Optimal stopping is a fundamental problem in optimization with applications in risk management, finance, robotics, and machine learning. We extend the standard framework to a multi-agent setting, named multi-agent optimal stopping (MAOS), where agents cooperate to make optimal stopping decisions in a finite-space, discrete-time environment. Since solving MAOS becomes computationally prohibitive as the number of agents is very large, we study the mean-field optimal stopping (MFOS) problem, obtained as the number of agents tends to infinity. We establish that MFOS provides a good approximation to MAOS and prove a dynamic programming principle (DPP) based on mean-field control theory. We then propose two deep learning approaches: one that learns optimal stopping decisions by simulating full trajectories and another that leverages the DPP to compute the value function and to learn the optimal stopping rule using backward induction. Both methods train neural networks to approximate optimal stopping policies. We demonstrate the effectiveness and the scalability of our work through numerical experiments on 6 different problems in spatial dimension up to 300. To the best of our knowledge, this is the first work to formalize and computationally solve MFOS in discrete time and finite space, opening new directions for scalable MAOS methods. Code is available at https://github.com/yuchen-zhu-zyc/Learning-to-Stop

Mean field games (MFGs) have emerged as a powerful framework for modeling interactions in large-scale multi-agent systems. Despite recent advancements in reinforcement learning (RL) for MFGs, existing methods are typically limited to finite spaces or stationary models, hindering their applicability to real-world problems. This paper introduces a novel deep reinforcement learning (DRL) algorithm specifically designed for non-stationary continuous MFGs. The proposed approach builds upon a Fictitious Play (FP) methodology, leveraging DRL for best-response computation and supervised learning for average policy representation. Furthermore, it learns a representation of the time-dependent population distribution using a Conditional Normalizing Flow. To validate the effectiveness of our method, we evaluate it on three different examples of increasing complexity. By addressing critical limitations in scalability and density approximation, this work represents a significant advancement in applying DRL techniques to complex MFG problems, bringing the field closer to real-world multi-agent systems.