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

arXiv: https://arxiv.org/abs/2606.28824 · HTML: https://mlsanity.com/paper/2606.28824/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 42 references that are in the map.
- First posted on arXiv 27 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [On the Resilience of Traffic Networks under Non-Equilibrium Learning](https://mlsanity.com/paper/2210.03214/) (2022)
- [Distributed Learning Dynamics for Coalitional Games](https://mlsanity.com/paper/2309.06705/) (2023)
- [On Poisoned Wardrop Equilibrium in Congestion Games](https://mlsanity.com/paper/2209.00094/) (2022)
- [Evolutionary Poisson Games for Controlling Large Population Behaviors](https://mlsanity.com/paper/1503.08085/) (2015)
- [Herd Behaviors in Epidemics: A Dynamics-Coupled Evolutionary Games Approach](https://mlsanity.com/paper/2106.08998/) (2021)

## Abstract (arXiv)

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-Dr`eze 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.