Implementing Ranking-Based Semantics in ConArg: a Preliminary Report

Carlo Taticchi; Francesco Faloci; Stafano Bistarelli

REVIEW 5 major objections 4 minor 28 references

Implementing Ranking-Based Semantics in ConArg: a Preliminary Report

T0 review · 5 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper defines a ranking-based semantics that turns Dung labellings into cooperative games and ranks arguments by one of four power indexes, proving the property profile for the Shapley case.

desk verdict A useful but overreaching preliminary report: the Shapley-based ranking is a good idea, but the Deegan-Packel and Johnston formulas are wrong as printed and the acceptance theorems lack valid proofs. read the letter →

arxiv 1908.07784 v2 pith:BKO6A5L2 submitted 2019-08-21 cs.AI

Stafano Bistarelli , Francesco Faloci , Carlo Taticchi This is my paper · ORCID

classification cs.AI

keywords ranking-basedsemanticsabstractargumentationpowerindexesShapleyvalueBanzhafindexDeegan-PackelJohnstonDung

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Ranking-based semantics in abstract argumentation usually sort arguments by graph-theoretic criteria with no tie to classical Dung acceptability. This paper claims that the classical semantics themselves can drive the ranking: for any chosen Dung semantics $\sigma$ and any power index $\pi$, it builds a two-sided cooperative game over the arguments, ranking each argument by $\pi$'s value on the 'in' extensions and breaking ties by the value on the 'out' extensions. The resulting PI-based semantics is total for the Shapley value and satisfies Abstraction, Independence and Totality for five standard Dung semantics; it satisfies Self-contradiction only for conflict-free and Non-attacked Equivalence only for complete, preferred, and stable semantics, and never satisfies Cardinality Precedence or Quality Precedence. The paper also reports an implementation of all four indexes in the ConArg web tool and shows the rankings on a worked example. If correct, this gives a parameterised family of argument rankings that inherit meaning from classical acceptance criteria, useful when a large framework must be reduced to a shortlist.

What carries the argument

The engine is the pair of characteristic functions $v^I_{\sigma,F}$ and $v^O_{\sigma,F}$ (Definition 9) together with the lexicographic comparison in Definition 10. Instead of introducing a new graph heuristic, the semantics turns the set of extensions of a chosen Dung semantics into a simple voting game, then asks how often and how decisively each argument belongs to winning coalitions. The Shapley value weights marginal contributions by the number of orderings in which a player joins a coalition; Banzhaf counts critical voters without ordering; Deegan-Packel averages over minimal winning coalitions; Johnston gives critical voters a fractional $1/\kappa(S)$ share. The implementation computes $\pi$ only for coalitions $S$ where either $S$ or $S\cup\{i\}$ is an extension, which is exactly where the marginal contribution can be non-zero.

What would settle it

For the AF in Figure 10, compute the Deegan-Packel and Johnston values from the standard definitions (minimal winning coalitions for $\rho$; sum of $1/\kappa(S)$ over winning coalitions for each critical voter for $\gamma$) using the listed extensions, and compare with Tables III and IV: any mismatch shows the implemented indexes are not those named. For Theorem 1, a finite AF giving a violation of any listed property row would refute that row.

Watch

Extended reading notes

Core claim

The central claim is that a ranking-based semantics can be defined by taking the in/out labellings of a Dung semantics $\sigma$ as the winning coalitions of a simple game over the arguments: $v^I_{\sigma,F}(S)=1$ exactly when $S$ is an in-set of some $\sigma$-labelling, and $v^O_{\sigma,F}(S)=1$ exactly when $S$ is an out-set of some $\sigma$-labelling. A power index $\pi$ (Shapley, Banzhaf, Deegan-Packel, or Johnston) is then evaluated on these two characteristic functions, and two arguments are compared lexicographically: higher $\pi(v^I_{\sigma,F})$ is better, and in case of a tie, lower $\pi(v^O_{\sigma,F})$ is better. The paper proves, for the Shapley value, that Abstraction, Independence and Totality hold for conflict-free, admissible, complete, preferred and stable semantics; Self-contradiction holds only for conflict-free; Non-attacked Equivalence holds only for complete, preferred and stable; and neither Cardinality Precedence nor Quality Precedence ever holds. It further claims that sceptically accepted arguments receive positive power-index value, credulously rejected arguments receive negative value, and consequently the semantics satisfies $\delta$-Sceptical Precedence and $\delta$-Credulous Precedence for every $\delta$ among the five Dung semantics.

