An Extension-Based Argument-Ranking Semantics:
                         Social Rankings in Abstract Argumentation
                         Lars Bengel1,† , Giovanni Buraglio2,† , Jan Maly3,2,† and Kenneth Skiba1,†
                         1
                           Artificial Intelligence Group, University of Hagen, Hagen, Germany
                         2
                           Institute of Logic and Computation, TU Wien, Wien, Austria
                         3
                           Institute of Data, Process and Knowledge Management, Vienna University of Economics and Business, Wien, Austria


                                        Abstract
                                        In this paper, we introduce a new family of argument-ranking semantics which can be seen as a refinement of the classification of
                                        arguments into skeptically accepted, credulously accepted and rejected. To this end we use so-called social ranking functions which
                                        have been developed recently to rank individuals based on their performance in groups. We provide necessary and sufficient conditions
                                        for a social ranking function to give rise to an argument-ranking semantics satisfying the desired refinement property. Moreover, we
                                        analyse the properties of the argument-ranking semantics induced by the most prominent social ranking function that satisfies all of
                                        these conditions by investigating the satisfaction of principles known from the argument-ranking literature.

                                        Keywords
                                        Argumentation, Social Rankings, Extension-ranking semantics, Argument-ranking semantics



                         1. Introduction                                                                                           tion to construct a fine-grained ranking of arguments by
                                                                                                                                   applying a social ranking function.
                         One of the problems of computational models of argumen-                                                      Closer to our needs, Skiba et al. [11] recently introduced
                         tation is to classify the quality of arguments in the context                                             so-called extension-ranking semantics that refine and ex-
                         of the larger discussion. In abstract argumentation, this                                                 tend classical argumentation semantics by providing a par-
                         is usually achieved by checking whether an argument is                                                    tial ranking over sets of arguments.
                         contained in a set of arguments, called extensions, that are                                                 We show that, by applying the right social ranking func-
                         acceptable according to one of several well-established se-                                               tions to an extension-ranking semantics, we can define
                         mantics. While these semantics provides a natural way to                                                  argument-ranking semantics that are a refinement of the tra-
                         rank arguments based on the larger context of the argu-                                                   ditional skeptical/credulous acceptance of arguments, both
                         mentation framework, it only allows us to distinguish three                                               in spirit and in a strict technical sense. More precisely, we
                         types of arguments: the ones that are skeptically accepted,                                               show that by applying the lexicographic excellence operator
                         i. e. that are contained in every extension; the ones that are                                            introduced by Bernardi et al. [9] to the extension-ranking
                         credulously accepted, i. e. that are contained in at least one                                            semantics of Skiba et al. [11] we generate an argument rank-
                         extension; and the ones that are not contained in any exten-                                              ing such that all skeptically accepted arguments are ranked
                         sion. For this reason, more fine-grained ways of comparing                                                before all credulously accepted arguments, which are, in
                         arguments have been proposed, the so called argument-                                                     turn, ranked before all non-accepted arguments. More gen-
                         ranking semantics [1, 2, 3, 4, 5]. However, generally, these                                              erally, we show which axiomatic properties are sufficient
                         argument-ranking semantics are technically quite distinct                                                 and necessary for a social ranking operator to give rise to
                         from the extension-based classifications of arguments that                                                such a ranking (Section 4). We conclude by studying the
                         are more commonly used.                                                                                   axiomatic properties of the argument-ranking semantics
                             In this paper, we propose a new way of ranking argu-                                                  induced by the lexicographic excellence operator (Section
                         ments which can be seen as a true refinement of the more                                                  5) and conclude that it is a well-behaved argument rank-
                         common classification in skeptically, credulously and not                                                 ing semantics even beyond the refinement properties that
                         accepted arguments. To this end, we combine two strands                                                   motivated our work (Sections 6 and 7).
                         of literature that have emerged recently, namely extension-
                         ranking semantics and social ranking functions, in a novel
                         way. Intuitively, social ranking functions allow us to rank                                               2. Preliminaries
                         elements based on the quality of sets they are contained in.
                         These functions were first introduced in the economics lit-                                               In this section, we introduce the basics of abstract argumen-
                         erature [6], in order to judge the performance of individuals                                             tation literature that are necessary for our work. We will
                         based on the success of groups that they were involved in,                                                start with the standard model of abstract argumentation, be-
                         and has received significant attention from economists and                                                fore introducing argument-ranking and extension-ranking
                         computer scientists [7, 8, 9, 10]. Unfortunately, extension                                               semantics.
                         semantics in formal argumentation only distinguish sets
                         of arguments that are accepted and the ones that are not                                                  Abstract Argumentation Frameworks An abstract ar-
                         accepted. This approach does not provide enough informa-                                                  gumentation framework (𝐴𝐹 ) is a directed graph 𝐹 =
                                                                                                                                   (𝐴, 𝑅) where 𝐴 is a (finite) set of arguments and 𝑅 ⊆ 𝐴×𝐴
                                                                                                                                   is an attack relation among them [12]. An argument 𝑎 is said
                         22nd International Workshop on Nonmonotonic Reasoning, November 2-4,
                         2024, Hanoi, Vietnam                                                                                      to attack an argument 𝑏 if (𝑎, 𝑏) ∈ 𝑅. We say that an argu-
                         †
                           These authors contributed equally.
                                                                                                                                   ment 𝑎 is defended by a set 𝐸 ⊆ 𝐴 if every argument 𝑏 ∈ 𝐴
                         $ lars.bengel@fernuni-hagen.de (L. Bengel);                                                               that attacks 𝑎 is attacked by some 𝑐 ∈ 𝐸. For 𝑎 ∈ 𝐴 we
                         giovanni.buraglio@tuwien.ac.at (G. Buraglio); jan.maly@tuwien.ac.at                                       define 𝑎− 𝐹 = {𝑏 | (𝑏, 𝑎) ∈ 𝑅} and 𝑎𝐹 = {𝑏 | (𝑎, 𝑏) ∈ 𝑅}
                                                                                                                                                                           +

