Entropy Regularization in Mean-Field Games of Optimal Stopping

We study mean-field games of optimal stopping (OS-MFGs) and introduce an entropy-regularized framework to enable learning-based solution methods. By utilizing randomized stopping times, we reformulate the OS-MFG as a mean-field game of singular stochastic controls (SC-MFG) with entropy regularization. We establish the existence of equilibria and prove their stability as the entropy parameter vanishes. Fictitious play algorithms tailored for the regularized setting are introduced, and we show their convergence under both Lasry-Lions monotonicity and supermodular assumptions on the reward functional. Our work lays the theoretical foundation for model-free learning approaches to OS-MFGs.