Load-bearing premise

The load-bearing premise is that the Deegan-Packel and Johnston power indexes are faithfully transcribed and implemented (Equations 3 and 4), yet those formulas as written are not well-formed: Equation 3 sums over a set-of-coalitions expression that does not parse, and Equation 4 divides by the critical-voter count of coalitions that do not contain the player, so a mistranscription would make the reported $\rho$ and $\gamma$ rankings something other than the named indexes.

Editorial extensions

If this is right

For any framework and any of the five Dung semantics, the Shapley-based semantics returns a total ordering of arguments, so no argument is left incomparable.
Under complete, preferred and stable semantics, all non-attacked arguments receive the same rank; under conflict-free semantics, self-attacking arguments are strictly worse than non-self-attacking ones.
More numerous or higher-ranked direct attackers do not by themselves make an argument worse: Cardinality Precedence and Quality Precedence fail because an argument's value includes how many other acceptable arguments it defends.
Sceptically accepted arguments are ranked above merely credulous or rejected arguments, and credulously accepted arguments above rejected ones, for all four power indexes and all five Dung semantics.
Unlike the Cat, Dbs and Bds ranking semantics, the PI-based semantics satisfies Credulous Precedence for admissible, complete, preferred and stable semantics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because $v^I$ and $v^O$ are defined only on exact extension sets, an argument's power-index value is determined entirely by the collection of extensions, not by graph paths; a natural testable extension is to weight coalitions by defence chains or attack distances and see whether Cardinality or Quality Precedence begin to hold.
Only the Shapley case receives a theorem in this paper; the Banzhaf, Deegan-Packel and Johnston rankings are demonstrated through tables, so a full property profile for those three indexes remains open rather than established.
The 'out' tie-breaker rewards arguments that are defeated in fewer ways, not just accepted more often, so the semantics can distinguish arguments that look identical when only in-extensions are compared; this could be exploited in frameworks with multiple preferred extensions.
The implementation's pruning of coalitions to those that are extensions may avoid the generic NP-hardness of power-index computation, and combining the Shapley value with restricted-coalition game values would let the ranking ignore arguments that are not even credulously accepted.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two quick things. The central idea—rank arguments by Shapley/Banzhaf/Deegan-Packel/Johnston power indexes computed over the extensions of a chosen Dung semantics—is a natural and genuinely useful transfer. The paper is the first place I've seen it worked out beyond the authors' own Shapley-only CEUR sketches. The ConArg tool with the greyscale output is a nice practical addition, and the property analysis in Theorem 1, informal as it is, starts to map the space properly.

The problem is that the paper overreaches as written. Equations 3 and 4 are not the Deegan-Packel and Johnston indexes. Equation 3 sums over subsets of a family of coalitions and uses marginal contributions where the standard DP index uses none; Equation 4 divides by κ(S) for coalitions that don't contain i, which inverts the standard construction. The implementation note in Section IV-A says the tool computes 'π only with respect to those sets S such that either S or S∪{i} is an extension', which is the right shortcut for Shapley and Banzhaf but not for DP and Johnston. The negative 'Johnston' values in Table IV confirm something is off: the standard Johnston index is nonnegative. So the rankings in Tables III and IV for ρ and γ are not the rankings promised by the names. That undercuts the parametric-family claim, though the Shapley-based results stand.

The other soft spot is the acceptance theorems. Proposition 1's proof shows that a credulously accepted argument has at least one positive marginal contribution; it does not show its total index is higher than that of a rejected argument. The proof of Theorem 2 is just declared 'straightforward'. These claims may be true, but they are unproven in this text. Theorem 1's property proofs lean on figures; for a preliminary report that is tolerable, but a full paper would need real case analysis or machine-checked proofs.

The citation pattern is fine—the authors build on their own prior work and credit the relevant game-theory and argumentation literature. Nothing circular.

