A Fuzzy-Core Extension of Scarf Theorem and Related Topics

Authors

  • Valery A. Vasil’ev Sobolev Institute of Mathematics

Abstract

The paper deals with a generalization of Scarf (Scarf, 1967) and Bondareva-Shapley (Bondareva, 1962; Shapley, 1967) theorems on the core of cooperative game to the case of fuzzy domination. The approach proposed is based on the concept of balanced collection of fuzzy coalitions, introduced by the author (Vasil’ev, 2012). This extension of classic notion of balanced collection of standard coalitions makes it possible to present a natural analog of balancedness for so-called fuzzy TU cooperative games. Moreover, it turns out that similar to the standard situation the new balancedness-like assumption is a necessary and sufficient condition for the non-emptyness of the core of fuzzy cooperative game with side payments.

Keywords:

F-balanced collection, F-balanced fuzzy NTU game, V-balanced fuzzy TU game, core, S*-representation

Downloads

Download data is not yet available.

References

Aubin, J.-P. (1993). Optima and Equilibria. Springer-Verlag: Berlin-Heidelberg.

Bondareva, O. N. (1962). Core theory for n-person game. Vestnik Leningrad. Univer., ser. matem., 17, 141–142.

Ekeland, I. (1979). Éléments d'Économy Mathématique. Hermann: Paris.

Hildenbrand, W. and A. P. Kirman (1991). Equilibrium Analysis. North-Holland: Amsterdam-New York-Oxford-Tokyo.

Peleg, B. and P. Sudhölter (2003). Introduction to the Theory of Cooperative Games. Kluver Academic Publishers: Boston/Dordrecht/London.

Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press: Princeton, NJ.

Scarf, H. E. (1967). The core of an n-person game. Econometrica, 35, 50–69.

Shapley, L. S. (1967). On balanced sets and cores. Naval Research Logistics Quarterly, 14, 453–460.

Vasil'ev, V. A. (1987). Generalized von Neumann-Morgenstern solutions and accessibility of cores. Soviet Math. Dokl., 36, 374–378.

Vasil'ev, V. A. (2006). Cores and generalized NM-solutions for some classes of cooperative games. In: Russian Contributions to Game Theory and Equilibrium Theory (Driessen, T. S. H., G. van der Laan, V. A. Vasil'ev and E. B. Yanovskaya, eds), pp. 91–150, Springer-Verlag:Berlin-New York.

Vasil'ev, V. A. (2012). An extension of Scarf theorem on the nonemptiness of the core. Discussion Paper, 283, 41p. Sobolev Inst. Math.: Novosibirsk.

Downloads

Published

2022-05-02

How to Cite

Vasil’ev, V. A. (2022). A Fuzzy-Core Extension of Scarf Theorem and Related Topics. Contributions to Game Theory and Management, 8. Retrieved from https://gametheory.spbu.ru/article/view/13467

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.