Split-Merge Dynamics for Shapley-Fair Coalition Formation

2026-03-17

AI Summary

This work addresses the dynamic trade-off between individual fairness and collective efficiency in coalition formation by proposing a control-theoretic split-and-merge mechanism: agents split when their Shapley value becomes negative and merge only when doing so yields a strict surplus improvement. For the first time, the Shapley value is embedded as a fairness signal within a dynamic coalition formation process. The paper defines and proves the existence and finite-time convergence to a Shapley-Fair and Merge-Stable (SFMS) equilibrium. Stability of fairness deficits and surpluses is rigorously analyzed using a vector Lyapunov function and the discrete-time LaSalle invariance principle. Empirical validation in a 10-agent game demonstrates the method’s effectiveness in mitigating fairness conflicts and confirms its theoretical and algorithmic soundness for endogenous coalition formation in dynamic environments.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Humans and AI: Learning Human Values and Preferences

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Coalition formation is often modeled as a static equilibrium problem, neglecting the dynamic processes governing how agents self-organize. This paper proposes a dynamic split-and-merge framework that balances two conflicting economic forces: individual fairness and collective efficiency. We introduce a control-theoretic mechanism where topological operations are driven by distinct signals: splits are triggered by fairness violations (specifically, negative Shapley values representing "agent-responsible inefficiency"), while merges are driven by strict surplus improvements (superadditivity). We prove that these dynamics converge in finite time to a specific class of steady states termed Shapley-Fair and Merge-Stable (SFMS) partitions. Convergence is established via a vector Lyapunov function tracking aggregate fairness deficits and system surplus, leveraging a discrete-time LaSalle invariance principle. Numerical case studies on a 10-player game demonstrate the algorithm's ability to resolve fairness tensions and reach stable configurations, providing a rigorous foundation for endogenous coalition formation in dynamic environments.

Problem

• coalition formation
• fairness
• efficiency
• Shapley value
• dynamic process

Innovation

• Shapley fairness
• split-merge dynamics
• coalition formation
• Lyapunov stability
• superadditivity