Miranda Menéndez, Pedro and Grabisch, Michel (2010) k-Balanced games and capacities. European journal of operational research, 200 (2). pp. 465-472. ISSN 0377-2217
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In this paper, we present a generalization of the concept of balanced game for finite games. Balanced games are those having a nonempty core, and this core is usually considered as the solution of the game. Based on the concept of k-additivity, we define the so-called k-balanced games and the corresponding generalization of core, the k-additive core, whose elements are not directly imputations but k-additive games. We show that any game is k-balanced for a suitable choice of k, so that the corresponding k-additive core is not empty. For the games in the k-additive core, we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion. Moreover, we look for necessary and sufficient conditions for a game to be k-balanced. For the general case, it is shown that any game is either balanced or 2-balanced. Finally, we treat the special case of capacities.
|Uncontrolled Keywords:||Cooperative games; k-Additivity; Balanced games; Capacities; Core|
|Subjects:||Sciences > Mathematics > Operations research|
O. Bondareva, Some applications of linear programming to the theory of cooperative games, Problemy Kibernet (10)(1963) 119–139.
A. Chateauneuf, J.-Y. Jaffray, Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion, Mathematical Social Sciences (17) (1989) 263–283.
G. Choquet, Theory of capacities, Annales de l’Institut Fourier (5) (1953) 131–295.
A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, The Annals of Mathematical Statististics (38) (1967) 325–339.
D. Denneberg, Non-Additive Measures and Integral, Kluwer Academic, 1994.
T. Driessen, Cooperative Games, Kluwer Academic, 1988.
P. Dubreil, Algèbre, Equivalences, Opérations, Groupes, Anneaux, Corps, vol. 1, Gauthiers-Villars, 1946 (in French).
M. Grabisch, k-Order additive discrete fuzzy measures, in: Proceedings of Sixth International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Granada, Spain, 1996, pp. 1345–1350.
M. Grabisch, Alternative representations of discrete fuzzy measures for decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997) 587–607.
M. Grabisch, k-Order additive discrete fuzzy measures and their representation, Fuzzy Sets and Systems (92) (1997) 167–189.
M. Grabisch, P. Miranda, On the vertices of the k-additive core, Discrete Mathematics (308) (2008) 5204–5217.
J.C. Harsanyi, A simplified bargaining model for the n-person cooperative game, International Economic Review 4 (1963) 194–220.
G. Owen, Game Theory, Academic Press, 1995.
G.C. Rota, On the foundations of combinatorial theory. I: Theory of Möbius functions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete (2) (1964) 340–368.
G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ, USA, 1976.
L.S. Shapley, A value for n-person games, in: H.W. Kuhn, A.W. Tucker (Eds.),Contributions to the Theory of Games, Annals of Mathematics Studies, vol. 2,Princeton University Press, 1953, pp. 307–317.
L.S. Shapley, Cores of convex games, International Journal of Game Theory 1 (1971) 11–26.
M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, 1974.
P. Walley, Coherent lower (and upper) probabilities, Technical Report 22,University of Warwick, Coventry, UK, 1981.
|Deposited On:||31 Oct 2012 10:35|
|Last Modified:||11 Dec 2012 16:58|
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