The equivalence of the cost share equilibria and the core of a voting game in a public goods economy
type: Item
instance of: scholarly article
title: The equivalence of the cost share equilibria and the core of a voting game in a public goods economy
zbMATH Open document ID: 0756.90009
DOI: 10.1007/BF00177673
author: Midori Hirokawa
published in: Social Choice and Welfare
publication date: 1992-06-28T00:00:00Z
review text: The author shows the equivalence of the cost share equilibria and the core of a voting game in a public goods economy. For individual \(i\), a consumption vector is a pair \((x,y^ i)=(x_ 1,\dots,x_ m;y^ i)\), where \(x_ j\) \((j\in\{1,\dots,m\})\) is the person's consumption of public good \(j\) and \(y^ i\) is her consumption of the private good. Individual \(i\)'s initial endowment is \(I^ i\); \(i\)'s payment for the production of the public goods is specified by a cost share function \(T^ i\) \((i\in\{1,\dots,n\})\). Given \(T=(T^ 1,\dots,T^ n)\), an allocation \((x_ 0,y_ 0)\) is said to be a cost share equilibrium iff \(y^ i_ 0=I^ i-T^ i(x_ 0)\) for all \(i\in\{1,\dots,n\}\), and \(u^ i(x_ 0,y^ i_ 0)\geq u^ i(x,I^ i-T^ i(x))\) for all \(x\in X\) and for all \(i\). Given any \(T\), a voting game \(G(T,W)\) introduces a structure of winning coalitions \(W\), where the set of simple majority coalitions is an example of \(W\). The core of the voting game \(G(T,W)\) is the set of all allocations which cannot be improved upon by any feasible allocation, denoted by \(C(G(T,W))\). The author proves two theorems, viz. (a) that any cost share equilibrium with respect to a given \(T=(T^ 1,\dots,T^ n)\) belongs to \(C(G(T,W))\), (b) under the assumptions of differentiability and that \((x_ 0,y_ 0)\) is an interior solution, if \((x_ 0,y_ 0)\) belongs to \(C(G(T,W))\), then it is a cost share equilibrium. These results are a generalization of those of \textit{M. Kaneko} [J. Econ. Theory 16, 123-136 (1977; Zbl 0399.90019)] where the cost share functions were restricted to be proportional (ratio equilibrium) to the total cost of providing the public goods, \(C(x)\). \textit{M. Kaneko} [Econometrica 45, 1589-1594 (1977; Zbl 0367.90046)] introduced a game in which the ratio was determined endogenously. In the present paper, it is not explained how the cost share functions \(T^ 1,\dots,T^ n\) are chosen.
reviewed by: Wulf Gaertner
Mathematics Subject Classification ID: 91B14
Mathematics Subject Classification ID: 91A12
Mathematics Subject Classification ID: 91B18
zbMATH DE Number: 35338
zbMATH Keywords: voting game
zbMATH Keywords: public goods economy
zbMATH Keywords: winning coalitions
zbMATH Keywords: simple majority coalitions
zbMATH Keywords: cost share equilibrium
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00177673
OpenAlex ID: W2013996828
description: scientific article; zbMATH DE number 35338