Collective argumentation: A survey of aggregation issues around argumentation frameworks

1.1.Organization of the paper

In Section 2 we briefly describe Dung’s argumentation frameworks, the encompassing formalism of our study. In Section 3 we overview the motivations leading to the study of argument aggregation. In Section 4 we analyze the contributions in the literature of collective argumentation, distinguishing between those that are based on the interaction of different sets of arguments and those on different attack relations. Section 5 discusses the application of ideas and methods from Game Theory and Social Choice Theory to collective argumentation. Section 6 analyzes the literature on dynamic argumentation. Section 7 analyzes the literature in which the conflicts of collective argumentation are solved by means of the use of numerical weights attached to components of the aggregate system. Some prominent works are detailed in Section 8. Section 9 covers the works that do not precisely fit in the previous taxonomy. We conclude in Section 10 with some considerations about possible future work.

4.1.The framework-wise approach

The pioneering work of Coste-Marquis et al. [25,57] on merging argumentation systems was motivated by the problem of deriving sensible information up from a collection of argumentation systems belonging to different agents. These authors focus on scenarios in which some agents are able to consider arguments not known by other agents and disagree on the attack relation. As pointed out above, these works discard argument-wise methods and propose instead a three-step process. In the first one, each attack relation is consensually expanded to become a partial system over the entire set of arguments considered by the group of agents. The second step amounts to merge the expanded systems, generating a class of argumentation systems that are at the shortest “distance” of the ones in the profile. The final step consists in selecting the acceptable arguments from the argumentation systems determined in the previous step (this work is detailed in Section 8.1).

Consensual expansions are also considered by Bromuri and Morge [18]. They present a multi-party argumentation game using event calculus [50]. The authors describe their work as an “individual-based approach where the cross-fertilization of argumentations emerge from the interactions between the agents”. In fact, the authors are not aimed at finding the aggregation of a profile of individual argumentation frameworks into a collectively supported one. They just define a way by which each agent could obtain a partial argumentation framework (PAF)44 on the basis of its own initial argumentation framework through a consensual expansion. The “gameboard” is a common environment consisting basically of an initial set of arguments and an attack relation, in which the agents can act by adding arguments, and either adding or deleting attacks. At the end of the game, each agent obtains her own expansion by confronting her initial framework with the common PAF obtained in the game (this differs from the approach by Coste-Marquis et al., where the consensual expansions are obtained by comparing each individual PAF with the entire profile). The game has two subgames, the argument game and the attack game. In the argument game, the agents add arguments until no agent makes any further move; in the attack game, every agent either adds or delete attacks until every agent withdraws. When this happens, the whole game ends.

Gabbay [37] deals with several issues involved in the interaction among argumentation frameworks. The merging problem is conceived as a particular instance in fibered argumentation systems (see also [36]). Gabbay considers the case in which the merged system ⟨AR,⇀⟩ is simply such that AR=⋃i∈NARi and ⇀=⋃i∈N⇀i, where ARi and ⇀i are, respectively, the set of arguments and the attack relation of agent i∈N. He poses the problem of finding the complete extensions of the fibered system up from the known extensions of the agents’ systems (it is not clear whether this involves also an argument-wise procedure). In Gabbay’s approach a priority order among agents is defined which can be used in the merging process. Furthermore, it gives more weight to some agents in the case that a voting system is implemented.

The work by Toni and Torroni [72] can be seen as a particular instance of Gabbay’s fibered systems. The authors propose a methodology for analyzing, in computational terms, the exchanges produced in social networks, technical fora and e-commerce sites. They observe that most of the existing work considering online systems and argumentation focuses on extracting argumentation frameworks of one form or another manually or semi-automatically from these exchanges, while computational argumentation has focused on the dialectical acceptance of arguments (or claims supported by arguments) with respect to a statically defined argumentation framework. Given that, the authors propose to fill the gap between those two lines of work. To this end, they envisage that the active participants in the exchange make very simple and graphical annotations, cataloguing texts as comments or opinions and drawing links to indicate source, support or objection. The annotations are expected to be dynamically added by users as the exchanges are performed over time in existing online systems. The “bottom-up argumentation” refers to the way the argumentation framework is built from users’ opinions, comments and suggested links. The authors choose the Assumption-Based Argumentation model [31] – an instance of Dung’s argumentation frameworks – for an automated mapping of the annotations in order to get the automatic computation of their dialectical acceptance. Clearly, though indirectly, the argumentation framework is the result of the aggregation (addition) of different users’ annotations. An open problem posed by the authors is whether the model would lead to different results when applied to the independent markings of other groups of users. This may call for a specific aggregation protocol to overcome the possible differences of opinion.

Pedersen and Dyrkolbotn [61] introduce a stepwise merging mechanism to model a deliberative process satisfying faithfulness, a condition requiring that every attack in the merged system must be supported by at least one agent. The authors are concerned with the conciliation of opinions in the argumentative process involving several agents. A novelty of the approach is the use of a dynamic modal logic language (previously used to model argumentation dynamics, e.g. Grossi [42]) to express the possible ways in which agents can deliberate in order to reach an agreement on the attacks among arguments.

The work of Airiau, Bonzon, Endriss, Maudet, and Rossit [1] presents an interesting variation. Instead of focusing on how to aggregate a profile of individual argumentation frameworks into one, they ask how a profile can be rationalized up from a “master” argumentation framework. The intuition behind this is as follows. Assume each agent i elaborates her own argumentation framework AFi considering the values that the arguments promote and her subjective assessment of those values, expressed by a preference order ⪰i. Consider now an argumentation framework AF, the “master” argumentation framework. If AF yields AFi under ⪰i, then, in a sense, AFi can be explained by AF (and ⪰i). If that can be done for every single individual argumentation framework in the profile, then AF rationalizes (provides a rationale for) the profile.

