Generalized Nucleoli and Generalized Bargaining Sets for Games with Restricted Cooperation

Abstract

A generalization of the theory of the bargaining set, the kernel, and the nucleolus for cooperative TU-games, where objections and counter-objections are permitted only between the members of a collection of coalitions $\mathcal{A}$ and can use only the members of a collection of coalitions $\mathcal{B}\supset \mathcal{A}$, is considered. Four versions of generalized bargaining set are possible. Three versions of generalized kernel and two versions of generalized nucleolus are defined. One generalized kernel, one generalized nucleolus, and the corresponding generalized bargaining sets were examined in Naumova (Contributions to Game Theory and Management, vol. 5, pp. 230–242. Graduate School of Management, St. Petersburg University, St. Petersburg, 2012; Contributions to Game Theory and Management GTM2014, vol. 8, pp. 231–242. St. Petersburg State University, St. Petersburg, 2015). Conditions on $\mathcal{A}$ and $\mathcal{B}$ that ensure existence of the second generalized kernel are obtained. Weakly mixed collections of coalitions are defined. For such collections of coalitions, the second generalized nucleolus is contained in the second generalized kernel and in two generalized bargaining sets. If $\mathcal{A}$ does not contain singletons such inclusion is valid for all games only if $\mathcal{A}$ is a weakly mixed collection of coalitions. For weakly mixed collection of coalitions $\mathcal{A}$ an iterative procedure that converges to a point in the second generalized kernel is described.