Bottom line: this is a preliminary report with a good idea and a flawed execution. It deserves a serious referee, and I would send it to review, but with a request for major revision: fix the DP and Johnston formulas and re-run the tables, and prove or refute Theorems 2 and Proposition 1. If the authors do that, this becomes a solid contribution to ranking-based semantics. I wouldn't cite it as it stands; I'd point students to it as an interesting problem statement.

Referee Report

5 major / 4 minor

Summary. The paper proposes a ranking-based semantics for abstract argumentation, called PI-based semantics, in which each argument is ranked by the lexicographic comparison of two power-index values computed from characteristic functions vI and vO that mark, respectively, the in-labellings and out-labellings of a selected Dung semantics. The authors claim to instantiate this parametric semantics with four cooperative-game power indexes (Shapley, Banzhaf, Deegan-Packel, Johnston), state an axiom profile for the Shapley instance (Theorem 1), state theorems connecting power-index sign with credulous/sceptical acceptance (Theorem 2 and Proposition 1), describe an implementation inside the ConArg web tool, and illustrate the rankings on a worked example.

Significance. The underlying idea is appealing and potentially useful: it connects ranking-based argumentation with well-studied cooperative-game power indexes and makes classical Dung semantics a parameter of the ranking. The Shapley and Banzhaf columns in Tables I and II are plausible and suggest that the tool works for those indexes. If the formal claims were properly proved and all four indexes were implemented faithfully, the paper would be a useful bridge between computational social choice and abstract argumentation. However, the manuscript as submitted does not support the claims for Deegan-Packel and Johnston: the defining equations are not mathematically well-formed, the reported numerical values are inconsistent with the named indexes, and Theorem 2 is contradicted by the paper's own Table III. No machine-checked proofs, reproducible code, or independent verification is provided for the property theorems.

major comments (5)

[Section II-B, Eq. (3)] The Deegan-Packel formula as printed is not well-formed: the summation ranges over S⊆M_i(v)\setminus{i}, but M_i(v) is a family of coalitions, so the subtraction of the player i from this family is undefined and S would range over subsets of a set of coalitions rather than over coalitions. The standard Deegan-Packel index is ρ_i(v) = (1/|M(v)|) Σ_{S∈M_i(v)} 1/|S|, with no marginal term v_{Si}. The values reported in Table III (for example, the ρ-COM row 0.5, 0, 0.5, 0, 0) are consistent with the standard formula using M(v) = {{a,c}} but do not follow from Eq. (3) as written. The paper must correct Eq. (3), state which definition is actually implemented, and reconcile the implementation note in Section IV-A, where the computation is described in terms of marginal contributions v(S∪{i})−v(S), a construction appropriate for Shapley and Banzhaf but not for Deegan-Packel.
[Section II-B, Eq. (4) and Table IV] The Johnston index is mis-defined. The standard Johnston index credits player i only in winning coalitions that contain i, dividing by the number of critical voters in that coalition, so it is nonnegative. Equation (4), by contrast, sums marginal contributions over S⊆N\setminus{i} and divides by κ(S), where κ(S) is the number of critical voters in a coalition that does not contain i. The implementation note in Section IV-A confirms that the tool computes v(S∪{i})−v(S) for all four indexes, and Table IV reports impossible negative Johnston values, e.g., γ-CF v_I(b) = −3.16667 and γ-ADM v_I(b) = −6.16667. Therefore the γ rankings and all γ-related property claims in Sections III and V are not about the Johnston index as conventionally defined. Both the equation and the implementation must be corrected before the parametric-family claim can be assessed.
[Section III, Theorem 2] Theorem 2 is false for the Deegan-Packel index under the standard (nonnegative) definition. If an argument is never accepted, it belongs to no minimal winning coalition, so its Deegan-Packel value is 0, not strictly negative as the theorem claims. This is visible in the paper's own Table III: the ρ-COM row gives b the value 0.00000 even though b is never in a complete extension. The statement 'if a is credulously rejected then π_a(v_I) < 0' can hold at best for the Shapley and Banzhaf instances, and the theorem should be re-stated index by index after the Deegan-Packel and Johnston definitions are fixed.
[Section III, Proposition 1 and its proof] The proof of Proposition 1 is not a valid derivation. From the fact that a credulously accepted argument i has at least one positive marginal contribution, it does not follow that the aggregated index value of i is higher than that of a rejected argument j; the aggregate is a weighted sum over all coalitions and may include many negative terms. A correct proof must compare the full weighted sums, or restrict to indexes for which the comparison can be established. The same gap affects the 'straightforward' proof of Theorem 2. Given that these results are presented as theorems and propositions, the authors need to supply complete proofs from the actual definitions.
[Section III, Theorem 1] The proof of Theorem 1 is informal and figure-based, and it is not possible to check the claimed property profile from the text. For example, the Self-contradiction case asserts an inequality involving E and v_I without deriving the Shapley comparison, and the other cases are delegated to Figures 2–6 with no numerical evaluations. Since Theorem 1 is the main theoretical contribution, the authors should provide either formal derivations or a machine-checkable verification script that reproduces each claim. If the paper is intended strictly as a tool report, the property statements should be presented as experimental findings rather than as proven theorems.

minor comments (4)

[Abstract and Introduction] The phrase 'well know properties' should be 'well-known properties'.
[Definition 4] The definition of a strict ranking contains a typo: 'a ≻F b is a shortcut for a ≽F b and b⁄≽F b' should read 'b⁄≽F a'.
[Table III] The entries '0, 16667' use a decimal comma instead of a decimal point, which is inconsistent with the other tables and should be corrected.
[Section II-B] The citation for the Johnston index is imprecise: the text refers to Johnston via reference [21], a general encyclopedia, while the formal definition is attributed to the Durán et al. paper [18]; the citation should be aligned with the actual source of the formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the PI-based semantics is explicitly defined in Definitions 9 and 10, and the claimed properties are derived from that definition and standard Dung-semantics facts; no fitted parameter is renamed as a prediction.

full rationale

The paper's load-bearing content is a definition of a ranking-based semantics in terms of power indexes over characteristic functions built from Dung extensions, followed by property statements proven from that definition. The core ranking is not fitted to the reported rankings or property outcomes; Tables I–IV are illustrative outputs of the implemented formulas, not predictions used to define the semantics. The tie-breaking use of vO is an explicit definitional choice, not an input disguised as a result. The self-citations to [6] and [7] are contextual (preliminary ideas and tool implementation) and do not carry the proof burden: Definition 10 is given in this paper and the proofs are self-contained, albeit concise. The potentially malformed Deegan-Packel and Johnston equations (Eqs. 3 and 4) and the negative Johnston values in Table IV are correctness or implementation-fidelity concerns about whether the tool computes the standard indexes; they are not circularity, because no claim reduces to its own input by construction. The abbreviated proof of Theorem 2 is an expositional omission rather than a circular step. Overall, no load-bearing derivation collapses into its inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper introduces a new semantic construct (PI-based semantics) and two characteristic functions, so the ledger records definitional assumptions rather than fitted parameters. There are no numerically fitted constants; the Dung semantics σ and power index π are user-selected parameters of the framework. The main assumptions are that extension membership can be treated as the winning condition of a simple game, that standard power-index formulas transfer without modification, and that the vO tie-breaker preserves the intended ranking. The Deegan-Packel treatment also assumes ∅∈M(v), which is an ad hoc convention.

assumptions (4)

domain assumption The in-sets of a Dung semantics σ define the winning coalitions of a simple game
Load-bearing encoding in Definition 9; no independent evidence is given that extension membership behaves like coalition value in a cooperative game.
domain assumption Standard power-index formulas (Shapley, Banzhaf, Deegan-Packel, Johnston) apply unchanged to the extension game
Section II-B transfers these indices from computational social choice to argumentation without proving that the game-theoretic framework remains valid.
ad hoc to paper Lexicographic tie-breaking through vO gives a meaningful refinement of the ranking
Definition 10 and Section IV-A introduce the vO tie-breaker to distinguish arguments with equal vI values; this is a design choice with no prior literature support.
ad hoc to paper The empty set is treated as a (minimal) winning coalition for Deegan-Packel
Section II-B states 'always assuming ∅∈M(v)'; this convention changes the index and is not standard in cooperative game theory.

invented entities (2)

PI-based ranking semantics
purpose: Defines a total pre-order over arguments of an AF by lexicographic order on power-index values over extension games
Introduced in Definitions 9 and 10; its only support is the internal property analysis, with no external data or falsifiable prediction.
Characteristic functions vI and vO
purpose: Map each subset of arguments to 1 if it is an in-set (or out-set) of a labelling of semantics σ, used to build the simple game
Introduced ad hoc in Definition 9; vO is specifically added to break ties and has no independent justification outside this paper.

