AI 中文总结
该研究为最优停止平均场博弈提出新概率公式，用耦合反射正倒向随机微分方程刻画均衡，证明其与均衡的等价性，通过不动点定理建立均衡存在性与唯一性，还给出序理论方法及与分析方法的联系，以及对多人博弈的作用。
AI 中文摘要
我们为具有随机策略的最优停止平均场博弈（OS - MFGs）提出了一种新颖的概率公式。通过一类新的耦合正倒向系统，即耦合反射正倒向McKean - Vlasov随机微分方程（MKV - RFBSDEs）来刻画平均场均衡。均衡由五元组$(X,Y,Z,A,L)$表示，其中$L$是一个适应的、取值于$[0,1]$的非增右连左极过程，表示随机停止策略。通过涉及$L$的两个新颖的Skorokhod型条件来刻画随机停止策略的最优性。我们严格证明了MKV - RFBSDE系统的解与随机策略中的OS - MFG均衡之间的等价性。通过将Kakutani - Fan - Glicksberg不动点定理应用于集值最佳响应对应来建立均衡的存在性，依赖于耦合MKV - RFBSDE系统的新稳定性、紧致性和连续性结果。在适当条件下还证明了唯一性。在替代单调性假设下，基于Tarski不动点定理开发了一种新的序理论方法，得到了极值均衡的存在性以及最小和最大解的构造方案。我们还表明平均场均衡为相关的$N$人停止博弈诱导出一个近似纳什均衡。最后，我们将我们的概率公式与以耦合约束偏微分方程组为特征的分析方法联系起来。
英文摘要
We propose a novel probabilistic formulation for optimal stopping mean field games (OS-MFGs) with randomized strategies. We characterize mean field equilibria through a new class of coupled forward-backward systems, termed coupled reflected forward-backward McKean--Vlasov stochastic differential equations (MKV-RFBSDEs). An equilibrium is represented by a quintuple $(X,Y,Z,A,L)$, where $L$ is an adapted, $[0,1]$-valued, non-increasing càdlàg process representing the randomized stopping strategy. The optimality of randomized stopping strategies is characterized through two novel Skorokhod-type conditions involving $L$. This characterization is new even for classical optimal stopping problems without mean field interactions. We rigorously prove an equivalence between solutions of the MKV-RFBSDE system and OS-MFG equilibria in randomized strategies. We establish the existence of equilibria by applying the Kakutani--Fan--Glicksberg fixed-point theorem to a set-valued best-response correspondence, relying on new stability, compactness, and continuity results for the coupled MKV-RFBSDE system. We also prove uniqueness under suitable conditions. Under alternative monotonicity assumptions, we develop a new order-theoretic approach based on Tarski's fixed-point theorem, yielding the existence of extremal equilibria and constructive schemes for the minimal and maximal solutions. We further show that a mean field equilibrium induces an approximate Nash equilibrium for the associated $N$-player stopping game. Finally, we connect our probabilistic formulation with the analytical approach characterized by a coupled system of constrained partial differential equations.
