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Split-Merge Dynamics for Shapley-Fair Coalition Formation

Quanyan Zhu, Zhengye Han

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.

Split-Merge Dynamics for Shapley-Fair Coalition Formation

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.
Paper Structure (12 sections, 5 theorems, 8 equations, 2 figures, 1 algorithm)

This paper contains 12 sections, 5 theorems, 8 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shapley value
  4. Negative Shapley mass and notation
  5. Merge-Split Dynamics
  6. Split Operator $F^{\mathrm{S}}$
  7. Merge Operator $F^{\mathrm{M}}$
  8. Steady States: SFMS Partitions
  9. Invariance Principle for Coalition Structures
  10. Invariance for the Merge-Split Algorithm
  11. Case Study: Merge-Split Dynamics
  12. Conclusion

Key Result

Lemma 1

Let $Y \subset \mathbb{R}^{k}$ be a finite set, and consider the strict lexicographic order $\succ$ restricted to $Y$. Then there is no infinite strictly increasing sequence in $Y$. Equivalently, any sequence $\left\{y_{t}\right\}_{t \geq 0} \subset Y$ that is nondecreasing with respect to $\succeq$

Figures (2)

  • Figure 1: Coalition evolution under the merge-split dynamics. After a single split iteration, the system enters the invariant set and the coalition structure remains unchanged.
  • Figure 2: Evolution of the Lyapunov components. The vector Lyapunov function stabilizes after finite time, confirming finite-time entry into the invariant set.

Theorems & Definitions (9)

  • Definition 1: SFMS Partition
  • Definition 2: Lexicographic order
  • Lemma 1: Finite lexicographic order has no infinite ascent
  • Definition 3: Weak invariance
  • Theorem 1: Generalized discrete LaSalle invariance principle
  • Definition 4: Vector Lyapunov Function
  • Lemma 2: Fairness monotonicity under Split Rule
  • Lemma 3: Monotonicity of the Merge-Split Update
  • Theorem 2: Invariance for Merge-Split Dynamics