Abstract
Many game-theoretic solution notions have been defined or can be defined not only with reference to the all-player coalition, but also with reference to an arbitrary coalition structure. In this paper, theorems are established that connect a given solution notion, defined for a coalition structure ℬ with the same solution notion applied to appropriately defined games on each of the coalitions in ℬ. This is done for the kernel, nucleolus, bargaining set, value, core, and thevon Neumann-Morgenstern solution. It turns out that there is a single function that plays the central role in five out of the six solution notions in question, though each of these five notions is entirely different. This is an unusual instance of a game theoretic phenomenon that does not depend on a particular solution notion but holds across a wide class of such notions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aumann, R. J., and M.Maschler: The Bargaining Set for Cooperative Games, in: Advances in Game Theory, M.Dresher, L. S.Shapley, and A. W.Tucker, eds., Annals of Mathematics Studies, No. 52, pp. 443–476, Princeton 1964.
Davis, M., and M.Maschler: Existence of Stable Payoff Configurations for Cooperative Games, in: Essays in Mathematical Economics in Honor of Oskar Morgenstern, M. Shubik ed., pp. 39–52, Princeton 1967.
—: The Kernel of a Cooperative Game, Naval Research Logistics Quarterly12, 223–259, 1965.
Gillies, D. B.: Solutions to General Non-zero-sum Games, in: Contributions to the Theory of Games IV, R. D.Luce, and A. W.Tucker eds., Annals of Mathematics Studies, No. 40, pp. 47–85, Princeton 1959.
Kohlberg, E.: On the Nucleolus of a Characteristic Function Game, SIAM Journal of Applied Mathematics,20, 62–66, 1971.
Maschler, M., andB. Peleg: The Structure of the Kernel of a Cooperative Game, SIAM Journal of Applied Mathematics,15, 569–604, 1967.
Maschler, M., B. Peleg, andL. Shapley: The Kernel and Bargaining Set for Convex Games, International Journal of Game Theory1, 73–93, 1972.
Savage, L. J.: Foundations of Statistics; Wiley, New York 1954.
Schmeidler, D.: The Nucleolus of a Characteristic Function Game, SIAM Journal of Applied Mathematics17, 1163–1170, 1969.
Selten, R.: Equal Share Analysis of Characteristic Function Experiments, in: Contributions to Experimental Economics III, H.Sauermann, editor, J. C. B. Mohr, pp. 130–165, Tübingen 1972.
Shapley, L. S.: A Value forn-person Games, in: Contributions to the Theory of Games II, H. W.Kuhn, and A. W.Tucker, eds., Annals of Mathematics Studies no. 52, pp. 307–317, Princeton 1953.
Thrall, R., andW. F. Lucas: Games in Partition Function Form, Naval Research Logistics Quarterly10, 281–298, 1963.
von Neumann, J., and O.Morgenstern: Theory of Games and Economic Behaviour, Princeton 1944.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Aumann, R.J., Dreze, J.H. Cooperative games with coalition structures. Int J Game Theory 3, 217–237 (1974). https://doi.org/10.1007/BF01766876
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01766876