Contribution Functions for Quantitative Bipolar Argumentation Graphs: A Principle-based Analysis

TL;DR

This paper develops a principled framework to analyze how individual arguments contribute to the final strength of a target argument in acyclic Quantitative Bipolar Argumentation Graphs (QBAGs), by formalizing four contribution functions and four guiding principles. It shows that no single function satisfies all principles across all gradual semantics, and highlights the strengths and weaknesses of removal-based, restricted removal, Shapley-value, and gradient-based approaches for explainability. The Shapley-value-based contributions achieve broad quantitative contribution existence across studied semantics, while gradient-based contributions provide local faithfulness under differentiable semantics, and removal-based methods support counterfactual reasoning in many cases. The framework is illustrated with a case study on movie ratings, demonstrating practical interpretability and guidance for selecting suitable contribution functions in real-world applications, and sets the stage for extensions to cyclic QBAGs and further theoretical refinement.

Abstract

We present a principle-based analysis of contribution functions for quantitative bipolar argumentation graphs that quantify the contribution of one argument to another. The introduced principles formalise the intuitions underlying different contribution functions as well as expectations one would have regarding the behaviour of contribution functions in general. As none of the covered contribution functions satisfies all principles, our analysis can serve as a tool that enables the selection of the most suitable function based on the requirements of a given use case.

Figures (41)

• Figure 1: QBAG $\graph$ (Example \ref{['ex:intro-modular']}). A node labelled $\argnode{\argx}{i}{f}$ represents argument $\argx$ with initial strength $\is(\argx) = i$ and final strength $\fs(\argx) = \mathbf{f}$. Edges labelled $+$ and $-$, respectively, represent supports and attacks.
• Figure 2: A QBAG $\graph$ and its updates, with $\argb$ ($\graph'$) and incoming relationships to $\argb$ removed ($\graph"$).
• Figure 3: While our contribution functions can be intuitively expected to satisfy directionality with respect to modular semantics, this may not be the case with respect to other semantics: consider $\ctrbrempty$ and $\fs(\argx) = \is(\argx) - \sum_{\{\argy | \argx \in \Att(\argy) \}} \tau(\argy)$.
• Figure 4: $\ctrbrempty$ and $\ctrbriempty$ violate the contribution existence and quantitative contribution existence principles w.r.t. DFQuAD, SD-DFQuAD, and EBT semantics.
• Figure 5: The contributions of $\ctrbrempty$ and $\ctrbgempty$ converge to zero faster than the contributions of $\ctrbsempty$. Here, the contribution of $\argb_i$, $1 \leq i \leq n$ to $\arga$ depends on the number of $\arga$'s supporters and we apply QE semantics.

Theorems & Definitions (94)

• definition 1: Quantitative Bipolar Argumentation Graph (QBAG) Potyka:2019Baroni:Rago:Toni:2019
• definition 2: Gradual Semantics and Strength Functions Baroni:Rago:Toni:2019Potyka:2019
• proposition 1
• proof
• definition 3: QBAG Initial Strength Modification
• proposition 2
• proof
• proposition 3
• proof
• proposition 4