Other works on merging argumentation frameworks are devoted to specific issues, which will be discussed below. The works by Tohmé, Bodanza, and Simari [71], Dunne, Marquis, and Wooldridge [33], and Endriss and Grandi [35] (the latter is actually on the more general topic of graph aggregation) analyze the satisfiability of rationality conditions in consonance with principles drawn from SCT, so they will be discussed in Section 5. Gabbay and Rodrigues [38] try to define the weight or strength of arguments in a merged system according to the weights given by the agents associated to the local systems. Since the approach is numerical, it will be discussed in Section 7.

4.2.The argument-wise approach

4.3.The commutation problem

Beyond the question of which procedure is more suitable, Bodanza and Auday [12] analyze the commutation problem: to determine which conditions ensure that the extensions of an aggregate argumentation framework obtained through an argument-wise procedure are the same as the aggregate extensions of the profile of individual argumentation frameworks obtained through a framework-wise procedure. The problem can be illustrated by a slight modification of Example 1.

The authors find some (rather severe) restrictions preserving the equivalence between the mechanisms, for instance: the class of arguments must not have more than three arguments; the number of agents must be odd; the decision is obtained by absolute majority; the individual attack relations are irreflexive and asymmetric; for every pair of arguments a and b, all the agents agree on either, 1) a and b are in conflict (i.e. either a⇀b or b⇀a), or 2) a and b are not in conflict; and the agents unanimously agree on the attack relation on at least one pair of arguments. Auday [6] establishes other rather negative results: commutation is not possible if the aggregation procedure respects strict unanimity (i.e. a social outcome is obtained if and only if all the agents agree on it), not even in the most basic settings, i.e. with only two arguments, two agents and acyclic attack relations. This negative result extends to aggregation procedures in which the proportion of agents that have to agree to impose a collective outcome has to reach a certain quota. The same is true for qualified voting aggregation rules, in which only a selected group of agents has to agree to impose the outcome. On the other hand, Auday [7] finds a positive result for revealed individual attack relations (i.e. subsets of pairs of arguments – the attacks “revealed” by the agents – that yield the same extensions as the original attack relation) under grounded semantics. If the original attack relations are all acyclic, the grounded extension of the union of the corresponding revealed attack relations coincides with the intersection of the grounded extensions of the revealed attack relations.

5.1.Properties from SCT and JAT studied in argument aggregation

We list some common properties from SCT (cf. Arrow, Sen, and Suzumura [5]) and JAT (cf. List and Puppe [56]) discussed in the literature on argument aggregation. We make an informal presentation in order to make these concepts more intuitive.

From SCT:

From JAT:

5.2.Social-choice and judgment-aggregation theoretic analyses of argument aggregation

Let us see now how the current literature deals with these and related properties in the setting of argument aggregation.

Tohmé et al. [71] analyze the aggregation of different abstract attack relations over a common set of arguments. Each of those attack relations can be considered as the representation of a criterion of defeat (or the beliefs of an agent about defeat, etc.). It is well known in the field of SCT that if the conditions of unanimity, anonymity, independence of irrelevant alternatives and non-dictatorship are imposed over an aggregation of preferences, it may be impossible to satisfy them. But a positive result may ensue under the same conditions if the class of winning coalitions (i.e. sets of agents who can impose a choice by voting coordinately) in an aggregation process by voting is a proper prefilter. In the case analyzed by these authors, a proper prefilter is such that: 1) the set of all the attack criteria is a winning coalition; 2) every set containing a winning coalition is also a winning coalition; 3) at least one attack criterion belongs to every winning coalition, and 4) no winning coalition has only one attack criterion. The outcome may preserve some features of the competing attack relations, such as the highly desirable property of acyclicity (in terms of Dung, this corresponds to well-founded argumentation frameworks) which can be associated with the existence of a single extension of an argumentation system. The downside of this is that, in fact, the resulting attack relation must be a portion common to all the “hidden dictators” of the system, that is, all the attack criteria that belong to all the winning coalitions (this work is detailed in Section 8.2).

In Rahwan and Larson [65] and its sequels [51,65], agents have self-interested preferences, meaning that each agent is only interested in the final status of her own subset of arguments. The status is defined in terms of labellings (see Section 4.2.1). The preferences of the agents are preferences over labellings. An agent has acceptability-maximizing preferences if she prefers those labellings that maximize the number of arguments labeled in among those of her interest. An interesting property widely studied in SCT is the Pareto optimality of outcomes. In this setting, a collective labelling L is Pareto optimal if there is no other labelling L′ such that for every agent i, L′ is at least as preferred as L and, for some agent j, L′ is strictly preferred to L for j. The point is that neither the grounded extension nor the preferred extension(s) correspond in general with labellings that are Pareto optimal. Then the authors investigate the relationship between Pareto optimal outcomes and extension semantics under various sufficient conditions. They prove, for instance, that if agents have acceptability-maximizing preferences then every Pareto optimal labelling corresponds to a preferred extension, and that the grounded extension characterizes exactly the Pareto optimal outcome among a rejection-minimizing population, i.e. consisting of agents that prefer the labellings that minimize the label out on their own arguments (more details in Section 8.3).

Other works analyze the possibility of rational collective outcomes under a set of widely accepted social choice principles. Rahwan and Tohmé [69] deal with the strategic aspects of aggregating argument frameworks under universal domain (every possible profile is in the domain of the aggregation function), unanimity, anonymity and systematicity (a variant of independence of irrelevant alternatives according to which the collective judgment about two arguments a and b in any two different profiles should be the same if every individual judgment on them is the same in the two profiles). Of particular interest is the ability of manipulating declarations (the same concern as in Rahwan and Larson [65]) in order to obtain a collective framework in which some desired arguments are labeled in. An aggregation procedure that disallows this possibility is called strategy-proof. It is shown that when voting is restricted to a core set of arguments (the focal set), argument-wise plurality is strategy-proof. On the other hand, as discussed above, this aggregation procedure does not satisfy collective rationality. In this paper this limitation is circumvented when the individual labellings satisfy two very demanding conditions. The first condition, Condorcet defeat, indicates that when for any argument a the alternative out wins, there must exist another argument b attacking a for which a plurality of agents votes in. On the other hand, non-Condorcet indecision is obtained if for any argument a the plurality votes yields in while none of its attackers can have a plurality of in or undec votes. Bodanza [11] presents three sensible restrictions on the domain that ensure the collective rationality, at least for the plurality voting mechanism: the number of agents is odd (to avoid ties), the argumentation framework admits only three possible labellings (this leads to the satisfaction of Condorcet defeat), and the agents should minimize the number of arguments labeled undec66 (satisfying thus non-Condorcet indecision).

