I Introduction
Argumentation Theory is a field of Artificial Intelligence that provides formalisms for reasoning with conflicting information. Arguments from a knowledge base are modelled by Dung as nodes in a directed graph, called Abstract Argumentation Framework (AF in short), where edges represent a binary attack relation between arguments. In a framework, it is possible to select the sets of arguments that are not in conflict with each other. Many
semantics (i.e., criteria through which refine this selection) have been defined in order to establish different kinds of acceptability (see [4] for a survey). All these semantics return two disjoint sets of arguments: “accepted” and “not accepted”. The sets of accepted arguments with respect to a certain semantics are called “extensions” of that semantics. An additional level of acceptability is introduced by Caminada with the reinstatement labelling, a kind of semantics that marks as rejected elements attacked by accepted arguments, and undecided the arguments that can be neither accepted nor rejected.Dividing the arguments into just three partitions is still not sufficient when dealing with very large AFs, where one needs to limit the choice on a restricted number of selected arguments [19]. For this reason, a different family of semantics can be used for obtaining a broader range of acceptability levels for the arguments. Such semantics, called rankingbased, have been studied in many works, as [2, 5, 7, 19, 20], each focusing on a different criterion for identifying the best arguments in an AF. The idea is to assign a value to each argument through an evaluation function, so to obtain a total order over the arguments (a ranking indeed). However, to date, there is no relation between rankingbased semantics and classical ones by Dung/Caminada [15, 12].
In this work, we present a rankingbased semantics which exploits power indexes, together with a tool that implements it. Contrary to other rankingbased semantics, ours has a strong connection with classical semantics, that are used as parameters for the evaluation of the arguments. We rely on power indexes for establishing the ranking of the arguments since they are a very wellknown concept in the fields of economics and computational social choice, where they are successfully adopted in many applications involving the fair division of costs or benefits. This paper extends [6] and [7], where preliminary ideas of the Shapley Value semantics and its implementation was sketched. We provide here a general and deep study on the application of power indexes to AFs and a thorough description of the implementation of our rankingbased semantics inside the ConArg suite, together with an example of how the rankings are computed by the different power indexes. We also discuss the differences among the various indices and give properties that characterise our semantics. In Section II we report the background on labelling and rankingbase semantics, and then the necessary preliminary notions on power indexes. Section III is devoted to the definition of the rankingbased semantics we use in our tool. Afterwards, Section IV and SectionV describe the implementation of the aforementioned semantics and provide a detailed example of how the tool works, respectively. Section VI wraps up the paper with conclusive thoughts and ideas about future work
Ii Preliminaries
In this section, we introduce the concepts from Argumentation and Game Theory we use for developing our tool. Besides the main definitions for rankingbased semantics
[2], we also provide the notion of the labellingbased semantics [12] that we implement in order to identify the sets of acceptable arguments. Then, we recall the definitions of the four different power indexes used for evaluating the arguments. Those power indexes are named after the authors that formalised them and are known as the Shapley Value and the Banzhaf, DeeganPackel and Johnston Index [18].Iia Argumentation
An Abstract Argumentation Framework [15] (AF in short) consists of a set of arguments and the relations among them. Such relations, which we call “attacks”, are interpreted as conflict conditions that allow for determining the arguments in that are acceptable together (i.e., collectively).
Definition 1 (Af).
An Abstract Argumentation Framework is a pair where is a set of arguments and is a binary attack relation on .
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 extensionbased [15] and labellingbased [12] semantics. While providing the same outcome in terms of accepted arguments, labellingbased semantics permit to differentiate between three levels of acceptability, by assigning labels to arguments according to the conditions stated in Definition 2.
Definition 2 (Reinstatement Labelling).
Let be an AF and . A labelling of is a total function . We define , and . We say that is a reinstatement labelling if and only if it satisfies the following conditions:

, if and then ;

