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In-Depth Information
How to compute
p
[
k
+1]
,
φ
a
j
(
B,gd,p
[
k
+1])
,
v
φ
[
k
+1]
using
p
[
k
]
,
φ
a
j
(
B,gd,p
[
k
])
and
v
φ
[
k
]
The capacity
v
φ
[
k
]
is used to compute
c
i
(
k
+1)
, i.e. the aggregation conviction for
the inclination vector
i
at time
k
+1
:
c
i
(
k
+1)=
C
v
φ
[
k
]
(
c
a
1
(
k
)
,
, c
a
n
(
k
))
.
The time-varying probability
p
[
k
+1]
can then be defined as follows:
···
c
i
(
k
+1)
j∈I
c
j
(
k
+1)
.
∀
i
∈
I, p
(
i
)[
k
+1]=
It then becomes possible to compute:
-
the decisional power for any agent
a
j
at
k
+1
:
φ
a
j
(
B,gd,p
[
k
+1])
;
-
the capacity
v
φ
[
k
+1]
over
2
{a
1
,···,a
n
}
, at time
k
+1
, as proposed in Subsection
3.1.
We have thus defined a time-varying probability. Note that the proposed method seems
to be rather intuitive since it corresponds to the notion that an agent's social influence
depends on the degree of assurance in the convictions of the other agents when he
speaks.
3.3
Conviction State Equations
The aim of this section is to establish the state equations that serve to model the dynamic
relationship between convictions and influences. Let's consider
a
l
to be any listener-
agent and
a
s
a speaker-agent. Their convictions at time
k
for the alternative
+1
(resp.
−
1
)arethen
c
a
l
(
k
)
and
c
a
s
(
k
)
(resp.
c
a
l
(
k
)
and
c
a
s
(
k
)
).
Two variables are necessary to model the rhetorical quantity exchanged between
the two agents
a
l
and
a
s
, namely: the difference in their conviction and their relative
importance at time
k
, as modeled by the capacities
v
φ
[
k
](
a
s
)
and
v
φ
[
k
](
a
l
)
.
Four rhetorical exchanges can be distinguished. These four situations are presented
in the case when the agent
a
l
prefers alternative
+1
. Two sub-cases can then be iden-
tified for agent
a
s
: his preferred alternative is either the same as
a
l
's or the other one.
Each case can be divided once again into two sub-cases:
a
s
's conviction is either greater
or less than
a
l
's conviction. When agent
a
l
prefers alternative
1
, convictions
c
replace
convictions
c
in the formula. More precisely, the equations appearing in the computa-
tion of
c
a
l
(
k
+1)
when both agents express the same preference are the same as those
used to compute
c
a
l
(
k
+1)
in the case of opposite preferences and
viceversa
. Hence,
the rhetorical exchanges can be summarized by the following exchanges: synergistic
exchange, revisionist exchange, and antagonistic exchange. Let's take a closer look at
each of them.
−
Synergistic Exchange.
In this case, the preference of agent
a
l
is reinforced by the inter-
vention of agent
a
s
, who resolutely looks favorably upon the same alternative.
The conviction of agent
a
l
then increases, to an extent proportional to the difference
between both convictions as well as to the capacity of speaker-agent
a
s
.
This situation, as represented in figure 1, corresponds to the case when
a
l
and
a
s
have the same preference and moreover
c
a
s
>c
a
l
. The intuitive difference equation is
then written:
