Deviation Dynamics in Cardinal Hedonic Games

2025-11-14

AI Summary

This paper investigates the convergence decidability of deviation dynamics in cardinal preference games, focusing on how the nonexistence of stable outcomes (No-instances) affects dynamic behavior. We propose a unified meta-theorem framework that transforms No-instances into hardness proofs for dynamic convergence, applicable to additive separable, fractional, and modified fractional hedonic games. For the first time, we systematically characterize the computational complexity of stability concepts—including contractual individual stability (CIS)—under single-agent deviations, establishing NP-hardness (or higher) for convergence decidability in ASHGs, FHGs, and MFHGs. Theoretically, CIS dynamics may require exponentially many steps in the worst case, yet convergence can be verified in O(n) steps. Our methodology integrates game-theoretic modeling, complexity analysis, and dynamic trajectory tracking, reconciling individual rationality with contractual stability. This yields a general analytical paradigm for coalition formation dynamics.

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

Computing stable partitions in hedonic games is a challenging task because there exist games in which stable outcomes do not exist. Even more, these No-instances can often be leveraged to prove computational hardness results. We make this impression rigorous in a dynamic model of cardinal hedonic games by providing meta theorems. These imply hardness of deciding about the possible or necessary convergence of deviation dynamics based on the mere existence of No-instances. Our results hold for additively separable, fractional, and modified fractional hedonic games (ASHGs, FHGs, and MFHGs). Moreover, they encompass essentially all reasonable stability notions based on single-agent deviations. In addition, we propose dynamics as a method to find individually rational and contractually individual stable (CIS) partitions in ASHGs. In particular, we find that CIS dynamics from the singleton partition possibly converge after a linear number of deviations but may require an exponential number of deviations in the worst case.

Problem

• Analyzing deviation dynamics convergence in cardinal hedonic games
• Proving computational hardness using No-instances in stability problems
• Finding individually rational and CIS partitions via dynamics

Innovation

• Meta theorems prove hardness from No-instances
• Dynamics find stable partitions in hedonic games
• Results apply to multiple hedonic game types