                         (J. Maly); kenneth.skiba@fernuni-hagen.de (K. Skiba)                                                      as the sets of arguments attacking 𝑎 and the sets of argu-
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                   𝑎        𝑏        𝑐        𝑑                        Definition 3. An argument-ranking semantics 𝜌 is a func-
                                                                       tion which maps an AF 𝐹 = (𝐴, 𝑅) to a preorder1 ⪰𝜌𝐹 on
                                                                       𝐴.
Figure 1: Argumentation framework 𝐹1 from Example 1.
                                                                          Intuitively 𝑎 ⪰𝜌𝐹 𝑏 means that 𝑎 is at least as strong as 𝑏
                                                                       in 𝐹 . We define the usual abbreviations as follows; 𝑎 ≻𝜌𝐹 𝑏
                                                                       denotes strictly stronger, i. e. 𝑎 ⪰𝜌𝐹 𝑏 and 𝑏 ̸⪰𝜌𝐹 𝑎. Moreover,
ments that are attacked by 𝑎 in 𝐹 . For a set of arguments             𝑎 ≃𝜌𝐹 𝑏 denotes equally strong, i. e. 𝑎 ⪰𝜌𝐹 𝑏 and 𝑏 ⪰𝜌𝐹 𝑎.
𝐸 ⊆ 𝐴   ⋃︀ we extend  these definitions  to 𝐸𝐹− and 𝐸𝐹+ via            𝑎 ◁▷𝜌𝐹 𝑏 denotes incomparability so neither 𝑎 ⪰𝜌𝐹 𝑏 nor
𝐸𝐹 = 𝑎∈𝐸 𝑎𝐹 and 𝐸𝐹 = 𝑎∈𝐸 𝑎𝐹 , respectively. If the
  −             −        +             +
                              ⋃︀
                                                                       𝑏 ⪰𝜌𝐹 𝑎.
AF is clear in the context, we will omit the index.                       Traditionally the development of argument-ranking se-
  Most semantics [13] for abstract argumentation are rely-             mantics is guided by a principle-based approach [15]. Each
ing on two basic concepts: conflict-freeness and admissibility.        principle embodies a different property for argument rank-
                                                                       ings. We recall some of the most fundamental principles [4]
Definition 1. Given 𝐹 = (𝐴, 𝑅), a set 𝐸 ⊆ 𝐴 is: conflict-
                                                                       as well as newer ones, which are closer to the extension-
free iff ∀𝑎, 𝑏 ∈ 𝐸, (𝑎, 𝑏) ̸∈ 𝑅; admissible iff it is conflict-free,
                                                                       based reasoning process [16]. Before we start, we need addi-
and every element of 𝐸 is defended by 𝐸.
                                                                       tional notations. Let 𝐹 = (𝐴, 𝑅) be an AF with arguments
   For an AF 𝐹 we use 𝑐𝑓 (𝐹 ) and 𝑎𝑑(𝐹 ) to denote the sets            𝑎, 𝑏 ∈ 𝐴. A path of length 𝑙𝑃 = 𝑛 between two arguments
of conflict-free and admissible sets, respectively. In order           𝑎, 𝑏 is a sequence of arguments 𝑃 (𝑎, 𝑏) = (𝑎0 , 𝑎1 , ..., 𝑎𝑛 )
to define the remaining semantics proposed by Dung [12]                with (𝑎𝑖 , 𝑎𝑖+1 ) ∈ 𝑅 for all 𝑖 with 𝑎0 = 𝑎 and 𝑎𝑛 = 𝑏. The
and in addition the semi-stable semantics [14] we use the              connected components 𝑐𝑐(𝐹 ) of an AF 𝐹 are the maximal
characteristic function.                                               subgraphs 𝐹 ′ = (𝐴′ , 𝑅′ ), where for every pair of argu-
                                                                       ments 𝑎, 𝑏 ∈ 𝐴′ there exists an undirected path 𝑃𝑢 (𝑎, 𝑏) =
Definition 2. For an AF 𝐹 = (𝐴, 𝑅) and a set of arguments              (𝑎 = 𝑎0 , 𝑎1 , ..., 𝑎𝑛−1 , 𝑎𝑛 = 𝑏) s.t. for every 𝑖 there is
𝐸 ⊆ 𝐴 the characteristic function ℱ𝐹 (𝐸) : 2𝐴 → 2𝐴 is                  either (𝑎𝑖 , 𝑎𝑖+1 ) ∈ 𝑅 or (𝑎𝑖+1 , 𝑎𝑖 ) ∈ 𝑅. For an extension-
defined via:                                                           based semantics 𝜎, an argument 𝑎 weakly 𝜎-supports 𝑏 if
                                                                       𝑏 ∈ 𝑐𝑟𝑒𝑑𝜎 (𝐹 ) and for all 𝐸 ∈ 𝜎(𝐹 ), with 𝑏 ∈ 𝐸 then
               ℱ𝐹 (𝐸) = {𝑎 ∈ 𝐴|𝐸 defends 𝑎}                            𝑎 ∈ 𝐸 and 𝑎 strongly 𝜎-supports 𝑏 if 𝑏 ∈ 𝑐𝑟𝑒𝑑𝜎 (𝐹 ) and for
                                                                       all 𝐸 ∈ 𝜎(𝐹 ), with 𝑏 ∈ 𝐸 then there is 𝐸 ′ ∈ 𝜎(𝐹 ) with
An admissible set 𝐸 ⊆ 𝐴 is a complete extension (𝑐𝑜) iff
                                                                       𝐸 ′ ⊆ 𝐸, 𝑎 ∈ 𝐸 ′ and 𝑏 ∈   / 𝐸′.
𝐸 = ℱ𝐹 (𝐸); a preferred extension (𝑝𝑟) iff it is a ⊆-maximal
complete extension; the unique grounded extension (𝑔𝑟) iff             Definition 4. An argument-ranking semantics 𝜌 satisfies
𝐸 is the least fixed point of ℱ𝐹 ; a stable extension (𝑠𝑡𝑏) iff        the respective principle iff for all AFs 𝐹 = (𝐴, 𝑅) and any
𝐸𝐹+ = 𝐴 ∖ 𝐸; a semi-stable extension (𝑠𝑠𝑡) iff it is a complete        𝑎, 𝑏 ∈ 𝐴:
extension, where 𝐸 ∪ 𝐸𝐹+ is ⊆-maximal.
                                                                       Abstraction (Abs). Names of arguments should not be rel-
   The sets of extensions of an AF 𝐹 for these five semantics                evant. For a pair of AFs 𝐹 = (𝐴, 𝑅) and 𝐹 ′ =
are denoted as (respectively) 𝑐𝑜(𝐹 ), 𝑝𝑟(𝐹 ), 𝑔𝑟(𝐹 ), 𝑠𝑡𝑏(𝐹 )                (𝐴′ , 𝑅′ ) and every isomorphism 𝛾 : 𝐹 → 𝐹 ′ , we
and 𝑠𝑠𝑡(𝐹 ). Based on these semantics, we can define the                     have 𝑎 ⪰𝜌𝐹 𝑏 iff 𝛾(𝑎) ⪰𝜌𝐹 ′ 𝛾(𝑏).
status of any argument, namely skeptically accepted (be-
longing to each 𝜎-extension), credulously accepted (belong-            Independence (In). Unconnected arguments should not in-
ing to some 𝜎-extension) and rejected (belonging to no                       fluence a ranking. For every 𝐹 ′ = (𝐴′ , 𝑅′ ) ∈ 𝑐𝑐(𝐹 )
𝜎-extension). Given an AF 𝐹 and an extension-based se-                       and for all 𝑎, 𝑏 ∈ 𝐴′ : 𝑎 ⪰𝜌𝐹 𝑏 iff 𝑎 ⪰𝜌𝐹 ′ 𝑏.
mantics 𝜎, we use (respectively) 𝑠𝑘𝜎 (𝐹 ), 𝑐𝑟𝑒𝑑𝜎 (𝐹 ) and
                                                                       Void Precedence (VP). Unattacked arguments should be
𝑟𝑒𝑗𝜎 (𝐹 ) to denote these sets of arguments.
                                                                             ranked better then attacked ones. If 𝑎−
                                                                                                                   𝐹 = ∅ and
Example 1. Consider the AF 𝐹1 = (𝐴, 𝑅) depicted as a                         𝑏−
                                                                              𝐹 ̸
                                                                                = ∅ then 𝑎  ≻ 𝜌
                                                                                              𝐹 𝑏.
directed graph in Figure 1, with the nodes corresponding to
                                                                       Self-Contradiction (SC). Self-attacking       arguments
arguments 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑}, and the edges corresponding to
                                                                             should be ranked worse than any other argument. If
attacks 𝑅 = {(𝑎, 𝑏), (𝑏, 𝑐), (𝑐, 𝑑), (𝑑, 𝑐)}. We see that 𝐹1
                                                                                    / 𝑅 and (𝑏, 𝑏) ∈ 𝑅 then 𝑎 ≻𝜌𝐹 𝑏.
                                                                             (𝑎, 𝑎) ∈
has three complete extensions {𝑎}, {𝑎, 𝑐} and {𝑎, 𝑑} only
the last two are preferred in addition. Also, we see that, 𝑎 ∈         Cardinality Precedence (CP). Two arguments are com-
𝑠𝑘𝑐𝑜 (𝐹1 ), 𝑐, 𝑑 ∈ 𝑐𝑟𝑒𝑑𝑐𝑜 (𝐹1 ), and 𝑏 ∈ 𝑟𝑒𝑗𝑐𝑜 (𝐹1 ).                        pared based on the number of attackers. If |𝑎−
                                                                                                                          𝐹| <
                                                                             |𝑏−           𝜌
                                                                               𝐹 | then 𝑎 ≻𝐹 𝑏.
   An isomorphism 𝛾 between two AFs 𝐹 = (𝐴, 𝑅) and
𝐹 ′ = (𝐴′ , 𝑅′ ) is a bijective function 𝛾 : 𝐹 → 𝐹 ′ such that         Quality Precedence (QP). Two arguments are compared
(𝑎, 𝑏) ∈ 𝑅 iff (𝛾(𝑎), 𝛾(𝑏)) ∈ 𝑅′ for all 𝑎, 𝑏 ∈ 𝐴.                           based on the strength of their attackers. If there is
                                                                             𝑐 ∈ 𝑏−                    −                  𝜌
                                                                                   𝐹 s.t. for all 𝑑 ∈ 𝑎𝐹 it holds that 𝑐 ≻𝐹 𝑑 then
                                                                                 𝜌
Argument-ranking Semantics Instead of reasoning                              𝑎 ≻𝐹 𝑏.
based on the acceptance of sets of arguments, argument-
ranking semantics (also know as ranking-based semantics) [2]           Counter-Transitivity (CT). Two arguments are compared
were introduced to focus on the strength of a single argu-                  based on the number and quality of their attackers. If
ment. Note that the order returned by an argument-ranking                   some injective 𝑓 : 𝑎−     −                     𝜌
                                                                                                𝐹 → 𝑏𝐹 exists s.t. 𝑓 (𝑥) ⪰𝐹 𝑥
                                                                                         −         𝜌
semantics is not necessarily total, i. e. not every pair of                 for all 𝑥 ∈ 𝑎𝐹 then 𝑎 ⪰𝐹 𝑏.
arguments is comparable.
                                                                       1
                                                                           A preorder is a (binary) relation that is reflexive and transitive.
Strict Counter-Transitivity (SCT). Strict version of CT.           𝐸 ′ ⊒𝜏𝐹 𝐸; 𝐸 and 𝐸 ′ are incomparable (denoted 𝐸 ≍𝜏𝐹 𝐸 ′ )
       If some injecitve 𝑓 : 𝑎−     −                    𝜌
                              𝐹 → 𝑏𝐹 exists s.t. 𝑓 (𝑥) ⪰𝐹 𝑥        if neither 𝐸 ⊒𝜏𝐹 𝐸 ′ nor 𝐸 ′ ⊒𝜏𝐹 𝐸.
                    −              −       −
       for all 𝑥 ∈ 𝑎𝐹 and either |𝑎𝐹 | < |𝑏𝐹 | or there exists        Skiba et al. [11] defined a family of approaches to de-
       some 𝑥 ∈ 𝑎−                𝜌
                   𝐹 with 𝑓 (𝑥) ≻𝐹 𝑥, then 𝑎 ≻𝐹 𝑏.
                                                  𝜌
                                                                   fine such extension-ranking semantics. Their semantics are
                                                                   generalisations of the classical extension-based semantics.
Defense Precedence (DP). For two arguments with the                Using these semantics we can state that a set is “closer” to
      same number of attackers, a defended argument is             being admissible, than another set. Before we define the
      ranked better than a non-defended argument. If               semantics, we recall the base relations, each of them gener-
      |𝑎−       −      − −             − −
        𝐹 | = |𝑏𝐹 |, (𝑎𝐹 )𝐹 ̸= ∅ and (𝑏𝐹 )𝐹 = ∅, then              alises one aspect of extension-based reasoning.
          𝜌
      𝑎 ≻𝐹 𝑏.
                                                                   Definition 6 (Base Relations [11]). Let 𝐹 = (𝐴, 𝑅) be
Distributed Defense precedence (DDP). The best de-                 an AF and 𝐸 ⊆ 𝐴 where the    ⋃︀ function  ℱ𝐹* : 𝒫(𝐴) →
       fender attacks exactly one attacker. If |𝑎−           −
                                                     𝐹 | = |𝑏𝐹 |   𝒫(𝐴) is defined as ℱ𝐹* (𝐸) = ∞     ℱ
                                                                                                  𝑖=1 𝑖,𝐹
                                                                                                         *
                                                                                                            (𝐸) over the pow-
               − −         − −                                                              *                      *
       and |(𝑎𝐹 )𝐹 | = |(𝑏𝐹 )𝐹 |, and if defense of 𝑎 is simple    erset 𝒫(𝐴) of 𝐴 with ℱ1,𝐹   (𝐸) = 𝐸 and ℱ𝑖,𝐹       (𝐸) =
       - every direct defender of 𝑎 directly attacks exactly       ℱ𝑖−1,𝐹 (𝐸) ∪ (ℱ𝐹 (ℱ𝑖−1,𝐹 (𝐸)) ∖ 𝐸𝐹 ). Each base relation
                                                                     *                  *              −

       one direct attacker of 𝑎 - and distributed - every direct   𝛼 ∈ {𝐶𝐹, 𝑈 𝐷, 𝐷𝑁, 𝑈 𝐴} is defined via:
       attacker of 𝑎 is attacked by at most one argument -
                                                                        • 𝐶𝐹𝐹 (𝐸) = {(𝑎, 𝑏) ∈ 𝑅|𝑎, 𝑏 ∈ 𝐸};
       and defense of 𝑏 is simple but not distributed, then
                                                                        • 𝑈 𝐷𝐹 (𝐸) = 𝐸 ∖ ℱ𝐹 (𝐸);
       𝑎 ≻𝜌𝐹 𝑏.
                                                                        • 𝐷𝑁𝐹 (𝐸) = ℱ𝐹* (𝐸) ∖ 𝐸;
Non-attacked Equivalence (NaE) Two unattacked argu-                     • 𝑈 𝐴𝐹 (𝐸) = {𝑎 ∈ 𝐴 ∖ 𝐸|¬∃𝑏 ∈ 𝐸 : (𝑏, 𝑎) ∈ 𝑅};
      ments should be equally ranked. If 𝑎−    −
                                          𝐹 = 𝑏𝐹 = ∅                  For every base relation, the corresponding 𝛼 base exten-
              𝜌
      then 𝑎 ≃𝐹 𝑏.
                                                                                  𝐹 for 𝐸, 𝐸 ∈ 𝐴 is given by:
                                                                                             ′
                                                                   sion ranking ⊒𝛼
Attack vs. Full Defense (AvsFD). Arguments       with-                                 ′                  ′
      out any unattacked indirect attackers should be                               𝐹 𝐸 iff 𝛼𝐹 (𝐸) ⊆ 𝛼𝐹 (𝐸 )
                                                                                 𝐸 ⊒𝛼
      ranked better than arguments only attacked by                  By combining these base relations, we denote the
      one unattacked argument. If 𝐹 acyclic and every              extension-ranking semantics.
      path 𝑃 (𝑢, 𝑎) in 𝐹 from unattacked 𝑢 to 𝑎 has
                                                                   Definition 7. Let 𝐹 = (𝐴, 𝑅) be an AF and 𝐸, 𝐸 ′ ⊆ 𝐴.
      𝑙𝑝 = 0 mod 2 and there exists unattacked 𝑣 ∈ 𝑏−
                                                    𝐹,             We define: Admissible extension-ranking semantics 𝑟-𝑎𝑑
      then 𝑎 ≻𝜌𝐹 𝑏.
                                                                   via 𝐸 ⊒𝑟-𝑎𝑑
                                                                             𝐹       𝐸 ′ iff 𝐸 ⊐𝐶𝐹𝐹   𝐸 ′ or (𝐸 ≡𝐶𝐹
                                                                                                                  𝐹   𝐸 ′ and
𝜎-Compatibility (𝜎-C). Credulously accepted arguments              𝐸 ⊒𝐹 𝐸 ). Complete extension-ranking semantics 𝑟-𝑐𝑜
                                                                       𝑈𝐷     ′

     should be ranked better than rejected arguments. For          via 𝐸 ⊒𝑟-𝑐𝑜
                                                                             𝐹       𝐸 ′ iff 𝐸 ⊐𝑟-𝑎𝑑
                                                                                                 𝐹    𝐸 ′ or (𝐸 ≡𝑟-𝑎𝑑
                                                                                                                  𝐹    𝐸 ′ and
     an extension-based semantics 𝜎 it holds that if 𝑎 ∈           𝐸 ⊒𝐷𝑁
                                                                       𝐹    𝐸  ′
                                                                                 ). Preferred extension-ranking semantics 𝑟-𝑝𝑟
     𝑐𝑟𝑒𝑑𝜎 (𝐹 ) and 𝑏 ∈ 𝑟𝑒𝑗𝜎 (𝐹 ), then 𝑎 ≻𝜌𝐹 𝑏.                   via 𝐸 ⊒𝑟-𝑝𝑟
                                                                             𝐹       𝐸 ′ iff 𝐸 ⊐𝑟-𝑎𝑑
                                                                                                 𝐹    𝐸 ′ or (𝐸 ≡𝑟-𝑎𝑑
                                                                                                                  𝐹    𝐸 ′ and
                                                                   𝐸 ′ ⊆ 𝐸). Grounded extension-ranking semantics 𝑟-𝑔𝑟
weak 𝜎-Support (w𝜎-S). If an argument 𝑎 is an unavoid-             via 𝐸 ⊒𝑟-𝑔𝑟
                                                                             𝐹       𝐸 ′ iff 𝐸 ⊐𝑟-𝑐𝑜
                                                                                                 𝐹    𝐸 ′ or (𝐸 ≡𝑟-𝑐𝑜
                                                                                                                  𝐹    𝐸 ′ and
      able side-effect of accepting another argument 𝑏, then       𝐸 ⊆ 𝐸 ). Semi-stable extension-ranking semantics 𝑟-𝑠𝑠𝑡
                                                                          ′

      𝑎 should be at least as acceptable as 𝑏. If 𝑎 weakly         via 𝐸 ⊒𝑟-𝑠𝑠𝑡
                                                                             𝐹       𝐸 ′ iff 𝐸 ⊐𝑟-𝑐𝑜
                                                                                                  𝐹   𝐸 ′ or (𝐸 ≡𝑟-𝑐𝑜
                                                                                                                  𝐹    𝐸 ′ and
      𝜎-supports 𝑏, then 𝑎 ⪰𝜌𝐹 𝑏.                                  𝐸 ⊒𝑈𝐹
                                                                         𝐴
                                                                            𝐸 ′
                                                                                ).
strong 𝜎-Support (s𝜎-S). If an argument 𝑎 is a prerequi-              In words, one set 𝐸 is at least as plausible to be accepted
       site for accepting another argument 𝑏 and 𝑏 is irrele-      as 𝐸 ′ with respect to the admissible extension-ranking se-
       vant for accepting 𝑎, then 𝑎 should be ranked better        mantics, if 𝐸 has less conflicts than 𝐸 ′ or if they have the
       then 𝑏. If 𝑎 strongly 𝜎-supports 𝑏, then 𝑎 ≻𝜌𝐹 𝑏.           same conflicts, then we look at the undefended arguments.

  Note that these principles are not always compatible with        Example 2. Continuing Example 1. Consider for instance the
each other, especially SC and CP are not compatible [2].           sets 𝐸1 = {𝑏}, 𝐸2 = {𝑏, 𝑑} and 𝐸3 = {𝑑}. For the complete
                                                                   extension-ranking semantics, we first of all have that all three
Extension-ranking Semantics Extension-ranking se-                  sets contain no conflicts. However, both 𝐸1 and 𝐸2 contain
mantics defined in Skiba et al. [11] are a generalisation of       the argument 𝑏 which they do not defend against 𝑎. It follows
extension-based semantics. These semantics are used to             that 𝐸3 ⊐𝑟-𝑐𝑜                  𝑟-𝑐𝑜
                                                                              𝐹1 𝐸1 and 𝐸3 ⊐𝐹1 𝐸2 . Furthermore, both 𝐸1

formalise whether a set 𝐸 is more plausible to be accepted         and 𝐸2 defend 𝑑 from 𝑐, but 𝑑 is not contained in 𝐸1 . Thus,
than another set 𝐸 ′ .                                             we have that 𝐸2 ⊐𝑟-𝑐𝑜
                                                                                       𝐹1 𝐸 1 .
                                                                      The relevant excerpt of the extension-ranking for 𝑟-𝑐𝑜 can
Definition 5. Let 𝐹 = (𝐴, 𝑅) be an AF. An extension                be found in Figure 2.
ranking on 𝐹 is a preorder over the powerset of arguments             Extension-ranking semantics also follow a principle-
2𝐴 . An extension-ranking semantics 𝜏 is a function that           based approach. Before we recall the principles defined
maps each 𝐹 to an extension ranking ⊒𝜏𝐹 on 𝐹 .                     in Skiba et al. [11], we need to introduce the notion of most
   For an AF 𝐹 = (𝐴, 𝑅), an extension-ranking semantics            plausible sets, i. e. sets for which we cannot find any other
𝜏 and two sets 𝐸, 𝐸 ′ ⊆ 𝐴 we say 𝐸 is at least as plausible        sets ranked strictly better.
to be accepted as 𝐸 ′ with respect to 𝜏 in 𝐹 if 𝐸 ⊒𝜏𝐹 𝐸 ′ . We     Definition 8 (Most plausible sets). Let 𝐹 = (𝐴, 𝑅) be an
define the usual abbreviations as follows: 𝐸 is strictly more      AF, 𝐸, 𝐸 ′ ⊆ 𝐴 two sets of arguments and 𝜏 an extension-
plausible to be accepted than 𝐸 ′ (denoted as 𝐸 ⊐𝜏𝐹 𝐸 ′ ) if       ranking semantics. We denote by 𝑚𝑎𝑥𝜏 (𝐹 ) the maximal (or
𝐸 ⊒𝜏𝐹 𝐸 ′ and not 𝐸 ′ ⊒𝜏𝐹 𝐸; 𝐸 and 𝐸 ′ are equally as plau-        most plausible) elements of the extension ranking ⊒𝜏𝐹 , i. e.
sible to be accepted (denoted as 𝐸 ≡𝜏𝐹 𝐸 ′ ) if 𝐸 ⊒𝜏𝐹 𝐸 ′ and      𝑚𝑎𝑥𝜏 (𝐹 ) = {𝐸 ⊆ 𝐴 | ∄𝐸 ′ ⊆ 𝐴 with 𝐸 ′ ⊐𝜏𝐹 𝐸}.
                                   ..                              Definition 11. Let 𝑥, 𝑦 ∈ 𝑆 be two elements of 𝑆. The lex-
                                    .
                                                                   cel ranking ⪰𝑙𝑒𝑥-c𝑒𝑙 is defined by 𝑥 ≻𝑙𝑒𝑥-𝑐𝑒𝑙
                                                                                                         ⊒       𝑦 if there exists
                                  {𝑏}
                                                                   a 𝑘 such that 𝑥𝑖,⊒ = 𝑦𝑖,⊒ for all 𝑖 < 𝑘 and 𝑥𝑘,⊒ > 𝑦𝑘,⊒
                      ..
                       .                                           and 𝑥 ∼𝑙𝑒𝑥-𝑐𝑒𝑙
                                                                            ⊒     𝑦 if 𝑥𝑖,⊒ = 𝑦𝑖,⊒ for all 𝑖.
                     {𝑐}         {𝑏, 𝑑}
                                                                      Intuitively, an object 𝑥 is ranked better than 𝑦 by the
                                                                   lexicographic excellence operator if 𝑥 is contained in more
                             ∅          {𝑑}                        highly ranked sets than 𝑦.

                                                                   Example 3. We continue Example 2 with the complete
                    {𝑎, 𝑐}        {𝑎}     {𝑎, 𝑑}                   extension-ranking as depicted in Figure 2. Then, we have
                                                                   three sets with rank 1, namely the complete extensions. The
Figure 2: Excerpt of the extension-ranking ⊒𝑟𝐹-1𝑐𝑜 for the com-    argument 𝑎 is contained in all three sets with rank 1, while 𝑐
plete semantics, where 𝐸 → 𝐸 ′ means 𝐸 ′ ⊐𝑟𝐹-1𝑐𝑜 𝐸 and 𝐸 − 𝐸 ′     and 𝑑 are only contained in one such set each. Consequently
means 𝐸 ′ ≡𝑟𝐹-1𝑐𝑜 𝐸                                                𝑎 ⪰lex-cel 𝑐 and 𝑎 ⪰lex-cel 𝑑. Now, the final admissible sets ∅
                                                                   and {𝑑} are dominated by all three complete extensions under
                                                                   the complete extension-ranking semantics, but dominate all
   The principle 𝜎-generalisation states, that the most plau-      non-admissible sets. Therefore, they are the only sets with
sible sets should coincide with the 𝜎-extensions.                  rank 2. It follows that 𝑑 ⪰lex-cel 𝑐 as both are contained in the
                                                                   same number of sets with rank 1, but 𝑑 is contained in more
Definition 9 (𝜎-Gen). Let 𝜎 be an extension-based se-              sets with rank 2.
mantics and 𝜏 an extension-ranking semantics. 𝜏 satis-
fies 𝜎-soundness iff for all 𝐴𝐹 : 𝑚𝑎𝑥𝜏 (𝐴𝐹 ) ⊆ 𝜎(𝐴𝐹 ).                Similarly to argument- and extension-ranking semantics,
𝜎-completeness iff for all 𝐴𝐹 : 𝑚𝑎𝑥𝜏 (𝐴𝐹 ) ⊇ 𝜎(𝐴𝐹 ).               social rankings have been studied axiomatically. Let us first
𝜎-generalisation iff 𝜏 satisfies both 𝜎-soundness and 𝜎-           introduce an axiom that has been part of a characterization
completeness.                                                      of the lex-cel function under the assumption that the ranking
                                                                   over sets is a total preorder [9]. As we generally do not
  Additional principles can be found in Skiba et al. [11].         assume the ranking over extensions to be a total preorder,
                                                                   the characterisation does not hold in our setting, but it is
                                                                   straightforward to see that the lex-cel function still satisfies
3. Social Ranking                                                  this axiom.
Let us now introduce the final piece of our puzzle, social         Definition 12 (Independence from the worst set). Let ⊒
rankings. Let 𝑆 be a set of arbitrary objects like players of a    be a preorder on 𝒫, let
sports team, employees of a company or arguments in an
AF and 𝒫(𝑆) its powerset. A social ranking function 𝜉, as                              𝑤 = max(rank⊒ (𝑋))
                                                                                            𝑋∈𝒫
introduced by Moretti and Öztürk [6], maps a preorder ⊒
on 𝒫(𝑆) to a partial order on 𝑆. In the context of this work,      and assume that ⊒* is another preorder on 𝒫 for which it
we consider the preorder to be an extension-ranking ⊒𝑟-𝜎    𝐹      holds
for some argumentation framework 𝐹 and a semantics 𝜎 as
defined above. The most prominent social ranking function               • rank⊒ (𝑋) = rank⊒* (𝑋) for all 𝑋 ∈ 𝒫 such that
is the lexicographic excellence operator (lex-cel), which was             rank⊒ (𝑋) < 𝑤.
first proposed by Bernardi et al. [9]. It ranks elements based          • rank⊒* (𝑋) ≥ 𝑤 for all 𝑋 ∈ 𝒫 such that
on the best sets they appear in, proceeding lexicographically             rank⊒ (𝑋) = 𝑤.
if there are ties. However, as proposed by Bernardi et al.
[9], the lex-cel operator requires a total order of the sets as    Then for any social ranking function that satisfies Indepen-
input, while the extension ranking semantics defined above         dence from the worst set, we must have that 𝑥 ≻⊒ 𝑦 implies
only provide a partial ranking. To circumvent this problem,        𝑥 ≻⊒* 𝑦.
we make use a measure of the quality of a set that allows us          Intuitively, this axiom states that if one element is already
to compare any two sets, the rank of a set.                        strictly worse than another, and we further subdivide the
Definition 10. Let 𝑋 ⊆ 𝑆 be a subset of 𝑆 and ⊒ a pre-             worst sets, this strict preference remains. As we will see
order on 𝒫(𝑆). Moreover, let 𝑋1 , 𝑋2 , . . . , 𝑋𝑘 be the longest   later, this axiom will be crucial for satisfying our desired
sequence such that 𝑋1 ⊐ 𝑋2 ⊐ · · · ⊐ 𝑋𝑘 ⊐ 𝑋. Then, we              refinement property. Next, we introduce a new, very weak
define the rank of 𝑋, as rank⊒ (𝑋) := 𝑘 + 1.                       axiom inspired by the classical Pareto-efficiency concept
  Moreover, for an element 𝑥 ∈ 𝑆, we define                        [17], that is satisfied by most reasonable rank-based social
                                                                   ranking functions.
    𝑥𝑘,⊒ := |{𝑋 ∈ 𝒫(𝑆) | rank⊒ (𝑋) = 𝑘, 𝑥 ∈ 𝑋}|,
                                                                   Definition 13 (Pareto-efficiency). Let ⊒ be a preorder on
as the number of rank 𝑘 subsets that contain 𝑥.                    𝒫 and let 𝑥, 𝑦 be elements such that

