A Tool For Ranking Arguments Through Voting-Games Power Indexes Stefano Bistarelli1 , Francesco Faloci1 , Francesco Santini1 , and Carlo Taticchi2 1 Dipartimento di Matematica e Informatica, Università degli studi di Perugia, Italia 2 Gran Sasso Science Institute, L’Aquila, Italy Abstract. Abstract Argumentation Frameworks allow to represent sets of arguments, together with possible relations among them, in form of oriented graphs. This paper gives a short overview of a plug-in function developed for ConArg, a solver of Abstract Argumentation related prob- lems. The web-based tool we present computes a ranking of arguments by applying different voting games power indexes, where the coalitions of individuals are defined by the extensions satisfying Dung’s semantics. At this stage of development, the tool can make use of both the Shapley Value and the Banzhaf Index. 1 Introduction ConArg is a suite of tools that was started to be developed with the purpose to facilitate research in the field of Argumentation in Artificial Intelligence [3], a discipline that copes with uncertainty and defeasible reasoning. In Abstract Argumentation, arguments have no internal structure and the attack relation is not defined; it provides means by which it is possible to distinguish acceptable and not acceptable arguments at an abstract , as its name suggests. In order for a set of arguments to be accepted, it has to be justified according to some criteria, that are called semantics. The sets of collectively-acceptable arguments according to a certain semantics are referred to as “extensions”. Recent works (as the ones presented in [5,6]) have been carried out with the help of ConArg3 . The project involves a series of components that address different aspects of argumentation, building on a constraint-based solver for ar- gumentation problems [7,9]. The tool has already been extended with two main additional features that allow for handling weighted [6,8] and probabilistic [5] argumentation. While the former relies on algebraic structures (c-semirings) for dealing with weights, the latter makes use of a probabilistic logic programming language. In this work, we present a new component of the ConArg suite, which inte- grates the possibility of managing ranking semantics. In classical argumentation, arguments can be either accepted or rejected according to their justification sta- tus, but no further distinction can be done beyond this division into these two 3 ConArg Website: http://www.dmi.unipg.it/conarg/. categories.4 On the other hand, ranking semantics permit to assign an individ- ual score to each argument so that an overall ranking of all arguments can be established by sorting the obtained set of scores. Carrying on the work in [4], we here propose an implementation of a ranking function based on the Shapley Value [13], a very well known concept in cooperative game theory, which we use to distribute the scores among the arguments: the more an argument contributes to the acceptability of an extension, the higher its score. In addition, we also take into account a different valuation scheme, the Banzhaf Index [2], and we implement it in order to study the differences with the results obtained through the Shapley Value. Given an argumentation framework, the tool computes the score of every argument over both the ranking schemes introduced above, and its output is a ranking of the arguments with respect to a given semantics. As previously introduced, this line of work commenced in [4] with the first theoretical results. This paper is instead dedicated to the description of the underlying tool, which also computes the Banzhaf Index, differently from [4]. In Section 2 we introduce the background information about Abstract Argu- mentation and Power Indexes. Section 3 describes the tool and its integration in ConArg, while Section 4 presents two examples of application on abstract frameworks. Finally, Section 5 wraps up the paper with final conclusions and ideas about future work. 2 Preliminaries We introduce our tool by first reporting the necessary background notions on labelling and ranking semantics in Abstract Argumentation, and successively we introduce Power Indexes in cooperative game theory. 2.1 Argumentation This work takes advantage on notions coming from two different fields: argumen- tation and cooperative games. In the following, we provide a brief introduction only to the concepts which are most relevant to us. An Abstract Argumentation Framework [12] (AF in short) consists of a pair hA, Ri where A is a set of ar- guments and R ⊆ A × A expresses the relations between pairs of arguments. Such relations, which we call “attacks”, are interpreted as conflict conditions that allow for determining the arguments in A are acceptable together (i.e., collectively). An argumentation semantics is a criterion that establishes which are the acceptable arguments by considering the relations among them. Two leading characterisations can be found in the literature, namely extension-based [12] and labelling-based [11] semantics. While providing the same outcome in terms of accepted arguments, labelling-based semantics can be used to differentiate 4 More than just two categories have been proposed in the literature, but still from a qualitative point of view. between three levels of acceptability, by assigning labels to arguments according to the conditions stated in Definition 1. Definition 1 (Reinstatement Labelling). Let F = hA, Ri be an AF and L = {in, out, undec}. A labelling of F is a total function L : A → L. We define in(L) = {a ∈ A | L(a) = in}, out(L) = {a ∈ A | L(a) = out} and undec(L) = {a ∈ A | L(a) = undec}. We say that L is a reinstatement labelling if and only if it satisfies the following conditions: – ∀a, b ∈ A, if a ∈ in(L) and (b, a) ∈ R then b ∈ out(L); – ∀a ∈ A, if a ∈ out(L) then ∃b ∈ A such that b ∈ in(L) and (b, a) ∈ R. The labelling obtained through the function in Definition 1 can be then analysed in terms of Dung’s semantics [12]. Definition 2 (Labelling-based semantics). A labelling-based semantics σ associates with an AF F a subset of all the possible labellings for F, denoted as Lσ (F ). Let L be a labelling of F = hA, Ri, then L is – conflict-free if and only if for each a ∈ A it holds that if a is labelled in then it does not have an attacker that is labelled in, and if a is labelled out then it has at least one attacker that is labelled in; – admissible if and only if the attackers of each in argument are labelled out, and each out argument has at least one attacker that is in; – complete if and only if for each a ∈ A, a is labelled in if and only if all its attackers are labelled out, and a is out if and only if it has at least one attacker that is labelled in; – preferred/grounded if L is a complete labelling where the set of arguments labelled in is maximal/minimal (with respect to set inclusion) among all complete labellings; – stable if and only if it is a complete labelling and undec(L) = ∅. The accepted arguments, with respect to a certain semantics σ, are those labelled in by σ. In order to further discriminate among arguments, ranking- based semantics [1] can be utilised for sorting the arguments from the most to the least preferred. Definition 3 (Ranking-based semantics). A ranking-based semantics asso- ciates with any F = hA, Ri a ranking