure-

ment is the most common way for the second view-

point. Since the conﬂict coefﬁcient k in Deﬁnition 2

cannot represent conﬂict reasonably, various conﬂict

measurements are proposed to quantify the opposi-

tion between mass function, such as the relative co-

efﬁcient Jousselme distance (Jousselme et al., 2001),

non-intersection correlation coefﬁcient (Jiang, 2018),

pignistic probability distance (Liu, 2006), plausibil-

ity conﬂict measurement (Daniel, 2014), and so on.

However, although the conﬂict measurements are var-

ious and fruitful, it is still inconclusive for what con-

ﬂict is and where it comes from. (Jiang, 2018)

All in all, by two major viewpoints to resolve the

conﬂict problem in D-S theory, we ﬁnd that the fun-

damental question is what does conﬂict mean among

evidence. Moreover, since the conﬂict problem in D-

S theory is due to criticisms on the counter-intuitive

result of applying Dempster’s combination rule (e.g.,

Example 1), there at least exists a type of conﬂict that

should be highly related to the counter-intuitive re-

sult. In this vein, we would like to make a claim

about the relationship between such type of conﬂict

and counter-intuitive combination result as follows:

Claim 1. For any two combination results with

Dempster’s combination rule, there at least exists

a type of conﬂict, such that its value of a counter-

intuitive combination result should be higher than its

value of a non-counter-intuitive combination result.

Since such type of conﬂict is highly related to the

counter-intuitive combination result, we would like to

call it essential conﬂict.

4 FORMAL DEFINITION OF

ESSENTIAL CONFLICT

Now, we show that the existing conﬂict measurements

have the limitations to address essential conﬂict.

First, we discuss the cases in which the existing

conﬂict measurements in D-S theory will infer a low

conﬂict while a counter-intuitive combination result

occurs. Consider the following example:

Example 2. let m

and m

be two mass functions pro-

vided by two distinct and totally reliable sources on a

frame of discernment Θ = {a, b, c} that:

({a})=m

({a})=0.8, m

({b})=m

({c})=0.2.

Clearly, by Deﬁnitions 1 and 2, the maximum dif-

ference of mass value for Example 2 is |m

({b}) −

({b})| = 0.2, the classical conﬂict coefﬁcient is

= 0.36, and the combination result is m

({a}) =

1. Following the idea of most current conﬂict mea-

surements, it should be a low conﬂict combination re-

sult. However, the combination result that a possible

state (i.e., state a) turns into a necessary true state and

some possible states (i.e., states b and c) turn into the

impossible states is somehow arbitrary. This combi-

nation result strongly violates our intuition.

Therefore, in Example 2, low difference of mass

value and low value of conﬂict coefﬁcient k cannot

indicate the Dempster’s combination rule is safe to be

applied. Thus, the essential conﬂict that highly related

to a counter-intuitive combination result in Example 2

cannot fully represented by the current methods.

Second, we discuss the cases that the existing con-

ﬂict measurements in D-S theory will infer a high

conﬂict while an acceptable justiﬁcation for the com-

bination result occurs by the following example:

Example 3. let m

and m

be two mass functions pro-

vided by two totally reliable sources on a frame of

discernment Θ = {a, b} that:

({a})=m

({b})=0.9, m

({b})=m

({a})=0.1.

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

1284

Clearly, the maximum difference of mass value

assign to set {a} (or {b}) between m

and m

({a}) − m

({a})| = 0.8 and classical conﬂict co-

efﬁcient is k

= 0.82 by Deﬁnition 2. Under the cur-

rent methods of conﬂict measurement, this difference

and conﬂict coefﬁcient value would warrant a verdict

that these two pieces of evidence are in high conﬂict

and Dempster’s rule should not be used.

However, the combination result of m

and m

with Dempster’s rule is m

({a}) = m

({b}) = 0.5.

This result seems to satisfy our intuition that as both

pieces of evidence are totally reliable and give differ-

ent judgements about an issue, in order to make an

agreement in the combination process, we make an

equally concession for the judgements of both orig-

inal beliefs. This phenomenon is common in real-

world applications. For example, a policeman thinks

that a suspect A should be the murderer of a murder

crime while another policeman thinks that suspect B

should be the murderer of this murder crime. If both

policemen are totally reliable, we prefer to consider-

ing that they make their judgements based on differ-

ent points of view and give equal attention to both

suspects.

