Given an argumentation framework and a group of agents, the individuals may have divergent opinions on the status of the arguments. If the group needs to reach a common po- sition on the argumentation framework, the question is how the individual evaluations can be mapped into a collective one. This problem has been recently investigated in [1]. In this paper, we study under which conditions these operators are Pareto optimal and whether they are manipulable.Collective decision making; Argumentation; Judgment aggregation; Social choice theory;

Caminada, Martin

Pigozzi, Gabriella

Podlaszewski, Mikolaj

English

Research Papers in Economics

Manipulation in group argument evaluation
(Extended Abstract)
Martin Caminada
Individual and Collective
Reasoning, University of
Luxembourg
martin.caminada@uni.lu
Gabriella Pigozzi
LAMSADE
Université Paris-Dauphine
France
gabriella.pigozzi@dauphine.fr
Mikołaj Podlaszewski
Individual and Collective
Reasoning, University of
Luxembourg
mikolaj.podlaszewski@gmail.com
ABSTRACT
Given an argumentation framework and a group of agents,
the individuals may have divergent opinions on the status of
the arguments. If the group needs to reach a common po-
sition on the argumentation framework, the question is how
the individual evaluations can be mapped into a collective
one. This problem has been recently investigated in [1]. In
this paper, we study under which conditions these operators
are Pareto optimal and whether they are manipulable.
Categories and Subject Descriptors
I.2.11 [Artiﬁcial Intelligence]: Distributed Artiﬁcial In-
telligence—multiagent systems
General Terms
Economics, Theory
Keywords
Collective decision making, Argumentation, Judgment ag-
gregation, Social choice theory
1. INTRODUCTION
Individuals can hold diﬀerent reasonable positions on the
information they share. In this paper we are interested
in group decisions where members share the same infor-
mation. One of the principles of argumentation theory is
that an argumentation framework can have several exten-
sions/labellings. If the information the group shares is rep-
resented by an argumentation framework, and each agent’s
reasonable position is an extension/labelling of that argu-
mentation framework, the question is how to aggregate the
individual positions into a collective one.
Caminada and Pigozzi [1] have studied this issue in ab-
stract argumentation and provided three aggregation opera-
tors. The key property of these operators is that the collec-
tive outcome is ‘compatible’ with each individual position.
That is, an agent who has to defend the collective position
in public will never have to argue directly against his own
private position.
Cite as: Manipulation in group argument evaluation (Extended Ab-
stract), Martin Caminada, Gabriella Pigozzi and Mikołaj Podlaszewski,
Proc. of 10th Int. Conf. on Autonomous Agents and Mul-
tiagent Systems (AAMAS 2011), Tumer, Yolum, Sonenberg and
Stone (eds.), May, 2–6, 2011, Taipei, Taiwan, pp. 1127-1128.
Copyright c  2011, International Foundation for Autonomous Agents and
Multiagent Systems (www.ifaamas.org). All rights reserved.
In this paper we focus on the behaviour of two of the
three aggregation operators of [1] and address the following
research questions:
(i) Are the social outcomes of the aggregation operators in
[1] Pareto optimal if preferences between diﬀerent outcomes
are also taken into account?
(ii) Do agents have an incentive to misrepresent their own
opinion in order to obtain a more favourable outcome? And
what are the eﬀects from the perspective of social welfare?
Due to page constraints, we refer the reader to [1] for an
outline of abstract argumentation theory and for the deﬁni-
tions of the sceptical and credulous aggregation operators.
2. PREFERENCES
In order to investigate Pareto optimality and strategy-
proofness we need to assume that agents have preferences
over the possible collective outcomes. We write L ≥i L
0
to denote that agent i prefers labelling L to L
0. We write
L ∼i L
0, and say that i is indiﬀerent between L and L
0, iﬀ
L ≥i L
0 and L
0 ≥i L. Finally, we write L >i L
0 (agent i
strictly prefers L to L
0) iﬀ L ≥i L
0 and not L ∼i L
0.
We assume that the labelling submitted by each agent is
his most preferred one and, hence, the one he would like to
see adopted by the whole group. The order over the other
possible labellings is generated according to the distance
from the most preferred one. For this purpose, we deﬁne
Hamming sets and Hamming distance among labellings.
Definition 1. Let L1 and L2 be two labellings of argu-
mentation framework. We deﬁne the Hamming set between
these labellings as L1 	 L2 = {A | L1(A) 6= L2(A)} and the
Hamming distance as L1 |	|L2 = |L1 	 L2|.
We are now ready to deﬁne an agent’s preference given by
the Hamming set and the Hamming distance as follows.
Definition 2. Let (Ar,def ) be an argumentation frame-
work, Labellings the set of all its labellings and ≥i the pref-
erence of agent i. We say that agent i’s preference is Ham-
ming set based (written as ≥i,	) iﬀ ∀L,L
0 ∈ Labellings,L ≥i
L
0 ⇔ L 	 Li ⊆ L
0 	 Li where Li is the agent’s most pre-
ferred labelling. Similarly, we say that agent i’s preference
is Hamming distance based (written as ≥i,|	|) iﬀ ∀L,L
0 ∈
Labellings,L ≥i L
0 ⇔ L|	|Li ≤ L
0 |	|Li where Li is the
agent’s most preferred labelling.
We now have the machinery to represent individual prefer-
ences over the collective outcomes. We can now turn to the
ﬁrst research question of the paper, i.e., whether the scepti-
cal and credulous aggregation operators are Pareto optimal.