, if then such that and .
The labelling obtained through the function in Definition 2 can be then analysed in terms of Dung’s semantics [15].
Definition 3 (Labellingbased semantics).
A labellingbased semantics associates with an AF a subset of all the possible labellings for F, denoted as . Let be a labelling of , then is

conflictfree if and only if for each it holds that if is labelled in then it does not have an attacker that is labelled in, and if 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 , is labelled in if and only if all its attackers are labelled out, and is out if and only if it has at least one attacker that is labelled in;

preferred if is a complete labelling where the set of arguments labelled in is maximal (with respect to set inclusion) among all complete labellings;

grounded if is a complete labelling where the set of arguments labelled in is minimal (with respect to set inclusion) among all complete labellings;

stable if and only if it is a complete labelling and .
The accepted arguments of , with respect to a certain semantics , are those labelled by . We refer to sets of arguments that are labelled in, out or undec in at least one labelling of with , and , respectively^{1}^{1}1We just write when the reference to is clear and unambiguous.. Given an argument , we say that is credulously accepted with respect to a semantics if it is labelled in in at least one extension of . We say that is sceptically accepted if it is labelled in in all extensions of . In order to further discriminate among arguments, rankingbased semantics [11] can be used for sorting the arguments from the most to the least preferred.
Definition 4 (Rankingbased semantics).
A rankingbased semantics associates with any a ranking on , where is a preorder (a reflexive and transitive relation) on . means that is at least as acceptable as ( is a shortcut for and , and is a shortcut for and ).
A rankingbased semantics can be characterised through some specific properties that take into account how couples of arguments in an AF are evaluated for establishing their position in the ranking. We provide a list of the properties suggested in [2] and that we use in Section V to discuss an example of how our tool can be used for both research and applicative purposes.
Definition 5 (Isomorphism).
An isomorphism between two AFs and is a bijective function such that , if and only if .
We can characterise the role of an argument with respect to another one according to the length of the path between them: an odd path represents an attack, while an even path is considered as a defence.
Definition 6 (Attackers and defenders [2]).
Let be an AF and and denote with a path from to . The multisets of defenders and attackers of are with length and with length , respectively. is the set of direct attackers of .
Besides arguments alone, also sets of arguments can be compared. Two rules apply: the greater the number of arguments, the more preferred the group; in case of two groups with the same size, the more preferred the arguments in a group, the more preferred the group itself.
Definition 7 (Group comparison [2]).
Let be a ranking on a set of arguments . For any , is a group comparison if and only if there exists an injective mapping such that . Moreover, is a strict group comparison if and only if and .
Below, we list the properties proposed in [2].
Definition 8 (Properties of rankingbased semantics).
Given a rankingbased semantics , an AF and two arguments , the following properties are defined.

Abstraction. For any isomorphism such that , if and only if .

Independence. Let be the set of connected components in . , then .

Selfcontradiction. and .

Cardinality Precedence. .

Quality Precedence. such that , .

Nonattacked Equivalence. We have that and .