Compared Example 3 with Example 1, although

two examples both have high difference of mass value

and high classical conﬂict coefﬁcient, we thinks the

combination result of Example 3 can be justiﬁed with-

out violating our intuition while that of Example 1 is

totally unacceptable. Thus, in Example 3, high differ-

ence of mass value and high value of k cannot indicate

high essential conﬂict value.

In other words, by Example 2 and 3, the differ-

ence of the mass values between two mass functions

and classical conﬂict coefﬁcient, which are two ma-

jor component parts of the current methods of conﬂict

measurement, is not the essential factor that prevents

us to use Dempster’s rule without causing the counter-

intuitive results. As a result, a formal deﬁnition and

conﬂict measurement for the essential conﬂict is re-

quired if we want to combine the evidence safely with

Dempster’s combination rule.

By analysing the set structures of Examples 1-3

before and after the combination result, we ﬁnd that

for Examples 1 and 2, there exists at least one focal el-

ement of one mass function, such that the intersection

of it and any focal element of another mass function

is an empty set. While for Example 3, such situation

does not exist. Following this idea, we infer that such

issue should be a main feature for deﬁning essential

conﬂicts. Formally, we have

Deﬁnition 4. Let m

and m

be two mass functions

over a frame Θ, F

and F

be the focal element sets

of m

and m

, respectively. Then ϒ

⊆ Θ is a set

of essential conﬂict elements with the mass function

and m

if and only if for any ω ∈ ϒ

, there exists

A ∈ F

∧ ω ∈ A, such that for any B ∈ F

, we have

A ∩ B =

0 (i 6= j and i, j ∈ {1, 2}).

In addition, if ϒ

0, then m

and m

are in es-

sential conﬂict.

In fact, by Deﬁnition 4, we can ﬁnd that if A ∈ F

and ∀B ∈ F

, A ∩ B =

0 (i 6= j and i, j ∈ {1, 2}), then

for any C ⊆ A, we have:

C ∩ B =

0, and

∑

Y =C

(X)m

(Y ) = 0.

In other words, the combined mass value of any sub-

set of any A ∈ ϒ

is zero by applying Dempster’s

combination rule (i.e., formula (1)). So, it means no

matter how many new mass function m

consider A

as a focal element and how high mass value m

(A) is,

if subset A belongs to the conﬂict element set (i.e.,

A ∈ ϒ

), the result of Dempster’s combination rule

still rules it and its more special subsets out of the set

of the possible states. This is the reason why Demp-

ster’s combination rule could cause some counter-

intuitive behaviours such as Example 1.

Now, we will apply Deﬁnition 4 to analyse Ex-

amples 1-3. For Example 1, after checking the focal

elements of each mass function (i.e., {a} and {b} for

, and {b} and {c} for m

), we can ﬁnd that

(1) m

({a}) > 0, and for any A ⊆ {a, b, c} ∧ A ∩

{a} 6=

0, we have m

(A) = 0, and

(2) m

({c}) > 0, and for any B ⊆ {a, b, c}∧B∩{c} 6=

0, we have m

(B) = 0.

Thus we have ϒ

= {a, c}. Since ϒ

0, m

and m

are in essential conﬂict in Example 1. Similarly, for

Example 2, we have ϒ

= {b, c} 6=

0. Thus, m

and

are in essential conﬂict in Example 2. Finally, for

Example 3, we ﬁnd that for all focal elements of m

(i.e., {a} and {b}), there exists a focal element of m

(i.e., {a} and {b}), such that the intersection of them

is not an empty set. Thus, we have ϒ

0. It means

and m

are not in essential conﬂict.

And such results of Examples 1-3 show that our

deﬁnition of essential conﬂict indeed reveals the rela-

tion between conﬂict and counter-intuitive combina-

tion result by applying Dempster’s combination rule.

