Exit-and-Join Dynamics and Equilibrium in Continuum Cooperative Games

2026-06-27

AI Summary

This study investigates the dynamic entry and exit behavior of a continuum of agents among coalitions in nonatomic cooperative games and characterizes the resulting stable equilibria. By developing a continuum coalition dynamics framework, the paper extends the Aumann–Shapley and Aumann–Dreze values to structured nonatomic games, where coalition choices are driven by marginal contribution densities. The analysis incorporates switching costs and endogenous acceptance rules into a constrained equilibrium concept. Leveraging a decentralized switching mechanism, the authors formulate a mean-field dynamical model that unifies value allocation in cooperative games with evolutionary and population game frameworks. Key contributions include establishing that, under an incentive-compatible switching rule strictly responsive to payoff differentials, equilibrium coincides with a mass-dynamics steady state; proving global convergence under strict concavity of utilities; and demonstrating equivalence between this equilibrium and the Wardrop equilibrium of an induced population game.

Abstract

This paper develops a continuum theory of exit-and-join coalition dynamics in nonatomic cooperative games. We extend the Aumann-Shapley value and the Aumann-Drèze value to coalition structures in which each coalition is treated as a restricted nonatomic game, yielding a marginal-contribution-based payoff density that governs incentives for agents to remain in, exit, or join coalitions. We derive deterministic mean-field dynamics from decentralized switching rules and show that payoff-difference switching recovers replicator dynamics as a special case. We characterize exit-and-join equilibrium by the absence of profitable positive-mass deviations and prove its equivalence with stationarity of the induced mass dynamics under incentive-compatible and strictly payoff-responsive switching rates. For mass-based cooperative games, we construct a Lyapunov function and establish global convergence under strict concavity. We further show that the equilibrium is equivalent to a Wardrop equilibrium of an induced nonatomic population game and admits a variational inequality formulation. The framework is extended to incorporate switching costs and endogenous coalition acceptance rules, leading to constrained equilibria characterized by quasi-variational inequalities. The proposed theory unifies cooperative value allocation, noncooperative coalition mobility, mean-field dynamics, evolutionary game theory, and population games within a common framework for analyzing coalition formation and adaptation in large-scale multi-agent systems.

Problem

• exit-and-join dynamics
• continuum cooperative games
• coalition formation
• equilibrium
• nonatomic games

Innovation

• continuum cooperative games
• exit-and-join dynamics
• mean-field dynamics
• Aumann-Shapley value
• variational inequality