Geodesic
Generalized Proportional Solutions to Games with Restricted Cooperation
Contributions to game theory and management, Tome 5 (2012), pp. 230-242

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In TU-cooperative game with restricted cooperation the values of characteristic function $v(S)$ are defined only for $S\in \mathcal{A}$, where $\mathcal{A}$ is a collection of some nonempty coalitions of players. If $\mathcal{A}$ is a set of all singletones, then a claim problem arises, thus we have a claim problem with coalition demands. We examine several generalizations of the Proportional method for claim problems: the Proportional solution, the Weakly Proportional solution, the Proportional Nucleolus, and $g$-solutions that generalize the Weighted Entropy solution. We describe necessary and sufficient condition on $\mathcal{A}$ for inclusion the Proportional Nucleolus in the Weakly Proportional solution and necessary and sufficient condition on $\mathcal{A}$ for inclusion $g$-solution in the Weakly Proportional solution. The necessary and sufficient condition on $\mathcal{A}$ for coincidence $g$-solution and the Weakly Proportional solution and sufficient condition for coincidence all $g$-solutions and the Proportional Nucleolus are obtained.
Keywords: claim problem, cooperative games, proportional solution, weighted entropy, nucleolus. 
@article{CGTM_2012_5_a20,
     author = {Natalia I. Naumova},
     title = {Generalized {Proportional} {Solutions} to {Games} with {Restricted} {Cooperation}},
     journal = {Contributions to game theory and management},
     pages = {230--242},
     publisher = {mathdoc},
     volume = {5},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CGTM_2012_5_a20/}
}
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Natalia I. Naumova. Generalized Proportional Solutions to Games with Restricted Cooperation. Contributions to game theory and management, Tome 5 (2012), pp. 230-242. http://geodesic.mathdoc.fr/item/CGTM_2012_5_a20/