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.
Highlights
We establish a fixed point theorem for the composition of measurable selection valued correspondences with Banach space valued selections.a We show that if the underlying probability space of states is nonatomic and if the selection correspondences in the composition are K -correspondences, the induced nonconvex, measurable selection valued composition correspondence is approximable and has fixed points.b
We apply our fixed point result to show that all nonatomic, uncountablecompact discounted stochastic games (DSGs) satisfying the assumptions of the Nowak– Raghavan DSG model have stationary Markov perfect equilibria (SMPE) – resolving a long standing open question in game theory.c
(2020) 2020:14 selection correspondences in the composition are K -correspondences (or equivalently, are weak∗ upper semicontinuous correspondences taking nonempty compact values (USCOs), as we show here), the composition correspondence takes contractible values in its set of selections where contractibility is with respect to the compatibly metrized weak∗ topologies.d It follows from results in Gorniewicz, Granas, and Kryszewski [2] that the composition correspondence is approximable
Summary
We note that if a DSG has a Nash payoff selection correspondence that is approximable, it will have fixed points, and by Blackwell’s theorem [3] the DSG will have stationary Markov perfect equilibria (e.g., see Page [8, 9]). Let X be a norm-bounded, weak∗-closed (i.e., w∗-closed), convex subset of E∗, the separable norm dual of a separable Banach space E, and equip X with metric ρX∗ compatible with the w∗-topology on X inherited from E∗.f let L∞ X denote the set of μequivalence classes of E∗-valued, Bochner integrable functions x(·), with xω ∈ X a.e. Equip the spaces Y and X with the Borel σ -fields B∗Y and B∗X , generated by the ρY∗ - and ρX∗ -open sets in Y and X, respectively
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