1127Sceptical Operator Credulous Operator
Hamming set Yes (Theorem 1) Yes (Theorem 3)
Hamming dist. Yes (Theorem 2) No (Observation 1)
Table 1: Pareto optimality of the aggregation oper-
ators depending on the type of preference.
3. PARETO OPTIMALITY
Pareto optimality is a fundamental social welfare principle
that guarantees that it is not possible to improve a social
outcome, i.e. it is not possible to make one individual better
oﬀ without making at least one other person worse oﬀ.
Definition 3. Let N = 1,...,n be a group of agents with
preferences ≥i,i ∈ N. L Pareto dominates L
0 iﬀ ∀i ∈ N,
L ≥i L
0 and ∃j ∈ N,L >j L
0.
A labelling is Pareto optimal if it is not dominated by any
other labelling.
Definition 4. Labelling L is Pareto optimal if there is
no L
0 6= L such that ∀i ∈ N, L
0 ≥i L and ∃j ∈ N,L
0 >j L.
We say that an aggregation operator is Pareto optimal if all
its outcomes are Pareto optimal.
Theorem 1. If individual preferences are Hamming set
based, then the sceptical aggregation operator is Pareto op-
timal when choosing from the admissible labellings that are
smaller or equal (w.r.t v) to each of the participants’ indi-
vidual labellings.
Theorem 2. If individual preferences are Hamming dis-
tance based, then the sceptical aggregation operator is Pareto
optimal when choosing from the admissible labellings that are
smaller or equal (w.r.t v) to each individual labellings.
Theorem 3. If individual preferences are Hamming set
based, then the credulous aggregation operator is Pareto op-
timal when choosing from the admissible labellings that are
compatible (≈) to each of the participants’ labellings.
Observation 1. The credulous aggregation operator is not
Pareto optimal when the preferences are Hamming distance
based. This can be shown with an example, not included due
to space constraints.
We summarise our results in Table 1.
4. STRATEGIC MANIPULATION
When an agent knows the positions of the other agents,
he may have an incentive to submit an insincere position.
If an aggregation rule is manipulable, an agent may obtain
a social outcome that is closer to his actual preferences by
submitting an insincere input. Hence, it is important to
study whether the aggregation operators are strategy-proof
(i.e. non-manipulable). Proﬁle PLk/L0
k is proﬁle P where
agent k’s labelling Lk has been changed to L
0
k.
Definition 5. Let P be a proﬁle and Lk ∈ P the most
preferred labelling of an agent with preference ≥k. Let O be
any aggregation operator. A labelling L
0
k such that
O(PLk/L0
k) >i O(P) is called a strategic lie.
Definition 6. An aggregation operator O is strategy-
proof if strategic lies are not possible.
Sceptical Credulous
Hamm. set No (Obs. 3) No
but benev. (Th. 4) and not benev. (Obs. 2)
Hamm. dist. No (Obs. 3) No
but benev. (Th. 4) and not benev. (Obs. 2)
Table 2: Strategy-proofness of operators depending
on the type of preference.
Observation 2. The credulous aggregation operator is not
strategy-proof (the example is omitted for space reasons).
Observation 3. The sceptical aggregation operator is not
strategy-proof (the example is omitted for space reasons).
Surprisingly, the lie under the sceptical operator does not
harm the other agent. On the contrary, it improves the social
outcome for both the agents. We call these lies benevolent.
Theorem 4. Under the sceptical aggregation operator and
Hamming distance or Hamming set based preferences, for
any agent, his strategic lies are benevolent.
We summarise our results in Table 2.
5. CONCLUSION AND RELATED WORK
The study of aggregation problems in abstract argumen-
tation is recent. For example, [2] presents an approach to
merge Dung’s argumentation frameworks.
Given an argumentation framework, [4] address the ques-
tion of how to aggregate individual labellings into a collec-
tive position. By drawing on a general impossibility theorem
from judgment aggregation, they prove an impossibility re-
sult and provide some escape solutions. Relevant for the
present paper is another work by [3], where they explore
welfare properties of collective argument evaluation.
In this paper we have analyzed the sceptical and credu-
lous aggregation operators from a social welfare perspective.
We have studied under which conditions these operators are
Pareto optimal and whether they are manipulable. In fu-
ture, we plan to consider focal set oriented agents, that is,
agents who care only about a subset of the argumentation
framework. We also plan to investigate distances that as-
sign higher values to in-out conﬂicts than to in-undec or
out-undec.
6. REFERENCES
[1] M. Caminada and G. Pigozzi. On judgment aggregation
in abstract argumentation. Autonomous Agents and
Multi-Agent Systems, 22(1):64–102, 2011.
[2] S. Coste-Marquis, C. Devred, S. Konieczny, M.-C.
Lagasquie-Schiex, and P. Marquis. On the merging of
dung’s argumentation systems. Artiﬁcial Intelligence,
171(10-15):730–753, 2007.
[3] I. Rahwan and K. Larson. Welfare properties of
argumentation-based semantics. In Proceedings of the
2nd International Workshop on Computational Social
Choice (COMSOC), 2008.
[4] I. Rahwan and F. Tohm´ e. Collective argument
evaluation as judgment aggregation. In Proc. of 9th
AAMAS, 2010.
1128

Collective argument evaluation as judgment aggregation.

On judgment aggregation in abstract argumentation.

On the merging of dung’s argumentation systems.

Welfare properties of argumentation-based semantics.

Manipulation in group argument evaluation.

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Manipulation in group argument evaluation.

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