Information Technology Reference
In-Depth Information
γ
k
· τ
k
(
˜
≥ γ
k
· τ
k
(
˜
X
t
n
−
2
a↔b
X
t
n
a↔b
U
a
1and2
[
k
])
[
k
]) and
γ
i
· τ
i
(
t
n
+1
)
≥ γ
i
· τ
i
(
t
n−
1
)if
is decreasing and all
γ
i
,γ
k
≥
0. Since each term of the sum in (3) at
t
n
is larger
x
t
n
+1
a→b
≥ x
t
n
−
1
than the corresponding term of the sum at
t
n−
2
it follows that
.
a→b
U
a
.
The same line of reasoning can be followed for increasing utility functions
Figure 1 shows the monotonic offers curves and the resulting monotonic utility
sequence when both agents use this mixing mechanism for Example 2 (dotted).
The outcome is changed in favour of the seller and agreement is reached earlier.
Agents using this mechanism do not expose the dynamic effects as described in
section 2.3. However, the mechanism does not force the agent to propose offers
in a monotonic manner. For instance, if the opponent still proposes offers in a
non-monotonic sequence, an imitative tactic in the mix may still copy it to some
degree. The agent can choose to strictly ensure monotonicity by applying a con-
straint
τ
jk
(
˜
X
t
a↔b
,x
t
n
−
1
a→b
C
to the imitative tactic:
C
(
[
j, k
])
[
j, k
]) where
C ≡
min
U
a
increasing. The individual imitative thread
used by this method does not represent the actual negotiation thread. This seems
counter-intuitive as the offer curve and the outcome of the individually applied
imitative tactics might indeed be different from the mixed strategy.
U
a
decreasing and
if
C ≡
max if
3.2 Mixing Based on Single Concessions
This mixing type calculates individual next concessions for each tactic to mix
behaviour-dependent and -independent tactics as defined in Section 2.2:
l
i
=1
γ
ji
·
(
τ
ji
(
t
n
+1
)
− τ
ji
(
t
n−
1
)) +
...
x
t
n
+1
a→b
[
j
]=
x
t
n
−
1
a→b
[
j
]+
(4)
m
τ
jk
(
˜
t
n
−
1
a→b
X
t
n
a↔b
+
+1
γ
jk
·
(
[
j
])
− x
[
j
])
k
=
l
m
l
with
denoting the total number and the number of behaviour-indepen-
dent tactics respectively. In order to use concessions at least two offers of the
opponent are necessary. Any of the former mechanisms can be used for ini-
tial offers as they propose the same offers in the first round. Concessions for
behaviour-independent tactics are, since they do not depend on opponent' of-
fers, the difference
τ
ji
(
t
n
+1
)
−τ
ji
(
t
n−
1
) between the calculated offer at
t
n
+1
and
the previous individual offer at
and
t
n−
1
. For the imitative tactic we can not follow
the same line of reasoning because, as described in the previous section, the last
offer of the individually applied imitative tactic is unknown. However, suppose
that the agent changed its strategy to the pure imitative tactic at time
t
n
+1
the
x
t
n
−
1
a→b
τ
jk
(
˜
X
t
a↔b
last offer is still be
]). We
can hence calculate the behaviour-dependent concession by the difference be-
tween the proposed imitative offer and the last offer of the agent. This approach
provides monotonic offer curves similar to the negotiation thread-based mixing
and also avoids non-monotonic aggregated utilities over time. The major advan-
tage, however, is that a monotonic sequence of utilities is also never introduced
if the agent changes weights for tactics dynamically.
and hence the next offer is given by
[
j