Totality. or .
The rankingbased semantics we present has been implemented in ConArg^{2}^{2}2ConArg website: http://www.dmi.unipg.it/conarg/, a web tool that implements various work we conduct in the field of Abstract Argumentation. The core component of the whole suite is the computational framework [10], based on Constraint Programming, that is able to solve different problems related to AFs. ConArg can be used for many purposes, as computing semantics, visualising Argumentation Frameworks (AF) together with the computed extensions, programming user application using a predefined AF library, and studying properties of semantics and AFs. Additional modules allow for dealing with weighted [9] and probabilistic [8] AFs. Our semantics evaluates the arguments of an AF by using the notion of power index, that we describe in the following section.
IiB Power Indexes
In game theory, cooperative games are a class of games where groups of players (or agents) are competing to maximise their goal, through one or more specific rules. Voting games are a particular category of cooperative games in which the profit of coalitions is determined by the contribution of each individual player. In order to identify the “value” brought from a single player to a coalition, power indexes are used to define a preference relation between different agents, computed on all the possible coalitions. In our work, we use four among the most commonly used power indexes, namely the Shapley Value [21, 22], the Banzhaf Index [3], the DeeganPackel Index and the Johnston Index [18].
Every power index relies on some characteristic function
that, given the set of players, associates each coalition with a real number in such a way that describes the total gain that agents in can obtain by cooperating with each other. The expected marginal contribution of a player , given by the difference of gain between and , is .The Shapley Value of the player , given a characteristic function , is computed as:
(1) 
The formula considers a random ordering of the agents, picked uniformly from the set of all possible orderings. The value
expresses the probability that all the agents in
come before in a random ordering.The second fair division scheme we use is the Banzhaf Index , which evaluates each player by using the notion of critical voter: given a coalition , a critical voter for is a player such that is a winning coalition, while alone is not. In other words, is a critical voter if it can change the outcome of the coalition it joins.
(2) 
The difference between the Shapley Value and the Banzhaf index is that the latter does not take into account the order in which the players form the coalitions.
Deegan and Packel assume that only minimal winning coalitions are formed, that they do so with equal probability, and that if such a coalition is formed it divides the (fixed) spoils of victory equally among its members. In order to avoid divisions by zero in the formula, we use the interpretation of [1]: let’s call the set of minimal winning coalitions of the game (always assuming ), and the subset of formed by coalitions such that . The DeeganPackel index of a player is computed as follows.
(3) 
The last index we implement is the Johnston index [16]. Based on the principle of critical vote, it differs from Banzhaf’s for the fact that critical voters in winning coalitions are rewarded with a fractional score instead of one whole unit (that is the score is equally divided among all critical members of the coalition). Let denote the number of critical voters in a winning coalition . The Johnston index of a player is computed as follows.
(4) 
Note that the summation is only done on the coalitions in which there is at least one critical voter. Below, we give the definition of our rankingsemantics based on power indexes, that we call “PIbased semantics”.
Iii PIBased Semantics
In order to rank arguments of a framework through the use of a power index, we need, first of all, to define the characteristic function that evaluates the coalition formed by the arguments.
Definition 9 (Characteristic function).
Let be an AF, a Dung semantics and the set of all possible labellings on satisfying . For any , the labellingbased characteristic functions and are defined as:
The function takes into account the acceptability of a set of arguments with respect to a certain semantics , assigning to such set a score equal to if there exists a labelling in which all and only the arguments of are labelled in. In other words, a set is positively evaluated by only if it represents an extension for the semantics , and the higher the score of the power index, the better the rank of an argument. A second characteristic function, , is then introduced to break possible ties in the final ranking. In the case two arguments of F have the same power index with respect to the function , we compare the evaluations obtained through , that considers the sets of arguments labelled out by . This further evaluation has a negative interpretation: the higher the score according to , the worse the rank.
Definition 10 (PIbased semantics).
Let be an AF, a Dung semantics, {} a power index, and , the characteristic functions. The PIbased semantics associates to a ranking on , defining a lexicographic order on the pairs such that , if and only if

, or

and
and that if and only if

and .
We study which properties among those in Definition 6 are satisfied by the PIsemantics obtained through the Shapley Value, and which are not.
Theorem 1.
Consider an AF , two arguments , a Dung semantics and the power index . The PIbased semantics satisfies the following properties:

Abstraction, Independence and Totality for any conflictfree, admissible, complete, preferred, stable

Selfcontradiction only for = conflictfree

Nonattacked Equivalence only for complete, preferred, stable
For any conflictfree, admissible, complete, preferred, stable, the PIbased semantics does not satisfy Cardinality Precedence and Quality Precedence.
Proof.
For each power index, characteristic function and semantics, we state if the properties are satisfied.

Abstraction. Any extension of every semantics is computed starting from the set of attack relations among arguments, thus the ranking is preserved up to isomorphisms of the framework.

