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

Natalia Naumova, Irina Korman

St. Petersburg State University, Faculty of Mathematics and Mechanics Universitetsky pr. 28, Petrodvorets, St. Petersburg, 198504, Russia

Abstract: 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 $\mathcal{A}$.

Keywords: cooperative games, kernel, bargaining set, limited communication.

Language: English