A fixed point theorem for measurable selection valued correspondences induced by upper Caratheodory correspondences

We show that any measurable selection valued correspondence induced by the composition of an m-tuple of real-valued Caratheodory functions with an upper Caratheodory (uC) correspondence has fixed points if the underlying uC correspondence in the composition contains a continuum valued uC sub-correspondence. As an application, we show that all uncountable-compact discounted stochastic games (DSGs) satisfying the usual assumptions have Nash payoff selection correspondences having fixed points provided of course that the uC Nash correspondence contains a continuum valued uC Nash sub-correspondence. Fu and Page [8] have shown that all such DSGs, in fact, have uC Nash correspondences containing continuum valued uC Nash sub-correspondences—implying, therefore, that all DSGs satisfying the usual assumptions have stationary Markov perfect equilibria.