A fixed point theorem for measurable-selection-valued correspondences arising in game theory

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 correspondence. We show that, in general, for any composition of Caratheodory functions and an upper Caratheodory correspondence, if the upper semicontinuous part of the underlying upper Caratheodory correspondence contains an upper semicontinuous sub-correspondence taking contractible values, then the induced measurable-selection-valued correspondence has fixed points. An excellent example of such a composition, from game theory, is provided by the Nash payoff correspondence of the parameterized collection of one-shot games underlying a discounted stochastic game. The Nash payoff correspondence is gotten by composing players’ parameterized collection of state-contingent payoff functions with the upper Caratheodory Nash equilibrium correspondence (i.e., the Nash correspondence). As an application, we use our fixed point result to establish existence of a stationary Markov equilibria in discounted stochastic games with uncountable state spaces and compact metric action spaces.