Li [53] establishes an impossibility result for aggregate attack relations, similar to the Paretian liberal dilemma in SCT [70]. This dilemma captures the inherent tension between individual rights and collective consensus. The introduction of rights in collective argumentation can be justified, for instance, by the fact that some agents can have a relevant expertise in the subject matter at stake that other agents lack, and thus they are assigned the right to determine the collective defeat relation on some pairs of arguments (cf. Kontarinis, Bonzon, Maudet, and Moraitis [48]). Li shows that an aggregation function yielding a collective defeat relation satisfying the conditions of universal domain and unanimity is incompatible with the condition of minimal liberalism, i.e. the assignment of rights to at least two agents, each of them on at least one pair of arguments.

Some works are concerned with a key problem derived from judgment aggregation, the Discursive Dilemma or Doctrinal Paradox. This is a widely discussed problem arising in settings where many conflicting judgments are aggregated to yield a collective judgment. More specifically, the paradox occurs when different sets of judgments satisfying desirable rationality conditions are aggregated yielding a collective set of judgments that does not satisfy those same conditions. For instance, consider a consistent set of propositions which are premises from which a particular conclusion can be logically derived. The discursive dilemma arises when a voting mechanism applied both on the premises and the conclusion yields an aggregate set of premises which is inconsistent with the aggregate conclusion. This paradox has motivated important developments in the literature on judgment aggregation (List and Pettit [55]; etc.).

Pigozzi [62] argues that the frequently suggested escape-routes for the discursive dilemma in the judgment aggregation literature, through the so called premise-based and conclusion-based procedures, are not satisfactory methods for group decision-making. She proposes a new aggregation procedure for the case in which the outcome is a set of arguments, combining features of the premise and the conclusion-based procedures. In this setting, an argument is a consistent assignment of truth-values to both the conclusion and its premises, instead of one or another. The premise-based procedure and the conclusion-based procedure are therefore included in this unitary approach that the author calls argument-based procedure.

Pigozzi and van der Torre [63] analyze the same problem but looking at the axioms that characterize the aggregation procedure. Noting that majority voting creates also problems in the aggregation of preferences, like the famous Condorcet Paradox, in which cycles of preference arise from the aggregation of acyclic individual preferences, they look for conditions that could help to get rid of the paradox. Following List and Pettit [55], they consider four axioms that, given a profile of interpretations (I1,…,In), should be satisfied by the aggregate interpretation IS, where S={1,…,n}: universal domain, collective rationality (IS is consistent), anonymity and systematicity (if in a profile (I1,…,In) a proposition A has the same votes as proposition B in an alternative profile (I1′,…,In′), the truth value or A in IS should be the truth value of B in IS′). List and Petit have shown that no aggregation process can satisfy these axioms simultaneously. Pigozzi and van der Torre propose to weaken systematicity, replacing it with a condition they call premise independence of irrelevant propositional alternatives, according to which if in a profile (I1,…,In) the proposition A has the same votes as in the alternative profile (I1′,…,In′), the truth value of X∈C(A) in IS should be the same in IS′, where C(A) is the set of propositions in the class L of logical consequences of A. This condition does yield a positive result, namely the existence of consistent ways of aggregating judgments.

Caminada and Pigozzi [20] note that the discursive dilemma can arise in a labelling aggregation setting, since statements such as ‘argument a is in’, ‘argument b is out’, etc., are just judgments. They seek ways to ensure the compatibility of a collective outcome with the individual judgments, in the sense that any individual member must be able to defend the collective decision. These authors define a partial ordering ⊑ of labellings: given two labellings L and L′, L⊑L′ iff in(L)⊆in(L′) and out(L)⊆out(L′). Several aggregation operators can be defined, avoiding the discursive dilemma. So, for instance, consider the largest admissible labelling Lso such that Lso⊑Li for each labelling in the profile (L1,…,Ln). The authors prove that it includes the skeptical labelling obtained from the profile (any argument a is labeled in in the aggregate labelling iff a is labeled in in each Li and is labeled out iff a is labeled out in each Li). Lso is free of the discursive dilemma for argument systems, as well as credulous and super credulous aggregate labellings obtained in terms of other extension concepts. The credulous aggregate labelling Lco is motivated by the idea that the group acceptance/rejection of an argument is justified if each individual either agrees on it or, in the worst case, is indifferent. The super credulous aggregate labelling Lsco is obtained through an “expansion” of the credulous outcome by relabelling illegal undecs to ins and outs. (More details of this work in Section 8.4).

Subsequently, Caminada, Pigozzi, and Podlaszewski [21] analyze the skeptical and credulous aggregation operators from a social welfare perspective (intuitively, through the credulous operator the group assigns in (resp. out) to argument A if there is someone who believes that A is in (resp. out) and nobody thinks that A is out (resp. in), and A is labeled undec in all other cases). The authors study under which conditions these operators are Pareto optimal and whether they are manipulable. To define those conditions it is assumed that individuals have preferences over the possible collective outcomes and the labelling submitted by each agent is her most preferred one (indeed, the one she would like to see adopted by the whole group). Hence, each agent’s preference order can be defined according to how “close” is each possible outcome to her own labelling. To this end, the authors define the notions of Hamming set and Hamming distance (see Section 7). The following results are obtained: if individual preferences are either Hamming set or Hamming distance based, then the skeptical aggregation operator is Pareto optimal when choosing from the admissible labellings that are smaller or equal (w.r.t. ⊑) to each of the participants’ individual labellings. Credulous aggregation operators are also Pareto optimal when choosing from the admissible labellings that are compatible with each individual labelling if individual preferences are Hamming set based, but not if they are Hamming distance based.

