An elementary proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley theorem
type: Item
instance of: scholarly article
title: An elementary proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley theorem
zbMATH Open document ID: 0811.54032
DOI: 10.1007/BF01215384
author: Stefan Krasa
author: Nicholas C. Yannelis
published in: Economic Theory
publication date: 1995-05-03T00:00:00Z
review text: The Knaster-Kuratowski-Mazurkiewicz-Shapley (K-K-M-S) theorem is formulated as follows: For every integer \(n \geq 1\), let \(N\) denote \(\{1,2,\dots,n\}\). If \(S \subseteq N\), then \(\Delta^ S\) means the simplex formed by all points \((x_ 1,\dots, x_ n)\) for which \(x_ i \geq 0\), \(\sum x_ i = 1\), and \(x_ i = 0\) for \(i \notin S\). The central point of the simplex \(\Delta^ S\) (with \(x_ i = 1/| S|\) for \(i \in S\)) is denoted by \(m^ S\). Let \(C_ S\), \(S \subseteq N\), be a family of closed subsets of \(\Delta^ N\) such that for every \(T \subseteq N\), and for every \(x \in \Delta^ T\), there exists a subset \(S \subseteq T\) for which \(x \in C_ S\). The theorem states that there exists a family \(\mathcal B\) of subsets of \(N\) for which all sets \(C_ S\), \(S \in {\mathcal B}\), have a common point, and for which \(m^ N\) is a convex combination of \(m^ S\) for \(S \in {\mathcal B}\). One possible direct game-theoretic interpretation of this result is as follows: Suppose that we have a situation in which \(n\) persons have different possible strategies. The largest total utility that they can gain is \(U\). The problem is: how to divide that utility between them, i.e., how to assign fractions \(x_ i \geq 0\), \(1 \leq i \leq n\), \(\sum x_ i = 1\) so that \(i\)-th person gets \(x_ i \cdot U\). For every coalition \(S\), let \(C_ S\) denote the set of all allocation vectors that members of this coalition considers fair to them. If \(x \in \Delta^ T\) (i.e., if money goes only to members of \(T\)), then either this \(x\) is fair to all persons from \(T\), or some members of \(T\) believe that they do not get their fair share, in which case this distribution \(x\) is fair only to some members of \(T\) (i.e., \(x \in C_ S\) for some \(S \subset T\)). In this setting, the theorem says that there exist a family \(\mathcal B\) of coalitions such that, first, there exists an allocation vector \(\tilde x\) that is acceptable by all these coalitions, and s…
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 707358
zbMATH Keywords: NTU game
zbMATH Keywords: Knaster-Kuratowski-Mazurkiewicz-Shapley theorem
zbMATH Keywords: K-K-M-S theorem
zbMATH Keywords: non-transverable utility
zbMATH Keywords: core
reviewed by: Olga Kosheleva
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 707358