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Market games with a continuum of indivisible commodities. (English) Zbl 0651.90099

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 L. S. Shapley and 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: \[ v_ P(S)=\max \{\sum_{i\in S}\nu_ i(E_ i):\quad \cup_{i\in S}E_ i=\cup_{i\in S}A_ i,\quad E_ i\cap E_ j=\emptyset \quad for\quad i\neq j,\quad E_ i\in \Sigma \}. \]

MSC:

91A12 Cooperative games
91B24 Microeconomic theory (price theory and economic markets)
91A40 Other game-theoretic models
91B50 General equilibrium theory
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