Economic equilibrium theorems of Shafer-Sonnenschein version and nonempty intersection theorems in interval spaces
type: Item
instance of: scholarly article
title: Economic equilibrium theorems of Shafer-Sonnenschein version and nonempty intersection theorems in interval spaces
zbMATH Open document ID: 0832.90012
author: Xian Wu
author: Zifei Shen
author: Shihsen Chang
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-03-05T00:00:00Z
review text: The main concerns of this paper are (I) the existence of equilibria in an abstract economy, and (II) the nonempty intersection theorem for sets with convex sections. Usually both problems heavily depend upon the concept of convexity. The authors try to extend known results to the case formulated in topological spaces without linear structure making use of a few newly designed concepts such as interval space (a topological space in which something analogous to a segment in a vector space is defined, so that a \(W\)-convex set in an interval space is analogous to a convex set in a linear space). (I) Let \(I\) be a set of agents, \(X_i\) a topological space (choice set), \(A_i\), \(B_i: \prod_{k\in I} X_k\to 2^{X_i}\) set- valued mappings which express certain constraints, and \(P_i: \prod_{k\in I} X_k\to 2^{X_i}\) a set-valued mapping which represents a preference of the agent \(i\) \((i\in I)\). A family \({\mathcal E}= \{(X_i, A_i, B_i, P_i), i\in I\}\) is called an abstract economy. Finally, a point \(x^*\in X\equiv \prod_{i\in I} X_i\) is called an equilibrium of \(\mathcal E\) if \(x^*_i\in B_i(x^*)\) and \(P_i(x^*)\cap A_i(x^*_i)= \emptyset\) for each \(i\in I\). Then we can state the first problem: under what assumptions does an equilibrium exist? Several versions of the full answer to this question in the case each \(X_i\) is an interval space are given. (II) Let \(\{X_i\mid i\in I\}\) be a family of Hausdorff compact interval spaces and let \(X= \prod_{i\in I} X_i\) be strongly Dedekind complete (new concept to be defined). We denote \(X_{- i}= \prod_{k\neq i} X_k\). For any \(x\in X\), we also use the notations \(x_{- i}= \text{proj}_{x_{-i}}x\), \(x_i= \text{proj}_{x_i} x\), and \(x= x_i\otimes x_{-i}\). Given \(A\subset X\), \(x_i\in X_i\) and \(y_{-i}\in X_{-i}\), the sets \(A(x_i)\) and \(A(y_{-i})\) are defined by \(A(x_i)= \{x_{- i}\in X_{-i}\mid x_i\otimes x_{- i}\in A\}\) and \(A(y_{-i})= \{y_i\in X_i\mid y_i\otimes y_{- i}\in A\}\). Let \(\{A_i\mid …
Mathematics Subject Classification ID: 91B50
Mathematics Subject Classification ID: 26B25
zbMATH DE Number: 743899
zbMATH Keywords: existence of equilibria in an abstract economy
zbMATH Keywords: interval space
zbMATH Keywords: strongly Dedekind complete
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1017
OpenAlex ID: W1969743367
DOI: 10.1006/JMAA.1995.1017
description: scientific article; zbMATH DE number 743899