论文标题
熵依赖路径的最佳计划
Entropic optimal planning for path-dependent mean field games
论文作者
论文摘要
在平均现场游戏的背景下，通过控制扩散系数，我们考虑了P.L.引入的计划问题的路径依赖性版本。狮子：给定一对边缘分布$（μ_0，μ_1）$，从初始分配$μ_0$开始找到游戏问题的规范，并在平均野外游戏平衡处诱导目标分配$μ_1$。我们的主要结果将依赖路径的计划问题减少到一个嵌入式问题中，也就是说，构建具有给定边缘$（μ_0，μ_1）$的McKean-Vlasov动力学。提供了$（μ_0，μ_1）$的一些足够条件，以保证存在解决方案。我们还表征了计划问题的最低熵解决方案。特别是，由于在我们的路径依赖性环境中唯一性不再存在，因此人们自然会引入最佳计划问题，该问题将简化为最佳运输问题以及受控的McKean-Vlasov Dynamics。
In the context of mean field games, with possible control of the diffusion coefficient, we consider a path-dependent version of the planning problem introduced by P.L. Lions: given a pair of marginal distributions $(μ_0, μ_1)$, find a specification of the game problem starting from the initial distribution $μ_0$, and inducing the target distribution $μ_1$ at the mean field game equilibrium. Our main result reduces the path-dependent planning problem into an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals $(μ_0,μ_1)$. Some sufficient conditions on $(μ_0,μ_1)$ are provided to guarantee the existence of solutions. We also characterize, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in our path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along with controlled McKean-Vlasov dynamics.
