# Split-Merge Dynamics for Shapley-Fair Coalition Formation

arXiv: https://arxiv.org/abs/2603.17153 · HTML: https://mlsanity.com/paper/2603.17153/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2026.

## Built on by (most cited first, 2 of 2)

- [Exit-and-Join Dynamics for Decentralized Coalition Formation](https://mlsanity.com/paper/2606.19683/) (2026)
- [Multiscale Exit-Join Dynamics: Tactical Consensus and Strategic Coalition Formation](https://mlsanity.com/paper/2606.26139/) (2026)

## Abstract (arXiv)

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.