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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - Natalia I. Naumova
TI  - Generalized Proportional Solutions to Games with Restricted Cooperation
JO  - Contributions to game theory and management
PY  - 2012
SP  - 230
EP  - 242
VL  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CGTM_2012_5_a20/
LA  - en
ID  - CGTM_2012_5_a20
ER  - 
%0 Journal Article
%A Natalia I. Naumova
%T Generalized Proportional Solutions to Games with Restricted Cooperation
%J Contributions to game theory and management
%D 2012
%P 230-242
%V 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CGTM_2012_5_a20/
%G en
%F CGTM_2012_5_a20
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/

[1] Soviet Math. Dokl., 30, 583–587 | MR

[2] Bregman L. M., Naumova N. I., “Goal programming solutions generated by utility functions”, Lecture Notes in Economic and Math. Systems, 510, 2002, 495–514 | DOI | MR | Zbl

[3] Bregman L. M., Romanovskij J. V., “Allotment and optimization in allocation problems”, Operations Research and Statistical Modeling, 3, ed. J. V. Romanovskij, Izdat. Leningrad. Univ., Leningrad, 1975, 137–162 (in Russian) | MR

[4] Moulin H., “Axiomatic cost and surplus sharing”, Handbook of Social Choice and Welfare, Chapter 6, v. 1, eds. K. J. Arrow, A. K. Sen, K. Suzumura, 2002, 289–357 | DOI

[5] Vestnik Leningrad. Univ. Math., 11 (1983), 67–73 | MR | Zbl | Zbl

[6] Naumova N. I., “Associated consistency based on utility functions of coalitions”, Game Theory and Applications, 13, Nova Publishers, New York, 2008, 115–125 | MR

[7] Naumova N., “Claim problems with coalition demands”, The Fourth International Conference Game Theory and Management, Collected Papers (June 28–30, 2010, St. Petersburg, Russia), Contributions to Game Theory and Management, 4, Graduate School of Management St. Petersburg University, St. Petersburg, 2011, 311–326 | MR | Zbl

[8] Thomson W., “Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey”, Mathematical Social Sciences, 45 (2003), 249–297 | DOI | MR | Zbl

[9] Yanovskaya E., “Consistency for proportional solutions”, International Game Theory Review, 4 (2002), 343–356 | DOI | MR | Zbl