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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 having the 3M property - a property reintroduced here from Page and Resende (2011). We show that, in general, for any composition of Caratheodory functions and a 3M upper Caratheodory correspondence (i.e., a 3M uc correspondence), the induced measurable-selection-valued correspondence has fixed points. An excellent example of a 3M uc correspondence is the Nash correspondence belonging to any parameterized state-contingent game (i.e., a PSG). Under mild conditions on primitives, we have shown that the Nash correspondence belonging to any PSG is 3M, and as a consequence, contains an R_δ-valued minimal uc correspondence.
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