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3.1
Decisional Power and Capacities
One of the new ideas presented in this paper is the ability to design a capacity based on
the decisional power included in the above model.
For any inclination vector i in I , Bi is the decision vector obtained from i whose
influence is modeled by B . gd ( Bi ) is the final decision of the group, whereby the group
decision function is modeled by gd .Forany i
,which
implies that the modified decisional power for any agent a j as denoted φ a j ( B,gd,p )
lies in the interval [
I , gd ( Bi ) belongs to
{−
1 , +1
}
1 , 1] .
Note that if the decisional power of an agent is close to
1 , this means that the agent
only rarely chooses the alternative to what the collective body ultimately chooses: he
fails most of the time ( FAIL ). On the other hand, when his decisional power is close
to 1 , the agent is most often successful ( SUCC ); his decisional power therefore is
strong. Hence, for any agent a j , we can normalize φ a j ( B,gd,p ) in order to obtain his
importance.
As an example, without any further information, the importance of agent a j ,i.e.his
capacity v ( a j ) , can be defined as follows:
Definition 6. The importance of agent a j for a given B , gd and p is v φ ( a j )= 2 φ a j ( B,
gd,p )+ 2 .
Note that for any agent a j , v ( a j ) [0 , 1] with v ( a j )=0 if and only if φ a j ( B,gd,p )=
1 and v ( a j )=1 if and only if φ a j ( B,gd,p )=1 .
A capacity v φ can then be generated over 2 {a 1 ,···,a n } , with constraints,
A, A
A
v ( A ) . Without any further knowledge, it may be
{
a 1 ,
···
,a n }
, A
v ( A )
stated: v φ ( A )=max
)=1 . This last
condition is necessary because it is uncertain that an agent can be found whose capacity
is equal to 1 .
Let's conclude this section with the following remark. The decisional power of indi-
viduals a j on which v φ :2 {a 1 ,···,a n }
a j ∈A v ( a j ) ,
A
⊂{
a 1 ,
···
,a n }
and v φ (
{
a 1 ,
···
,a n }
[0 , 1] is based, measures those cases where the
final decision of a j matches the group decision. An agent with considerable decisional
power is expected to sway several other agents; thus, decisional power is construed as
an estimation of his influence within the group, although this is not an influence index
in the sense of [3].
3.2
Time-Varying Probabilities
This subsection focuses on the design of probability p as a time-varying function, to
be denoted p [ k ] at time k . Along with this time-varying probability, a time-varying
extended decisional power, as presented in [3], can be computed. The following method
proposes basing the probability computation on the convictions of agents with respect
to the available alternatives. In this part therefore, the conviction vectors are assumed
to be known. c ( k ) (resp. c ( k ) ) denotes the conviction vector of agents w.r.t. alternative
+1 (resp.
1 ) at time k :
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 .
 
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