Generalized Nucleolus, Kernels, and Bargainig Sets for Cooperative Games with Restricted Cooperation

Authors

  • Natalia Naumova Saint Petersburg State University

Abstract

Generalization of the theory of the bargaining set, the kernel, and the nucleolus for cooperative TU-games, where objections and counter-objections are permited only between the members of a family of coalitions A and can use only the members of a family of coalitions B ⊃ A, is considered. Two versions of objections and two versions of counter-objections generalize the definitions for singletons. These definitions provide 4 types of generalized bargaining sets. For each of them, necessary and sufficient conditions on A and B for existence these bargaining sets at each game of the considered class are obtained. Two types of generalized kernels are defined. For one of them, the conditions N that ensure its existence generalize the result for B = 2 of Naumova (2007). Generalized nucleolus is not single-point and its intersection with nonempty generalized kernel may be the empty set. Conditions on A which ensure that the intersections of the generalized nucleolus with two types of generalized bargaining sets are nonempty sets, are obtained. The generalized nucleolus always intersects the first type of the generalized kernel only if A is contained in a partition of the set of players.

Keywords:

cooperative games, nucleolus, kernel, bargaining set

Downloads

Download data is not yet available.

References

Aumann, R. J. Maschler, M. (1964). The bargaining set for cooperative games. Annals of Math. Studies 52, Princeton Univ. Press, Princeton, N.J., 443–476.

Davis, M., Maschler, M. (1967). Existence of stable payoff configurations for cooperative games. Bull. Amer. Math. Soc., 69, 106–108.

Davis, M., Maschler, M. (1967). Existence of stable payoff configurations for cooperative games. In: Essays in Mathematical Economics in Honor of Oskar Morgenstern, M. Shubic ed., Princeton Univ. Press, Princeton, 39–52.

Maschler, M., Peleg, B. (1966). A characterization, existence proof and dimension bounds of the kernel of a game. Pacific J. of Math., 18, 289–328.

Naumova, N. I. (1976). The existence of certain stable sets for games with a discrete set of players. Vestnik Leningrad. Univ. N 7 (Ser. Math. Mech. Astr. vyp. 2), 47–54; English transl. in Vestnik Leningrad. Univ. Math., 9, 1981.

Naumova, N. I. (1978). M-systems of relations and their application in cooperative games. Vestnik Leningrad. Univ. N 1 (Ser. Math. Mech. Astr. ), 60–66; English transl. in Vestnik Leningrad. Univ. Math., 11, 1983, 67–73.

Naumova, N. (2007). Generalized kernels and bargaining sets for families of coalitions. Contributions to game theory and management GTM2007 Collected papers, Ed. by L. A. Petrosjan and N. A. Zenkevich, St.Petersburg, 346–360.

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

Peleg, B. (1967). Existence theorem for the bargaining set Mi1. In: Essays in Mathematical Economics in Honor of Oskar Morgenstern, M. Shubic ed., Princeton Univ. Press, Princeton, 53–56.

Peleg, B. (1967). Equilibrium points for open acyclic relations. Canad. J. Math., 19, 366–369.

Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on Applied Mathematics, 17(6), 1163–1170.

Downloads

Published

2022-05-24

How to Cite

Naumova, N. (2022). Generalized Nucleolus, Kernels, and Bargainig Sets for Cooperative Games with Restricted Cooperation. Contributions to Game Theory and Management, 8. Retrieved from https://gametheory.spbu.ru/article/view/13461

Issue

Vol. 8 (2015)

Section

Articles

License

Articles of "Contributions to Game Theory and Management" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.