Single-Deviation Stability in Additively Separable Hedonic Games with Constrained Coalition Sizes

2025-10-14

AI Summary

This paper investigates the existence and computational complexity of four classical stability concepts—Nash, individual, contractual Nash, and contractual individual stability—in additive, separable hedonic games where coalition sizes are constrained by fixed upper and lower bounds. Two models are distinguished: one requiring post-deviation coalitions to satisfy size constraints, and another relaxing this requirement. The work provides the first complete characterization of existence conditions for each stability notion. Under only an upper bound constraint, it fully classifies computational complexity: a polynomial-time algorithm is given for contractual individual stability; contractual Nash stability is shown polynomial-time decidable when the maximum coalition size is at most two; and Nash stability becomes tractable when the minimum coalition size is at least two. The analysis integrates hedonic game modeling, combinatorial optimization, and computational complexity theory, revealing how size constraints and preference structure fundamentally influence stability properties.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

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

Abstract

We study stability in additively separable hedonic games when coalition sizes have to respect fixed size bounds. We consider four classic notions of stability based on single-agent deviations, namely, Nash stability, individual stability, contractual Nash stability, and contractual individual stability. For each stability notion, we consider two variants: in one, the coalition left behind by a deviator must still be of a valid size, and in the other there is no such constraint. We provide a full picture of the existence of stable outcomes with respect to given size parameters. Additionally, when there are only upper bounds, we fully characterize the computational complexity of the associated existence problem. In particular, we obtain polynomial-time algorithms for contractual individual stability and contractual Nash stability, where the latter requires an upper bound of 2. We obtain further results for Nash stability and contractual individual stability, when the lower bound is at least 2.

Problem

• Analyzing stability in hedonic games with constrained coalition sizes
• Examining four single-agent deviation stability notions under size constraints
• Characterizing computational complexity for stability existence with upper bounds

Innovation

• Analyzes stability in hedonic games with constrained coalition sizes
• Compares four stability notions under two deviation constraints
• Provides computational complexity and polynomial-time algorithms for stability