5 PROPERTIES OF ESSENTIAL

CONFLICT

In this section, we will reveal two properties (i.e.,

belief absolutization and uncorrectable assertion) of

the essential conﬂict that make it as a main factor

A Measurement for Essential Conﬂict in Dempster-Shafer Theory

1285

to cause the counter-intuitive combination result by

Dempster’s combination rule.

Theorem 1 (Belief Absolutization). Let m

and m

be two mass functions over a frame of discernment

Θ that are in essential conﬂict, m

be the combina-

tion result of m

and m

with Dempster’s combination

rule, and BetP

, BetP

and BetP

be the pignistic

probability function of m

, m

and m

, respectively.

Then there exists ω ∈ Θ, such that BetP

(ω) > 0 or

BetP

(ω) > 0 but BetP

(ω) = 0.

Proof. Suppose F

, F

and F

are the focal element

sets of m

, m

and m

, respectively. Then by Deﬁ-

nition 4, if m

and m

are in essential conﬂict, there

exists A ∈ F

∧ ω ∈ A, such that for any B ∈ F

, we

have A ∩ B =

0 (i 6= j and i, j ∈ {1, 2}). Without loss

of generality, we assume A ∈ F

, then by Deﬁnitions 1

and 3, we have m

(A) > 0 and BetP

(ω) > 0. Since

∀B ∈ F

, A ∩ B =

0 and ω ∈ A, we have for any C ∈ F

and any B ∈ F

, ω 6∈ C ∩ B. Thus, ω 6∈ F

. Then, by

Deﬁnitions 2 and 3, we have BetP

(ω) = 0.

Since the pignistic probability function BetP

(ω)

tells what is the total mass value that a state ω can

carry for decision making based on the corresponding

evidence referred by mass function m, BetP

(ω) > 0

for i ∈ {1, 2} means one of the original mass functions

and m

support the claim that there exists some

chance that the state ω turns out to be the real state,

while BetP

(ω) = 0 means it is impossible that ω

turns out to be the real state by the combination result

of m

and m

. Therefore, Theorem 1 means that if the

essential conﬂict exists for two mass functions with

Dempster’s combination rule, then there at least ex-

ists a possible state ω supported by the original mass

function turns out to be an impossible state after the

combination result. To make matters worse, if only

one possible state of a frame does not belong to the set

of essential conﬂict elements deﬁned by Deﬁnition 4

for two mass functions, then such possible state will

turn out to be a necessary true state after combination

of the two mass functions by Dempster’s combina-

tion rule. And this is the exactly reason why Zadeh’s

counter-example (i.e., Example 1) occurs.

Before discussing the property of uncorrectable

assertion, we ﬁrst deﬁne the concept of correctable in

information fusion by Dempster’s combination rule.

In real world application of intelligent surveillance,

for the reason of limited surveillance, time pressure,

the scotomas of cameras, the deﬁnition and sharp-

ness of images, disturbance of unknown factors (such

as signal interference, a sudden jarring or jerking),

and so on, a sensor might produce a fault evidence

for a given target that disagrees with the other in-

formation sources. And the evidence combination

with such fault evidence will lead to a wrong judge-

ment about the surveillance target. However, in some

cases, such wrong judgement can be corrected by

the further information fusion of the additional ev-

idences. For example, suppose the mass functions

and m

in Example 3 are provided by two in-

formation sources in an intelligent surveillance sys-

tem and the true state is b. Thus, the combination

result m

({a}) = m

({b}) = 0.5 somehow devi-

ated from the correct judgement made by m

. How-

ever, if we have a new mass function m

provided by

an additional information source with m

({a}) = 0.1

and m

({b}) = 0.9, then the combination result of

and m

for the state b is exactly the same as

({b}) = 0.9. Thus, we can say the mistake or the

deviation caused by m

is corrected by additional evi-

dence combination. Formally, we can deﬁne the con-

cept of correctable for the original combination result

as follows:

Deﬁnition 5. Let m

and m

be two mass functions

over a frame of discernment Θ, m

be the combina-

tion result of m

and m

with Dempster’s combination

rule, and BetP

and BetP

be the pignistic proba-

bility functions of m

and m

, respectively. Then the

combination result m

is correctable if for any ω ∈ Θ

and BetP

(ω) (i ∈ {1, 2}), we can always construct

an additional mass function m

, such that for the com-

bination result m

123

of the mass functions m

, m

and

, we have

BetP

123

(ω) = BetP

(ω).