   As we will see later, a rank-based approach to social rank-          • rank⊒ (𝑍 ∪ {𝑥}) ≤ rank⊒ (𝑍 ∪ {𝑦}) for all 𝑍 ∈ 𝒫
ing provides many desirable properties, at least in our con-              with 𝑥, 𝑦 ∈
                                                                                    / 𝑍;
text of ranking arguments. With the definition of a rank                • rank⊒ (𝑍 ∪ {𝑥}) < rank⊒ (𝑍 ∪ {𝑦}) for at least one
at hand, we can now define our rank-based version of the                  𝑍 ∈ 𝒫 with 𝑥, 𝑦 ∈
                                                                                          / 𝑍.
lex-cel social ranking function.                                   A social ranking function 𝜉 satisfies Pareto-efficiency, iff
                                                                   𝑥 ≻𝜉⊒ 𝑦.
  Furthermore, we establish the novel Dominating set ax-           define argument-ranking semantics based on an extension-
iom which captures the intuition that if there exists a set        ranking.
containing the object 𝑥 that is ranked better than every set
that contains some other object 𝑦, then 𝑥 must be ranked           4.1. The Singleton Approach
better than 𝑦 by the social ranking function.
                                                                   The most immediate way of ranking objects based on a
Definition 14 (Dominating set). Let ⊒ be a preorder on 𝒫           ranking over sets of objects is to restrict the ranking over
and let 𝑥, 𝑦 be elements such that ∃𝑋 ⊆ 𝒫 with 𝑥 ∈ 𝑋 such          sets of objects to the singleton sets. The behaviour of these
that ∀𝑌 with 𝑦 ∈ 𝑌 then 𝑋 ⊐ 𝑌 . A social ranking function          singleton sets then gives us insight into the relationship
𝜉 satisfies Dominating set iff 𝑥 ≻𝜉⊒ 𝑦.                            between the objects. If {𝑎} is ranked better than {𝑏} then
   Crucially, Independence from the Worst Set and Pareto-          𝑎 is also ranked better than 𝑏 in the restricted ranking.
efficiency together imply Dominating set.
                                                                   Definition 15. Let 𝐹 = (𝐴, 𝑅) be an AF and 𝜏 any
Theorem 1. Any social ranking function that satisfies Inde-        extension-ranking semantics. For any two arguments 𝑎, 𝑏 ∈
pendence from the worst set and Pareto-efficiency also satisfies   𝐴, the singleton argument-ranking semantics 𝒮𝒯 𝜏 is de-
Dominating set.                                                    fined via 𝑎 ⪰𝒮𝒯
                                                                                𝐹
                                                                                   𝜏
                                                                                     𝑏 iff {𝑎} ⊒𝜏𝐹 {𝑏}.
Proof. Let ⊒ be a preorder on 𝒫 and let 𝑥, 𝑦 be elements             Bernardi et al. [9] have already discussed that a rank-
such that ∃𝑋 𝑑 ⊆ 𝒫 with 𝑥 ∈ 𝑋 𝑑 such that ∀𝑋 ′ with 𝑦 ∈            ing based solely on singleton sets is too simplistic, as it
𝑋 ′ then 𝑋 𝑑 ⊐ 𝑋 ′ . Furthermore, let 𝑤 := rank⊒ (𝑋 𝑑 ) + 1.       ignores all the information provided by rankings over sets
We consider the preorder ⊒* that is defined as follows: For        with cardinality larger than one. In the context of abstract
any two sets 𝑋, 𝑌 ∈ 𝒫 we have 𝑋 ⊒* 𝑌 if and only if                argumentation, this is also the case.
𝑋 ⊒ 𝑌 and either rank⊒ (𝑋) < 𝑤 or rank⊒ (𝑌 ) < 𝑤. We
claim that                                                         Example 4. Consider the AF 𝐹1 from Example 1. We use 𝑟-
                 max(rank⊒* (𝑋)) = 𝑤.                              𝑎𝑑 as the underlying extension-ranking semantics, then since
                                                                                                             𝒮𝒯
                  𝑋∈𝒫                                              {𝑎} and {𝑑} are admissible we have 𝑎 =𝐹1 𝑟-𝑎𝑑 𝑑 and both
   First, to see that max𝑋∈𝒫 (rank⊒* (𝑋)) ≤ 𝑤 we assume            {𝑏} and {𝑐} are conflict-free and not defended, so
for the sake of a contradiction that there is a set 𝑋 with
                                                                                   𝒮𝒯              𝒮𝒯             𝒮𝒯
rank⊒* (𝑋) = 𝑤* > 𝑤. Then, by definition, there is a                           𝑎 =𝐹1 𝑟-𝑎𝑑 𝑑 ≻𝐹1 𝑟-𝑎𝑑 𝑏 =𝐹1 𝑟-𝑎𝑑 𝑐
sequence 𝑋1 ⊒* 𝑋2 ⊒* · · · ⊒* 𝑋𝑤* ⊒* 𝑋. As every
preference in ⊒* is also valid in ⊒, the same sequence exists         The example shows that 𝒮𝒯 𝑟-𝑎𝑑 has a limited expressive-
for ⊒, i. e. 𝑋1 ⊒ 𝑋2 ⊒ · · · ⊒ 𝑋𝑤* ⊒ 𝑋. However, this              ness, since 𝒮𝒯 𝑟-𝑎𝑑 has at most three ranks. The first rank
means rank⊒ (𝑋𝑤* ) ≥ 𝑤* − 1 ≥ 𝑤 and rank⊒ (𝑋) ≥                    contains arguments for which the singleton set is admissi-
𝑤* > 𝑤, which contradicts 𝑋𝑤* ⊒* 𝑋.                                ble and the lowest rank are all self-attacking arguments, in
  To see that max𝑋∈𝒫 (rank⊒* (𝑋)) ≥ 𝑤 we first ob-                 between are the non-admissible sets, but conflict-free single-
serve that as rank⊒ (𝑋 𝑑 ) = 𝑤 − 1 there is a sequence             ton sets. Observe also that this approach does not refine the
𝑋1 ⊒ 𝑋2 ⊒ · · · ⊒ 𝑋𝑤−1 ⊒ 𝑋. As this sequence is                    classical skeptical/credulous acceptance classification, as in
maximal, rank⊒ (𝑋𝑖 ) < 𝑤 for all elements 𝑋𝑖 of the se-            Example 4 the credulously accepted argument 𝑐 is ranked
quence. Hence the same sequence exists in ⊒* . Finally,            the same as the rejected argument 𝑏.
as 𝑋 𝑑 is a dominating set, we know 𝑋 𝑑 ⊒ {𝑦} and as
rank⊒ (𝑋 𝑑 ) < 𝑤, we also have 𝑋 𝑑 ⊒* {𝑦}. Therefore,              4.2. Generalised Social Ranking
𝑋1 ⊒* 𝑋2 ⊒* · · · ⊒* 𝑋𝑤−1 ⊒* 𝑋 ⊒* {𝑦} witnesses                         Argument-ranking Semantics
that rank⊒* ({𝑦}) ≥ 𝑤.
                                    *
   Next, we claim that 𝑥 ≻⊒ 𝑦 for all social rank-                 In the literature, a number of different social ranking func-
ing functions that satisfy Pareto-efficiency: By definition,       tions that are more complex than the singleton approach
rank⊒* (𝑋 𝑑 ) = 𝑤 − 1. Furthermore, we have 𝑋 𝑑 =                  can be found [18, 9, 8, 7]. To understand what constitutes
(𝑋 𝑑 ∖ {𝑥}) ∪ {𝑥} ⊒* (𝑋 𝑑 ∖ {𝑥}) ∪ {𝑦}, and thus                   a good social ranking function in this context, we define a
rank⊒* ((𝑋 𝑑 ∖ {𝑥}) ∪ {𝑦}) > 𝑤 − 1. This shows that                general argument-ranking semantics using social ranking
rank⊒* (𝑋 𝑑 ) < rank⊒* ((𝑋 𝑑 ∖ {𝑥}) ∪ {𝑦}). On the other           solutions with respect to an extension ranking.
hand, there can be no 𝑍 such that rank⊒* (𝑍 ∪ {𝑦}) <               Definition 16. Let 𝐹 = (𝐴, 𝑅) be an AF and 𝜉 a social
rank⊒* (𝑍 ∪{𝑥}): As 𝑍 ∪{𝑦} is dominated by 𝑋 𝑑 , we know           ranking function with respect to extension ranking 𝜏 . For any
rank⊒ (𝑍 ∪{𝑦}) ≥ 𝑤 and thus rank*⊒ (𝑍 ∪{𝑦}) ≥ 𝑤. Thus,             𝑎, 𝑏 ∈ 𝐴 we call 𝜉𝜏 the Social ranking argument-ranking
the claim follows directly from max𝑋∈𝒫 (rank⊒* (𝑋)) = 𝑤            semantics such that:
   Finally, if ⪰ also satisfies Independence from the worst
set, if follows that also 𝑥 ≻⊒ 𝑦, as ⊒ is just a refinement of                           𝑎 ⪰𝜉𝐹𝜏 𝑏 iff 𝑎 ⪰𝜉𝜏 𝑏
the worst set of ⊒* .
                                                                      In words, an argument 𝑎 is at least as strong as argument
                                                                   𝑏 if the social ranking function 𝜉 applied to the extension
4. Defining Argument-ranking                                       ranking ⊒𝜏 returns that 𝑎 is at least as strong as 𝑏.
   Semantics via Social Rankings                                   Example 5. In Example 3 the social ranking argument rank-
The idea of combining extension-ranking semantics with             ing lex-celr-co was applied to the AF 𝐹1 from Example 1 where
argument-ranking semantics was briefly discussed by Skiba          lex-cel is used and the underlying extension-ranking semantics
et al. [11], where, based on a ranking over sets of argu-          is r-co. Thus, the resulting argument ranking is:
ments, a ranking over arguments was defined. In this sec-
tion, we take a more general view on this approach and                         𝑎 ≻lex-cel
                                                                                  𝐹1
                                                                                         r-co
                                                                                              𝑑 ≻lex-cel
                                                                                                 𝐹1
                                                                                                        r-co
                                                                                                             𝑐 ≻lex-cel
                                                                                                                𝐹1
                                                                                                                        r-co
                                                                                                                             𝑏
   Any social ranking function can be used to rank argu-              Next, we investigate principles for social ranking based
ments. Skiba et al. [11] have used a variation of the lex-cel      argument-ranking semantics from a general point of view.
social ranking function in their definitions, where an argu-       In particular, we are interested in understanding which com-
ment 𝑎 is ranked better than another argument 𝑏 if we can          binations of axioms for extension-ranking semantics 𝜏 and
find a set 𝐸 containing 𝑎 which is ranked better than any          social ranking functions 𝜉 represent necessary and sufficient
set containing 𝑏.                                                  conditions for the corresponding social ranking argument-
                                                                   ranking semantics 𝜉𝜏 to satisfy fundamental principles of
Definition 17 ([11]). Let 𝐹 = (𝐴, 𝑅) be an AF, 𝑎, 𝑏 ∈
                                                                   argument rankings, chiefly among them our desired refine-
𝐴, and 𝜏 be an extension-ranking semantics. We define an
                                                                   ment property. This translates to the following research
argument-ranking semantics ⪰𝜏𝐹 via 𝑎 ⪰𝜏𝐹 𝑏 iff there is a
                                                                   questions:
set 𝐸 with 𝑎 ∈ 𝐸 s.t. for all sets 𝐸 ′ with 𝑏 ∈ 𝐸 ′ we have
𝐸 ⊒𝜏𝐹 𝐸 ′ .                                                        RQ1 What properties of 𝜉 and 𝜏 are adequate to ensure
Example 6. Continuing with Example 1. Using 𝑟-𝑎𝑑 as the                that 𝜉𝜏 satisfies a specific principle for argument-
underlying extension-ranking semantics, we see that {𝑎, 𝑐}             ranking semantics?
and {𝑎, 𝑑} are admissible sets, hence also among the most          RQ2 What properties of 𝜉𝜏 are adequate to ensure that
plausible sets. Since 𝑟-𝑎𝑑 satisfies 𝑎𝑑-generalisation there           𝜉 satisfies a specific principle for social ranking
cannot be any set containing 𝑏 ranked strictly better, than            functions when combined with a certain extension-
these two sets. This observation result in the ranking 𝑎 ≃𝑟-𝑎𝑑
                                                          𝐹1           ranking semantics 𝜏 ?
𝑐 ≃𝑟-𝑎𝑑
     𝐹1   𝑑 ≻𝑟-𝑎𝑑
               𝐹1   𝑏. Since {𝑎, 𝑐}, {𝑎, 𝑑} ∈ 𝜎(𝐹1 ) for 𝜎 ∈
{𝑐𝑜, 𝑝𝑟, 𝑠𝑡𝑏, 𝑠𝑠𝑡} the ranking is the same for any 𝑟-𝜎. Only       Next, we address RQ1 and RQ2 for a selected number of
for 𝑟-𝑔𝑟 the induced ranking differs:                              principles for argument ranking semantics.
                    𝑟-𝑔𝑟
                 𝑎 ≻𝐹1
                         𝑐 ≃𝑟-𝑔𝑟  𝑟-𝑔𝑟
                            𝐹1 𝑑 ≻𝐹1 𝑏                             4.2.1. Sufficient Conditions for Social Ranking
   The previous examples show that where lex-celr-co can                  Argument-ranking semantics
differentiate 𝑎, 𝑏, 𝑐, and 𝑑, the argument ranking of Defini-      We start by considering 𝜎-Compatibility. For this we show
tion 17 under 𝑟-𝑐𝑜 does not allow to distinguish among 𝑎, 𝑐        that Independence from the worst set together with the quite
and 𝑑. Indeed, lex-cel is more informative than the operator       weak condition Pareto-efficiency, is sufficient for satisfying
of Skiba et al. [11].                                              𝜎-C.
Proposition 1. Let 𝐹 = (𝐴, 𝑅) be an AF, 𝑎, 𝑏 ∈ 𝐴 and 𝜏             Theorem 2. Let 𝐹 = (𝐴, 𝑅) be an argumentation frame-
an extension ranking. If 𝑎 ⪰lex-cel
                            𝐹
                                    𝜏
                                      𝑏, then 𝑎 ⪰𝜏𝐹 𝑏.             work, 𝜏 an extension-ranking semantics, satisfying 𝜎- gen-
Proof. Let 𝐹 = (𝐴, 𝑅) be an AF, 𝑎, 𝑏 ∈ 𝐴 and 𝜏 an exten-           eralisation for extension semantics 𝜎 and 𝜉 a social ranking
sion ranking. Assume 𝑎 ⪰lex-cel  𝜏
                                   𝑏, then there is an 𝑘 s.t for   function that satisfies Independence from the worst set and
                            𝐹
all 𝑖 < 𝑘 we have 𝑎𝑖,𝜏 = 𝑏𝑖,𝜏 and 𝑎𝑘,𝜏 ≥ 𝑏𝑘,𝜏 .                    Pareto-efficiency. Then 𝜉𝜏 satisfies 𝜎-C.
   If 𝑏𝑗,𝜏 ̸= 0 for 1 ≤ 𝑗 ≤ 𝑘, then there is one 𝑌 ⊆ 𝐴 with
                                                                   Proof. Consider first the extension ranking ⊒𝜎 defined by
𝑟𝑎𝑛𝑘𝜏 (𝑌 ) = 𝑗 and 𝑏 ∈ 𝑌 . W.l.o.g. let 𝑗 be the smallest
                                                                   𝑋 ⊒𝜎𝐹 𝑌 if and only if 𝑋 ∈ 𝜎(𝐹 ) and 𝑌 ̸∈ 𝜎(𝐹 ). Fur-
number s.t. 𝑏𝑗,𝜏 ̸= 0. Then 𝑌 ⊒𝜏𝐹 𝑋 for all 𝑋 ⊆ 𝐴 with
                                                                   thermore, let 𝑥 ∈ 𝑐𝑟𝑒𝑑     𝜎 (𝐹 ) and 𝑦 ∈ 𝑟𝑒𝑗𝜎 (𝐹 ). Then,
𝑎 ∈ 𝑋, therefore 𝑏 ⪰𝜏𝐹 𝑎. Since, 𝑏𝑗,𝜏 ≤ 𝑎𝑗,𝜏 , there has                                  𝜎
                                                                   we claim that 𝑥 ≻⊒ 𝑦 for any social ranking function
to be an 𝑋 ′ ⊆ 𝐴 with 𝑟𝑎𝑛𝑘𝜏 (𝑋 ′ ) = 𝑗 and 𝑎 ∈ 𝑋 ′ s.t.
                                                                   𝜉 that satisfies Pareto-efficiency: As 𝑥 is credulously ac-
𝑋 ′ ≡𝜏𝐹 𝑌 , so 𝑎 ⪰𝜏𝐹 𝑏.
                                                                   cepted, there exists a 𝑋 ∈ 𝜎(𝐹 ) with 𝑥 ∈ 𝑋 and as
   If 𝑏𝑗,𝜏 = 0 for all 𝑗 ∈ {1, . . . , 𝑘} and 𝑎𝑘,𝜏 > 0, then
                                                                   𝑦 is rejected, we have 𝑌 ̸∈ 𝜎(𝐹 ) for all 𝑦 ∈ 𝑌 . It fol-
there is at least one 𝑋 ⊆ 𝐴 with 𝑎 ∈ 𝑋 and 𝑟𝑎𝑛𝑘𝜏 (𝑋) = 𝑘
                                                                   lows that rank⊒𝜎 (𝑋 ∖ {𝑥}) ∪ {𝑥}) = 1 < rank⊒𝜎 ((𝑋 ∖
s.t. 𝑋 ⊐𝜏𝐹 𝑌 for all 𝑌 ⊆ 𝐴 with 𝑏 ∈ 𝑌 , and therefore
                                                                   {𝑥}) ∪ {𝑦}). On the other hand, there can be no 𝑆 such that
𝑎 ≻𝜏𝐹 𝑏.
                                                                   rank⊒𝜎 (𝑆 ∪ {𝑦}) < rank⊒𝜎 (𝑆 ∪ {𝑥}) as, due to the fact
   In particular, lex-celr-co allows us to distinguish among       that 𝑤 = max𝑋⊆𝐴 (rank⊒𝜎𝐹 (𝑋)) = 2, this would imply
skeptically and credulously accepted arguments (𝑎 is ranked        rank⊒𝜎 (𝑆 ∪ {𝑦}) = 1 and therefore 𝑆 ∪ {𝑦} ∈ 𝜎(𝐹 ).
before 𝑐 and 𝑑). To capture this, we define a skeptical varia-        Furthermore, as 𝜏 satisfies 𝜎-generalisation, we know that
tion of 𝜎-Compatibility. Skeptical accepted arguments are          rank⊒𝜎𝐹 (𝑋) = 1 if and only if rank⊒𝜏𝐹 (𝑋) = 1. There-
part of every 𝜎-extension, therefore they should be ranked         fore, it follows from Independence from the worst set that
                                                                       𝜉
better than any other argument.                                    𝑥 ≻𝐹⊒𝜎 𝑦 implies 𝑥 ≻𝜉𝐹𝜏 𝑦. Consequently, we know that
                                                                   𝜉𝜏 satisfies 𝜎-C.
Definition 18. Let 𝐹 = (𝐴, 𝑅) be an AF, 𝑎, 𝑏 ∈ 𝐴, and let 𝜎
be a extension-based semantics. Argument-ranking semantics           Next, we show that Independence from the worst set and
