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Theorem 2.
The mixing mechanism based on single concessions of pure tactics
results in a monotonic offer curve (and therefore preserves a monotonic sequence
of utilities) if monotonic tactics from Definitions 1 and 2 are used.
X
t
n
a↔b
x
t
n
b→a
Proof.
Let
be the negotiation thread at time
t
n
with
being the last
x
t
n
+1
a→b
offer and
being the next counteroffer of agent
a
then according to Definition
τ
ji
− τ
ji
1 the behaviour-independent concession
(
t
n
+1
)
(
t
n−
1
)isalwaysgreater
zero if
U
a
is increasing. The offer proposed by the pure behaviour-dependent
x
t
n
−
1
a→b
(
˜
X
t
a↔b
τ
jk
tactics
]if
monotonic tactics from Definition 2 are used and the opponent never introduces
non-monotonicity. The behaviour-dependent concession
[
j
]) for issue
j
is greater than the previous offer
[
j
(
˜
− x
t
n
−
1
a→b
X
t
a↔b
τ
jk
[
j
])
[
j
]
is therefore always greater zero. For all weights
γ
i
,γ
k
≥
0 follows that each term
x
t
n
+1
a→b
≥ x
t
n
−
1
of the sum in Eq. (4) is greater zero and hence
. The same line of
a→b
reasoning can be followed for an increasing scoring function
U
a
.
Similar to the previous method the agent can strictly avoid imitating a non-
monotonic sequence of opponent's offers by applying a constraint
C
to each imi-
τ
jk
(
˜
−x
t
n
−
1
a→b
X
t
n
a↔b
tative concession in (4) written as
C
(
[
j
])
[
j
]
,
0) where
C ≡
min if
U
a
decreasing or
U
a
increasing. In contrast to the thread-based mix-
ing this mechanism needs no separate negotiation threads and produces mono-
tonic offer curves even for dynamically changing weights.
C ≡
max if
4
Evaluation
We present the results of a comparative evaluation of the mixing mechanisms
with respect to their non-monotonic behaviour and its respective effects in dif-
ferent bilateral single- and multi-issue negotiation settings. As the number of
possible mixes of tactics is infinite, we focus on a mix of two tactics from [3], one
behaviour- and one time-dependent, for each agent with static weights through-
out the encounter and settings as follows:
-
Time-dependent (polynomial)
: (C)onceder:
β ∈{
,
,
}
β ∈
3
5
7
; (B)oulware:
}
-
Behaviour-dependent
: (a)bsolute tft:
{
.
,
.
,
.
0
1
0
2
0
3
δ
=1
,R
(
M
)= 0; (r)elative tft:
δ
=1
}
Initial letters indicate strategies, for example, 'CaS' denotes the strategy group
containing conceder time-dependent and absolute tit-for-tat tactics mixed by
small weights. Before considering a multi-issue scenario we are interested in when
and to what degree non-monotonic behaviour occurs in static mixed strategies.
For that reason, we consider first a single-issue scenario with two agents, a buyer
b
-
Weights
: (S)mall:
γ ∈{
0
.
1
,
0
.
2
,
0
.
3
}
; (L)arge:
γ ∈{
0
.
7
,
0
.
8
,
0
.
9
, who negotiate about issue 1 from example 1. To enable a more
realistic setting deadlines of both agents can be different with
and a seller
s
t
b
max
=20and
t
s
max
∈{
. The tables in figure 2 illustrates the rate (%) of negotia-
tions with non-monotonic offer curves in the case of both agents applying the
traditional linear weighted combination for a particular strategy group. Num-
bers below the rate are the maximum variation in terms of non-monotonicity
10
,
20
,
30
}
