Abstract
The aim of this paper is to study a new class of cooperative games called interior operator games. These games are additive games restricted by antimatroids. We consider several types of cooperative games as peer group games, big boss games, clan games and information market games and show that all of them are interior operator games. Next, we analyze the properties of these games and compute the Shapley, Banzhaf and Tijs values.
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Algaba, E., J.M. Bilbao, R. van den Brink, and A. Jiménez-Losada. (2004). “Cooperative Games on Antimatroids.” Discrete Math. 282, 1–15.
Aumann, R., and M. Mashler. (1964). “The Bargaining Set for Cooperative Games.” Advances in Game Theory, M. Dresher, L.S. Shapley and A.W. Tucker (Eds.), Princeton, New Jersey: Princeton University Press, pp. 443–476.
Banzhaf, J.F. (1965). “Weighted Voting Doesn't Work: A Mathematical Analysis.” Rutgers Law Review 19, 317–343.
Borm, P., G. Owen, and S. Tijs. (1992). “On the Position Value for Communication Situations.” SIAM J. Discrete Math. 5,305–320.
Brânzei, R., V. Fragnelli, and S. Tijs. (2002).“Tree-Connected Peer Group Situations and Peer Group Games.” Math. Meth.Oper. Res. 55, 93–106.
R. van den Brink. (1997). “An Axiomatization of the Disjuntive Permission Value for Games with a Permission Structure.” Int. J.Game Theory 26, 27–43.
Dilworth, R.P. (1940). “Lattices with Unique Irreducible Descompositions.” Ann. Math. 41, 771–777.
Driessen, T.S.H. (1988). Cooperative Games, SolutionsApplications, Kluwer, Dordrecht, The Netherlands.
Dubey, P. and L.S. Shapley. (1979). “Mathematical Properties of the Banzhaf Power Index.” Math. Oper. Res. 4, 99–131.
Edelman, P.H. (1980). “Meet-Distributive Lattices and the Anti-Exchange Closure.” Algebra Universalis 10, 290–299.
Edelman, P.H. and R.E. Jamison. (1985). “The Theory of Convex Geometries.” Geometriae Dedicata 19, 247–270.
Faigle, U. and W. Kern. (1993). “The Shapley Value for Cooperative Games Under Precedence Constraints.” Int. J. Game Theory21, 249–266.
Gilles, R.P., R. Owen, and R. van den Brink. (1992).“Games with Permission Estructures: The Conjuntive Approach.” Int. J. GameTheory 20, 277–293.
Gillies, D.B. (1953). “Some Theorems on n-Person Games.” Ph. D. Thesis. Princeton University Press, Princeton, New Jersey.
Goecke, O., B. Korte, and L. Lovász. (1986).“Examples and Algorithmic Properties of Greedoids.” CombinatorialOptimization, B. Simone (Ed.), Springer-Verlag, New York.
Harsanyi, J.C. and R. Selten. (1988). A General Theory of Equilibrium Selection in Games. Cambridge, Massachusets: The MIT Press.
Jiménez-Losada, A. (1998). “Valores para juegos sobre Estructuras Combinatorias.” Ph. D. Thesis, Universidad de Sevilla,Spain.
Korte, B., L. Lóvasz, and R. Schrader. (1991).Greedoids. Berlin, Heidelberg, New York: Spinger-Verlag.
Meggido, N. (1978). “Computational Complexity of the Game Theory Approach to Cost Allocation for a Tree.” Math. Oper. Res. 3, 189–196.
Muto, S., J. Potters, and S. Tijs. (1986). “InformationMarket Games.” Int. J. Game Theory 18, 209–226.
Muto, S., M. Nakayama, J. Potters, and S. Tijs. (1987). “On Big Boss Games.” Economic Studies Quarterly 39, 303–321.
Myerson, R.B. (1977). “Graphs and Cooperation in Games.”Math. Oper. Res. 2, 225–229.
Owen, G. (1986). “Values of Graph-Restricted Games.” SIAM J. Alg. Disc. Meth. 7, 210–220.
Potters, J., R. Poos, S. Muto, and S. Tijs. (1989). “Clan Games.” Games and Economic Behavior 1, 275–293.
Shapley, L.S. (1953). “A Value for n-Person Games.” H.W. Kuhn and A.W. Tucker (Eds.), Contributions to the theory of games, vol. 2, Princeton University Press, pp. 307–317.
S.H. Tijs. (1981). “Bounds for the Core and the Τ-Value.” O. Moeschlin and D. Pallaschke (Eds.), Game Theory and Mathematical Economics, North-Holland Publishing Company, Amsterdam, The Netherlands, pp. 123–132.
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Bilbao, J.M., Jiménez-Losada, A., Lebrón, E. et al. Values for Interior Operator Games. Ann Oper Res 137, 141–160 (2005). https://doi.org/10.1007/s10479-005-2251-x
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DOI: https://doi.org/10.1007/s10479-005-2251-x