We establish a fixed point theorem for the composition of nonconvex, measurable selection valued correspondences with Banach space valued selections. We show that if the underlying probability space of states is nonatomic and if the selection correspondences in the composition are K-correspondences (meaning correspondences having graphs that contain their Komlos limits), then the induced measurable selection valued composition correspondence takes contractible values and therefore has fixed points. As an application we use our fixed point result to show that all nonatomic uncountable-compact discounted stochastic games have stationary Markov perfect equilibria – thus resolving a long-standing open question in game theory.

中文翻译：

---

折扣随机游戏中出现的K-信函，USCO和定点问题

我们建立了一个非定点定理，用于与Banach空间值选择的非凸，可测量选择值对应的组成。我们表明，如果状态的基本概率空间是非原子的，并且组成中的选择对应关系是K对应关系（意味着对应关系的图形包含其Komlos极限），则诱导可测量选择值的组成对应关系将采用可收缩值，因此具有定点。作为应用程序，我们使用定点结果来证明所有非原子不可数紧致的打折随机游戏都具有平稳的马尔可夫完美均衡-从而解决了博弈论中一个长期存在的开放性问题。