Dimensions of Power: A Systematic Guide to Power Indices for Explainable AI

Abstract

Power indices, originating in cooperative game theory, quantify each player's influence on the outcome of a given game. Originally designed to distribute profits or costs among players and to analyse the fairness of voting systems, power indices have recently gained prominence as methods for attributing outputs of AI-based systems to inputs, thus facilitating explainability. However, selecting the appropriate power index for a given explanation task is an understudied problem. To address this, we organise power indices along three attribution dimensions: single-player, set-based, and cardinality-based. For each dimension, we review the corresponding power indices, generalise existing ones where applicable, and analyse which formal principles they satisfy. We provide proofs for properties that are missing in the literature and show that moving to the cardinality-based setting removes player-identity information while preserving some index-level distinctions. Using concrete examples, we illustrate how the choice of dimension and index affects the resulting attributions in practice, and offer guidance for practitioners seeking to select a suitable power index for a given application context.