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Pith. "Pith review of Implementing Ranking-Based Semantics in ConArg: a Preliminary Report." pith.science (2026). https://pith.science/paper/BKO6A5L2

@misc{pith2026190807784,
  author       = {Pith},
  title        = {Pith review of: Implementing Ranking-Based Semantics in ConArg: a Preliminary Report},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BKO6A5L2}},
  note         = {Machine review of arXiv:1908.07784}
}

read the original abstract

ConArg is a suite of tools that offers a wide series of applications for dealing with argumentation problems. In this work, we present the advances we made in implementing a ranking-based semantics, based on computational choice power indexes, within ConArg. Such kind of semantics represents a method for sorting the arguments of an abstract argumentation framework, according to some preference relation. The ranking-based semantics we implement relies on Shapley, Banzhaf, Deegan-Packel and Johnston power index, transferring well know properties from computational social choice to argumentation framework ranking-based semantics.

Figures

Figures reproduced from arXiv: 1908.07784 by the authors.

**Figure 1.** Counterexample for Self-contradiction: we have c φ F b when σ ∈ {admissible, complete, preferred}, and b ' φ F c when σ = stable. 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 Self-contradiction property en… view at source ↗

**Figure 6.** Counterexample for property Non-attacked Equivalence of PI-based semantics. For σ = conflict-free, d φ F a, and when σ = admissible, a φ F d. based 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 com… view at source ↗

**Figure 3.** A counterexample for Quality Precedence of PI-based semantics: when σ = conflict-free, a φ F c and b φ F d. If σ = admissible, we have instead c φ F a and d φ F b. lower ranking than the others. Indeed, the conflict-free 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 low… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: A counterexample for property φφ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: A counterexample for property Quality Precedence of PI-based semantics. When σ ∈ {preferred, stable}, a φ F c and b ' φ F e holds [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 8.** Figure 8: Example of two AFs with different rebuttal/restatement configurations. [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 9.** Figure 9: A screenshot of the ConArg web interface. The highlighted elements are: 1. Options menu, 2. Semantics selection panel, 3. Canvas where the AF is [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

**Figure 10.** Figure 10: Example of an AF. The sets of extensions for the conflict-free, [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

J. M. Alonso-Meijide, M. ´Alvarez-Mozos, and M. G. Fiestras-Janeiro. Power Indices and Minimal Winning Coalitions in Simple Games with Externalities. UB Economics Working Papers E15/328, University of Barcelona - Department of Actuarial, Financial and Economic Mathe- matics, 2015

work page 2015

[2] [2]

Amgoud and J

L. Amgoud and J. Ben-Naim. Ranking-Based Semantics for Argumen- tation Frameworks. In Proc. SUM 2013 - 7th International Conference , volume 8078 of LNCS, pages 134–147. Springer, 2013

work page 2013

[3] [3]

J. F. Banzhaf. Weighted V oting Doesn’t Work: A Mathematical Analysis. Rutgers Law Rev , 19(2):317–343, 1965

work page 1965

[4] [4]

Baroni, M

P. Baroni, M. Caminada, and M. Giacomin. An introduction to argumentation semantics. Knowl Eng Rev , 26(4):365–410, 2011

work page 2011

[5] [5]

Besnard and A

P. Besnard and A. Hunter. A logic-based theory of deductive arguments. Artif Intell, 128(1-2):203–235, 2001

work page 2001

[6] [6]

Bistarelli, F

S. Bistarelli, F. Faloci, F. Santini, and C. Taticchi. A Tool For Ranking Arguments Through V oting-Games Power Indexes. In Proceedings 34th CILC, volume 2396 of CEUR Workshop Proceedings , pages 193–201. CEUR-WS.org, 2019

work page 2019

[7] [7]

Bistarelli, P

S. Bistarelli, P. Giuliodori, F. Santini, and C. Taticchi. A Cooperative- game 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. CEUR-WS.org, 2018

work page 2018

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