Booth, Awad, and Rahwan [15] elaborate on this result, pointing out that the aggregate labellings that are obtained by means of down-admissibility (the g.l.b. of admissible labellings L⊑Li for each i) as well as of up-completeness (the l.u.b. of complete labellings Li⊑L) fail to satisfy independence (analogous to Premise Independence of Irrelevant Propositional Alternatives for labellings and labels in and out). To solve this problem they introduce interval aggregation: consider the number of agents that vote out for an argument a, say n− and those that vote in, n+. Then (n−,n+), belongs to a previously defined family of intervals in S, a is labeled in if n+⩾n− and out if n+⩽n−. If the interval does not belong to the allowed family, a is labeled undec. This procedure satisfies many desirable properties, but it cannot simultaneously satisfy independence and collective rationality. To achieve this, they resort to a labelling L↓, which is the g.l.b. among the admissible labellings L such that L⊑Lint (where Lint is the aggregate labelling obtained through interval aggregation). Then they obtain the l.u.b. of the complete labellings L such that L↓⊑L. The resulting labelling L↓↑ satisfies collective rationality as well as a weaker form of independence.

Awad et al. [9] note that labelling aggregation, with its tree of “truth values” is closer to 3-value judgment aggregation. In fact, argumentation frameworks are closer to be a representation of contradictory information than of propositional satisfiability. In any case, majority voting is no longer a reasonable procedure when three alternatives are at play. But it can be replaced by plurality voting, in which the alternative with more votes wins. Applied argument-wise, this procedure ensures the satisfaction of several conditions, in particular independence, weak systematicity and non-dictatorship but violates universal domain (it becomes restored when ties are disallowed) and collective rationality. The only way in which this voting procedure yields collective rationality is when the voters are restricted to vote only for arguments in the grounded extension of their respective argument frameworks.

Endriss and Grandi [35] analyze a related problem, namely the aggregation of graphs with more than 3 edges. As a graph is fully determined by its set of edges, a graph aggregator is a function that maps a profile of sets of edges to a set of edges. They show that there does not exist a way to obtain a graph with some desired properties if a graph-theoretic version of independence is required jointly with the non-existence of a dictator, i.e. an individual whose proposed graph becomes always the aggregate one.

Dunne et al. [33] study the computational complexity of checking SCT properties in the context of argument aggregation. They consider Boolean circuits as a “natural” representation of argument aggregation procedures, where the Boolean values ⊤ and ⊥ stand for the presence and absence of an attack between a pair of arguments, respectively. The complexity of determining whether a given aggregation function has a specific property is reduced to that of finding the value of a propositional formula over some specified basis (e.g. the set of logical connectives {∧,∨,¬}). Using this method the authors show, for example, that the problem of checking anonymity is coNP-complete.

Delobelle et al. [27] analyze their distance-based merging operators in terms of the aggregation axioms, as defined in Dunne et al. [33]. These authors prove that these operators satisfy anonymity, identity (if all the argumentation frameworks in the input coincide, the aggregation result must be identical to them), unanimity, and majority (for majoritarian aggregation functions).

Rahwan and Larson [65] present a game-theoretical model of argumentation aggregation in which agents can act strategically revealing or hiding their arguments in such a way that the collective outcome can be manipulated. Their goal is the design of a mechanism which cannot be manipulated by the agents. Then the authors offer a strategy-proof direct argumentation mechanism that makes the agents to reveal their arguments simultaneously. Strategy-proofness is guaranteed under the condition that the agents use a skeptical criterion of argument justification (grounded semantics) and their sets of arguments do not include odd-cycles of attack. Rahwan, Larson, and Tohmé [67], moreover, consider more realistic scenarios in which agents can also lie presenting arguments that they do not have in their argument sets. Strategy-proofness is obtained under two conditions: (a) an agent cannot benefit from hiding any of its own arguments, because its arguments cannot “harm” its focal argument, and (b) an agent cannot benefit from revealing any argument it does not have, because it cannot “benefit” its focal argument.

Bonzon, Maudet, and Moretti [14] investigate the uncertainty inherent to debates and apply concepts from cooperative game theory to account for the decision-problem faced by an agent in this context. The authors assume that agents have cardinal preference relations over single arguments, assessing the relevance of each argument with respect to her/his own goals, and facing the uncertainty about the outcome of the debate. The aim is to measure the relative importance (the “worth”) of arguments for an agent, taking into account her own preferences as well as the information provided by the attack relations among arguments. The authors develop a game theoretic coalitional model in which a classical power indexes for coalitional games (the Shapley value) is used to measure the impact of a single argument in a debate.

