A structure theorem for coupled balanced games without side payments(Nonlinear Analysis and Convex Analysis)
Shyh-Nan Lee, Mau-Hsiang Shih · Institutional Repositories DataBase (IRDB) · 2006
The notion of 'core' for a game was defined as an independent solution concept by Gillies and Shapley[l] [2].The most basic issue is the core to be non-empty.The work of Bondereve [3] and Shapley[4] were on the balancedness condition for the non-emptiness of the core of a TU game and Scarf's[5] work was on balancedness in NTU games.In this report, we will study a general problem: Given $nNTU$ games, is there a tight coupling between $nNTU$ games so that the common core has non-empty?Let $N=\{1,2, \ldots, n\},$ $\eta$ the collection of all non-empty subsets of $N$ , and for $S\in\eta$ ,Let $X\subset E^{N},$ $S\in\eta,$ $\alpha=(\alpha_{1}, \ldots, \alpha_{n}),$ $\beta=(\beta_{1}, \ldots, \beta_{n})$ .Denote by $\alpha^{S}$ the projection of $\alpha$ to $E^{S}$ .Denote by" $\leq$ " the natural order on $E^{N}$ .$X$ is comprehensive if $\alpha\in X,$ $\beta\leq\alpha$ then $\beta\in X$ .Denote by $\hat{X}$ the comprehensive $h\tau dl$ of $X$ , that is, the smallest comprehensive set containing X.An NTU game (game without side payments) is an ordered triple $(N, F, D)$ .Here $F$ is a closed subset of $E^{N}$ , and $D$ is a function from $\eta$ to open, comprehensive non-empty, proper subsets of $E^{N}$ that satisfiesis non-empty and bounded.Here $\overline{D(S)}$ denotes the closure of $D(S)$ .