Here, the pignistic probability function BetP

(ω)

works as a probability measurement to represent the

total mass value of m that a state ω can carry for de-

cision making (Smets, 2005). Thus, BetP

123

(ω) =

BetP

(ω) for i ∈ {1, 2} means no matter what judge-

ment the combination result m

makes about the

chance that a state ω turn out to be a true state, with

the combination of an additional mass function m

the judgement of m

can be corrected and m

123

will

share the same judgement with the mass functions

about the chance that a state ω turns out to be a

true state. Thus, even the mass function m

( j 6= i

and j ∈ {1, 2}) makes a wrong judgement about the

chance that a state ω turns out to be a true state, if

the combination result is correctable, the inﬂuence

of m

in the combination process can be eliminated.

Such property of correctable for the combination re-

sult is desirable for the real-world application, since

the wrong judgement of a small portion of the infor-

mation resource cannot prevent the convergence to-

ward truth in the combination process.

Unfortunately, such property of correctable for

the combination result is non-universal. Consider

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

1286

Zadeh’s counter-example in Example 1, the combi-

nation result of the mass functions m

and m

({b}) = 1. In this situation, by Deﬁnition 2, for

any mass function m

with k

123

< 1, the combination

result of m

and m

will be m

123

({b}) = 1. Thus, by

Deﬁnition 5, the combination result in Example 1 is

uncorrectable.

Now, we will show that if two mass functions are

in essential conﬂict, the combination result of them is

uncorrectable by the following theorem.

Theorem 2 (Uncorrectable Assertion). Let m

and

be two mass functions over a frame of discernment

Θ that are in essential conﬂict, m

be the combina-

tion result of m

and m

with Dempster’s combination

rule, and BetP

and BetP

be the pignistic proba-

bility function of m

and m

, respectively. Then there

exists a state ω ∈ Θ and BetP

(ω) (i ∈ {1, 2}), such

that for any mass function m

, we have

BetP

123

(ω) 6= BetP

(ω).

Proof. Suppose F

, F

and F

are the focal element

sets of m

, m

and m

, respectively. Then by Deﬁ-

nition 4, if m

and m

are in essential conﬂict, there

exists A ⊂ Θ, such that A ∈ F

∧ ω ∈ A, and for any

B ∈ F

, we have A ∩ B =

0 (i 6= j and i, j ∈ {1, 2}).

Without loss of generality, we assume A ∈ F

, then

by Deﬁnitions 1 and 3, we have m

(A) > 0 and

BetP

(ω) > 0. Since ∀B ∈ F

, A ∩ B =

0 and ω ∈ A,

we have for any C ∈ F

and any B ∈ F

, ω 6∈ C ∩ B.

Thus, ω 6∈ F

. Then, by Deﬁnitions 2 and 3, we have

BetP

(ω) = 0.

By Theorem 1 and the fact that the mass functions

and m

are in essential conﬂict, we can ﬁnd a state

ω ∈ Θ, such that BetP

(ω) > 0 or BetP

(ω) > 0

but BetP

(ω) = 0. Without loss of generality, we

assume BetP

(ω) > 0 and BetP

(ω) = 0. Then

by Deﬁnition 3 and the fact that BetP

(ω) = 0, for

any T ⊆ Θ that satisﬁes ω ∈ T , we have m

(T ) = 0.

Moreover, by Deﬁnition 2, for any mass function m

the combination mass value m

123

(T ) that obtained by

applying Dempster’s combination rule for mass func-

tions m

and m

should satisfy

123

(T ) =

∑

B=T

(A)m

(B)

1 −

∑

(A)m

(B)

= 0

Thus, by Deﬁnition 3, we have

BetP

123

(ω) =

∑

T ⊆Θ,ω∈T

|T |

m(T )

1 − m(

= 0.

By BetP

(ω) > 0 and BetP

123

(ω) = 0, we have

BetP

(ω) 6= BetP

123

(ω) and prove the theorem.

