Overview
This research presents a systematic guide to power indices, which originated in cooperative game theory, for their application in Explainable AI (XAI). The guide organizes these indices along three attribution dimensions: single-player, set-based, and cardinality-based. For each dimension, the work reviews corresponding power indices, generalizes existing ones where applicable, and analyzes the formal principles they satisfy. It also includes proofs for properties previously unaddressed in the literature and illustrates how the choice of dimension and index influences attribution outcomes through concrete examples. The research aims to provide practitioners with guidance for selecting appropriate power indices in specific application contexts.
Research Context
Power indices quantify the influence of individual players on the outcome of a given game. Historically, these indices were developed to facilitate the distribution of profits or costs among players and to assess the fairness of voting systems. More recently, power indices have been adopted as methods for attributing the outputs of AI-based systems to their inputs, thereby contributing to explainability in AI. Despite this emerging application, the problem of selecting an appropriate power index for a specific explanation task has remained understudied.
Approach
The research approach involved organizing power indices based on three distinct attribution dimensions:
- Single-player dimension: This dimension focuses on attributing influence to individual players.
- Set-based dimension: This dimension considers the influence of sets of players.
- Cardinality-based dimension: This dimension involves attribution where player-identity information is removed, while certain index-level distinctions are preserved.
For each dimension, the researchers undertook several steps:
- Reviewing existing power indices relevant to that dimension.
- Generalizing existing indices where the context allowed for broader applicability.
- Analyzing the formal principles satisfied by the reviewed and generalized indices.
- Providing proofs for properties that were not found in the existing literature.
- Illustrating the practical implications of selecting different dimensions and indices using concrete examples to demonstrate their effect on resulting attributions.
Findings
- Power indices can be systematically categorized along three attribution dimensions: single-player, set-based, and cardinality-based.
- Existing power indices can be reviewed and generalized within these dimensions.
- Formal principles satisfied by various power indices were analyzed.
- Proofs were provided for properties of power indices that were missing in the existing literature.
- Moving to the cardinality-based setting removes player-identity information while preserving some index-level distinctions.
- The choice of attribution dimension and specific power index affects the resulting attributions in practical applications.
Why This Matters
This research matters because it addresses the understudied problem of selecting appropriate power indices for explainable AI. By systematically organizing power indices and analyzing their properties, it provides a structured framework for practitioners. The guidance offered aims to help in making informed decisions about which power index is suitable for a given AI explanation task, thereby facilitating more effective and reliable attribution of AI system outputs.