Intersection theorems and their applications to Berge equilibria

In game theory, the existence of a solution is a key theoretical issue. One of the most powerful techniques, historically, has been the use of fixed point theorems. Here we explore a related theoretical treatment, the consideration of intersection of compact, convex sets. We follow the work of Ma [T.-W. Ma, On sets with convex sections, Journal of Mathematical Analysis and Applications, 27 (1969) 413-416], who extended the earlier results of Fan [K. Fan, Applications of a theorem concerning sets with convex sections, Mathematische Annalen, 163 (1966) 189-203]. Our results utilize an intersection theorem for an indexed family of sets, finite or infinite. With this theorem, combinatorial generalization of Ma's results are derived. Following the style of Fan and Ma, proofs and game theory applications are given and novel existence theorems for Berge and Nash equilibrium result.