A simple proof of the Shapley-Folkman theorem
type: Item
instance of: scholarly article
title: A simple proof of the Shapley-Folkman theorem
zbMATH Open document ID: 0819.90017
DOI: 10.1007/BF01212924
author: Lin Zhou
published in: Economic Theory
publication date: 1994-11-27T00:00:00Z
review text: \textit{R. M. Starr} [Econometrica 37, 25-38 (1969; Zbl 0172.446)] first introduced into the literature the Shapley-Folkman theorem, which later proves to be extremely important in dealing with non-convexity in economic models of large but finite agents. Most published of the Shapley-Folkman theorem use the Krein-Milman theorem or the Minkowski theorem. In this note I provide a simple proof that uses the following fact from linear algebra: Assume that \(u,u^ 1,\dots, u^ p\) are vectors in \(\mathbb{R}^ q\). If \(u\) is a nonnegative combination of \(\{u^ 1,\dots, u^ p\}\), then it must be a nonnegative combination of no more than \(q\) vectors from \(\{u^ 1,\dots, u^ p\}\). The proof of this fact can be found in any book on linear programming. This fact has been used in a standard proof of the Caratheodory Theorem. I now show that it can also be used to prove the Shapley-Folkman theorem.
Mathematics Subject Classification ID: 91B50
zbMATH DE Number: 695676
zbMATH Keywords: Shapley-Folkman theorem
zbMATH Keywords: non-convexity in economic models
MaRDI profile type: Publication
cites work: Discrete and Continuous Bang-Bang and Facial Spaces Or: Look for the Extreme Points
cites work: Quasi-Equilibria in Markets with Non-Convex Preferences
full work available at URL: https://doi.org/10.1007/bf01212924
OpenAlex ID: W2033340349
description: scientific article; zbMATH DE number 695676