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

• URL: http://arxiv.org/abs/2606.28824v1
• Date: Sat, 27 Jun 2026 09:17:07 GMT
• Title: Exit-and-Join Dynamics and Equilibrium in Continuum Cooperative Games
• Abstract summary: We develop a continuum theory of exit-and-join coalition dynamics in nonatomic cooperative games.<n>We derive mean-field dynamics from decentralized switching rules and show that payoff-difference switching recovers replicator dynamics.<n>The proposed theory unifies cooperative value allocation, noncooperative coalition mobility, mean-field dynamics, evolutionary game theory, and population games.
• Score: 20.036142577511455
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

Related papers

• Parametric Open Source Games [78.29352305480285]
  We introduce parametric open-source games, a continuous analogue of program equilibria.<n>We derive an exact coupling threshold at which selfish gradient ascent in symmetric $2times2$ games switches from defection toward cooperation.<n>We extend the framework to a neural semantics class whose first-order cooperation condition is governed by the ratio of cross-player to self-player sensitivity.
  arXiv  Detail & Related papers  (2026-06-25T14:14:24Z)
• Exit-and-Join Dynamics for Decentralized Coalition Formation [20.036142577511455]
  We study coalition formation as a decentralized dynamical process driven by unilateral exit-and-join decisions.<n>Agents evaluate local moves using the Aumann-Dreze value, so payoffs are computed within the agent's current coalition.<n>A terminal partition is precisely a coalition structure with no admissible, individually profitable exit-and-join deviation.
  arXiv  Detail & Related papers  (2026-06-18T01:19:31Z)
• Phi-Actor-Critic: Steering General-Sum Games to Pareto-Efficient Correlated Equilibria [3.061219970798378]
  Real-world multi-agent systems are often modeled as general-sum games where individual incentives conflict with collective welfare.<n>Standard deep multi-agent reinforcement learning (MARL) methods struggle with this problem.<n>We propose $$-Actor-Critic ($$-AC), a framework that leverages swap regret minimization to steer learning toward high-welfare correlated equilibria.
  arXiv  Detail & Related papers  (2026-06-09T16:40:26Z)
• A Collective Variational Principle Unifying Bayesian Inference, Game Theory, and Thermodynamics [1.9365949510386082]
  We introduce the Game-Theoretic Free Energy Principle, showing that multi-agent systems implicitly implement a game.<n>We show that stationary points of collective free energy correspond to approximate Nash equilibria of an induced game.<n>This yields a predictive theory of cooperation, including a falsifiable non-monotonic relationship between sensory precision and agent influence.
  arXiv  Detail & Related papers  (2026-04-30T14:42:23Z)
• Toward a Sustainable Federated Learning Ecosystem: A Practical Least Core Mechanism for Payoff Allocation [71.86087908416255]
  We introduce a payoff allocation framework based on the least core (LC) concept.<n>Unlike traditional methods, the LC prioritizes the cohesion of the federation by minimizing the maximum dissatisfaction.<n>Case studies in federated intrusion detection demonstrate that our mechanism correctly identifies pivotal contributors and strategic alliances.
  arXiv  Detail & Related papers  (2026-02-03T11:10:50Z)
• Integrating upstream and downstream reciprocity stabilizes cooperator-defector coexistence in N-player giving games [1.1381558444077822]
  We show how pay-it-forward chains and reputation systems can jointly maintain social including cooperation despite cognitive limitations and group size challenges.<n>This framework demonstrates how pay-it-forward chains and reputation systems can jointly maintain social including cooperation despite cognitive limitations and group size challenges.
  arXiv  Detail & Related papers  (2025-09-05T01:49:26Z)
• Vairiational Stochastic Games [1.6703448188585752]
  We propose a novel variational inference framework tailored to decentralized multi-agent systems.<n>Our framework addresses the challenges posed by non-stationarity and unaligned agent objectives.<n>We demonstrate theoretical convergence guarantees for the proposed decentralized algorithms.
  arXiv  Detail & Related papers  (2025-03-08T03:21:23Z)
• Learning to Balance Altruism and Self-interest Based on Empathy in Mixed-Motive Games [47.8980880888222]
  Multi-agent scenarios often involve mixed motives, demanding altruistic agents capable of self-protection against potential exploitation.<n>We propose LASE Learning to balance Altruism and Self-interest based on Empathy.<n>LASE allocates a portion of its rewards to co-players as gifts, with this allocation adapting dynamically based on the social relationship.
  arXiv  Detail & Related papers  (2024-10-10T12:30:56Z)
• On Tractable $Φ$-Equilibria in Non-Concave Games [53.212133025684224]
  We study tractable $Phi$-equilibria in non-concave games.<n>We show that when $Phi$ is finite, there exists an efficient uncoupled learning algorithm that converges to the corresponding $Phi$-equilibria.
  arXiv  Detail & Related papers  (2024-03-13T01:51:30Z)
• Exploiting hidden structures in non-convex games for convergence to Nash equilibrium [62.88214569402201]
  A wide array of modern machine learning applications can be formulated as non-cooperative Nashlibria. We provide explicit convergence guarantees for both deterministic and deterministic environments.
  arXiv  Detail & Related papers  (2023-12-27T15:21:25Z)
• Optimistic Policy Gradient in Multi-Player Markov Games with a Single Controller: Convergence Beyond the Minty Property [89.96815099996132]
  We develop a new framework to characterize optimistic policy gradient methods in multi-player games with a single controller. Our approach relies on a natural generalization of the classical Minty property that we introduce, which we anticipate to have further applications beyond Markov games.
  arXiv  Detail & Related papers  (2023-12-19T11:34:10Z)
• How Bad is Selfish Driving? Bounding the Inefficiency of Equilibria in Urban Driving Games [64.71476526716668]
  We study the (in)efficiency of any equilibrium players might agree to play. We obtain guarantees that refine existing bounds on the Price of Anarchy. Although the obtained guarantees concern open-loop trajectories, we observe efficient equilibria even when agents employ closed-loop policies.
  arXiv  Detail & Related papers  (2022-10-24T09:32:40Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.