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

Summary 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.The results presented here were first reported in Shapley (1987) and Vohra (1987). A version of our proof of Theorem 1 has also been presented in a recent book by C.D. Aliprantis, D.J. Brown and O. Burkinshaw,Existence and Optimality of Competitive Equilibria (1989) Springer-Verlag. We are grateful to Ky Fan, Wanda Gorgol, Tatsuro Ichiishi and Ali Khan for comments on earlier drafts. Vohra's research has been supported in part by NSF grant SES-8605630.