The optimistic stability of the core of mixed markets
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 \(C\in F\) the set of its competitive equilibrium allocations, is not stable even if the core and competitive allocations coincide in the original market. The main result may also be rephrased as saying that in mixed markets the graph of the core mapping is the unique von Neumann and Morgenstern abstract stable set.
scientific article; zbMATH DE number 700860
scientific article