8.1.Coste-Marquis, Devred, Konieczny, Lagasquie-Schiex, and Marquis: Merging argumentation systems

In [57], the authors offered the first approach to argument aggregation using Dung’s argumentation frameworks. Subsequent developments were introduced in [25]. They propose a procedure that starts by making each agent aware of the arguments considered by the rest of the agents. Then each agent expands her framework by incorporating all those arguments. The set of arguments is now common to all the agents, becoming the status quaestionis on the basis of which they start discussing. Given that, the agents add new attacks in a consensual way. To do this, each agent keeps her initial attacks and adds all the attacks (a,b) accepted by every agent who had a and b in his original set of arguments. Similarly, if there is a consensus on the non-existence of attacks between a and b, no such attacks are added to the expanded argument systems. In this way, each agent ends up with a consensual expansion of her own argumentation framework. Each of these expansions constitutes a partial argumentation framework (PAFs), in which, given a pair of arguments a and b, the agent determines either that (a,b) belongs already to her own attack relation or that there is no attack between them or, finally, that she ignores whether an attack exists. More formally, a PAF is a quadruple ⟨A,R,I,N⟩ where A is a finite set of arguments and R, I and N are binary relations on A: R is the attack relation, I is the ignorance relation and is such that R∩I=∅, and N=A×A∖R∪I is the non-attack relation (since N can be deduced from A, R and I, a PAF can be fully specified by a triple ⟨A,R,I⟩). A PAF can be “completed” just by transferring any (a,b) from the ignorance class to any of the other two. That is, AF=⟨A,S⟩ is a completion of PAF=⟨A,R,I⟩ iff R⊆S⊆R∪I.

Next, a fusion process will select a class of argumentation frameworks representing the profile P. To do this, a pseudo-distance between PAFs is defined, that is, a number which represents how far or close is one PAF to another one. A pseudo-distance d between PAFs over A is a mapping which associates a non-negative real number to each pair of PAFs over A and satisfies the properties of symmetry (d(x,y)=d(y,x)) and minimality (d(x,y)=0 if and only if x=y). The pseudo-distance can be defined in many ways, one of them, for example, being the number of pairs (a,b) that belong to the attack relation of one of the PAFs but do not belong to the attack relation of the other one. Then the authors define the merging of P as any set of argumentation frameworks that minimizes the distances among the PAFs of the profile. An aggregation function is a mapping ⊗ from R+n (each value i⩽n representing the distance of the i-est PAF to the entire profile) to R+ that satisfies non-decreasingness (i.e. if xi⩾xi′ then ⊗(x1,…,xi,…,xn)⩾⊗(x1,…,xi′,…,xn)), minimality (i.e. ⊗(x1,…,xn)=0 if ∀ixi=0), and identity (i.e. ⊗(x)=x). The merging of a profile is defined as a set of argumentation frameworks. Given a profile P=⟨AF1,…,AFn⟩, a pseudo-distance d between PAFs, an aggregation function ⊗ and n expansion functions exp1,…,expn, the merging of P is the set of AFs

The contribution of this approach is twofold: on one hand, the authors present a general model for merging argumentation systems; on the other, they define concrete ways in which the model can be instantiated. In the general model, a common set of arguments for all the agents is prescribed; in the instantiation, the union of all the individual sets of arguments is taken, representing the common set obtained in information exchanges. In the general model, an agreement on the attack relation is needed; in the instantiation, since the union of all the subjective attack relations leads to counterintuitive outcomes, consensual expansions are adopted. In the general model, a merging operator is defined in terms of a distance relation among PAFs; in the instantiation, a pseudo-distance called edit distance is introduced, counting the number of pairs of arguments that belong/do not belong to the attack relations of the individual systems. In the general model, an aggregation operator yields the argumentation frameworks closer to the rest of them in a given profile; in the instantiation sum, max and leximax aggregation functions are considered.

8.2.Tohmé, Bodanza, and Simari: Social choice-theoretical analysis of attack aggregation

SCT is concerned with the rationality of aggregation operations on individual preferences. Similarly, the work by Tohmé et al. [71] is concerned with the rationality of aggregation operations on individual attack criteria among arguments. Arrow’s Impossibility Theorem [4] claims that four quite natural constraints that capture abstractly the properties of a democratic aggregation process (to wit, the Pareto condition (i.e. unanimity), positive responsiveness, independence of irrelevant alternatives and non-dictatorship) cannot be simultaneously satisfied. That is true for the case of reflexive and transitive preference relations over the alternatives. Once those constraints become incorporated in the framework of argumentation, we could expect something like Arrow’s theorem to ensue. But attack relations and preference relations are different in many respects. While preferences are usually formalized as reflexive, transitive and complete relations, attack relations are free to adopt any configuration. This is a reason why an Arrow-like result may not be a necessary outcome for the aggregation of argumentation systems. Indeed, the authors obtain a possibility result, though it is achieved at the cost of accepting the existence of “hidden dictators”.

The approach considers an extended argumentation framework AFn=⟨AR,⇀1,…,⇀n⟩, for a given n where each ⇀i (1⩽i⩽n) is a particular attack relation among arguments of AR representing different criteria according to which arguments are evaluated one against another. An aggregate argumentation framework is understood as a framework AF∗=⟨AR,F(⇀1,…,⇀n)⟩ where F(⇀1,…,⇀n)=⇀ is a function of the attack relations of AFn (F may be applied over any extended argumentation framework with n attack relations). This is of course analogous to a social system in which a unified criterion must by reached. A paradigmatic instance is that of majority voting, which in this setting can be expressed as follows:

The authors offer a more or less realistic example:

Another way of aggregating attack relations is by restricting majority voting to a qualified voting aggregation function. It fixes a class U⊂{⇀1,…,⇀n} such that the outcome of majority voting over a pair of arguments is imposed on the aggregate only if every ⇀j∈U belongs to the majority (i.e. each member of U has veto power). Otherwise, in the attack relation none of the arguments attacks the other. In the example above, the qualified voting aggregation function with U={⇀2,⇀3} yields ⇀={(a,c)} (there is no consensus on the interaction between b and c among the members of U). Both majority and qualified voting (with U⩾2) satisfy the four mentioned Arrovian properties (Proposition 1, p. 14). Majority voting, on the other hand, is not free of yielding controversial outcomes. One issue is the Condorcet’s Paradox, i.e. the case in which acyclic individual preference relations lead to cycles in the aggregate relation. Nevertheless, if F is a qualified voting aggregation function and each ⇀i is acyclic, then the aggregate ⇀ is acyclic (Proposition 2, pp. 15–16). Though attack relations can adopt any configuration, the focus on acyclic relations is justified by the fact that they characterize well-founded argumentation frameworks, and this has the consequence that all Dung’s semantics coincide in a single extension [30].

