Solving Four Open Problems about Core Stability in Altruistic Hedonic Games

2025-11-27

AI Summary

This paper resolves four long-standing open problems concerning core stability verification in altruistic hedonic games, focusing on two preference variants—“equal treatment” and “altruistic treatment”—based on average and minimum utility aggregation. Using a novel integration of computational social choice and cooperative game theory, the authors construct carefully engineered reduction instances featuring intricate friendship structures. They establish, for the first time, that core stability verification is coNP-complete for all four variants. This result precisely characterizes the computational complexity boundary for coalition formation under altruistic preferences, thereby closing a fundamental theoretical gap. Moreover, it provides a critical complexity benchmark and a reusable reduction paradigm for future work, significantly advancing the rigorous formalization of altruistic preference modeling in hedonic game theory.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science

Abstract

Hedonic games -- at the interface of cooperative game theory and computational social choice -- are coalition formation games in which the players have preferences over the coalitions they can join. Kerkmann et al. [13] introduced altruistic hedonic games where the players' utilities depend not only on their own but also on their friends' valuations of coalitions. The complexity of the verification problem for core stability has remained open in four variants of altruistic hedonic games: namely, for the variants with average- and minimum-based "equal-treatment" and "altruistic-treatment" preferences. We solve these four open questions by proving the corresponding problems coNP-complete; our reductions rely on rather intricate gadgets in the related networks of friends.

Problem

• Analyzes core stability complexity in altruistic hedonic games
• Solves four open verification problems for specific preference variants
• Proves coNP-completeness using intricate network gadget reductions

Innovation

• Proved coNP-complete for four altruistic hedonic game variants
• Used intricate gadgets in friend networks for reductions
• Solved open verification problems for core stability