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∗
a
(ii) In the case of
DoT
XY
τ
>
(DoT
XX
τ
+
B) (with B
=
0 and
A
<
a
<
1
,A
<
1) X's degree of
trust in Y is not sufficient for delegation
.
Note that even if in both the cases
X's trust in Y is smaller than X's selftrust, only in one of
them is
it possible to delegate: so we can say that (for utilities reasons)
it is possible to delegate
to agents which X trusts less than herself
.
Considering now also the term
B(B
=
0)
,
If
U(X)
p
−
−
U(X)
d
−
>
0
, then a positive term is added to the
A: A
+
B
>
A
,
i.e.,
if the utility of the failure in case of X's direct performance is bigger than the utility of
the failure in case of delegation, then - in order to delegate - the trust of X in Y about
τ
must
be greater than in the case in which the right part of
(3.5)
is constituted by A alone
.
Vice versa,
If
U(X)
p
−
−
U(X)
d
−
<
0
, then
A
+
B
<
A
,
i.e.,
if the utility of the failure in the case of non-delegating is smaller than the utility of the
failure in the case of delegation, then - in order to delegate - the trust of X in Y about
τ
must be smaller than in the case in which the right part of the formula
(3.5)
is constituted
by just A alone
.
10
DoT
XX
τ
∗
A
Since
DoT
XY
τ
≤
1
, from the formula
(3.5)
we can obtain (starting from
1
>
+
B)
:
DoT
XX
τ
<
(
U
(
X
)
d
+
−
/
(
U
(
X
)
p
+
−
U
(
X
)
p
−
)
U
(
X
)
p
−
)
(3.6)
From the formula
(3.6)
we have two consequences in the dynamics of trust; to delegate
X
to
Y
the task
τ
, as the selftrust (
DoT
XX
τ
) grows either:
1) the difference between the utility of the success in delegation and the utility of the failure
in the direct performance increases; or
2) it reduces the difference between the utility of the success and of the failure in direct
performance.
Because
DoT
XX
τ
≥
0
,from
(3.6)
we obtain (starting from
0
<
(U(X)
d
+
−
U(X)
p
−
)/(U(X)
p
+
−
U(X)
p
−
)
:
U
(
X
)
d
+
>
U
(
X
)
p
−
(3.7)
(consider that for definition we have
U(X)
p
+
>
U(X)
p
−
).
In practice, for delegating, a necessary (but not sufficient) condition is that the utility of
the success in delegation is greater than the utility of the failure in
direct performance
(as
intuitively rational).
Let us conclude this section by underlining the fact that (in our model and in real life)
we
do not necessarily delegate to the most trustworthy agent
; we do not necessarily choose the
alternative where trust is greater. We might prefer to choose a partner or to rely on a device
that is not the most reliable one, simply because there are other parameters involved in our
10
Both for
A
and
B
there is a normalization factor
(U(X)
d
+
−
U(X)
d
-
)
: the more its value increases, the more the
importance of the terms is reduced.
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