\(K\)-correspondences, USCOs, and fixed point problems arising in discounted stochastic games. (English) Zbl 1505.54030
Summary: 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.
MSC:
| 54C60 | Set-valued maps in general topology |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 91A15 | Stochastic games, stochastic differential games |
Keywords:
\(K\)-correspondences; USCOs with contractible values; approximable USCOs; fixed points of nonconvex measurable selection valued correspondences; composition correspondences; Komlos limits; weak\(^\ast \)-compact sets of Banach-space-valued Bochner integrable selections; stationary Markov perfect equilibria; discounted stochastic gamesReferences:
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