We generalize the well-known Scarf theorem on the nonemptiness of the core to the case of generalized fuzzy cooperative games without side payments provided that the set of blocking coalitions is extended by the so-called fuzzy coalitions. The notion of a balanced family is extended to the case of an arbitrary set of fuzzy blocking coalitions, owing to which it is possible to introduce a natural analogue of balancedness of a fuzzy game for the characteristic function with an arbitrary efficiency domain. Based on an appropriate approximation of a fuzzy game by finitely-generated games, together with the seminal combinatorial Scarf lemma on ordinal and admissible bases, we obtain rather general conditions of the existence of unblocked imputations for F-balanced fuzzy cooperative games. Bibliography: 15 titles.
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V. A. Vasil’ev, An Extension of Scarf Theorem on the Core of an n-person Game [in Russian], Preprint, No. 283, IM SO RAN USSR, Novosibirsk (2012).
H. Scarf, “The core of an n-person game,” Econometrica35, No. 1, 50–69 (1967).
Ch. D. Aliprantis, D. J. Brown, O. Burkinshaw, Existence and Optimality of Competitive Equilibria, Springer, Berlin etc. (1995).
V. A. Vasil’ev, “A fuzzy-core extension of Scarf theorem and related topics,” In: Contributions to Game Theory and Management, Vol. VIII, pp. 300–314, St.Petersburg State Univ., St. Petersburg (2015).
O. N. Bondareva, “Core theory for n-player game” [in Russian], Vestn. Leningr. Univ.13, No. 3, 141–142 (1962).
J. Rosenmüller, The Theory of Games and Markets, North-Holland, Amsterdam etc. (1974).
T. Hansen and H. Scarf, On the Applications of a Recent Combinatorial Algorithm, Cowles Foundation Discussion Paper No. 272 (1969).
J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton Univ. Press, Princeton, NJ (1953).
I. Ekeland, Éléments d’ Économie Mathématique, Hermann, Paris (1979).
V. A. Vasil’ev, “On Edgeworth equilibria for some types of nonclassic markets,” Sib. Adv. Math.6, No. 3, 96–150 (1996).
V. A. Vasil’ev, Mathematical Models of General Economic Equilibrium [in Russian], Novosibirsk State Univ. Press, Novosibirsk (2009).
V. A. Vasil’ev and V. I. Suslov, “On Edgeworth equilibrium in a model of interregional economic relationships” [in Russian], Sib. Zh. Ind. Mat.13, No. 1, 18–33 (2010).
J.-P. Aubin, Optima and Equilibria. An Introduction to Nonlinear Analysis, Springer, Berlin etc. (1993).
S. L. Pechersky and E. B. Yanovskaya, Cooperative Games: Solutions and Axioms [in Russian], SPb-Europ. Univ. Press, St.-Petersb. (2004).
L. Billera, “Some theorems on the core of an n-person game without side payments,” SIAM J. Appl. Math.18, No. 3, 567–579 (1970).
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Translated from Sibirskii Zhurnal Chistoi i Prikladnoi Matematiki18, No. 1, 2018, pp. 35-53.
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Vasil’ev, V.A. Unblocked Imputations of Fuzzy Games. I: Existence. J Math Sci 246, 828–845 (2020). https://doi.org/10.1007/s10958-020-04785-2
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DOI: https://doi.org/10.1007/s10958-020-04785-2