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Shapley Revisited: Tractable Responsibility Measures for Query Answers

Meghyn Bienvenu, Diego Figueira, Pierre Lafourcade

TL;DR

The paper challenges the dominance of the drastic Shapley value for non-numeric, monotone queries by introducing WSMS, a family of responsibility measures based on weighted sums of minimal supports. It shows WSMS are equivalent to Shapley values for suitably defined wealth functions, and proves tractability of WSMS computation for UCQs and many bounded query classes, while identifying hardness for certain CQ subclasses. The authors provide a comprehensive complexity landscape, including data- and combined-complexity results, reductions to counting homomorphisms, and tractable cases under bounded generalized hypertree width and self-join width. They also connect WSMS to related measures in inconsistency, homomorphism counting, and SHAP explainability, and discuss extensions to explainable AI and ontology-mediated query answering. Overall, WSMS offer a principled, computable, and interpretable alternative to drastic Shapley for explaining query answers, with broad implications for semantic explanations and data-provenance tools.

Abstract

The Shapley value, originating from cooperative game theory, has been employed to define responsibility measures that quantify the contributions of database facts to obtaining a given query answer. For non-numeric queries, this is done by considering a cooperative game whose players are the facts and whose wealth function assigns 1 or 0 to each subset of the database, depending on whether the query answer holds in the given subset. While conceptually simple, this approach suffers from a notable drawback: the problem of computing such Shapley values is #P-hard in data complexity, even for simple conjunctive queries. This motivates us to revisit the question of what constitutes a reasonable responsibility measure and to introduce a new family of responsibility measures -- weighted sums of minimal supports (WSMS) -- which satisfy intuitive properties. Interestingly, while the definition of WSMSs is simple and bears no obvious resemblance to the Shapley value formula, we prove that every WSMS measure can be equivalently seen as the Shapley value of a suitably defined cooperative game. Moreover, WSMS measures enjoy tractable data complexity for a large class of queries, including all unions of conjunctive queries. We further explore the combined complexity of WSMS computation and establish (in)tractability results for various subclasses of conjunctive queries.

Shapley Revisited: Tractable Responsibility Measures for Query Answers

TL;DR

The paper challenges the dominance of the drastic Shapley value for non-numeric, monotone queries by introducing WSMS, a family of responsibility measures based on weighted sums of minimal supports. It shows WSMS are equivalent to Shapley values for suitably defined wealth functions, and proves tractability of WSMS computation for UCQs and many bounded query classes, while identifying hardness for certain CQ subclasses. The authors provide a comprehensive complexity landscape, including data- and combined-complexity results, reductions to counting homomorphisms, and tractable cases under bounded generalized hypertree width and self-join width. They also connect WSMS to related measures in inconsistency, homomorphism counting, and SHAP explainability, and discuss extensions to explainable AI and ontology-mediated query answering. Overall, WSMS offer a principled, computable, and interpretable alternative to drastic Shapley for explaining query answers, with broad implications for semantic explanations and data-provenance tools.

Abstract

The Shapley value, originating from cooperative game theory, has been employed to define responsibility measures that quantify the contributions of database facts to obtaining a given query answer. For non-numeric queries, this is done by considering a cooperative game whose players are the facts and whose wealth function assigns 1 or 0 to each subset of the database, depending on whether the query answer holds in the given subset. While conceptually simple, this approach suffers from a notable drawback: the problem of computing such Shapley values is #P-hard in data complexity, even for simple conjunctive queries. This motivates us to revisit the question of what constitutes a reasonable responsibility measure and to introduce a new family of responsibility measures -- weighted sums of minimal supports (WSMS) -- which satisfy intuitive properties. Interestingly, while the definition of WSMSs is simple and bears no obvious resemblance to the Shapley value formula, we prove that every WSMS measure can be equivalently seen as the Shapley value of a suitably defined cooperative game. Moreover, WSMS measures enjoy tractable data complexity for a large class of queries, including all unions of conjunctive queries. We further explore the combined complexity of WSMS computation and establish (in)tractability results for various subclasses of conjunctive queries.

Paper Structure

This paper contains 25 sections, 36 theorems, 42 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Differences with Respect to Conference Paper
  4. Preliminaries
  5. Responsibility Measures and the Drastic Shapley Value
  6. Responsibility Measures
  7. Shapley Value and Shapley-Based Responsibility Measures
  8. Shapley Axioms
  9. Desirable Properties
  10. Discussion on Desired Properties of Responsibility Measures
  11. Formally Defining Desired Properties
  12. Alternative Responsibility Measures
  13. Weighted Sums of Minimal Supports
  14. WSMSs are Shapley-like Scores
  15. Other Notable WSMSs
  16. ...and 10 more sections

Key Result

theorem 1

The "Shapley value" is the unique function that satisfies Sh:1, Sh:2, Sh:4 and Sh:3.

Figures (8)

  • Figure 1: Pathological examples for some "responsibility measures". The databases are represented as graphs with nodes being constants and edges being $R$-tuples.
  • Figure 2: Instance for \ref{['ex:aotbe']}. $\Dn\vcentcolon= \{n_1,\dots, n_{\mathrm{x}}\}$, $\Dx\vcentcolon=\{x_1,x_2\}$, and the query $q$ is such that the "minimal supports" are the colored regions (the red ones contain $e_0$, the blue ones $e_5$ and the green ones neither).
  • Figure 3: A visual representation of the setting considered by \ref{['MS:t']}. The $A_i,B_j$ are the "minimal supports" of $q$.
  • Figure 4: Example of intersecting minimal supports. The database $\D$ is the set $\{e_0\dots e_9,e_\mathrm{x}\}$, and the query $q$ is such that the "minimal supports" are the colored regions (the red ones contain $e_0$ and the blue ones contain $e_5$).
  • Figure 5: Illustration of the graph database $\mathcal{D}_G$. Each arrow represents a path with the given label.
  • ...and 3 more figures

Theorems & Definitions (79)

  • remark 1
  • theorem 1
  • proof
  • lemma 1: shapley1953value
  • remark 2
  • remark 3
  • proof
  • proposition 1
  • proof
  • proposition 2
  • ...and 69 more