Every aggregation function (or voting system) can be represented by its class of decisive sets, i.e. the winning coalitions that can impose an outcome. In our setting, U⊆{⇀1,…,⇀n} is a decisive set iff, if a⇀jb for every ⇀j∈U then a⇀b (for example, the decisive sets corresponding to majority voting are those coalitions with half of the voters plus one). Then the authors show (Proposition 3, pp. 16–17) that if for every profile (⇀1,…,⇀n) of individual acyclic attack relations F yields an acyclic ⇀, then the Pareto condition, positive responsiveness, independence of irrelevant alternatives and non-dictatorship are guaranteed if and only if the class of decisive sets satisfies the algebraic properties of a proper prefilter, to wit: 1) the class formed by all the agents is a decisive set, 2) any set containing a decisive set is also decisive, 3) at least one agent belongs to every decisive set, and 4) no decisive set has exactly one member. More formally, let Ω¯={Ωj}j∈J the class of decisive sets of F, then

Finally, if each ⇀i is acyclic and F is such that its class of decisive sets form a proper prefilter, then the resulting collective attack relation ⇀ is acyclic, determining a unique extension which is grounded, preferred and stable (Proposition 5, p. 19).

8.3.Rahwan and Larson: Game-theoretic argumentation mechanism design

In [65], Rahwan and Larson observe that the outcome of argumentation is determined not only by the rules of argument acceptability, but also by the strategies employed by the agents who present these arguments. Since agents can be self-interested and may have different preferences over the arguments, the design of the argument acceptability rule should take the mechanism design perspective (a well-known concept from Economics and GT). The authors pose the question: what game rules guarantee a desirable social outcome when each self-interested agent selects the best strategy for itself? In a multi-agent setting where agents introduce the arguments, manipulation can arise if by hiding an argument an agent can change the status of another argument, such that the result is better according to her own interests. Then Rahwan and Larson apply the tools of GT and mechanism design to abstract argumentation frameworks, finding argument evaluation criteria ensuring strategy-proofness.

A set of self-interested agents is denoted by I. The type θi∈Θ of agent i is drawn from the set of possible types Θ. The type represents the preferences and private information of an agent. The preferences of an agent are defined over a set O of possible outcomes. In the context of an argumentation framework AF=⟨AR,⇀⟩, the type of agent i, Ai∈AR, is the set of arguments that the agent is capable of putting forward (note that agents can operate on the set of arguments but not on the attack relation, which is fixed). Given the agents’ types (argument sets) a social choice function f maps a type profile into a subset of arguments, f:2AR×⋯×2AR→2AR. Given a semantics S, and given a type profile (A1,…,AI), the argument acceptability social choice function f is defined as the set of acceptable arguments given the semantics S, in symbols, f(A1,…,AI)=Acc(⟨A1∪⋯∪AI,⇀⟩,S). The set of possible outcomes is O=2AR, and the preferences of the agents are expressed using utility functions u where ui(o,Ai) denotes agent i’s utility of outcome o when her type is the argument set Ai.

Agents may not have incentive to reveal their true type because they are able to influence the final argument status assignment by lying, and thus obtain higher utility. There are two ways that an agent can lie in the model. On one hand, an agent might create new arguments that it does not have in her argument set. Another way is by hiding some of her arguments. Either by presenting arguments that the agent does not subscribe or by refusing to reveal certain arguments, an agent might be able to break defeat chains in the argumentation framework, thus changing the final set of acceptable arguments. A strategy of an agent specifies a complete plan that describes what action the agent takes at every stage at which she has to make a decision. In this model, the actions available to an agent involve announcing sets of arguments. Thus a strategy si∈Σi for agent i would specify, for each possible subset of arguments that could define its type, what set of arguments to reveal. Then, a direct argumentation mechanism is defined as MAFS=(Σ1,…,ΣI,g(·)), where Σi=2Ai and g:Σ1×⋯×ΣI→2AR. That is, the mechanism determines a subset of acceptable arguments for each profile of agents’ strategies (revealed arguments).

Consider now that agents have acceptability maximizing preferences in the sense that, given two possible outcomes o1 and o2, if |Ai∩o1|⩾|Ai∩o2| then ui(o1,Ai)⩾ui(o2,Ai) for every agent i (i.e. agents prefer those outcomes that maximize the number of arguments of their respective types). And consider a skeptical direct argumentation mechanism, to wit, a mechanism in which S is the grounded semantics. The following example shows how the mechanism can be manipulated.

The authors find a sufficient condition to make the skeptical direct argumentation mechanism strategy-proof when agents have acceptability maximizing preferences: each agent’s type is a conflict-free set of arguments which does not include indirect attacks (i.e. odd-length sequences of attack among its members) (Theorem 32, p. 1037). Conversely, strategy-proof mechanisms implementing skeptical social choice functions lead to agents’ strategies free of indirect attacks (Theorem 33, ibid.).

Subsequently, in [66] and [51], Rahwan and Larson focus on the property of Pareto optimality, which measures whether an outcome can be improved for one agent without harming other agents. In this setting, the authors change the methodology from using extension semantics to using labellings. These works were commented in Section 5.2.

