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current environment and belief of the agent. The above method of using pure
or mixed tactics represent decision functions which an agent uses to make
con-
cessions
such that
t
n
+1
a→b
t
n
−
1
a→b
U
a
(
<U
a
(
). In multi-issue negotiations, an agent
can also make trade-offs where the next offer has the same utility as its previous
offer (both are on the same indifference curve) with
x
)
x
t
n
+1
a→b
t
n
−
1
a→b
U
a
(
x
U
a
(
x
). In
this paper, we focus on the concession-making mechanisms as detailed above
and refer to [11,5] for well-discussed trade-off mechanisms.
)=
2.2 Definition of Monotonic Tactics
To determine if a mixed strategy generates a monotonic offer sequence we distin-
guish between monotonic behaviour-dependent and -independent pure tactics:
Definition 1.
Given a negotiation between agents
a
and
b
,a
monotonic beha-
τ
j
viour-independent tactic
(
t
k
)
of agent
a
for issue
j
is a function generating
τ
j
≥ τ
j
U
a
is decreasing or
offers at any times
t
k
,t
i
∈ T
n
such that
(
t
k
)
(
t
i
)
if
τ
j
≤ τ
j
U
a
is increasing under the condition that
(
t
k
)
(
t
i
)
if
k, i ∈{
1
,
2
,...,n}
and
.
Definition 2.
Given a negotiation between agents
k>i
a
and
b
at time
t
n
,a
mono-
τ
j
(
X
t
n
tonic behaviour-dependent tactic
)
generates an offer using any se-
a↔b
X
t
a↔b
T
n
T
n
⊆ T
n
x
a↔b
quence
=(
)
where
=
∅
and
=
{t
1
,...,t
n
}
under
t∈ T
n
x
t
b→a
∈ D
j
the conditions that there exists at least one offer
of agent
b
in the
sequence such that
-
(
X
t
a↔b
(
X
t
n
−
2
a↔b
τ
j
≥ τ
j
x
b→a
)
)
if the sequence of opponent's offers
(
)
and
t∈ T
n
U
a
is monotonic decreasing or
(
X
t
a↔b
(
X
t
n
−
2
b↔a
τ
j
≤ τ
j
x
b→a
-
)
)
if the sequence of opponent's offers
(
)
and
t∈ T
n
U
a
is monotonic increasing.
Definition 1 typically represents tactics depending on a particular resource which
state may change over time. Throughout the paper we denote this class of tactics
with
τ
j,
time
for issue
j
. In the simplest case the tactic may depend on time or
the number of negotiation rounds. For instance, the polynomial and exponen-
tial time-dependent decision functions proposed by Faratin et al [4] represent
such tactics as they generate offers in a monotonically decreasing or increasing
manner. In the case of a resource-dependent tactic, however, the resource may
diminish and increase over time such that a monotonic sequence of offers is not
guaranteed. An imitative tactic according to Definition 2 uses historical offers
from the opponent to propose counteroffers by preserving a monotonic offer se-
quence as long as the opponent's sequence is monotonic as well. We refer to
such imitative tactics as
τ
j,
beh
. For instance, the imitative tit-for-tat tactics in
[4] fulfil this definition. Once non-monotonicity is introduced by one partner it
can in turn cause a non-monotonic offer sequence of the opponent depending on
the degree of how much the concessions are copied. As a result, if monotonic tac-
tics are mixed together, non-monotonic behaviour can emerge even when both
agents apply monotonic tactics as we demonstrate in the next section.
