Parameterized state-contingent games, 3M minimal Nash correspondences, and connectedness

Under mild assumptions on primitives, we show that all parameterized state-contingent games (PSGs) have upper Caratheodory (uC) Nash (equilibrium) correspondences which contain minimal uC Nash correspondences having the 3M property (defined here). This implies that all PSGs have Nash correspondences made up of minimal uC Nash correspondences taking closed, connected, and essential Nash equilibrium values (essential in the sense of Fort, 1950). It then follows from Fu and Page (2022b), that because all PSGs have continuum valued minimal Nash correspondences, all PSGs have Caratheodory approximable Nash payoff correspondences - which in turn implies that all PSGs have approximable Nash payoff selection correspondences, and therefore have Nash payoff selection correspondences with fixed points.