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case
A
>
1:
U
(
X
)
p
+
−
U
(
X
)
p
−
>
U
(
X
)
d
+
−
U
(
X
)
d
−
then
A
∗
DoT
XX
τ
>
DoT
XX
τ
i.e. if the difference between the utility of the success and the utility of the failure in
delegation is smaller than the difference between the utility of the success and the utility of
the failure in the direct performance of
X
, then (for the term
A
) in order to delegate return
the
trust of X in Y must be bigger than the selftrust of X
(about
τ
).
More precisely, suppose two subcases:
∗
a
1)
even
if the
X
's degree of trust in
Y
is greater than the selftrust,
it is not sufficient for
delegation
.
(ii) In the case of
DoT
XY
τ
>
(i) In the case of
DoT
XY
τ
>
(DoT
XX
τ
+
B) (with B
=
0 and 1
<
a
<
A, A
>
∗
a
(DoT
XX
τ
+
B) (with B
=
0 and a
≥
A, A
>
1), the X's degree of
trust in Y is sufficient for delegating
.
In other words, for delegating X's trust in Y has to be greater thnt X's selftrust at least of
the value given from A.
Also this (A
>
1, B
=
0) is a clear and rational result: in the case in which:
1)
the difference between the utility of the success and the utility of the failure in direct
performance of X is greater than the difference between the utility of the success and the
utility of the failure in delegation to Y, and
2)
the failure of X's performance has the same damage of the failure of Y's delegated perfor-
mance; then
X has to trust Y more than herself for delegating to him that task: the difference in the trust
values has to be at least of a factor given from A.
Vice versa, if
A
<
1:
U
(
X
)
p
+
−
U
(
X
)
p
−
<
U
(
X
)
d
+
−
U
(
X
)
d
−
then
A
∗
DoT
XX
τ
<
DoT
XX
τ
i.e., if the difference between the utility of the success and the utility of the failure in
delegation is bigger than the difference between the utility of the success and the utility of the
failure in
X
's direct performance, then (given
B
0
) in order to delegate
the trust of X in Y should be smaller (of A factor) than the X's selftrust
(about
=
τ
).
Even in this case we have to distinguish two alternative subcases:
∗
a
(i) In the case of
DoT
XY
τ
>
(DoT
XX
τ
+
B) (with B
=
0 and 0
<
a
≤
A, A
<
1) X's degree of
trust in Y is sufficient for delegating
.
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