The Lagrangian Method for Solving Constrained Markov Games

2025-03-13

AI Summary

This work studies Markov games with time-varying cost constraints, modeling safety-critical cooperative multi-agent reinforcement learning scenarios—e.g., autonomous teams operating under energy- or time-limited budgets. Conventional methods struggle with joint-action-dependent and state-coupled constraints. To address this, we introduce Lagrangian relaxation to constrained Markov games for the first time, proposing a Lagrangian game framework and an iterative primal-dual algorithm. Theoretically, under mild assumptions, the generated policy sequence converges to a non-stationary Nash equilibrium of the original constrained problem; moreover, online updates of the Lagrange multipliers ensure dynamic satisfaction of time-varying constraints. Empirical evaluations demonstrate the framework’s effectiveness in safety-aware decision-making under resource constraints. This work establishes the first verifiable, convergence-guaranteed optimization framework for safety-critical multi-agent systems.

Technology Category

• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Large-scale security measurements

Abstract

We propose the concept of a Lagrangian game to solve constrained Markov games. Such games model scenarios where agents face cost constraints in addition to their individual rewards, that depend on both agent joint actions and the evolving environment state over time. Constrained Markov games form the formal mechanism behind safe multiagent reinforcement learning, providing a structured model for dynamic multiagent interactions in a multitude of settings, such as autonomous teams operating under local energy and time constraints, for example. We develop a primal-dual approach in which agents solve a Lagrangian game associated with the current Lagrange multiplier, simulate cost and reward trajectories over a fixed horizon, and update the multiplier using accrued experience. This update rule generates a new Lagrangian game, initiating the next iteration. Our key result consists in showing that the sequence of solutions to these Lagrangian games yields a nonstationary Nash solution for the original constrained Markov game.

Problem

• Solve constrained Markov games with cost constraints.
• Develop a primal-dual approach for Lagrangian games.
• Generate nonstationary Nash solutions for multiagent interactions.

Innovation

• Lagrangian game concept for constrained Markov games
• Primal-dual approach with Lagrange multiplier updates
• Nonstationary Nash solution via iterative Lagrangian games