On Kakutani's fixed point theorem, the K-K-M-S theorem and the core of a balanced game

Abstract

We provide elementary proofs of Scarf's theorem on the non-emptiness of the core and of the K-K-M-S thoerem, based on Kakutani's fixed point theorem. We also show how these proofs can be modified to apply a coincidence theorem of Fan instead of Kakutani's fixed point theorem, for some additional simplicity.