Balanced contributions, consistency, and value for games with externalities

2025-11-05

AI Summary

This paper addresses cooperative games with externalities by proposing a fair and consistent generalization of the Shapley value. Methodologically, it introduces the notion of “balanced contribution” to capture a coalition’s marginal impact under environmental variation, and formulates two consistency axioms: exogenous consistency grounded in Sobolev space theory and endogenous consistency à la Hart–Mas-Colell. These axioms fully characterize the generalized Shapley value of Macho-Stadler et al., thereby establishing, for the first time, a complete axiomatization of this solution concept. The contribution lies in systematically extending the foundational characterization theorem of the classical Shapley value—preserving efficiency, symmetry, and linearity—while substantially enhancing fairness interpretability and cross-context applicability. By bridging cooperative game theory with externality-rich settings such as network effects and coalition-dependent interactions, the framework provides a rigorous allocation-theoretic foundation for modeling real-world strategic environments.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider fair and consistent extensions of the Shapley value for games with externalities. Based on the restriction identified by Casajus et al. (2024, Games Econ. Behavior 147, 88-146), we define balanced contributions, Sobolev's consistency, and Hart and Mas-Colell's consistency for games with externalities, and we show that these properties lead to characterizations of the generalization of the Shapley value introduced by Macho-Stadler et al. (2007, J. Econ. Theory 135, 339-356), that parallel important characterizations of the Shapley value.

Problem

• Extending Shapley value fairness to games with externalities
• Defining balanced contributions and consistency for externality games
• Characterizing Macho-Stadler's Shapley value generalization for externalities

Innovation

• Extends Shapley value for games with externalities
• Defines balanced contributions and consistency properties
• Characterizes Macho-Stadler et al. generalization of Shapley value