Stationary Markov equilibria for approximable discounted stochastic games

We identify a new class of uncountable-compact discounted stochastic games for which existence of stationary Markov equilibria can be established and we prove two new existence results for this class. Our approach to proving existence in both cases is new – with both proofs being based upon continuous approximation methods. For our first result we use approximation methods involving measurable-selection-valued continuous functions to establish a new fixed point result for Nash payoff selection correspondences - and more generally for measurable-selection-valued correspondences having nonconvex values. For our second result, we again use approximation methods, but this time involving player action-profile-valued continuous functions to establish a new measurable selection result for upper Caratheodory Nash payoff correspondences. Because conditions which guarantee approximability - the presence of sub-correspondences taking contractible values (or more generally, Rd-values) - are the very conditions which rule out Nash equilibria homeomorphic to the unit circle, we conjecture that for uncountable-compact discounted stochastic games, the approximable class is the widest class for which existence of stationary Markov equilibria can be established.