Independence. The semantics we propose computes the ranking starting from the sets of extensions of a chosen semantics . Since the labelling of each argument is determined by the other arguments in the same connected component of , also the ranking between every pair of arguments and is independent of any other argument outside the connected component of and .

Selfcontradiction. Consider = conflictfree. If and , we can state that
Thus from which we conclude that when = conflictfree. In Figure 1 we show a counterexample for the other cases.

Cardinality Precedence. The property is not satisfied for any {conflictfree, admissible, complete, preferred, stable} with respect to . Counterexample in Figure 2.

Nonattacked Equivalence. Nonattacked arguments are labelled in in every complete extension, thus, if two arguments are nonattacked, then we have and . Hence when = complete. Since all the preferred and stable extensions are also complete, Nonattacked Equivalence holds for complete, preferred, stable. On the other hand, for the property is not satisfied (see the counterexample in Figure 6).

Totality. The Shapley Value associates a real number to every arguments of an AF, thus all pairs of arguments can be compared through the order of .
∎
Abstraction, Independence and Totality are desirable properties, since they guarantee that a total order can always be established over the arguments of an AF, only considering the structure of the underlying graph and the relations among the arguments. The Selfcontradiction property ensures, for the conflictfree semantics, that selfattacking arguments have a lower ranking than the others. Indeed, the conflictfree semantics only takes into account whether there are attacks among the arguments. For the other semantics, it may happen that an argument attacked by another argument with a high value is ranked lower than a selfattacking argument: in other words, when the notion of defence is taken into account, an argument which is defeated by a solid counterargument has less value than a contradictory argument. For the complete, preferred and stable semantics, which always label in the arguments that do not receive attacks, the Nonattacked Equivalence property allows for knowing the value of all the nonattacked arguments just by computing the value for one of them. Cardinality Precedence and Quality Precedence never hold. In fact, the ranking of an argument does not only depend on either the number of attackers of or their position in the ranking, but also on how many other arguments are defended by . We plan to study such property when extending our work to rankingbased semantics over weighted AFs.
Given a ranking, we can correlate the value given to each argument to its credulous/sceptical acceptance. Looking at the acceptability of an argument, we can have in advance some information about the value of its evaluation, without even computing the power index.
Theorem 2.
Let be an AF, {} a power index, and a characteristic function. Then

if is sceptically accepted ;

