K-Correspondences, USCOs, and fixed point problems arising in discounted stochastic games

Abstract

Abstract 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.

Keywords

• K-correspondences
• USCOs with contractible values
• Approximable USCOs
• Fixed points of nonconvex measurable selection valued correspondences
• Composition correspondences
• Komlos limits