In fact, Theorem 2 reveals an undesirable property

of essential conﬂict that a wrong judgement about the

true state might be always remained in the combina-

tion process no matter how many correct evidence we

have collected.

The property of belief absolutization in Theorem

1 and the property of uncorrectable assertion in The-

orems 2 show the insight of essential conﬂict that

the combination process with two essential conﬂict

mass functions will always lead to an extreme judge-

ment (i.e., necessary true or impossible) of a possi-

ble state that cannot be corrected by any further ev-

idence. Such insight reveals the relation of conﬂict

and counter-intuitive and this is the reason why we

call such type of conﬂict as essential conﬂict.

6 A MEASUREMENT OF

ESSENTIAL CONFLICT

In this section, we will ﬁrst propose a measurement

for essential conﬂict that we discover in the previous

section. After that, we will reveal its properties. Fi-

nally, we will illustrates the advantage of our mea-

surement by analysing some examples.

Deﬁnition 6. Let m

and m

be two mass functions

over a frame Θ, ϒ

⊆ Θ be a set of essential conﬂict

elements with the mass functions m

and m

. Then

the degree of the essential conﬂict with two mass func-

tions m

and m

, denoted as κ(m

, m

), is given by:

κ(m

, m

) =

∑

A,B⊆ϒ

(A) +m

(B) −m

(A)m

(B).

Here, a state set A ⊆ ϒ

means for any state ω ∈

A, we have ω ∈ ϒ

, thus

∑

A⊆ϒ

(A) for i ∈ {1, 2}

means the total support degree of the evidence repre-

sented by mass function m

for the states that ruled

out in the combination process. Moreover, since the

mass value of

∑

A,B⊆ϒ

(A)m

(B) has been double

counted in

∑

A⊆ϒ

(A) for i ∈ {1, 2}, −m

(A)m

(B)

is required for the degree of the essential conﬂict with

two mass functions m

and m

Moreover, we ﬁnd that the essential conﬂict mea-

surement κ(m

, m

) has some good properties ap-

proved in (Destercke and Burger, 2012).

Theorem 3. Let m

and m

be two mass functions

over a frame of discernment Θ, F

and F

be the focal

element sets of m

and m

, ϒ

⊆ Θ be a set of essen-

tial conﬂict elements with the mass function m

and

, and κ(m

, m

) be the essential conﬂict measure-

ment of m

and m

. Then we have:

(i) Symmetry. κ(m

, m

) = κ(m

, m

A Measurement for Essential Conﬂict in Dempster-Shafer Theory

1287

(ii) Extreme consistency. κ(m

, m

) = 0 if and only

if m

and m

are not in essential conﬂict, while

κ(m

, m

) = 1 if and only if A ∩ B =

0 for any

A ∈ F

and B ∈ F

(iii) Bounded. 0 ≥ κ(m

, m

) ≥ 1.

(iv) Ignorance is bliss. If m

(Θ) = 1, then

κ(m

, m

) = 0.

Proof. By Deﬁnition 6, we have

κ(m

, m

) =

∑

A,B⊆ϒ

(A) +m

(B) − m

(A)m

(B)

= κ(m

, m

Thus, item (i) of the theorem holds.

If m

and m

are not in essential conﬂict, by Deﬁ-

nition 4, we have ϒ

0. Hence, by Deﬁnition 6, we

have κ(m

, m

) = m

(

0)+m

(

0) = 0. If A∩B =

0 for

any A ∈ F

and B ∈ F

, then

κ(m

, m

∑

A∈F

,B∈F

(A)+m

(B) −m

(A)m

(B)=1.

Thus, item (ii) of the theorem holds.

By Deﬁnition 1, for any A, B ⊆ ϒ

⊆ Θ, we have

(A) ∈ [0, 1], m

(B) ∈ [0, 1], and m

(A) + m

(B) ≥

(A)m

(B). Hence by the fact that

∑

A⊆Θ

(A) = 1,

∑

B⊆Θ

(B) = 1, and ϒ

⊆ Θ, we have

0 ≥

∑

A⊆ϒ

(A) ≥ 1, 0 ≥

∑

B⊆ϒ

(B) ≥ 1, and

0 ≥

∑

A∈F

,B∈F

(A)m

(B) ≥ 1.