if is credulously rejected ;
Proof.
The proof is straightforward and can be derived from the definition of the power indexes. ∎
Sceptically accepted arguments are ranked higher than credulously accepted and rejected arguments. Analogously, credulously accepted arguments are ranked higher that rejected arguments.
Definition 11.
Let be an AF, two arguments, a rankingbased semantics and a Dung’s semantics. We define the following properties.
Sceptical Precedence (ScP). If is sceptically accepted with respect to and is not, than .
Credulous Precedence (CrP). If is credulously accepted with respect to and is always rejected, than .
The following proposition holds.
Proposition 1.
The PIbased semantics satisfies ScP and CrP for any conflictfree, admissible, complete, preferred, stable.
Proof.
Given an AF , a credulously accepted argument with respect to and an evaluation function , there exists at least one subset of arguments such that . Thus the value of will always be higher than that of any rejected argument , for which for any , and CrP holds. We can make the same consideration for SkP, showing that skeptically accepted arguments have higher value than the others. ∎
As a first step for comparing our semantics with other from the literature, we have checked if property CrP is satisfied by some of the rankingbased semantics surveyed in [11] that return a total ranking, namely Cat, Dbs, Bds.
Proposition 2.
The rankingbased semantics Cat, Dbs and Bds do not satisfy the CrP property when {admissible, complete, preferred, stable}.
Indeed, considering the example in Figure 7, we have that argument is always preferred to in the rankings obtained by using Cat, Dbs and Bds.
The graded semantics proposed in [17] also takes into account extensions of classical semantics in order to determine an ordering between arguments of an AF. The two principles on which the semantics is based are: having fewer attackers is better than having more; having more defenders is better than having fewer. Since the authors assume the sceptical definition for the justification of the arguments, the graded semantics satisfies both ScP and CrP. However, the used order relation is only partial (and thus some of the arguments may be incomparable). Moreover, the ranking being built on the two principles mentioned above does not allow to catch the real contribution of the arguments in forming the extensions, that, instead, is the intention of the PIbased semantics.
Finally, we discuss the ranking semantics, based on subgraphs analysis, introduced in [14]. This semantics sorts the arguments of an AF by establishing a lexicographical ordering between the values of a tuple that contains, for each argument , the label assigned to by a certain semantics, and the number of times is labelled over the total number of subgraphs, for in, out and undec, respectively. The semantics satisfies ScP and CrP. The main difference with our approach is that, while we only consider acceptable extensions for obtaining the evaluation of an argument, the semantics in [14] uses all the possible subsets of arguments for computing the ranking, leading to results that do not fit the definition of the chosen Dung semantics. For instance, the ranking returned by the PIbased semantics for the AF in Figure (a)a, with respect to the preferred semantics, is , since the only preferred extension is . Similar considerations also hold for the example in Figure (b)b for the preferred semantics, where the PIbased semantics returns for the functions and , and for . Those rankings reflects the meaning of preferred semantics: arguments and are both necessary for obtaining a preferred extension, so they are ranked the same. According to the Johnston Index , argument is the best one since it is the only critical voter of its coalition (i.e., the extension ). On the other hand, for the admissible semantics, and return the ranking . In fact, , that is admissible alone, also reinstate . The tie between and is broken when considering the function , for which .
In the following section, we show how the PIbased ranking semantics have been implemented in ConArg.
Iv Tool Description
The ConArg Web Interface (see Figure 9 for an overview) allows one to easily perform complex argumentation related tasks. Below, we describe the main features of the tool, highlighting those introduced more recently.
Menu. Positioned to the left side of the interface, it allows for choosing among different options for both visualising AFs and solving argumentation problems.
Semantics selection panel. Here it is possible to set the parameters for the resolution of several problems. First of all, one is required to select a Dung semantics through the dedicated dropdown menu. For each semantics, four different kinds of problem can be solved.
Canvas. This area of the interface has a twofold purpose. On one hand, it is possible to define an AF by drawing nodes and edges. On the other hand, after the calculation of a solution for a certain problem, the canvas allows for visualising the output directly on the displayed AF, through a specific colouration of the arguments.
AF in input. AFs can be entered in this panel. Changes to the canvas also affect this area, that maintains a coherent representation of the AF.
Output panel. The solutions for the various problems solved by ConArg are displayed here. It is also possible to download a text file containing the output.
Iva Implementation of the PIBased Semantics
