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U
2
increases monotonically.
The expected value of
U
2
is
2
w
j
∈X
w
j
so the algorithm is expected to terminate
when
either
2
w
j
∈X
w
j
>U
1
or
2
w
j
∈X
w
j
>
w
j
∈W
w
j
−
U
1
.
The RHS of each inequality is constant, so the best strategy is to maximise the LHS at
each step, that is, to choose the largest remaining weight. But this is strategy
S
as in
Theorem 1.
Finally we consider the case where the persuadee, agent
B
, has initial beliefs about
attributes relating to both options, but has varying degrees of confidence in these beliefs,
and wishes to confirm them during the dialogue. Theorem 3 shows
to be an optimal
strategy on the assumption that the currently preferred option is at least as likely as the
alternative proposed by the persuader to satisfy the criteria valued by the persuadee.
This will be the case where the persuadee is quite sure that its preferred option satisfies
some desirable attributes, but has only tenuous beliefs about the other option.
S
remains an optimal strategy if the probabilities
p
ij
are not equal, pro-
vided that
p
1
j
≥
S
Theorem 3:
p
2
j
for each attribute
j
,where
p
ij
=
P
(
τ
ij
=1)
.
Sketch of Proof of Theorem 3:
After examining attributes with weights in
X
⊆
W
,
(expected value of
U
i
)=
w
j
∈X
p
ij
w
j
. So the algorithm is expected to terminate when
p
2
j
)
w
j
>
w
j
∈W
(
p
1
j
−
w
j
−
w
j
.
w
j
∈X
w
j
∈X
Arguing as before, an optimal strategy is, at each step, to choose from the remaining
weights so as to maximise
w
j
(
p
1
j
−
p
2
j
+1)
.
6
Concluding Remarks
We have identified the distinctive features of a sub-type of persuasion dialogue where
one agent is trying to convince another that its currently preferred option is not as
good as some other possibility known to the persuading agent. We have also drawn
attention to the pragmatic meanings of utterances in these persuasion dialogues, as re-
vealed by considering what the utterances conversationally imply, and have used these
pragmatic meanings to develop a protocol and strategy for agent persuasion dialogues,
which we have shown to be an optimal strategy in a range of representative scenar-
ios for these persuasion dialogues. In future work we intend to consider dialogues with
three or more participants, and dialogues which attempt to change the preferences of the
participants.
