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v
(
a
l
)
v
(
a
s
)
v
(
{a
l
,a
s
}
)
v
a
l
,a
s
(
a
l
)=
v
(
{a
l
,a
s
}
)
,
v
a
l
,a
s
(
a
s
)=
and
v
a
l
,a
s
(
{
a
l
,a
s
}
)=1
.
The change of conviction can then be modeled using the symmetric Choquet integral,
which is also called the Sipos integral. The definition of the Choquet and Sipos integrals
will now be provided.
[0
,
1]
n
be a vector of convictions,
()
be a per-
Definition 4.
Let
c
=(
c
a
1
,...,c
a
n
)
∈
c
a
(
n
)
and
v
a capacity on
2
{a
1
,···,a
n
}
.
The Choquet integral of
c
with respect to
v
is expressed as:
mutation on
{
1
,...,n
}
such that
c
a
(1)
≤
...
≤
n
c
a
(
i
)
−
c
a
(
i
−
1)
v
(
C
v
(
c
)=
{
(
i
)
,
···
,
(
n
)
}
)
with
c
a
(0)
=0
.
i
=1
1
,
1]
n
be a vector capable of assuming nega-
Definition 5.
Let
c
=(
c
a
1
,
···
,c
a
n
)
∈
[
−
tive values,
()
be the permutation on
{
1
,
···
,n
}
such that
c
a
(1)
≤
c
a
(
p
)
<
0
≤
c
a
(
p
+1)
≤
c
a
(
n
)
and
v
a capacity on
2
{a
1
,···,a
n
}
.
The symmetric Choquet Integral of
c
with respect to
v
is given by:
···≤
p−
1
C
v
(
c
)=
[
c
a
(
i
)
−
c
a
(
i
+1)
]
v
(
{
(1)
,
···
,
(
i
)
}
)+
c
a
(
p
)
v
(
{
(
i
)
,
···
,
(
p
)
}
)
i
=1
n
+
c
a
(
p
+1)
v
(
{
(
p
+1)
,
···
,
(
n
)
}
)+
[
c
a
(
i
)
−
c
a
(
i
−
1)
]
v
(
{
(
i
)
,
···
,
(
n
)
}
)
.
i
=
p
+2
and denoted
C
v
a
l
,a
s
.
The changes of conviction proposed can then be summarized as follows:
In [7] the Sipos integral is defined on the set of agents
{
a
l
,a
s
}
If agents
a
l
and
a
s
have the same preference, then one of them is more convinced, and
this situation entails two possible cases.
-
If
c
a
s
>c
a
l
then the new conviction of agent
a
l
becomes:
C
v
a
l
,a
s
(
c
a
s
,c
a
l
)=
c
a
l
+(
c
a
s
−
c
a
l
)
v
a
l
,a
s
(
a
s
)
.
-
If
c
a
l
>c
a
s
then the new conviction of agent
a
l
becomes:
C
v
a
l
,a
s
(
c
a
s
,c
a
l
)=
c
a
s
+(
c
a
l
−
c
a
s
)
v
a
l
,a
s
(
a
l
)
.
If agents
a
l
and
a
s
have different preferences, then the new conviction of agent
a
l
is:
-
C
v
a
l
,a
s
(
c
a
s
,c
a
l
)=
−c
a
s
v
a
l
,a
s
(
a
s
)+
c
a
l
v
a
l
,a
s
(
a
l
)
.
The main drawback to this model is its lack of semantic justification with regard to
capacity
v
(i.e. influence is merely a normalized relative importance); in addition, the
concept of conviction has not been formally defined and the revision equations are
not provided in an appropriate formalism, in which time would appear explicitly (i.e.
dynamic aspects).
3
Presentation of Our Dynamic Model
This section presents our dynamic model for simulating a debate outcome. To begin,
let's note that within the framework of this paper, the influence function used in [3] is
perceived as a disturbance function applied to the set of all possible inclination vectors.
