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to its Occurrence Probability (
OP
), measured by multiplying the possibility weight of
the trust values,
PW
(
τ
i
)
, that are used to build
Π
a
S
→a
D
(
τ
)
,
T
.
In the second approach, we only consider the trust ratings,
τ
∀
τ
∈
∈
T
such that
Π
a
S
→a
(
τ
)=1
. In other words, we only consider the trust ratings that have the highest
weight of
PW
in the
T
P
OS
a
T
distributions
derived from these trust values have the highest
OP
value which makes them the most
expected distributions. We denote by
μ
a
the number of trust ratings,
τ
set. Consequently, the
Π
a
S
→a
D
(
τ
)
,
∀
τ
∈
T
P
OS
a
∈
that sat-
isfy
Π
a
S
→a
(
τ
)=1
for agent
a
. In this approach, we only select the trust ratings in
μ
a
for each agent
a
∈
A
and build the possibility distributions of
Π
a
S
a
D
(
τ
)
,
∀
τ
∈
T
out
→
of those trust ratings. After building
M
=
a∈A
μ
a
different possibility distributions of
Π
a
S
a
D
(
τ
)
, we compute their average, since all of them have equal
OP
weight.
→
Proposition 1.
In both approaches, the conditions of the general case described in the
previous section are satisfied.
Proof.
Proof can be easily done by enumerating different cases.
Due to the computational burden of the first approach (which requires building
K
dis-
tributions of
Π
a
S
→a
D
(
τ
)
,
T
), we used the second approach in our experiments as
it only requires building
M
distributions.
To conclude this section, we would like to comment on the motivation behind us-
ing possibility distribution rather than probability distributions. Indeed, if probability
distributions were used instead of possibility distributions, in order to consider uncer-
tainty a confidence interval should be measured in place of each possibility value. Con-
sequently, in order to represent the trust of agent
a
S
∀
τ
∈
in an agent
a
, each possibility
value
Π
a
S
→a
(
τ
)
should be replaced with a confidence interval. The same representa-
tion should be applied for each agent
a
A
's trust in
a
D
. Now, in order to estimate the
probability distribution of agent
a
D
's trust with respect to its uncertainty, we need to find
some tools for merging the confidence intervals of
a
S
's trust in
A
with the confidence
intervals of
A
's trust in
a
D
. To the best of our knowledge, no work addresses this issue,
except for the following related works. In [25], the number of the occurrences of each
element in the domain, which in our model is equivalent to the number of observance
of each
τ
value in the interactions between agent
a
and
a
D
, is reported by agents in
A
to
a
S
and then, the confidence intervals on the trust of agent
a
D
is built by
a
S
.Thework
of [26] measures the confidence intervals of
a
D
's trust out of several confidence inter-
vals provided by agents in
A
. In both works, the manipulation of information by the
agents in
A
is not considered and for building the confidence intervals of
a
D
, the trust
of agent
a
S
in
A
is neglected. Despite lack of proper tool in the probability domain,
we employed possibility distributions as they can address the same problem in a much
simpler approach.
∈
7
Experiments
We first introduce two metrics for evaluating the outcomes of our experiments and then
present the experimental results.
