We establish a new fixed point result for measurable-selection-valued correspondences with nonconvex and possibly disconnected values arising from the composition of Caratheodory functions with an upper Caratheodory (uC) correspondence. Using Caratheodory approximation methods, we show that for any such upper Caratheodory composition correspondence, if in each state, the upper semicontinuous part of the underlying upper Caratheodory correspondence contains an upper semicontinuous sub-correspondence taking contractible values, then the underlying upper Caratheodory correspondence is Caratheodory approximable, further implying that the induced measurable-selection-valued correspondence has fixed points—all accomplished without the induced selection correspondence being convex-valued or upper semicontinuous in the appropriate topologies (in the case the weak star topologies). An excellent example of such a composition correspondence is provided by discounted stochastic games (DSG). In particular, the Nash payoff selection correspondence of the parameterized collection of state-contingent one-shot games underlying a discounted stochastic game is gotten by composing players’ parameterized collection of state-contingent Caratheodory payoff functions with the upper Caratheodory Nash equilibrium correspondence (i.e., the uC Nash correspondence). We are able to conclude via our fixed point result that if the uC Nash correspondence has an upper semicontinuous part containing a contractibly valued upper semicontinuous sub-correspondence, implying that the uC Nash correspondence is Caratheodory approximable, then the Nash payoff selection correspondence induced by the uC Nash correspondence has fixed points. It then follows from Blackwell’s Theorem (1965–extended to games) that the DSG to which the selection correspondence belongs has stationary Markov perfect equilibria.
Fu et al. (Wed,) studied this question.