𝜌 satisfies 𝜎-skeptical-Compatibility (𝜎-sk-C) iff 𝑎 ∈ 𝑠𝑘𝜎 (𝐹 )    Pareto-efficiency together also imply that every skeptically
and 𝑏 ∈/ 𝑠𝑘𝜎 (𝐹 ) then 𝑎 ≻𝜌𝐹 𝑏.                                    accepted argument is ranked before any argument that is
   Crucially, a well-behaved argument ranking semantics            not skeptically accepted.
should be able to rank skeptically accepted arguments before       Theorem 3. Let 𝐹 = (𝐴, 𝑅) be an AF, 𝜏 an
all credulously accepted ones, which should be, in turn,           extension-ranking semantics satisfying 𝜎-generalisation for
ranked before all non-accepted arguments. This translated          an extension-based semantics 𝜎, then if social ranking func-
to the following refinement property.                              tion 𝜉 satisfies Pareto-efficiency and Independence from the
Definition 19 (𝜎-Refinement). Argument-ranking seman-              worst set then 𝜉𝜏 satisfies 𝜎-sk-C.
tics 𝜌 satisfies 𝜎-Refinement if 𝜌 satisfies 𝜎-C and 𝜎-sk-C for
extension-based semantics 𝜎 for all AFs 𝐹 .
Proof. Let 𝐹 = (𝐴, 𝑅) be an AF, 𝜏 an extension-ranking se-               However, then, by our choice of ℓ we know
mantics satisfying 𝜎-generalisation for an extension-based
                                                                             |{𝑍 ∈ 𝒫 | 𝑥, 𝑦 ̸∈ 𝑍 ∧ |𝑍| ≥ 2}| > 𝑘 = 𝑘 · |{∅}|.
semantics 𝜎, and 𝜉 a social ranking function satisfying
Pareto-efficiency and Independence from the worst set.                   It follows that rank 𝑘-super majority is violated.
Since 𝜎-generalisation is satisfied by 𝜏 we can view 𝜏 as a
                                                                            Next, consider the axiom SC. Here, we can find a property
refinement of the extension-ranking semantics 𝜏 ′ defined
          ′                                                              of social ranking functions that guarantees that 𝜉𝜏 satis-
by 𝑋 ⊒𝜏𝐹 𝑌 iff 𝑋 ∈ 𝜎(𝐹 ) and 𝑌 ∈    / 𝜎(𝐹 ) for 𝑋, 𝑌 ⊆ 𝐴.
                                                                         fies SC under the assumption that 𝜏 satisfies the following
   Now consider two arguments, 𝑎, 𝑏 ∈ 𝐴, such that 𝑎 ∈
                                                                         principle:
𝑠𝑘𝜎 (𝐹 ) and 𝑏 ∈/ 𝑠𝑘𝜎 (𝐹 ). Assume there exists a 𝑍 ⊆ 𝐴 ∖
{𝑎, 𝑏} s.t. rank⊒𝜏 ′ (𝑍 ∪ {𝑏}) < rank⊒𝜏 ′ (𝑍 ∪ {𝑎}). Since               Definition 21 (Respects Conflicts). For AF 𝐹 = (𝐴, 𝑅) and
                   𝐹                          𝐹
                                                                         𝐸, 𝐸 ′ ⊆ 𝐴 extension-ranking semantics 𝜏 satisfies respects
𝜏 only has two levels, this implies 𝑍 ∪ {𝑏} ∈ 𝑚𝑎𝑥𝜏 ′ (𝐹 )
 ′
                                                                         conflicts if 𝐸 ∈ 𝑐𝑓 (𝐹 ) and 𝐸 ′ ∈
                                                                                                          / 𝑐𝑓 (𝐹 ), then 𝐸 ⊐𝜏𝐹 𝐸 ′ .
and thus 𝑍 ∪ {𝑏} ∈ 𝜎(𝐹 ). As 𝑎 ∈ 𝑠𝑘𝜎 (𝐹 ), we must have
𝑎 ∈ 𝑍 ∪ {𝑏}. However, as 𝑎 ∈          / 𝑍 we know that also                 To show that 𝜉𝜏 satisfies SC we also need the Dominating
𝑎∈ / 𝑍 ∪ {𝑏}. This is a contradiction and hence such a 𝑍                 set property from Definition 14. With these two properties
cannot exist.                                                            we can then show when SC is satisfied.
   Since 𝑏 ∈
           / 𝑠𝑘𝜎 (𝐹 ) we know there has to exists 𝑌¯ ⊆ 𝐴 s.t.            Theorem 4. For AF 𝐹 = (𝐴, 𝑅) if extension-ranking seman-
𝑌¯ ∈ 𝑚𝑎𝑥𝜏 ′ (𝐹 ) and 𝑦 ∈ / 𝑌¯ . Then because 𝑎 ∈ 𝑠𝑘𝜎 (𝐹 ) we             tics 𝜏 satisfies respects conflicts and social ranking function 𝜉
know that (𝑌¯ ∖ {𝑎}) ∪ {𝑏} ∈     / 𝑚𝑎𝑥𝜏 ′ (𝐹 ). Consequently,            satisfies Dominating set, then 𝜉𝜏 satisfies SC.
                                 𝜉𝜏 ′
Pareto-efficiency implies 𝑎 ≻𝐹 𝑏.
          𝜉 ′
   As 𝑎 ≻𝐹𝜏 𝑏 holds for 𝜏 ′ , and 𝜏 is a refinement of 𝜏 ′ such          Proof. For AF 𝐹 = (𝐴, 𝑅), let 𝑎, 𝑏 ∈ 𝐴, (𝑏, 𝑏) ∈ 𝑅 and
that 𝑚𝑎𝑥𝜏 ′ (𝐹 ) = 𝑚𝑎𝑥𝜏 (𝐹 ) it follows from Independence                       / 𝑅, then {𝑎} ∈ 𝑐𝑓 (𝐹 ) and for all 𝐸 ′ with 𝑏 ∈ 𝐸 ′
                                                                         (𝑎, 𝑎) ∈
for the worst set that the same holds for 𝜏 , i. e. 𝑎 ≻𝜉𝐹𝜏 𝑏.            it holds that 𝐸 ′ ∈
                                                                                           / 𝑐𝑓 (𝐹 ). Because of respects conflicts we
                                                                         have {𝑎} ⊐ 𝐸 ′ and therefore because of Dominating set
  Additionally, observe that Independence from the worst                 we have 𝑎 ≻𝜉𝐹𝜏 𝑏.
set means that we might have to ignore most of the infor-
mation that is available to us. The following result shows               4.2.2. Necessary Conditions for Social Ranking
that, at least for the rank information, this is essentially                    Argument-ranking semantics
unavoidable if we want to satisfy 𝑐𝑓 -C. Let us first intro-
                                                                         Let us try to go the other way, that is finding necessary con-
duce an axiom that encodes the idea that we cannot ignore
                                                                         ditions for the social ranking functions to satisfy desirable
overwhelming, rank based evidence.
                                                                         properties. First observe it is not possible to formulate any
Definition 20 (Rank 𝑘-super majority). Let 𝑘 ∈ N be a                    necessary conditions that also hold for any ranking that
natural number. Then we say a social ranking function 𝜉                  cannot be realised by any AF, i. e., we cannot find an AF that
satisfies rank 𝑘-super majority if for all 𝑥 and 𝑦 such that             induces this ranking. This is because any property of the
                                                                         argument-ranking only restricts the social ranking function
     |{𝑍 ∈ 𝒫 | 𝑥, 𝑦 ̸∈ 𝑍∧rank(𝑍∪{𝑥}) < rank(𝑍∪{𝑦})}| >                   on realisable rankings. Therefore, we need to define the
                                                                         following concept in a similar vain to Dunne et al. [19].
𝑘·|{𝑍 ∈ 𝒫 | 𝑥, 𝑦 ̸∈ 𝑍∧rank(𝑍∪{𝑦}) < rank(𝑍∪{𝑥})}|.
                                                                         Definition 22. Let 𝑋 be a set and let ⊒ be a preorder on
we have 𝑥 ⪰ 𝑦.                                                           𝒫(𝑋). Then, we say that ⊒ is 𝜏 -realisable for a extension-
   In words, if there are 𝑘-times as many sets 𝑍 such that the           ranking semantics 𝜏 if there is an AF 𝐹 with 𝐴 = 𝑋 such
rank of 𝑍 ∪ {𝑥} is strictly better than the rank of 𝑍 ∪ {𝑦},             that ⊒𝜏𝐹 =⊒.
than the other way round, then 𝑥 must be (weakly) preferred
                                                                           For example, for a set {𝑎, 𝑏} any preorder containing
to 𝑦.
                                                                         {𝑎, 𝑏} ⊒ {𝑎} is not 𝑟-𝑐𝑓 -realisable. The conflicts in {𝑎, 𝑏}
Proposition 2. Any social ranking function 𝜉𝑟-𝑐𝑓 satisfies               must be a strict super-set of the conflicts in {𝑎}. On the
𝑐𝑓 -C and violates rank 𝑘-super majority for every 𝑘.                    other hand, the preorder containing exactly the relations
                                                                         {𝑎} ⊒ {𝑎, 𝑏} and {𝑏} ⊒ {𝑎, 𝑏} is realised by the AF
Proof. Let 𝑘 be an arbitrary natural number, ℓ a natural                 ({𝑎, 𝑏}, {(𝑎, 𝑏)}).
number such that ℓ ≥ 𝑘 and ℓ ≥ 3. Furthermore con-
                                                                         Theorem 5. Let 𝜉 be a social ranking function such that
sider an argumentation framework 𝐹 with the arguments
                                                                         𝜉𝑟-𝑐𝑓 satisfies 𝑐𝑓 -𝐶. Then, 𝜉 satisfies Dominating set for all
𝑎, 𝑏, 𝑐1 , . . . 𝑐ℓ and the attacks (𝑏, 𝑏) and (𝑐𝑖 , 𝑎) for all 𝑖 ≤ ℓ.
                                                                         𝑟-𝑐𝑓 -realisable preorders ⊒.
Then, 𝑎 ∈ 𝑐𝑟𝑒𝑑𝑐𝑓 (𝐹 ), as witnessed by the conflict free set
{𝑎}, but 𝑏 ∈ 𝑟𝑒𝑗𝑐𝑓 (𝐹 ), as it is self-attacking. It follows             Proof. Let ⊒ be a 𝑐𝑓 -realisable preorder and let 𝐹 be an
from the fact that 𝑎 ≻ 𝑏, because ⪯ satisfies 𝑐𝑓 -C. However,            𝐴𝐹 that realises it. Assume further that there are 𝑥, 𝑦 ∈ 𝐴
observe that                                                             such that there exists a 𝑋 with 𝑥 ∈ 𝑋 for which we have
                                                                         𝑋 ⊐ 𝑌 for all 𝑌 such that 𝑦 ∈ 𝑌 .
     {𝑍 ∈ 𝒫 | 𝑥, 𝑦 ̸∈ 𝑍 ∧ rank𝑟-cf (𝑍 ∪ {𝑎})}                               As 𝑋 contains 𝑥, its set of conflicts must be a strict super-
                                 < rank𝑟-cf (𝑍 ∪ {𝑏}) = {∅}              set of the conflicts in {𝑥}. It follows that {𝑥} ⊒ 𝑋 ⊐ 𝑌
                                                                         and hence by transitivity also {𝑥} ⊐ 𝑌 for all 𝑌 such
while                                                                    that 𝑦 ∈ 𝑌 . In particular, it follows that {𝑥} ⊐ {𝑦}. By
                                                                         definition, this means 𝐶𝐹𝐹 ({𝑥}) ⊂ 𝐶𝐹𝐹 ({𝑦}), which can
     {𝑍 ∈ 𝒫 | 𝑥, 𝑦 ̸∈ 𝑍                                                  only hold if 𝑦 is self-attacking and 𝑥 is not. However, then 𝑥
                                                                         is credulously accepted in the under conflict-free semantics
          ∧ rank𝑟-cf (𝑍 ∪ {𝑎}) < rank𝑟-cf (𝑍 ∪ {𝑏})}
                                                                         while 𝑦 is not. Consequently, it follows from 𝑐𝑓 -𝐶 that
                         = {𝑍 ∈ 𝒫 | 𝑥, 𝑦 ̸∈ 𝑍 ∧ |𝑍| ≥ 2}.                𝑥 ≻ 𝑦. Hence, dominating set is satisfied.
  It follows that dominating set is a necessary and sufficient     Proof. First, consider sets 𝑍1 , . . . , 𝑍𝑛 ∈ 𝒫 for which condi-
condition for a social ranking function to satisfy 𝑐𝑓 -𝐶 when      tion (2) of Pareto-efficiency holds. Among these, take those
combined with 𝑟-𝑐𝑓 .                                               𝑍1 , . . . , 𝑍𝑚 (with 𝑚 ≤ 𝑛) for which rank⊒ (𝑍𝑗 ∪ {𝑥}) = 𝑘
  A similar result can be found for admissible semantics.          (with 1 ≤ 𝑗 ≤ 𝑚) is minimal. At this level in the ranking,
                                                                   we have that rank⊒ (𝑍 ∪ {𝑥}) = rank⊒ (𝑍 ∪ {𝑦}) for each
Theorem 6. Let 𝜉 be a social ranking s.t. 𝜉𝑟-𝑎𝑑 satisfies          𝑍 ̸= 𝑍𝑗 . Hence, for every 𝑍 ∪ {𝑥} there is exactly one
𝑎𝑑-𝐶. Then 𝜉 satisfies Dominating set for all 𝑟-𝑎𝑑-realisable      corresponding set 𝑍 ∪ {𝑦}, except for each 𝑍𝑗 ∪ {𝑥} (be-
preorders ⊒.                                                       cause rank⊒ (𝑍𝑗 ∪ {𝑦}) > 𝑘). Thus, for each 𝑍 ∈ 𝒫 with
                                                                   𝑥, 𝑦 ∈/ 𝑍:
Proof. Let ⊒ be a 𝑟-𝑎𝑑-realisable preorder and AF 𝐹 =
(𝐴, 𝑅) induces ⊒. Assume 𝑥, 𝑦 ∈ 𝐴 such that there exists
𝑋 ⊆ 𝐴 with 𝑥 ∈ 𝑋 for which we have 𝑋 ⊐ 𝑌 for all 𝑌                    |{𝑍 ∪ {𝑥} ∈ 𝒫 | rank⊒ (𝑍 ∪ {𝑥}) = 𝑘}| >
such that 𝑦 ∈ 𝑌 .                                                                  |{𝑍 ∪ {𝑦} ∈ 𝒫 | rank⊒ (𝑍 ∪ {𝑦}) = 𝑘}|.
   Assume that the set 𝑋 is not admissible. That means one
of the following two cases must apply                              At level 𝑘, there are more sets containing 𝑥 than those
                                                                   containing 𝑦, i. e. 𝑥𝑘,⊒ > 𝑦𝑘,⊒ by Definition 10. To prove
     (1)   𝐶𝐹𝐹 (𝑋) ̸= ∅       or,   (2)   𝑈 𝐷𝐹 (𝑋) ̸= ∅            𝑥 ≻lex-cel
                                                                        ⊒     𝑦 it remains to show that 𝑥𝑖,⊒ = 𝑦𝑖,⊒ for all 𝑖 < 𝑘.
                                                                   By construction, for all 𝑖 < 𝑘 and 𝑍 ∈ 𝒫 ∖ {𝑥, 𝑦}, we know
to (1): Then, there is some attack (𝑎, 𝑏) ∈ 𝐶𝐹𝐹 (𝑋) for            that rank⊒ (𝑍 ∪ {𝑥}) = rank⊒ (𝑍 ∪ {𝑦}). Hence, for each
𝑎, 𝑏 ∈ 𝑋. From 𝑋 ⊐ 𝑌 it follows that 𝐶𝐹𝐹 (𝑋) ⊆                     set containing 𝑥 there is exactly one set containing 𝑦. By
𝐶𝐹𝐹 (𝑌 ) and thus (𝑎, 𝑏) ∈ 𝐶𝐹𝐹 (𝑌 ). Now, if 𝑦 = 𝑎 or              Definition 10, we obtain 𝑥𝑖,⊒ = 𝑦𝑖,⊒ , as desired.
𝑦 = 𝑏 it follows that 𝑦 ∈ 𝑋 which directly contradicts
our assumption because of 𝑋 ≡ 𝑌 ′ for 𝑌 ′ = 𝑋 with                    Bernardi et al. [9] have shown that lex-cel satisfies Inde-
𝑦 ∈ 𝑌 ′ . However, if 𝑦 ̸= 𝑎 and 𝑦 ̸= 𝑏 we can con-                pendent from worst set for total orders and it is straightfor-
struct 𝑌 ′ = 𝑌 ∖ {𝑎, 𝑏}. Clearly, that means we either             ward to see that this also holds for our setting. By Theorem 1
have 𝐶𝐹𝐹 (𝑌 ′ ) = ∅ which means 𝑌 ⊐ 𝑋 or we have                   this means lex-cel also satisfies Dominating set, which im-
𝐶𝐹𝐹 (𝑌 ′ ) ̸= ∅ which implies 𝑋 ≍ 𝑌 ′ . Because of 𝑦 ∈ 𝑌 ′         plies that lex-cel𝜏 satisfies 𝜎-C and 𝜎-sk-C if 𝜏 satisfies
both cases contradict the initial assumption, hence we must        𝜎-generalisation. Since both 𝜎-C and 𝜎-sk-C are satisfied
have that 𝐶𝐹𝐹 (𝑋) = ∅, i. e. the set 𝑋 is conflict-free.           the resulting argument ranking has a quite interesting pat-
   to (2): Then, there exists an argument 𝑎 ∈ 𝑈 𝐷𝐹 (𝑋)             tern. The argument ranking can be split into three groups,
which is not defended by 𝑋. Consider now the set 𝑌 ′ =             first the skeptically accepted arguments wrt. 𝜎 then the
{𝑦} for which we either have that 𝑈 𝐷𝐹 (𝑌 ′ ) = ∅ or               credulously accepted wrt. 𝜎 and finally the rejected argu-
𝑈 𝐷𝐹 (𝑌 ′ ) = {𝑦}. If 𝑈 𝐷𝐹 (𝑌 ′ ) = ∅, it follows directly         ments wrt. 𝜎. Inside all these groups the arguments can still
that 𝑌 ′ ⊐ 𝑋, contradicting our initial assumption. On the         be differentiated, so the resulting ranking is a generalisation
other hand, for 𝑈 𝐷𝐹 (𝑌 ′ ) = {𝑦} we distinguish between           of the acceptance problems for abstract argumentation.
two cases:                                                            The following result summarises the compliance of lex-
                                                                   cel𝜏 with the argument-ranking principles.
             (2.1) 𝑦 = 𝑥,           (2.2) 𝑦 ̸= 𝑥
                                                                   Theorem 8. lex-cel𝜏 satisfies the respective principles as
Clearly, if 𝑥 = 𝑦 we contradict our initial assumption be-         stated in Table 1 for 𝜏 ∈ {𝑟-𝑎𝑑, 𝑟-𝑐𝑜, 𝑟-𝑔𝑟, 𝑟-𝑝𝑟, 𝑟-𝑠𝑠𝑡}.
cause 𝑋 ≡ 𝑌 ′′ for 𝑌 ′′ = 𝑋. Consider now the case 𝑦 ̸= 𝑥.
                                                                     We want to discuss the following counterexample show-
That means, we have that 𝑈 𝐷𝐹 (𝑋) ≍ 𝑈 𝐷𝐹 (𝑌 ′ ) and
                                                                   ing that VP is violated by lex-cel𝜏 in particular.
thus 𝑋 ≍ 𝑌 ′ . Therefore, it follows that we must have
𝑈 𝐷𝐹 (𝑋) = ∅, i. e. 𝑋 defends all its elements.                    Example 7. We examine the following AF 𝐹 =
  That means 𝑋 is admissible and thus it follows directly          ({𝑎, 𝑏, 𝑐, 𝑑, }, {(𝑎, 𝑏), (𝑎, 𝑑), (𝑏, 𝑐), (𝑐, 𝑏)}). Consider, for in-
that 𝑥 ∈ 𝑐𝑟𝑒𝑑𝑎𝑑 (𝐹 ).                                              stance, the 𝑟-𝑎𝑑 extension-ranking for 𝐹 :
  From 𝑈 𝐷𝐹 (𝑋) = ∅ and 𝑋 ⊐𝑈       𝐹
                                     𝐷
                                       𝑌 for all 𝑌 it follows
that 𝑈 𝐷𝐹 (𝑌 ) ̸= ∅. Since ⊒ satisfies ad-generalisation it fol-                0 : {𝑎, 𝑐}, {𝑎}, {𝑐}, ∅             1 : {𝑐, 𝑑}
lows that 𝑌 ∈ / 𝑎𝑑(𝐹 ) for all 𝑌 and thus also 𝑦 ∈ 𝑟𝑒𝑗𝑎𝑑 (𝐹 ).
Consequently, it follows from 𝑎𝑑-𝐶 that 𝑥 ≻ 𝑦. Hence,              The lex-cel𝑟-𝑎𝑑 argument-ranking is then:
Dominating set is satisfied.                                                       lex-cel𝑟-𝑎𝑑      lex-cel𝑟-𝑎𝑑      lex-cel𝑟-𝑎𝑑
                                                                              𝑐 ≻𝐹               𝑎 ≻𝐹             𝑑 ≻𝐹             𝑏