Behind the web interface, ConArg has several modules (like the solver and the ranking script) that allow one to access different functionalities to cope with argumentation problems. In this section, we discuss, in particular, the component of the tool that concerns rankingbased semantics, putting attention on implementation aspects.
When we start the computation of the ranking over the arguments of an AF , the interface calls the ConArg solver that returns the set of extensions for the chosen Dung semantics . These extensions represent the sets of in arguments with respect to and are formatted as sets of strings (e.g., , where and are arguments). Together with the set of extensions, also the framework and the power index that we want to use are passed to the ranking script. The script, then, computes the specified power index for each of the argument in . The obtained values are approximated to the nearest fifth decimal digit. The four functions that implement the equations of Section IIB share a common part, namely , that represents the evaluation of the contribution of the argument in forming acceptable extensions. For the sake of efficiency, we compute only with respect to those sets such that either or is an extension for . In any other case, the value of is zero, so we don’t need to do the calculation.
We distinguish between two different characteristic functions: and . As stated in Definition 9, the former function takes into account the set of in arguments. Given a set of arguments that does not contain , if is an extension with respect to and alone is not, then brings a positive contribution to the coalition, and its own rank will be higher according to . On the other hand, the latter function () only considers arguments that are labelled out by . In detail, gets a positive value by when is a set of out arguments and alone is not. The set is obtained by computing the sets of arguments that are attacked by the extensions of the semantics . At this point, each argument of is associated with the values of the two functions; the resulting structure has the format of an array , where the three components are: the identifier of the argument, the value of the power index obtained through , and the value of obtained through , respectively.
In order to establish the preference relation between two arguments, the PIbased semantics considers the value pi_in first: the greater the score of an argument with respect to , the higher its position in the ranking. In case of a tie, i.e., when the value of pi_in is the same for both the arguments that we want to compare, we perform a further control looking at the value of . Following the principle that accepted arguments are better than rejected ones, the greater the value of an argument with respect to pi_out, the lower its position in the ranking. Consider, for example, two arguments and , belonging to , with the following evaluations obtained through : and . The value pi_in is equal for both and , therefore we proceed to confront the values for pi_out. Since , we have that . The motivation to this kind of ranking is that, while and have the same contribution in forming acceptable extensions, belongs to fewer sets of out arguments, that is, is defeated less times than . Hence, it is reasonable to prefer to .
Finally, when all the arguments are sorted according to the semantics and the power index that we have selected, the results of the computation are displayed in the output panel (frame 5 of Figure 9). Along with the overall ranking, we show the pi_in and pi_out values of each argument. For providing a visual hint about which arguments are the most preferred and which ones the least, we assign a colour to each node of the AF visualised in the canvas. The assigned colours vary in a greyscale, according to the position of the corresponding argument in the obtained ranking: the lighter the colour, the higher the rank (as depicted in the frame 3 of Figure 9).
V An Example with ConArg
In this section, we provide an example of how the ConArg web interface can be used for dealing with ranking semantics. We show the procedure for obtaining a ranking among the arguments of a given AF through one of the implemented power indexes. We also compare the results for all the different power indexes, highlighting the differences in terms of final ordering of the arguments. For our example, we consider the AF in Figure 10, that has an initiator (i.e., the argument , which is not attacked by any other argument), a symmetric attack (between and ), a selfattack (in ), and a cycle involving and .
Given an AF in input, there are two prerequisites for the calculation of the ranking over the arguments. First of all, since the PIbased semantics is parametric to a Dung semantics, this latter must be selected in the semantics panel (frame 2 of Figure 9). Then, we need to choose a power index among the four implemented. At this point, we can run the computation from the start button. Below, we report the output provided by ConArg for , , and (that correspond to the functions for computing the Shapley Value and the Banzhaf, DeeganPackel and Johnston Index, respectively), and the semantics conflictfree, admissible, complete and preferred. We omit the stable one since, in this example, it returns the same set of extensions as the preferred. For each semantics, the values of the power index obtained with respect to the sets of in and out arguments are alternated in each row. Tables I, II, III, IV show the results for the aforementioned indexes.
a  b  c  d  e  Semantics  Ranking  

