论文标题
一阶平均现场游戏的通用属性
Generic Properties of First Order Mean Field Games
论文作者
论文摘要
我们考虑一类确定性平均野战游戏，其中与每个玩家相关联的状态根据线性W.R.T.控件。从通用理论的角度研究了解决方案的存在，独特性和稳定性。在适当的动态和成本功能的拓扑空间中，我们证明，对于几乎所有平均野外游戏（在Baire类别意义上），最佳答复映射是对A.E的单一值。球员。结果，平均野外游戏承认了强大的（不是随机）的解决方案。举例说明了一组公开的游戏，这些游戏承认单个解决方案，其他开放式设置承认多个解决方案。进一步的示例表明，存在一组开放的MFG，该集合具有独特的解决方案，该解决方案是渐近稳定的W.R.T.最佳答复地图和另一组开放式MFG具有不稳定的唯一解决方案。我们以一个终端约束的MFG为例，这些示例没有任何解决方案，甚至没有随机策略的温和意义。
We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of view of generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for nearly all mean field games(in the Baire category sense) the best reply map is single valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.