5. Investigating Principles for                                    However, 𝑎 is unattacked, while 𝑐 is attacked and therefore VP
                                                                   is violated. Since all other sets that contain 𝑎 are not conflict-
   lex-cel𝜏                                                        free, that means that {𝑐, 𝑑} is always ranked better than these
In the previous section, we looked at social ranking solutions     sets. Therefore 𝑐 is ranked better than 𝑎 wrt. lex-cel𝜏 for all
from a general perspective and were able to characterise           other 𝜏 ∈ {𝑟-𝑐𝑜, 𝑟-𝑔𝑟, 𝑟-𝑝𝑟, 𝑟-𝑠𝑠𝑡}
𝜎-C and present sufficient conditions for 𝑆𝐶 and 𝜎-sk-C,              At first glance it might seem unintuitive that VP is vio-
however a number of principles are still to be investigated.       lated. Both arguments 𝑎 and 𝑐 are skeptically accepted wrt.
In this section, we take a closer look at lex-cel𝜏 and analyse     complete semantics, so there is no reason to reject either ar-
which principles it satisfies.                                     gument. However, argument 𝑎 is involved in more conflicts
   As the results from the previous section suggest we should      than 𝑐 and thus 𝑐 is compatible with more arguments than
start with checking if lex-cel satisfies Pareto-efficiency.        𝑎. Therefore we reason that 𝑐 should be ranked better than
Theorem 7. lex-cel satisfies Pareto-efficiency.                    𝑎. In general, if we think back to the motivation of social
                                                                   ranking functions then employees who can work together
              Abs    In    VP     SC     CP    QP     CT     SCT     DP         DDP     NaE     AvsFD     w𝜎 -S     s𝜎 -S    𝜎 -C   𝜎 -sk-C
  lex-cel𝜏    ✓      ✓     X      ✓      X      X      X      X       X          X       X        ✓            X        ✓    ✓        ✓
    Cat       ✓      ✓     ✓      X      X      X      ✓      ✓       ✓          X       ✓        X            X        X    X        X
     ser      ✓      ✓     X      X      X      X      X      X       X          X       ✓        ✓            ad       ad   ad       X