a  b  c  d  e  Semantics  Ranking  

a  b  c  d  e  Semantics  Ranking  

a  b  c  d  e  Semantics  Ranking  

We now analyse the differences between the obtained rankings, following two levels of detail: we first compare, for each power index, the ranking obtained for all the Dung semantics. Then, for each Dung semantics, we consider the ranking obtained with respect to the different power indexes.
In this example, the Shapley Value (Table I), provides a rankling without indifferences when the conflictfree semantics is considered. While , and return the same output, where in particular and , the ranking for the admissible semantics gives an opposite interpretation, that is and . This happens because both and are admissible extensions, while is not. Hence, when the admissible semantics is taken into account, and are better arguments than .
When Banzhaf Index is used (Table II), such an inversion of preferences never occurs: there is no semantics for which or . Looking at the formulas of the Shapley Value (Equation 1) and the Banzhaf Index (Equation 2), we can see that the only difference is the factor by which the gain is multiplied. Contrary to Shapley, Banzhaf does not consider the order in which the coalitions form; since the acceptability of the arguments does not depend on how the extensions are formed, produces more consistent results and, therefore, is a more appropriate index to be used for building a rankingbased semantics.
Using the DeeganPackel Index for computing the ranking with respect to the conflictfree and the admissible semantics is not meaningful. Indeed, for such semantics, the empty set is always an extension, and it also represents the only minimal winning coalitions. Since relies on the set of minimal winning coalitions , when is the only element of , all the arguments receive a value of , according to Equation 3. For this reason, we omit to include and in Table III.
The ranking obtained through all power indexes share some common features, that we discuss below. The argument , that is not attacked by any other, is always in the first position of the rank, for every power index. Consequently, the argument , that is attacked by , always results to be the worst argument in the AF, excepted for indifferences. For the complete, preferred and stable semantics, the ranking does not distinguish between and , and between and . Indeed, the set corresponds to the grounded semantics, that is and are equally “important” and should be evaluated the same. Similarly, and , that only appear in two distinct maximal admissible sets, receive the same value from all the power indexes. Finally, since the extensions of the preferred and the stable coincide, these two semantics always provide the same final ranking.
Vi Conclusion and Future Work
We have presented an online tool capable of dealing with rankingbased semantics. The tool implements the definition of the PIbased semantics [7] with respect to four power indexes, namely the Shapley Value and the Banzhaf, DeeganPackel and Johnston Index. Differently from other rankingbased semantics defined in the literature, our approach allows for distributing preferences among arguments taking into account classical Dung/Caminada semantics. In this way, we obtain a more accurate ranking with respect to the desired acceptability criterion. We have also provided an applicative example in which an AF is studied from the point of view of the different power indexes that can be used for extracting the ranking over the arguments; the tool can be used for studying properties of the rankingbased semantics, and in particular the PIbased ones, where also notions from cooperative games converge.
In the future, we plan to implement other indexes in the tool, or combinations of them: our aim is to understand which ranking properties (or families of them, i.e., local or global) listed in [11] such indexes can successfully capture. With the comparison of different indexes, we aim to determine if there is a link between ties on rankings and the possible resolution of ambiguities. So far, we have only captured properties that are local to an argument, i.e., they can be checked by inspecting the immediate neighbourhood of an argument. Global properties derive, instead, from the whole framework structure (e.g., full attacking or defending paths), and could be exploited for further refining the ranking returned by our semantics. We are also interested in extending our work on weighted AFs, where a different notion of defence is used.
Another direction we plan to investigate concerns the ranking function we use for evaluating the arguments. Similarly to what is done in [13] for studying coalitions with particular properties, we want to restrict the set of possible extensions by considering only the subsets of that are in a given semantics, that is to exclude arguments that are not even credulously accepted. For instance, we could devise a PIbased semantics where the arguments are evaluated with respect to the stable semantics and the only coalitions to be taken into account are the admissible ones.
References
 [1] J. M. AlonsoMeijide, M. ÁlvarezMozos, and M. G. FiestrasJaneiro. Power Indices and Minimal Winning Coalitions in Simple Games with Externalities. UB Economics Working Papers 2015/328, Universitat de Barcelona, Facultat d’Economia i Empresa, UB Economics, 2015.
 [2] L. Amgoud and J. BenNaim. RankingBased Semantics for Argumentation Frameworks. In Scalable Uncertainty Management  7th International Conference, SUM 2013, volume 8078 of LNCS, pages 134–147. Springer, 2013.
 [3] J. F. Banzhaf. Weighted Voting Doesn’t Work: A Mathematical Analysis. Rutgers Law Review, 19(2):317–343, 1965.
 [4] P. Baroni, M. Caminada, and M. Giacomin. An introduction to argumentation semantics. Knowledge Eng. Review, 26(4):365–410, 2011.
 [5] P. Besnard and A. Hunter. A logicbased theory of deductive arguments. Artif. Intell., 128(12):203–235, 2001.
 [6] S. Bistarelli, F. Faloci, F. Santini, and C. Taticchi. A Tool For Ranking Arguments Through VotingGames Power Indexes. In Proceedings of the 34th Italian Conference on Computational Logic, volume 2396 of CEUR Workshop Proceedings, pages 193–201. CEURWS.org, 2019.
 [7] S. Bistarelli, P. Giuliodori, F. Santini, and C. Taticchi. A Cooperativegame Approach to Share Acceptability and Rank Arguments. In Proceedings of the 2nd Workshop on Advances In Argumentation In Artificial Intelligence, AI3@AI*IA, volume 2296 of CEUR Workshop Proceedings, pages 86–90. CEURWS.org, 2018.
 [8] S. Bistarelli, T. Mantadelis, F. Santini, and C. Taticchi. Probabilistic Argumentation Frameworks with MetaProbLog and ConArg. In IEEE 30th International Conference on Tools with Artificial Intelligence, ICTAI, pages 675–679. IEEE, 2018.
 [9] S. Bistarelli, F. Rossi, and F. Santini. ConArg: A Tool for Classical and Weighted Argumentation. In Computational Models of Argument  Proceedings of COMMA, volume 287 of Frontiers in Artificial Intelligence and Applications, pages 463–464. IOS Press, 2016.
 [10] S. Bistarelli and F. Santini. ConArg: A ConstraintBased Computational Framework for Argumentation Systems. In IEEE 23rd International Conference on Tools with Artificial Intelligence, ICTAI, pages 605–612. IEEE Computer Society, 2011.
 [11] E. Bonzon, J. Delobelle, S. Konieczny, and N. Maudet. A Comparative Study of RankingBased Semantics for Abstract Argumentation. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pages 914–920. AAAI Press, 2016.
 [12] M. Caminada. On the Issue of Reinstatement in Argumentation. In Logics in Artificial Intelligence, 10th European Conference, JELIA, volume 4160 of LNCS, pages 111–123. Springer, 2006.
 [13] J. Derks and H. Peters. A Shapley Value for Games with Restricted Coalitions. International Journal of Game Theory, 21(4):351–60, December 1993.
 [14] P. Dondio. Ranking Semantics Based on Subgraphs Analysis. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS, pages 1132–1140. International Foundation for Autonomous Agents and Multiagent Systems Richland, SC, USA / ACM, 2018.

