Alternative version of Shapley’s theorem on closed coverings of a simplex

Abstract

Sperner’s theorem as formulated by Ky Fan is dual to the KnasterKuratowski-Mazurkiewicz theorem. Shapley’s theorem is a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem. This paper points out that Shapley’s theorem is a generalization of Sperner’s theorem as well, by establishing an alternative version of Shapley’s theorem. Applications to the multiperson cooperative game theory are also discussed.