Since

κ(m

∑

A∈F

(A)+

∑

B∈F

(B)−

∑

A∈F

,B∈F

(A)m

(B),

we have 0 ≥ κ(m

, m

) ≥ 1. Thus, item (iii) of the

theorem holds.

If m

(Θ) = 1, by Deﬁnition 4, we have ϒ

Hence, by Deﬁnition 6, we have κ(m

, m

) = 0. Thus,

item (iv) of the theorem holds.

Now, we will use Examples 1-3 to illustrates the

effectiveness of our essential conﬂict measurement.

For Example 1, by Deﬁnition 4, we have ϒ

= {a, c}.

Thus, by Deﬁnition 6, we have κ(m

, m

) = 0.99.

Similarly, for Example 2, by ϒ

= {b, c} and Deﬁni-

tion 6, we have κ(m

, m

) = 0.36. And for Example 3,

by ϒ

0 and Deﬁnition 6, we have κ(m

, m

) = 0.

Thus, we have κ(m

, m

) > κ(m

, m

) > κ(m

, m

This result satisﬁes our intuitions discussed in Section

4. In other words, our measurement is highly related

to counter-intuitive combination result with Demp-

ster’s combination rule. Hence, compared with the

current conﬂict measurements, our measurement in-

deed gives a better explanation for the conﬂict.

7 CONCLUSION AND FUTURE

WORKS

Our goal is to study the notion of conﬂict in D-S

theory from a new perspective about the relation of

conﬂict and counter-intuitive combination result with

Dempster’s combination rule. After showing the lim-

itations of the existing conﬂict measurements in han-

dling the type of conﬂict (i.e., essential conﬂict) that

highly related to the counter-intuitive combination re-

sult, we give a formal deﬁnition of essential conﬂict

by the intersection relation of the focal elements of

the original mass functions during the combination

process. Moreover, by revealing two core properties

of the essential conﬂict: belief absolutization and un-

correctable assertion, we show the insight of essen-

tial conﬂict, that is the combination process with two

essential conﬂict mass functions will always lead to

an extreme judgement (i.e., necessary true or impos-

sible) of a possible state that cannot be corrected by

any further evidence. Thus, such type of conﬂict is the

reason that causes the counter-intuitive combination

result. Finally, by proposing a formal measurement

for essential conﬂict, analysing the properties of such

measurement, and applying such measurement to ad-

dress some examples, we show that our new conﬂict

measurement indeed gives a better explanation for the

relation of the conﬂict and the counter-intuitive com-

bination result between two mass functions.

There are many possible extensions to our work.

Maybe the most interesting one is to extend our con-

cept of essential conﬂict to speciﬁc needs and real-

world applications, such as multiple sensor surveil-

lance system (Hong et al., 2016) and automated e-

business negotiation (Zhan et al., 2018). Another

tempting avenue is to develop an alternative combina-

tion rule that can solve the conﬂict situation we men-

tioned in this paper. Since for the Dempster’s combi-

nation rule, we can only suggest to avoid the case of

essential conﬂict, it is interesting to ﬁnd out whether

there exists a combination rule that can solve the es-

sential conﬂict without losing the desirable proper-

ties of Dempster’s combination rule. Finally, it is

signiﬁcant to analyse more properties and rationali-

ties about our conﬂict concept in information fusion

as well as the theoretical comparison with other pro-

posed conﬂict concepts in (Deng, 2015; Liu, 2006;

Shafer, 1976).

ACKNOWLEDGEMENTS

The works described in this paper are supported by

the National Natural Science Foundation of China un-

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

1288

der Grant Nos. 61772210, 61806080, 62006085 and

U1911201; Guangdong Province Universities Pearl

River Scholar Funded Scheme (2018); Humanities

and Social Sciences Foundation of Ministry of Edu-

cation of China under Grant No. 18YJC72040002;

Doctoral Startup Project of Natural Science Founda-

tion of Guangdong Province of China under Grant

No. 2018A030310529; Project of Science and Tech-

nology in Guangzhou in China under Grant Nos.

201807010043 and 202007040006.

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