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Antagonistic Exchange.
In this situation, the two agents do not share the same prefer-
ence: agent
a
l
nevertheless understands the advantages of
a
s
preference. A convincing
intervention from
a
s
may contribute to making
a
l
dubious, whereas an unpersuasive
intervention might on the contrary strengthen
a
l
's preference.
(1
c
a
s
(
k
))
is a measure of
a
s
's hesitation and provides
a
l
with an estimation of the
strength of
a
s
's opposition. Depending on the strength of this hesitation, the previous
difference equations can again be used with
(1
−
c
a
s
(
k
))
, yielding two situations to be
−
distinguished (see figure 3).
An overly weak preference expressed by
a
s
implies weak opposition from
a
l
's point
of view and reinforces
a
l
's opinion, resulting in a likely strengthening of
a
l
's conviction.
The intuitive difference equation is then:
c
a
s
≥
Case 1:
1
−
c
a
l
c
a
s
(
k
))
c
a
l
(
k
+1)
−
c
a
l
(
k
) = ((1
−
−
c
a
l
(
k
))(
v
φ
[
k
](
a
s
))
, which is equivalent to
c
a
l
(
k
+1)=
c
a
l
(
k
)+(1
−
c
a
s
(
k
)
−
c
a
l
(
k
))(
v
φ
[
k
](
a
s
))
.
c
a
s
<c
a
l
.
In this case
a
l
's conviction weakens following
a
s
's intervention.
c
a
l
(
k
+1)
Case 2:
1
−
c
a
s
(
k
)))(1
−
c
a
l
(
k
)=
−
(
c
a
l
(
k
)
−
(1
−
−
v
φ
[
k
](
a
l
))
, which is equivalent
c
a
s
(
k
)) + (
c
a
l
(
k
)+
c
a
s
(
k
)
1)(
v
φ
[
k
](
a
l
))
.
All these various types of exchanges can be synthesized using a Sipos integral.
to
c
a
l
(
k
+1)=(1
−
−
Proposition 1.
If agents
a
s
and
a
l
express the same preference, then:
c
a
l
(
k
+1)=
C
v
φ
[
k
](
c
a
s
(
k
)
,c
a
l
(
k
));
If agents
a
s
and
a
l
do not share the same preference, then:
c
a
l
(
k
+1)=
C
v
φ
[
k
](1
c
a
s
(
k
)
,c
a
l
(
k
))
.
As a conclusion to this section of the paper, the decisional power
φ
provides a semantic
interpretation for the capacity
v
in the recurrence equations presented in [7], with con-
viction here being related to the probability an agent will choose one alternative over
the other (i.e. probability distribution over inclination vectors). The model in [7] thus
becomes interpretable within a game theory framework [3]. The revision equations for
conviction appear as input-output balances according to the alternatives assessment. In-
troducing time into the equations in [7] implies that revision equations of conviction
are now seen as state equations of agents' mental perception. This new interpretation
then provides a semantics for the debate model in [7]: it incorporates the notions of
influence and decisional power, as proposed in [3], with a formalism close to that of
dynamic models found in control theory, as suggested in [6].
−
4
Illustration
4.1
Preference Calculus
This section discusses how to compute preferences during the debate. Initially, each
agent
a
j
assesses both alternatives
+1
and
−
1
lying in the interval
[0
,
1]
.Theseas-
sessments are denoted
n
+1
a
j
and
n
−
1
a
j
, respectively. It is then possible to build initial
preferences and convictions as follows: Let
a
in
{−
1
,
1
}
