The optimistic stability of the core of mixed markets
type: Item
instance of: scholarly article
title: The optimistic stability of the core of mixed markets
zbMATH Open document ID: 0828.90013
DOI: 10.1016/0304-4068(94)90020-5
author: Joseph Greenberg
author: Benyamin Shitovitz
published in: Journal of Mathematical Economics
publication date: 1996-01-08T00:00:00Z
review text: Let \(T\) denote the set of traders, \(F\) a \(\sigma\)-field of measurable subsets of \(T\) (the coalitions) and \(\mu\) a positive and bounded measure on \(F\). For a coalition \(C\) in \(F\), a \(C\)-allocation is a integrable map \(\xi: C\to R^k_+\) s.t. \(\int_c \xi d\mu\leq \int_c wd\mu\), where the integrable map \(w\) from \(T\) to \(R^k_+\) denotes the initial commodity allocation to traders. \(X(C)\) denotes the set of all \(C\)-allocations. For each \(t\) in \(T\), \(u_t(x)\) is a continuous and strongly monotone utility function on \(R^k_+\) which is jointly measurable in \(t\) and \(x\). A map \(\sigma\) that assigns to each coalition \(C\in F\) a (possibly empty) subset \(\sigma(C)\subset X(C)\) is called stable iff it satisfies the following two properties: (i) (internally stable) if for all \(C\in F\), \(\xi\in \sigma(C)\) implies that there is no coalition \(B\in f\), \(B\subset C\) with \(\mu(B)> 0\) and \(\eta\in \sigma(B)\) s.t. \(u_t(\eta(t))> u_t(\xi(t))\) for almost all \(t\in B\). (ii) (externally stable) if for all \(C\in F\), \(\xi\) is in the complement of \(\sigma(C)\) in \(X(C)\) implies that there exist a coalition \(B\in F\), \(B\subset C\) with \(\mu(B)> 0\), and \(\eta\in \sigma(B)\) s.t. \(u_t(\eta(t))> u_t(\xi(t))\) for almost all \(t\in B\). A (non null) coalition \(B\) blocks a \(C\)-allocation \(\xi\), if \(B\subset C\) and there exists a \(B\)-allocation \(\eta\) s.t. \(u_t(\eta(t))> u_t(\xi(t))\) a.e. in \(B\). Let Core \((C)\) denote the set of all \(C\)- allocations that cannot be blocked. The main result established is that there is a unique map \(\sigma\) which is stable and this is the map Core (.). This extends earlier results, namely showing that the core is externally stable in games with a finite set of players as well as those with atomless traders, to mixed markets, i.e., for economies that have both `small' (atomless) and `large' (atoms) traders. An example is provided where the map, which assigns to each \(C\in F\…
Mathematics Subject Classification ID: 91B50
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 700860
zbMATH Keywords: stability
zbMATH Keywords: mixed markets
zbMATH Keywords: competitive equilibrium
zbMATH Keywords: core
zbMATH Keywords: stable set
MaRDI profile type: Publication
full work available at URL: 0304-4068(94)90020-5
OpenAlex ID: W2050600761
cites work: Markets with a Continuum of Traders
cites work: Q3378952
cites work: Q4121644
cites work: Credible coalitions and the core
cites work: Oligopoly in Markets with a Continuum of Traders
cites work: Theory of games and economic behavior.
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description: scientific article; zbMATH DE number 700860