Carathéodory-type selection and random fixed point theorems for discontinuous correspondences

Abstract: Research in Economics and Game theory has necessitated results on Carath\'eodory-type selections. In particular, one has to obtain Carath\'eodory type-selections from correspondences that need not be continuous (neither lower-semicontinuous nor upper-semicontinuous). We provide new theorems on Carath\'eodory type-selections that include as corollaries the results in Kim-Prikry-Yannelis \cite{KPY:87}. We also, obtain new random fixed-point theorems, random maximal elements, random (Nash) equilibrium and Bayesian equilibrium extending and generalizing theorems of Browder \cite{Browder:68}, Fan \cite{Fan:52} and Nash \cite{Nash}, among others.