A theorem on open coverings of a simplex and Scarf's core existence theorem through Brouwer's fixed point theorem
In the paper under review, a direct elementary proof if provided for Scarf's result that every balanced game with non-transferable utility has a core. An indirect elementary proof of this result is given in a paper by \textit{S. Krasa} and \textit{N. C. Yannelis} [ibid. 467-471 (1994; see the review above)]. In this review, we will use the denotations from the previous review. Similar technique is also used to get an elementary proof of the result that is in some sense ``dual'' to K-K-M-S theorem formulated in the previous review: the result about open coverings of a simplex (K-K-M-S theorem is about closed coverings). Namely, if \(O_ S\), \(S \subseteq N\), is an open covering of a simplex \(\Delta^ N\) (i.e., \(O_ S\) are open and \(\Delta^ N \subseteq \bigcup O_ S\)), and \(\Delta^{N \setminus\{i\}} \subseteq O_{\{i\}}\) for all \(i\), then there exists a family \(\mathcal B\) for which, first, all sets \(O_ S\), \(S \in {\mathcal B}\), have a common point, and second, \(m^ N\) is a convex combination of \(m^ S\), \(S \in {\mathcal B}\).
scientific article; zbMATH DE number 707359