Market games with a continuum of indivisible commodities
The author introduces a class of (side-payment) market games with a continuum of indivisible commodities, and shows that it coincides with the class of totally balanced games (which in turn coincides with the class of market games introduced by \textit{L. S. Shapley} and \textit{M. Shubik} [J. Econ. Theory 1, 9-25 (1969)]). That class of games is defined as follows. Given a measurable space \((X,\Sigma)\), let \(A_{\mu}\) be the space of all set functions \(f\circ \mu\), where \(\mu: \Sigma\to {\mathbb{R}}^ m \)is a nonatomic measure, and f is any real-valued concave function defined (at least) on the range of \(\mu\). A market with a continuum of indivisible commodities is \({\mathcal M}_ P=\{(\nu_ i,A_ i)\), \(i\in I\}\), where \(I=\{1,...,n\}\) is the set of traders, \(\nu_ i\in A_{\mu}\) and \(P=(A_ 1,...,A_ n)\) (the initial endowments) is an ordered measurable partition of X: traders in coalition S may exchange commodities to obtain a more preferred division of the set \(\cup_{i\in S}A_ i\). Given \({\mathcal M}_ P\), the game is so defined: \[ v_ P(S)=\max \{\sum_{i\in S}\nu_ i(E_ i):\quad \cup_{i\in S}E_ i=\cup_{i\in S}A_ i,\quad E_ i\cap E_ j=\emptyset \quad for\quad i\neq j,\quad E_ i\in \Sigma \}. \]
scientific article; zbMATH DE number 4062867