Mean-Field Games with Constraints

2025-10-13

AI Summary

This work studies constrained mean-field games (CMFGs), which model multi-agent decision-making under feasibility, safety, or regulatory constraints. We propose the Constrained Mean-Field Occupancy Measure Optimization (CMFOMO) framework, reformulating equilibrium computation as a single-objective convex-constrained optimization problem over occupancy measures—without requiring equilibrium uniqueness—and enabling approximation of the full equilibrium set. We establish necessary and sufficient conditions for existence and uniqueness of CMFG equilibria and, for the first time, prove that any CMFG equilibrium induces an $O(1/sqrt{N})$-Nash equilibrium in the corresponding $N$-player game. The method integrates constrained MDP theory with mean-field analysis, leveraging occupancy measure parameterization, convex optimization, and a strict feasibility assumption. Numerical experiments on a controlled susceptible–infected–susceptible (SIS) epidemic model demonstrate the framework’s effectiveness, flexibility, and high-accuracy scalability to large-scale constrained systems.

Technology Category

• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Constraint Optimization
• Game Theory and Economic Paradigms: Equilibrium

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

This paper introduces a framework of Constrained Mean-Field Games (CMFGs), where each agent solves a constrained Markov decision process (CMDP). This formulation captures scenarios in which agents' strategies are subject to feasibility, safety, or regulatory restrictions, thereby extending the scope of classical mean field game (MFG) models. We first establish the existence of CMFG equilibria under a strict feasibility assumption, and we further show uniqueness under a classical monotonicity condition. To compute equilibria, we develop Constrained Mean-Field Occupation Measure Optimization (CMFOMO), an optimization-based scheme that parameterizes occupation measures and shows that finding CMFG equilibria is equivalent to solving a single optimization problem with convex constraints and bounded variables. CMFOMO does not rely on uniqueness of the equilibria and can approximate all equilibria with arbitrary accuracy. We further prove that CMFG equilibria induce $O(1 / sqrt{N})$-Nash equilibria in the associated constrained $N$-player games, thereby extending the classical justification of MFGs as approximations for large but finite systems. Numerical experiments on a modified Susceptible-Infected-Susceptible (SIS) epidemic model with various constraints illustrate the effectiveness and flexibility of the framework.

Problem

• Extending mean-field games to incorporate agent constraints
• Establishing existence and uniqueness of constrained equilibria
• Developing optimization method to compute all possible equilibria

Innovation

• Constrained Mean-Field Games framework with agent constraints
• CMFOMO optimization method parameterizes occupation measures
• Approximates all equilibria without relying on uniqueness