Generalized Kernels and Bargainig Sets for Cooperative Games with Limited Communication Structure

For a fixed undirected connected graph $\varphi$ with a node set $N$, we study generalized kernels and bargaining sets for cooperative games $(N,v)$, where players are able to cooperate only if they can form a connected subgraph in graph $\varphi$. We consider generalizations of Aumann–Maschler theory of the bargaining set and the kernel, where objections and counter-objections are defined between coalitions from a fixed collection of coalitions $\mathcal{A}$.
Two problems are solved in this paper. Necessary and sufficient condition on $\mathcal{A}$, which ensures that each TU-game $(N,v)$ would have a nonempty $\varphi $-restricted generalized kernel $\mathcal{K}_\mathcal{A}(N,v)$ is obtained. For two different generalizations of bargaining sets, we obtained necessary and sufficient conditions on $\varphi$, which ensure that each game $(N,v)$ would have nonempty $\varphi $-restricted generalized $\mathcal{A}$-bargaining set for each $\varphi$-permissible collection $\mathcal{A}$.