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Yager [23]:
Π
a
S
→a
D
(
τ
)=min
a∈A
[
τ
a
S
×
Π
a→a
D
(
τ
)+1
−
τ
a
S
→a
]
.
(4)
→a
Dubois and Prade [24]:
Π
DP
a
S
→a
D
(
τ
)=min
a∈A
[max(
Π
a→a
D
(
τ
)
,
1
−
τ
a
S
→a
)]
.
(5)
In Yager's fusion rule, the possibility of each trust value
τ
moves towards a uniform dis-
tribution as much as
(1
a
)
, which is the extent to which the agent
a
is not trusted.
In Dubois and Prade's fusion rule, when an agent's trust declines, the
max
operator
would more likely select
1
−
τ
a
S
→
T
reported by
a
gets closer to a uniform distribution. Consequently, it has a less chance of
being selected by the
min
operator.
Once a fusion rule in this Section is applied, the resulted possibility distribution of
Π
a
S
→a
D
(
τ
)
,
−
τ
a
S
→a
and, hence, the information in
Π
a→a
D
(
τ
)
,
∀
τ
∈
∀
τ
∈
T
is normalized to represent the possibility distribution of agent
a
S
's
trust in
a
D
.
5
Merging Successive Possibility Distributions
In this section, we present the main contribution, i.e., a methodology for merging the
possibility distribution of
Π
a
S
→a
(
τ
)
,
∀
τ
∈
T,
∀
a
∈
A
(representing the trust of agent
a
S
A
(representing the trust of the agent set
A
in agent
a
D
). These two possibility distributions
are associated to the trust of entities at successive levels in a multi-agents systems and
hence giving it such a name.
In order to perform such a merging, we need to know how the distribution of
Π
a→a
D
(
τ
)
,
in its advisors) with the possibility distribution of
Π
a→a
D
(
τ
)
,
∀
τ
∈
T,
∀
a
∈
∀
τ
∈
T
changes, depending on the characteristics of the possibility distri-
bution of
Π
a
S
T
. We distinguish the following cases for a proper merging
of the successive possibility distributions.
a
(
τ
)
,
∀
τ
∈
→
5.1
Specific Case
Consider a scenario where
→a
(
τ
)=
1
,
τ
=
τ
0
,otherwise
i.e., only one trust value is possible in the domain of
T
and the possibility of all other
trust values is equal to 0. Then, trust of agent
a
S
!
τ
,τ ≤ τ
≤ τ
and
Π
a
S
∃
in agent
a
can be associated with a
single value of
τ
a
S
→a
=
τ
and the fusion rules described in section 4 can be applied to
get the possibility distribution of
Π
a
S
T
.
Considering the TM fusion rules, for each agent
a
, first the possibility distribution
of
Π
a→a
D
(
τ
)
,∀τ ∈ T
is transformed based on the trust value of
τ
a
S
→a
D
(
τ
)
,
∀
τ
∈
→a
=
τ
as dis-
cussed in Section 4. Then, an intersection of the transformed possibility distribution is
taken and the resulted distribution is normalized to get the possibility distribution of
Π
a
S
→a
D
(
τ
)
,
∀
τ
∈
T
.
