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swayed by the arguments of a particular agent or group (coalition) of agents, or that they
feel somewhat obliged, owing to a hierarchical, political or perhaps even more obscure
reason, to follow the opinion of that particular agent or coalition. Another reason may
be that they are acting in reaction against a given agent or coalition, by systematically
embracing the opposite opinion. We use the generic word ”influence” herein to refer to
all these types of phenomena [4].
Models have been introduced into game theory in order to represent influence in so-
cial networks. The point of departure is the concept of the Hoede-Bakker index, a notion
that computes the overall decisional power of an agent within a social network, which
in this case is a group of
n
agents. This index was developed in 1982 [5]; an extended
definition of decisional power was proposed in [3] and will now be summarized. The
reasons behind the existence of influence phenomena, i.e. why a given individual finally
changes his decision, is more a matter of the psychological sciences and lies beyond the
scope of such approaches.
Let's start by considering a set of agents
}
in order to simplify notations along with a power set denoted
2
{a
1
,···,a
n
}
. Each agent
is inclined to choose either
+1
or
{
a
1
,
···
,a
n
}
, denoted
N
=
{
1
,
···
,n
−
1
. An inclination vector, denoted
i
, is an n-vector
consisting of
+1
and
−
1
. The j-th coordinate of
i
is thus denoted
i
a
j
∈{−
1
,
+1
}
and
n
be the set of all inclination
represents the inclination of agent
a
j
.Let
I
=
{−
1
,
+1
}
vectors.
It can then be assumed that agents influence one another; moreover, due to influ-
ences arising in the network, the final decision of an agent may differ from his original
inclination. In other words, each inclination vector
i
∈
I
is transformed into a decision
vector
B
(
i
)
,where
B
:
I
→
I
,
i
→
B
(
i
)
is the influence function. The coordinates of
B
(
i
)
are expressed by
(
Bi
)
a
j
,
j
∈{
1
,
···
,n
}
and
(
Bi
)
a
j
is the decision of agent
a
j
.
Lastly,
gd
:
B
(
I
)
is a group decision function, assigned the value
+1
if
the group decision is
+1
and the value
→{−
1
,
+1
}
1
.
An influence function
B
may correspond to a common collective behavior. For ex-
ample, in [3] a majority influence function
Maj
[
t
]
−
1
for a group decision of
−
parametrized by a real
t
has been
introduced. More precisely, for a given
i ∈ I
,
Maj
[
t
]
i
=
1
N
if
i
+
|
|≥
t
i
+
−
1
N
if
|
|
<t
where
i
+
=
{
k
∈
N
|
i
k
=+1
}
and
1
N
(resp.
−
1
N
) is the vector equal to 1 (resp. -1)
everywhere.
This set-up corresponds to the intuitive collective human behavior: when a majority
of players have an inclination of
+1
, then all players decide
+1
. Many classifications of
potential collective behavior (polarization, groupthink, mass psychology, etc.) can thus
be described mathematically.
An influence function may also be defined as a simple rule. For example, the follow-
ing rule may be associated with the
Guru
function: “when
a
Guru
thinks
+1
,thenall
agents decide
+1
“. Another example would be the opportunistic behavior, i.e.: “when
most of my supervisors decide
+1
, then I decide
+1
”.
It can also be anticipated that mapping
B
:
I
I
is learned from experiment. The
identification of
B
may be perceived as a data-mining step using knowledge bases in
which collective decisions have been recorded as minutes of company meetings.
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