Information Technology Reference
In-Depth Information
Definition 1.
The Hoede-Bakker index of agent
a
j
is defined for a given
B
and a given
gd
by:
1
2
n−
1
GHB
a
j
=
gd
(
B
(
i
))
.
{i|i
a
j
=+1
}
The main drawback with the Hoede-Bakker index is that it blurs the actual role of the
influence function, by analyzing the final decision only in terms of success and failure.
The decision is successful for an agent once his inclination matches the group decision.
In [3], the authors distinguish the influence component from the group decision
component and moreover propose a first modified index of decisional power, whereby
the agent's decision must coincide with the group's decision to constitute a success for
the agent. Lastly, these authors provided a second modified decisional power, allowing
the inclination vectors to be assigned unequal probabilities.
Definition 2.
Let
p
:
I
[0
,
1]
be a probability distribution, with
p
(
i
)
being the prob-
ability of an
i
occurrence. The modified decisional power of agent
a
j
for given
B
,
gd
and
p
can then be expressed as:
φ
a
j
(
B,gd,p
)=
{i|
(
Bi
)
a
j
=+1
}
→
p
(
i
)
.gd
(
B
(
i
))
−
p
(
i
)
.gd
(
B
(
i
))
.
{i|
(
Bi
)
a
j
=
−
1
}
To conclude this summary section, for each agent
a
j
the probabilities of success and
failure are recalled as follows:
SUC
a
j
(
B,gd,p
)=
{b∈I|
(
b
)
a
j
=
gd
(
b
)
}
B
−
1
(
b
)
p
◦
FAIL
a
j
(
B,gd,p
)=
{b∈I|
(
b
)
a
j
=
−gd
(
b
)
}
B
−
1
(
b
)
.
p
◦
Note that:
φ
a
j
(
B,gd,p
)=
SUC
a
j
(
B,gd,p
)
−
FAIL
a
j
(
B,gd,p
)
.
2.2
Convictions and Preferences During a Debate
This section will present the dynamic model of the debate proposed in [7]. The influence
an agent may exert on the others in the debate is modeled by a capacity over
2
{a
1
,···,a
n
}
.
Definition 3.
A capacity
v
over
2
{a
1
,···,a
n
}
is a set function
v
:2
{a
1
,···,a
n
}
→
[0
,
1]
A, A
⊆{
A
⇒
such that
v
(
∅
)=0
,
v
(
{
a
1
,
···
,a
n
}
)=1
and
∀
a
1
,
···
,a
n
}
,A
⊆
v
(
A
)
.
v
(
A
)
≤
The profile of an agent
a
j
includes his preference, importance (i.e. his capacity
v
(
a
j
)
)
and preference intensity named conviction in the following (it is to be denoted
c
a
j
∈
[0
,
1]
).
It is an agreed rule of the debate that agents are to speak in turns. In the proposed
model set-up, the agent
a
s
(speaker-agent) who is currently speaking and any agent
a
l
(listener-agent) who is listening are formally isolated from the remainder of the group.
More precisely, a capacity
v
a
l
,a
s
, defined relative to the pair of agents
(
a
l
,a
s
)
,isintro-
duced as follows:
