Solidary Solutions to Games with Restricted Cooperation

Natalia Naumova

Contributions to game theory and management, Tome 6 (2013), pp. 316-337.

Résumé

In TU–cooperative game with restricted cooperation the values of characteristic function $v(S)>0$ are defined only for $S\in \mathcal{A}$, where $\mathcal{A}$ is a collection of some nonempty coalitions of players. We examine generalizations of both the proportional solutions of claim problem (Proportional and Weakly Proportional solutions, the Proportional Nucleolus, and the Weighted Entropy solution) and the uniform losses solution of claim problem (Uniform Losses and Weakly Uniform Losses solutions, the Nucleolus, and the Least Square solution). These generalizations are $U$–equal sacrifice solution, the $U$–nucleolus and $qU$–solutions, where $U$ and $q$ are strictly increasing continuous functions. We introduce Solidary (Weakly Solidary) solutions, where if a total share of some coalition in $\mathcal{A}$ is less than its claim, then the total shares of all coalitions in $\mathcal{A}$ (that don't intersect this coalition) are less than their claims. The existence conditions on $\mathcal{A}$ for two versions of solidary solution are described. In spite of the fact that the versions of the solidary solution are larger than the corresponding versions of the proportional solution, the necessary and sufficient conditions on $\mathcal{A}$ for inclusion of the $U$–nucleolus in two versions of the solidary solution coincide with conditions on $\mathcal{A}$ for inclusion of the proportional nucleolus in the corresponding versions of the proportional solution. The necessary and sufficient conditions on $\mathcal{A}$ for inclusion $qU$–solutions in two versions of the solidary solution coincide with conditions on $\mathcal{A}$ for inclusion of the Weighted Entropy solution in the corresponding versions of the proportional solution. Moreover, necessary and sufficient conditions on $\mathcal{A}$ for coincidence the $U$–nucleolus with the $U$–equal sacrifice solution and conditions on $\mathcal{A}$ for coincidence $qU$–solutions with the $U$–equal sacrifice solution are obtained.

Détail

Keywords: claim problem; cooperative games; proportional solution; weighted entropy; nucleolus.

Bibliographie

[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] Katsev I., Yanovskaya E., “The prenucleolus for games with restricted cooperation”, Mathematical Game Theory and Applications, 3:4 (2011) (in Russian) | Zbl

[5] Katsev I., Yanovskaya E., “The prenucleolus for games with restricted cooperation”, Mathematical Social Sciences, 2013 (to appear) | MR

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

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

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

[9] 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

[10] Naumova N., “Generalized proportional solutions to games with restricted cooperation”, The Fifth International Conference Game Theory and Management, Collected Papers (June 27–29, 2011, St. Petersburg, Russia), Contributions to Game Theory and Management, 5, Graduate School of Management St. Petersburg University, St. Petersburg, 2012, 230–242

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

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