Generalized nucleolus, kernels, and bargainig sets for cooperative games with restricted cooperation

Natalia Naumova

Contributions to game theory and management, Tome 8 (2015), pp. 231-242

Résumé

Generalization of the theory of the bargaining set, the kernel, and the nucleolus for cooperative TU-games, where objections and counter–objections are permited only between the members of a family of coalitions $\mathcal{A}$ and can use only the members of a family of coalitions $\mathcal{ B}\supset \mathcal{ A}$, is considered. Two versions of objections and two versions of counter–objections generalize the definitions for singletons. These definitions provide 4 types of generalized bargaining sets. For each of them, necessary and sufficient conditions on $\mathcal A$ and $\mathcal B$ for existence these bargaining sets at each game of the considered class are obtained. Two types of generalized kernels are defined. For one of them, the conditions that ensure its existence generalize the result for $\mathcal{ B}=2^N$ of Naumova (2007). Generalized nucleolus is not single–point and its intersection with nonempty generalized kernel may be the empty set. Conditions on $\mathcal A$ which ensure that the intersections of the generalized nucleolus with two types of generalized bargaining sets are nonempty sets, are obtained. The generalized nucleolus always intersects the first type of the generalized kernel only if $\mathcal A$ is contained in a partition of the set of players.

Détail

Keywords: cooperative games; nucleolus; kernel; bargaining set.