8.4.Caminada and Pigozzi: Judgment aggregation in abstract argumentation

In Caminada and Pigozzi [20], judgment aggregation is studied as the aggregation of individual labellings of a given argumentation framework. It constitutes the first approach to judgment aggregation from the point of view of argumentation theory, and the first in using abstract argumentation to that end. The authors are not particularly concerned with the discursive dilemma – a central subject of JAT – but with defining social outcomes that any individual participating in the decision could subscribe while guaranteeing collective rationality. Given a set of individuals N={1,…,n}, the aim is to define a general labelling aggregation operator FAF that assigns a collective labelling LColl to each profile {L1,…,Ln} of individual labellings. Note that the profiles here are defined as sets instead of as vectors. This is because though several individuals may submit the same judgments set (i.e. the same labellings), the motivation is to avoid situations in which any group member is forced to commit herself to a position that goes against her opinion. Therefore, cardinality considerations (regarding, for instance, minoritarian vs. majoritarian opinions) do not play any role in the authors’ approach. Consequently, a general labellings aggregation operator is defined as a function FAF such that, given an argumentation framework AF=⟨AR,⇀⟩ and the set Labellings of all the possible labellings of AF, FAF:2Labellings∖{∅}→Labellings and FAF({L1,…,Ln})=LColl. Then the authors define two kinds of outcomes: skeptical outcomes and credulous outcomes.

The skeptical outcome is illustrated as follows. Assume all the group members are gathered in a meeting with a chair who asks the opinion of the participants about each argument. Then the authors explain: “If all participants unanimously think the argument should be accepted, then the argument is initially accepted. If all participants unanimously think the argument should be rejected, then the argument is initially rejected. In all other cases, the group as a whole does not have an explicit opinion about the argument. After all arguments have been treated this way, the meeting goes to the second phase. The chairman evaluates whether each acceptation or rejection can be justified by the outcome found so far. An argument that is accepted without every defeater being rejected can no longer be accepted. Conversely, an argument rejected without an accepted defeater can no longer be rejected. In each of these two cases, the group has to abstain from stating any further opinion about those arguments. This is an iterative process, since once one abstains from stating an explicit opinion about a particular argument, opinions (acceptation or rejection) about related arguments can be no longer justified. The process does not stop until the group no longer has unjustified explicit opinions. After this second phase is over, the result will be a position that is “smaller or equal” (less or equally committed) than each individual position of the participants. That is, each argument that is accepted by the group is also accepted by each individual participant, and each argument that is rejected by the group is also rejected by each individual participant.” (p. 77). In this way, the skeptical outcome is the most committed position. Let us see the authors’ formalisms to capture these ideas. Given two labellings L1 and L2, L1 is less or equally committed as L2 (L1⊑L2) iff in(L1)⊆in(L2) and out(L1)⊆out(L2). The relation ⊑ is a partial order on labellings (i.e. reflexive, transitive and antisymmetric). As a result, the set of admissible labellings that are smaller or equal to a given labelling L has a (unique) biggest element (Theorem 5, p. 78). The biggest element of the set of all the admissible labellings that are less or equally committed as a labelling L is called the down-admissible labelling of L. Then, once the group has found the skeptical initial labelling Lsio in the first phase of the meeting, the second phase consists in finding the down-admissible labelling of Lsio. The skeptical initial labelling is found through the operator sioAF:2Labellings∖{∅}→Labellings such that sioAF({L1,…,Ln})={(a,in)∣∀i∈{1,…,n}:Li(a)=in}∪{(a,out)∣∀i∈{1,…,n}:Li(a)=out}∪{(a,undec)∣∃i∈{1,…,n}:Li(a)≠in∧Li(a)≠out}.

To find the down-admissible labelling of Lsio the authors define a series of contractions. First, a contraction function cAF:Labellings×AR→Labellings is such that cAF(L,a)=(L∖{(a,in),(a,out)})∪{(a,undec)}. Second, given a labelling L, a contraction sequence from L is a list of labellings [L1,…,Lm] such that:

As a counterexample the authors present the case of a floating defeat:

The credulous outcome is motivated by the idea that the group acceptance/rejection of an argument is justified if each individual either agrees on it or, in the worst case, is indifferent. A compatibility relation among labellings is defined: L1≈L2 iff in(L1)∩out(L2)=∅ and out(L1)∩in(L2)=∅ (this relation is reflexive, symmetric and non-transitive. Moreover, if L1⊑L2 then L1≈L2). The process of finding the aggregate labelling resembles the skeptical one, but in this case an opinion of the kind in/out finds collective acceptance if each individual either agrees on it or abstains subscribing undec. The credulous initial labelling is found through the operator cioAF:2Labellings∖{∅}→Labellings such that cioAF({L1,…,Ln})={(a,in)∣∃i∈{1,…,n}:Li(a)=in∧∄i∈{1,…,n}:Li(a)=out}∪{(a,out)∣∃i∈{1,…,n}:Li(a)=out∧∄i∈{1,…,n}:Li(a)=in}∪{(a,undec)∣∀i∈{1,…,n}:Li(a)=undec∨(∃i∈{1,…,n}:Li(a)=in∧∃i∈{1,…,n}:Li(a)=out)}. The credulous initial labelling Lcio=cioAF({L1,…,Ln}) is such that for every i∈N, Lcio≈Li (Theorem 9, p. 85). The credulous aggregation operator coAF is defined only on the domain of admissible labellings and it also only returns admissible labellings. As a result, the ensuing credulous aggregate labelling Lco is such that for every i∈N, Lco≈Li (Theorem 10, p. 86. In this case, the result cannot be extended to complete labellings).

The authors then apply a similar methodology to define a super-credulous aggregation operator. We make an informal presentation here. The idea is to apply an expansion function to “expand” the credulous outcome by relabelling illegal undecs to ins and outs. A third phase in the meeting is conceived in which each undecided argument is accepted if all its defeaters were previously rejected, and rejected if some of its defeaters was previously accepted. The process goes on (an expansion sequence is defined) until every argument that can be accepted is accepted and each argument that can be rejected is rejected (no illegal undec remains). It is proved that the ensuing sequence meets the super-credulous outcome Lsco (Theorem 12, p. 89), and that for every i∈N, Lsco≈Li (Theorem 13, p. 90).

In sum, this work proposes particular aggregation mechanisms oriented to get collective outcomes that any individual can defend without betraying her own opinions. Moreover, the collective labellings are guaranteed to keep rationality in terms of admissibility and related argumentation-theoretical semantical concepts.