[15]
P. M. Dung.
On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming and nPerson Games.
Artif. Intell., 77(2):321–358, 1995.  [16] E. A. Durán, J. M. Bilbao, J. R. F. García, and J. J. López. Computing power indices in weighted multiple majority games. Mathematical Social Sciences, 46(1):63–80, 2003.
 [17] D. Grossi and S. Modgil. On the Graded Acceptability of Arguments. In Proceedings of the TwentyFourth International Joint Conference on Artificial Intelligence, IJCAI, pages 868–874. AAAI Press, 2015.
 [18] D. Keith. Encyclopedia of Power. SAGE Publications, Inc., 2011.
 [19] J. Leite and J. Martins. Social Abstract Argumentation. In IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, pages 2287–2292. IJCAI/AAAI, 2011.
 [20] P. Matt and F. Toni. A GameTheoretic Measure of Argument Strength for Abstract Argumentation. In Logics in Artificial Intelligence, 11th European Conference, JELIA, volume 5293 of LNCS, pages 285–297. Springer, 2008.
 [21] L. S. Shapley. Contributions to the Theory of Games, volume II of AM28. Princeton University Press, 1953.
 [22] E. Winter. The shapley value. In Handbook of Game Theory with Economic Applications, volume 3, pages 2025–2054. Elsevier, 2002.
Comments
There are no comments yet.