     Table 1
     Principles satisfied by lex-cel𝜏 for 𝜏 ∈ {𝑟-𝑎𝑑, 𝑟-𝑐𝑜, 𝑟-𝑔𝑟, 𝑟-𝑝𝑟, 𝑟-𝑠𝑠𝑡} and other ranking semantics from the literature. Existing
     results for Cat and ser are taken from Bonzon et al. [4] and Blümel and Thimm [16].



with more employees are considered better, so this ranking
of 𝑎 and 𝑐 is in line with the idea behind social ranking                                         𝑎        𝑏        𝑐
functions.
   The remaining proofs and counterexamples can be found
in the supplementary material2 .                                          Figure 3: The AF 𝐹2 from Example 8.


6. Related Work
A number of social ranking functions are discussed in the                                                 𝑎         𝑏
literature. In the following, let 𝐴 be an arbitrary set of
objects and ⊒ is a preorder on the powerset 𝒫(𝐴).
   A prominent social ranking function is the Ceteris Paribus             Figure 4: The AF 𝐹3 from Example 9.
Majority Solution (CP), which was defined by Haret et al. [8]
as follows.
                                                                                                  𝑎
Definition 23. For the preorder ⊒ and for any 𝑥, 𝑦 ∈ 𝐴, we                                                     𝑐         𝑑
have that 𝑥 ⪰𝐶𝑃⊒ 𝑦 if and only if
                                                                                                  𝑏

  |{𝑆 ∈ 𝒫(𝐴 ∖ {𝑥, 𝑦})|𝑆 ∪ {𝑥} ⊐ 𝑆 ∪ {𝑦}}| ≥
             |{𝑆 ∈ 𝒫(𝐴 ∖ {𝑥, 𝑦})|𝑆 ∪ {𝑦} ⊐ 𝑆 ∪ {𝑥}}|                      Figure 5: The AF 𝐹4 from Example 10.

   Another relevant social ranking function is the Ordinal
Banzhaf Index Solution (BI) of Khani et al. [7]. For that, we
denote with 𝑈𝑖 = {𝑆 ∈ 𝒫 | 𝑖 ∈     / 𝑆} the set of subsets that            Example 8. The argument ranking ⪰CP𝜏 violates SC for
do not contain 𝑖 and with 𝑈𝑖𝑗 = {𝑆 ∈ 𝒫 | 𝑖, 𝑗 ∈      / 𝑆} the             𝜏 ∈ {𝑟-𝑎𝑑, 𝑟-𝑐𝑜, 𝑟-𝑔𝑟, 𝑟-𝑝𝑟, 𝑟-𝑠𝑠𝑡}. Consider the AF 𝐹2 in
set of subsets that contain neither 𝑖 nor 𝑗.                              Figure 3. Then we have that 𝑐 ⪰CP 𝜏
                                                                                                          𝐹2 𝑎, which contradicts SC.
   First, we define the notion of ordinal marginal contribution
as follows.                                                               Example 9. The argument ranking ⪰BI𝜏 violates SC for
                                                                          𝜏 ∈ {𝑟-𝑎𝑑, 𝑟-𝑐𝑜, 𝑟-𝑔𝑟, 𝑟-𝑝𝑟, 𝑟-𝑠𝑠𝑡}. Consider the AF 𝐹3 in
Definition 24. Let ⊒ be a preorder on 𝒫(𝐴). The ordinal                   Figure 4. Then we have that 𝑎 ⪰BI𝜏
                                                                                                          𝐹3 𝑏, which contradicts SC.
marginal contribution 𝑚𝑆   𝑖 (⊒) of element 𝑖 wrt. the set 𝑆
with 𝑖 ∈
       / 𝑆, for the preorder ⊒ is defined as:                                So, self-contradicting arguments are not necessarily the
                                                                          worst ranked arguments. Thus, these two social ranking
                                                                          functions are not suitable to rank arguments in the context
                      ⎧
                      ⎨ 1 if 𝑆 ∪ {𝑖} ⊐ 𝑆,
             𝑆
          𝑚𝑖 (⊒) =        −1 if 𝑆 ⊐ 𝑆 ∪ {𝑖},              (1)             of abstract argumentation and therefore we do not discuss
                      ⎩
                             0 otherwise.                                 them further.
                                                                             A number of other argument-ranking semantics were in-
  We denote with 𝑢+,⊒     (𝑢−,⊒  ) the set of subsets 𝑆 ∈ 𝑈𝑖              troduced in the literature (for an overview see Bonzon et al.
                     𝑖       𝑖
such that 𝑚𝑖 (⊒) = 1 (𝑚𝑆
             𝑆
                            𝑖 (⊒) = −1) respectively. Fur-
                                                                          [4]). However, the only known argument-ranking seman-
thermore, we refer to the difference 𝑠⊒        +,⊒
                                                   − 𝑢−,⊒ as              tics satisfying 𝑎𝑑-Compatibility is the serialisability-based
                                       𝑖 = 𝑢𝑖          𝑖
the ordinal Banzhaf score of 𝑖 wrt. ⊒.                                    argument-ranking semantics (ser) by Blümel and Thimm [16].
  Finally, we define the social ranking solution based on                 The serialisability-based argument ranking semantics ranks
the ordinal Banzhaf score as follows.                                     arguments according to the number of conflicts that need
                                                                          to be resolved to include these arguments in an admissible
Definition 25. For the preorder ⊒ and for any 𝑥, 𝑦 ∈ 𝐴, we                set. However, this semantics violates 𝑐𝑜-sk-C.
define that 𝑥 ⪰𝐵𝐼⊒ 𝑦 if and only if
                                                                          Example 10. Let 𝐹4 be the AF as depicted in Figure 5. Then
                           𝑠⊒    ⊒
                            𝑖 ≥ 𝑠𝑗
                                                                          argument 𝑑 ∈ 𝑠𝑘𝑐𝑜 (𝐹4 ). So, according to 𝑐𝑜-sk-C it should
                                                                          hold that 𝑑 ≻𝐹4 𝑎, however this is not the case for 𝑠𝑒𝑟, i. e.
   However, the corresponding Social ranking argument-                    𝑎 ≻𝑠𝑒𝑟
                                                                              𝐹4 𝑑. Thus 𝑐𝑜-sk-C is violated.
ranking semantics 𝐵𝐼𝜏 and 𝐶𝑃𝜏 do not generalise credu-
lous acceptance, because these two argument-ranking se-                     Similarly, we have that the categorizer ranking semantics
mantics do not satisfy the principle SC, as shown by the                  (Cat) violates 𝜎-sk-C.
following examples.

                                                                          2
                                                                              https://fernuni-hagen.sciebo.de/s/eTCZyHVIOzRtIsE
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