Abstract
A solution on a class of TU games that satisfies the axioms of the pre-nucleolus or -kernel except the equal treatment property and is single valued for two-person games, is a nonsymmetric pre-nucleolus (NSPN) or -kernel (NSPK). We investigate the NSPKs and NSPNs and their relations to the positive prekernel and to the positive core. It turns out that any NSPK is a subsolution of the positive prekernel. Moreover, we show that an arbitrary NSPK, when applied to a TU game, intersects the set of preimputations whose dissatisfactions coincide with the dissatisfactions of an arbitrary element of any other NSPK applied to this game. This result also provides a new proof of sufficiency of the characterizing conditions for NSPKs introduced by Orshan (Non-symmetric prekernels, discussion paper 60. Center for Rationality, The Hebrew University of Jerusalem, 1994). Any NSPN belongs to “its” NSPK. Several classes of NSPNs are presented, all of them being subsolutions of the positive core. We show that any NSPN is a subsolution of the positive core provided that it satisfies the equal treatment property on an infinite subset of the universe of potential players. Moreover, we prove that, for any game whose prenucleolus is in its anticore, any NSPN coincides with the prenucleolus.
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References
Aumann RJ, Maschler M (1985) Game theoretic analysis of a bankruptcy problem from the Talmud. J Econ Theory 36: 195–213
Davis M, Maschler M (1965) The kernel of a cooperative game. Nav Res Logist Q 12: 223–259
Davis M, Maschler M (1967) Existence of stable payoff configurations for cooperative games. In: Shubik M (eds) Essays in mathematical economics in honor of Oskar Morgenstern. Princeton University Press, Princeton, pp 39–52
Derks J, Kuipers J (2002) On the number of extreme points of the core of a transferable utility game. In: Borm P, Peters H (eds) Chapters in game theory in Honor of Stef Tijs, theory and decision library, series c: game theory, mathematical programming and operations research. Kluwer Academic Publishers, Dordrecht, pp 83–97
Hwang Y-A, Sudhölter P (2001) Axiomatizations of the core on the universal domain and other natural domains. Int J Game Theory 29: 597–623
Kohlberg E (1971) On the nucleolus of a characteristic function game. SIAM J Appl Math 20: 62–66
Maschler M, Peleg B, Shapley LS (1972) The kernel and bargaining set for convex games. Int J Game Theory 1: 73–93
Orshan G (1993) The prenucleolus and the reduced game property: Equal treatment replaces anonymity. Int J Game Theory 22: 241–248
Orshan G (1994) Non-symmetric prekernels. Discussion paper 60. Center for Rationality, The Hebrew University of Jerusalem
Orshan G, Sudhölter P (2003) Reconfirming the prenucleolus. Math Oper Res 28: 283–293
Orshan G, Sudhölter P (2010) The positive core of a cooperative game. Int J Game Theory 39: 113–136
Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15: 187–200
Peleg B, Sudhölter P (2003) Introduction to the theory of cooperative games. Kluwer Academic Publishers, Dordrecht
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17: 1163–1170
Sobolev AI (1975) The characterization of optimality principles in cooperative games by functional equations. In: Vorobiev NN (ed) Mathematical methods in the social sciences, vol 6. Academy of Sciences of the Lithuanian SSR, Vilnius, pp 95–151 (in Russian)
Sudhölter P (1993) Independence for characterizing axioms of the pre-nucleolus. Working paper 220. Institute of Mathematical Economics, University of Bielefeld
Sudhölter P, Peleg B (2000) The positive prekernel of a cooperative game. Int Game Theory Rev 2: 287–305
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Orshan, G., Sudhölter, P. Nonsymmetric variants of the prekernel and the prenucleolus. Int J Game Theory 41, 809–828 (2012). https://doi.org/10.1007/s00182-011-0294-6
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DOI: https://doi.org/10.1007/s00182-011-0294-6