Information Technology Reference
In-Depth Information
c
(
k
)=(
c
a
1
(
k
)
,
···
,c
a
j
(
k
)
,
···
,c
a
n
(
k
))
,where
c
a
j
(
k
)
is the conviction of agent
a
j
w.r.t alternative
1
at time
k
.
Their respective computations will be provided in the next section.
The conviction of an agent concerning a given alternative is correlated with the prob-
ability that this particular agent chooses this alternative, i.e. the probability of his incli-
nation as defined in [3].
Let
i
−
∈
I
be an inclination vector. Each coordinate
i
a
j
is the preference of agent
a
j
and constitutes one of the two alternatives.
Definition 7.
Let
i ∈ I
be an inclination vector. The conviction vector of
i
at time
k
is
c
(
i,k
)=(
c
a
1
(
k
)
,···, c
a
n
(
k
))
, where for any
j
,
c
a
j
(
k
)
is
c
a
j
(
k
)
if
i
a
j
=1
and is
c
a
j
(
k
)
if
i
a
j
=
−
1
.
I
be an inclination vector and let's define
c
i
(
k
)
Let
i
[0
,
1]
as an average conviction
at time
k
for
i
. This value summarizes the distributions of agents' convictions in
i
at
time
k
.
c
i
(
k
)
is an ”aggregated conviction” of the group of agents for
i
. This aggrega-
tion should take into account the relative importance of agents and their interactions.
Consequently, it seems only natural to state the following definition.
∈
∈
Definition 8.
Let
i
I
be an inclination vector and
v
[
k
]
be a capacity defined at time
k
on
2
{a
1
,···,a
n
}
,then
c
i
(
k
+1)=
C
v
[
k
]
(
c
a
1
(
k
)
,
∈
···
, c
a
n
(
k
))
,where
C
v
[
k
]
is the Choquet
integral with respect to
v
[
k
]
.
The time-varying probability is built by recurrence on
k
. We start at time
k
=0
and
will proceed by presenting how to compute
p
[
k
+1]
using
p
[
k
]
.
At time
k
=0
:
Each agent assigns a score to each alternative in the interval
[0
,
1]
. For each agent,
if we were to denote
n
+1
(resp.
n
−
1
) as the score of
+1
(resp.
−
1
), then the
n
+1
n
+1
+
n
−
1
n
−
1
n
+1
+
n
−
1
and
c
a
j
(0) =
convictions could be computed by
c
a
j
(0) =
.We
then have
c
a
j
(0) +
c
a
j
(0) = 1
. Initially, at time
k
=0
,if
i
a
j
is the preference
of
a
j
then the probabilities of the agent
a
j
regarding his preference and the other
alternative would be:
p
a
j
(
i
a
j
)[
k
=0]=
c
a
j
(0)
and
p
a
j
(
c
a
j
(0)
.
We assume that before the debate starts, the inclination of each agent does not
depend on the social network. The probability distribution associated with a
priori
probabilities is thus the product of the individual probabilities
p
a
j
−
i
a
j
)[
k
=0]=1
−
at
k
=0
, leading
to the following probability:
n
∀
i
∈
I,p
(
i
)[0] =
p
a
j
(
i
a
j
)[0]
.
j
=1
It is thus possible to compute the following
-
the decisional power for any agent
a
j
at
k
=0
:
φ
a
j
(
B,gd,p
[0])
;
-
the capacity
v
φ
[0]
over
2
{a
1
,···,a
n
}
,for
k
=0
, as proposed in Subsection 3.1:
v
φ
[0](
a
j
)=
2
φ
a
j
(
B,gd,p
[0])+
2
, and the capacity on a set
A
is the maximum
of the capacity of agents present in the considered coalition.