8.5.Bonzon and Maudet: On the outcomes of multiparty persuasion

In [13], Bonzon and Maudet study dynamical aspects of argumentation and propose a multiparty persuasion protocol. Close in spirit to the work of Rahwan and Larson [65,66], the authors consider that agents have a chance to influence the outcome of an argumentation game depending on how they play, and use game-theoretical concepts for their study. They also take for granted that agents’ argument moves should immediately improve their satisfaction with respect to the current situation of the debate. In this context, a number (n>2) of non-coordinated agents exchange arguments on a common gameboard (among the motivating applications they have in mind are, for example, online platforms allowing users to asynchronously modify the content of a collective debate). To keep things simple, the following assumptions are made: (i) all the agents are focused on the same single argument; (ii) all the agents make use of the same argumentation semantics, to wit, grounded semantics, to evaluate both their private argumentation system and the situation on the common gameboard; and (iii) all the agents share the same set of arguments, but they may have different views on the attack relations between these arguments (giving raise to a framework-wise approach).

The model considers a set N of agents, each one holding an argumentation framework AFi=⟨AR,⇀i⟩, sharing the same arguments AR. To tackle the problem of the collective view, a notion of merged system is considered, adopting a majority voting approach. Ties are broken in favor of the absence of an attack, which allows to ensure the existence of a single merged argumentation system MASN=⟨AR,⇀⟩, where (a,b)∈⇀ iff |{i∈N:(a,b)∈⇀i}|>|{i∈N:(a,b)∉⇀i}| . The corresponding merged outcome is the grounded extension (E) of MASN, denoted E(MASN). Agents contribute to the debate in a step-by-step process, guided by their individual assessment of the current state of the discussion, and without coordination with the other agents. The debate has a focused argument, called the issue d of the debate, and agents are concerned with the coincidence of the acceptability status of d in the merged system and in their individual systems. Then the debate can be seen as opposing two groups of agents, PRO={i:d∈E(AFi)} and CON={i:d∉E(AFi)}. The gameboard is a “common” weighted argumentation system, where the weight is simply a number equal in the difference between the number of agents who asserted a given attack and the number of agents who opposed it. The fact that the attack of argument a on argument b has a weight α is denoted a⇀αb. Let A(GB) be the set of all the arguments present on the gameboard. The collective outcome is obtained by applying the semantics used on the argumentation framework ⟨A(GB),⇀⟩, where ⇀∈A(GB)×A(GB) and ⇀={a⇀αb:α>0}. In words, the attacks retained are only those supported by a (strict) majority of agents having expressed their view on this relation.

The protocol proceeds in rounds which alternate between the two groups PRO and CON. Within these groups though, no coordination takes place: the agents may for instance play asynchronously and the authority simply picks the first permitted and relevant move before returning the token to the other side. Permitted moves are simply positive assertions of attacks a⇀b (with b being an element of the set of arguments A(GB) at round t, At(GB)), or contradiction of (already introduced) attacks (with (a,b) being an element of the attack relation ⇀ at round t, ⇀t). A move is relevant at round t for a PRO agent (resp. CON agent) if it puts the issue back in (resp. drops the issue from) E(AFt(GB)). Furthermore, the protocol prevents the repetition of similar moves from the same agent.

Then the protocol is defined as follows:

Example 7.

Let 1, 2, and 3 be three agents with their argumentation frameworks AF1, AF2 and AF3, respectively, and the merged argumentation framework MAS as follows:

AF1=⟨{a,b,c},{(a,b),(a,c)}⟩;E(AF1)={a}AF2=⟨{a,b,c},{(a,b),(b,c)}⟩;E(AF2)={a,c}AF3=⟨{a,b,c},{(b,c)}⟩;E(AF3)={a,b}MAS=⟨{a,b,c},{(a,b),(b,c)}⟩;E(MAS)={a,c}

The issue of the dialogue is the argument c. We have CON={1,3} and PRO={2}. At the beginning, we have AF0(GB)=⟨{c},{}⟩ with E(AF0(GB))={c}. A sequence of moves allowed by the protocol is the following (the second column records the provisional attack relation at each time t): At this point the gameboard is stable and the outcome E(AF(GB))={a,b} is obtained.

Note that the status of an issue in the merged argumentation system can contradict the opinion of the majority. Bonzon and Maudet agree with Coste-Marquis et al. that if agents vote on extensions, a lot of significant information is not exploited. (Moreover, this example replicates the commutation problem discussed in Section 4.3.)

In order to characterize the status of the issue obtained by the protocol the authors introduce the notion of global arguments-control graph (ACG). The idea is to gather the attacks of all agents in the same argumentation graph, and then determine which group, PRO or CON, has the control over some path of this graph, and thus a possible way to reach its preferred outcome. Let add(a,b)={i∈N:(a,b)∈⇀i} and rem(a,b)={i∈N:(a,b)∉⇀i}. Then X∈{PRO,CON} has constructive control of (a,b) in ⇀=⋃i=1n⇀i iff the number of agents in X who can add (a,b) is greater than the number of agents in X¯ ∈{PRO,CON}∖{X} who can remove it. X∈{PRO,CON} has destructive control of (a,b) in ⇀=⋃i=1n⇀i iff the number of agents in X who can remove (a,b) is greater than the number of agents in X¯∈{PRO,CON}∖{X} who can add it. Observe that the notion of destructive control intuitively says that a group has the control to overturn any possible attempt to establish a given relation. Then the global arguments-control graph ACGN=⟨AR,⇀⟩ is constructed as follow: (1) ⇀=⋃i=1n⇀i, (2) label each (a,b)∈⇀ by the information about control and playability for each group X∈{PRO,CON} ((a,b) is playable by X if (a,b)∈⇀i for some i∈X).

The authors establish some properties and then show conditions under which the MAS is reachable (though convergence is not guaranteed).